from fuzzy logic to fuzzy mathematics

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From Fuzzy Logic to Fuzzy Mathematics Petr Cintula Disertaˇ cn´ ı pr´ace v oboru Matematick´ e inˇ zen´ yrstv´ ı Praha, prosinec 2004

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Page 1: From Fuzzy Logic to Fuzzy Mathematics

From Fuzzy Logic to FuzzyMathematics

Petr Cintula

Disertacnı prace v oboru Matematicke inzenyrstvı

Praha, prosinec 2004

Page 2: From Fuzzy Logic to Fuzzy Mathematics

Disertacnı prace v oboru Matematicke inzenyrstvıFakulta jaderna a fyzikalne inzenyrskaCESKE VYSOKE UCENI TECHNICKE v Prazec© 2004 Petr Cintula

FJFI CVUT, Katedra matematikyTrojanova 13, 120 00 Praha 2

Doctoral dissertationFaculty of Nuclear Sciences and Physical EngineeringCZECH TECHNICAL UNIVERSITY in Praguec© 2004 Petr Cintula

FNSPE CTU, Department of MathematicsTrojanova 13, 120 00 Prague 2, Czech Republic

Under the supervision of: Prof. RNDr. Petr Hajek, DrSc.,Institute of Computer Science,ACADEMY OF SCIENCES OF THE CZECH REPUBLIC, Prague

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Preface

This work contains contributions I have made to the mathematical fuzzy logic during myPhD study. Although I was interested in many topics during this period, two main ones haveemerged quite recently.

The first one is a “universalistic” approach towards the variety of fuzzy logics studied inthe literature. The main imperative of this approach is to focus not on the particular logicsbut rather on their classes. The reader should be aware that I use the term fuzzy logic inrather narrow sense (see Chapter 1 for more details) and thus the term “universal” is to beunderstood as restricted to my particular setting. I was inspired by existing (usually moregeneral) approaches described in the literature, mainly by the so-called Abstract AlgebraicLogic (AAL). I tried to isolate the essential elements of these approaches and alter theirmethods to suit better the existing variety of fuzzy logics. This resulted in the definition ofthe class of the weakly implicative fuzzy logics, which is, in my view, a very good formalizationof the pre-theoretic notion of a fuzzy logic as a logic of “comparative degrees of truth”. I wasable to prove some interesting general theorems and apply them to obtain not only knownresults about existing fuzzy logic, but also new ones. Chapter 2 is dedicated to elaboratingthose general methods and Chapter 3 develops existing fuzzy logics in the proposed generalframework (to the extent needed for the subsequent chapters, the thorough development is tobe subject of my future work). These results are so far described only in my technical report[25]. However, my paper about this topic is nearly completed and will be submitted soon.There is also a submitted joint paper with Libor Behounek [8] concentrating on philosophical,methodological, and pragmatical reasons for using the term weakly implicative fuzzy logics asa formal delimitation of the existing informal class of fuzzy logic.

The second main issue can be described as a syntactic turn in fuzzy logic and fuzzymathematics especially. This work is a joint project with Libor Behounek. The main idea ofthis approach results from our belief that the formal mathematical fuzzy logic is developedenough to support the development of some formal foundations of the existing fuzzy mathe-matics and can be expressed by the following imperative: work in a formal axiomatic theoryover a fuzzy logic, rather than investigate particular models. We believe that following thisimperative can shelter us from the common problems of existing fuzzy mathematics, i.e.,ad hoc definitions of fuzzy concepts, which often suffer from arbitrariness and hidden crisp-ness, or even references to particular crisp models of fuzziness (e.g. membership functions).We propose one particular logical system to serve as the foundations of fuzzy mathematics,namely the Higher-order Fuzzy Logic. Formulation of this theory and some examples of itssuitability for the sketched task are to be found in Chapter 7 (it is based on our paper [6])and in the last Chapter the reader can find the so-called Methodological Manifesto which isbased on our submitted paper [5].

These two issues go far beyond the scope of this thesis and are subjects of the twoproposed research programs. Here I present only the starting points. However, they nicelydelimit my whole work and this thesis in particular and give it its name: From fuzzy logic tofuzzy mathematics.

The rest of this thesis is dedicated to my particular results in mathematical fuzzy logic.Chapter 4 is based on the joint work with Rostislav Horcık. Here we developed the notion ofthe so-called Product ÃLukasiewicz logic (roughly speaking, we add one additional connective

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into the ÃLukasiewicz logic). This chapter is derived from the paper [27] however it wascompletely rewritten because the general results proved in Chapters 2 and 3 simplified thispaper greatly. Later, Chapter 5 deals with the so-called Product logic and has two mainparts. The first one is based on my paper [20] and contains results about possible axiomaticsystems of this logic. The second part is based on a joint paper with Brunella Gerla [26] anddeals with the notion of (semi)normal form of formulae of this logic and with its functionalrepresentation. Finally, in Chapter 6 we examine the notion of compactness in fuzzy logic.Again, this chapter consists of two main parts. The first one is based on my submitted paper[19] and develops two different notions of compactness in the so-called Godel logics. Thesecond one is based on a joint paper with Mirko Navara [28] and extends the results on oneof the notions of compactness defined in the first part to the broader family of fuzzy logics.

There is another big research topic of mine—the �Π 12 logic (see my papers [17, 21, 23, 24]).

However, the majority of results from those papers was included already in my master thesis([22]) and there is a survey paper (in preparation) by Esteva, Godo, Montagna, and myselfabout �Π 1

2 logic. Thus I decided not to deal with this topic in this thesis.

¦

Now I would like to thank all the people who helped me to create this thesis. The linearityof the written text forces me to do it in some fixed order and although contribution of someparticularly persons are indeed greater than of some others, I value all of them greatly andlist them without order.

There are only the exceptions of this rule: Petra and Petr. Petr Hajek, my supervisor andgood friend helped me greatly during my study with his comments and suggestions and hedirected my scientific interest. Petra Ivanicova, my girlfriend, helped me very much duringthe long process of writing this thesis simply by being here, with me.

Then I would like to thank all my coauthors, namely: Libor Behounek, Brunella Gerla,Rostislav Horcık, and Mirko Navara. I also need to credit all the friends and colleagues whoread parts of this thesis and help improve them greatly, especially Carles Noguera, JosepMaria Font, Libor Behounek, Petr Hajek, and Zuzana Hanikova. I would especially like tothank to Mirko Navara for a very careful reading of some of my papers and helping me verymuch to improve my stylistical and writing skills. Special thanks also go to Libor Behounek,first for his efforts in creating a huge project in formalization of fuzzy mathematics (althoughmy original aims were more modest), second for helping me to create the notion of weaklyimplicative logics in the current general form (by his constant “Do you really need to assume. . . ”), and third for his belief (stronger than mine) that the weakly implicative fuzzy logicsare the fuzzy logics. I would also like to thank to all the colleagues from the Institute ofComputer Science for creating nice working environment. At last, but definitely not at least,I would like to thank my parents for their everlasting support in my life and work.

¦

At the end of this preface I would like to credit the grants which supported me during myPhD. study: the grant no. IAA1030004 of the Grant Agency of the Academy of Sciences ofthe Czech Republic, the PhD. grant no. GD401/03/H047 of the Grant Agency of the CzechRepublic, COST action 274 TARSKI, and the exchange program Net CEEPUS SK-042.

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Contents

Preface iii

1 From Fuzzy Logic . . . 1

2 Weakly implicative (fuzzy) logics 52.1 General theory for propositional logics . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.3 Fuzzy logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Special propositional logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.1 Adding rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.2 Adding connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.3 The connective ∨ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.4 The connective 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 First-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.3.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.3.2 Henkin and witnessed theories . . . . . . . . . . . . . . . . . . . . . . 292.3.3 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.3.4 Skolem functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 Known fuzzy logics 353.1 Standard completeness and Core fuzzy logics . . . . . . . . . . . . . . . . . . 363.2 Basic fuzzy logic and its expansions . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.1 Basic fuzzy logic and its extensions . . . . . . . . . . . . . . . . . . . . 393.2.2 Basic fuzzy logic with 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2.3 Strict Basic fuzzy logic with involutive negation . . . . . . . . . . . . . 45

3.3 Core fuzzy logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.3.1 ÃLukasiewicz logic and its expansions . . . . . . . . . . . . . . . . . . . 463.3.2 Godel logic and its expansions . . . . . . . . . . . . . . . . . . . . . . 483.3.3 Product logic and its expansions . . . . . . . . . . . . . . . . . . . . . 503.3.4 The ÃLΠ and ÃLΠ1

2 logics . . . . . . . . . . . . . . . . . . . . . . . . . . 513.4 Adding truth constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.4.1 Rational extension and basic definitions . . . . . . . . . . . . . . . . . 543.4.2 Pavelka-style extension . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.5 First-order logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.5.1 Basic facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.5.2 Crisp equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.5.3 Adding truth constants . . . . . . . . . . . . . . . . . . . . . . . . . . 63

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vi CONTENTS

4 Product ÃLukasiewicz logic 674.1 P� and P�′ logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.1.1 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.1.2 Semantics and completeness . . . . . . . . . . . . . . . . . . . . . . . . 694.1.3 More on P� and P�′-algebras . . . . . . . . . . . . . . . . . . . . . . . 71

4.2 P�4 and P�′4 logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.3 Pavelka style extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.4 The predicate logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5 New results in Product logic 795.1 On axioms of Product logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.1.1 New axiomatic system . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.1.2 More about possible axiomatic systems . . . . . . . . . . . . . . . . . 81

5.2 Semi-Normal Forms and Normal Forms . . . . . . . . . . . . . . . . . . . . . 825.2.1 Literals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.2.2 Semi-Normal Forms and Normal Forms . . . . . . . . . . . . . . . . . 845.2.3 Simplification of formulae in CsNF and DsNF . . . . . . . . . . . . . . 885.2.4 Theorem proving algorithm . . . . . . . . . . . . . . . . . . . . . . . . 90

5.3 Functional representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.3.1 Piecewise monomial functions . . . . . . . . . . . . . . . . . . . . . . . 935.3.2 Product functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6 Compactness in fuzzy logics 976.1 The notions of compactness in Godel logics . . . . . . . . . . . . . . . . . . . 97

6.1.1 Two notions of compactness . . . . . . . . . . . . . . . . . . . . . . . . 986.1.2 Compactness and the cardinality of VAR . . . . . . . . . . . . . . . . 1016.1.3 K-compactnessSat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.1.4 V -compactnessEnt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106.1.5 Correspondence between two notions of compactness . . . . . . . . . . 111

6.2 Compactness in other core fuzzy logics . . . . . . . . . . . . . . . . . . . . . . 1116.2.1 Satisfiability based compactness . . . . . . . . . . . . . . . . . . . . . 1126.2.2 ÃLukasiewicz logic and Product ÃLukasiewicz logic . . . . . . . . . . . . 1126.2.3 Product logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.2.4 Logics with 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.2.5 �Π and �Π1

2 logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

7 Fuzzy class theory 1177.1 Class theory over ÃLΠ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.1.1 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1187.1.2 Elementary class operations . . . . . . . . . . . . . . . . . . . . . . . . 1197.1.3 Elementary relations between classes . . . . . . . . . . . . . . . . . . . 1207.1.4 Theorems on elementary class relations and operations . . . . . . . . . 121

7.2 Tuples of objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237.3 Higher types of classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

7.3.1 Second-level classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1247.3.2 Simple fuzzy type theory . . . . . . . . . . . . . . . . . . . . . . . . . 125

7.4 Adding structure to the domain of discourse . . . . . . . . . . . . . . . . . . . 1267.5 Fuzzy mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

8 . . . to Fuzzy Mathematics 129

A T-norms 135

References 137

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Chapter 1

From Fuzzy Logic . . .

Vagueness pervades human language, perception, and reasoning since the beginning of itsexistence. Classical mathematics and logic can, however, model the concept of vaguenessonly indirectly. Although many natural-language predicates (young, tall, hot) show naturaldegrees of truth, all notions of classical logic and mathematics are bivalent. Even thoughmany-valued logics were developed (for other purposes) already during the first half of theXX century (ÃLukasiewicz [62], Godel [42]), a systematic study of vagueness by means of themany-valued approach began only after L.A. Zadeh [85] proposed to investigate fuzzy sets in1965. Since then, the notion of fuzziness spread to nearly all mathematical disciplines: fuzzyarithmetic, fuzzy logic, fuzzy probability, fuzzy relations, fuzzy topology, etc.

For a long time, however, fuzzy logic and fuzzy mathematics were engineering toolsrather than a well-designed mathematical theories. Driven just by applications, they lacked(meta)theoretical grounding and general results; developed mostly by engineers for particularpurposes, it suffered from arbitrariness in definitions and often even mathematical impreci-sion. Moreover, it could be objected that it was just a theory of [0,1]-valued functions andthus a part of real analysis. There have been many (more or less successful) attempts toformalize or even axiomatize some areas of fuzzy mathematics. However, these axiomaticsystems are designed ad hoc. The authors usually select some concepts in their area of in-terest and change them into vague ones. This selection is usually based solely on intuitionor on the desired application. Another problem with these axiomatic attempts lies in theirfragmentation; it is nearly impossible to combine two of them into one theory.

The great success, not suffering from the above-mentioned defects, is without doubt thearea of mathematical fuzzy logic developed by Gottwald [43], Hajek [44], Novak at al. [75],Mundici at al. [12], and others. Equipping a variety of fuzzy logics with algebraic semanticsand Hilbert-style calculi made them a legitimate part of the long tradition of many-valuedlogics. Both propositional and first-order logics were defined and developed. With [44], thiseffort has been completed to the point in which predicate fuzzy logics can serve as groundtheories for fuzzy mathematics, i.e., for constructing axiomatic theories over these logics,aimed to describe fuzzy arithmetic, fuzzy geometry, fuzzy analysis, etc.

However, there is one small problem: the multitude of existing fuzzy logics. It is nota big problem, it can be easily shown that this multitude is not only desirable but alsoindispensable for creating general theories of vagueness phenomena (see Chapter 8 for moredetails). The problem is rather methodological and pragmatical. The methodological natureof the problem can be expressed by the question: “What is fuzzy logic?” and the pragmaticalone with the question: “Is there any order or system in the class of fuzzy logics?”

The first question is practically unanswerable, mainly because it entails a question “Whatis logic?”. In order to get any answer we need to make some “design” choices (and forgetabout philosophy) to answer the “What is logic?” question. First of all we restrict to thepropositional level (in the beginning, later we move to the first-(and even higher)-order log-ics) and we retain the classical syntax (i.e., the formulae are created inductively from the

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2 CHAPTER 1. FROM FUZZY LOGIC . . .

set of atoms and the set of propositional connectives). In this work we understand the logicas an asymmetric consequence relation (following Dunn and Hardegree’s terminology,see [30]), i.e., a relation between sets of formulae and formulae closed under arbitrary sub-stitution. This is quite a departure from the common practice in the existing fuzzy logicliterature, where logic is usually seen as a unary consequence relation, i.e., just a set offormulae (theorems/tautologies). As the reader will see this forces us to make some termino-logical changes, however the author believes that the transition to asymmetric consequencerelation is necessary to increase generality of results and to cover a much broader class oflogics.

Thus the task of the day is to look at the class of logics we described in the previousparagraph and decide which of them should be called fuzzy and which not. Before we proceedwith answering this question we stress one thing. Many authors believe that the fuzzy logicshould be a logic directly dealing with some aspects of vagueness, such as vague predicates,linguistic expressions, etc. Fuzzy logic should, according to their opinion, be able to (at leastpartially) describe the process of reasoning under vagueness, formalize the notion of fuzzycontrol, etc. It is obvious that these tasks cannot by fulfilled by any propositional logic.Therefore, we believe that there is no such thing as the fuzzy logic and that the fuzzy logicssolving those tasks are numerous and that they are based on some appropriate propositionallogics. Thus we think that it is perfectly reasonable to speak about fuzzy logics among theclass of propositional logics (we mentioned before).

In order to answer that “What is fuzzy logic?” question we need one more restrictionon the notion of logic: we restrict ourselves to the class of the so-called weakly implicativelogics (a generalization of Rasiowa’s well-known notion of implicative logics, see [77]). A logicis weakly implicative iff its language contains a (definable) connective → that satisfies thefollowing conditions:

` ϕ → ϕ

ϕ, ϕ → ψ ` ψ

ϕ → ψ, ψ → χ ` ϕ → χ

ϕ → ψ, ψ → ϕ ` c(. . . , ϕ, . . .) → c(. . . , ψ, . . .) for all connectives c

Weakly implicative logics can be characterized as those which are complete w.r.t. a class ofordered matrices (in which the set D of designated values is upper), if the ordering of theelements of the matrix is defined as

x ≤ y ≡df x → y ∈ D

From this class we select the class of weakly implicative fuzzy logics as the class of thoseweakly implicative logics which are complete w.r.t. linearly ordered matrices. It turns outthat this class approximates well the bunch of logics studied in the so-called “fuzzy logic innarrow sense”.

As far as the author knows there is no weakly implicative logic studied as fuzzy logic inthe literature that is not fuzzy in our formal setting. We need to recall that we are selectingfuzzy logics from the class of weakly implicative logics only, we do not say anything aboutthose outside this class! However, for the logics that can be interpreted in this class, ourcriterion works well (logics with evaluated syntax come out fuzzy, most substructural logicsdo not).

On the other hand there are fuzzy logics in our setting, which are not considered fuzzyby some researchers. One example is the Relevance RM, which comes out fuzzy. We thinkthat such logics should be regarded and studied as fuzzy too, because they have a naturalinterpretation of comparative degrees of truth. We can say that they are both relevant andfuzzy (similarly, Godel logic is both intermediary and fuzzy). Other group of “problematic”fuzzy logics is constituted by logics lacking some structural rule, namely exchange or weak-ening. For example, some people claim that there are no natural real-life examples involving

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non-commutative conjunction (except perhaps temporal, but such phenomena should, theyclaim, be handled by temporal rather than fuzzy logics). We notice that most methods ofmetamathematics of fuzzy logic work with non-commutative linear matrices as well, so wethink that it is better to be as general as possible, especially if we realize that the lackingmotivation can be found later. We also think that the extra assumption would require anextra explanation (the proof burden is on the proponent of extra assumptions). Final objec-tion is aimed at the fact that classical logic comes out fuzzy too. In our opinion, fuzzy logicis a generalization of classical logic (to encompass also vague predicates). Thus classical logiccan be viewed as a fuzzy logic for those special occasions when by chance the realizations ofall predicates are crisp.

To demonstrate the usefulness of this definition we show that it is equivalent to someimportant properties studied in fuzzy literature. For finitary logics, the following are equiv-alent:

• L is weakly implicative fuzzy logic.

• Each L-matrix is a subdirect product of linear ones. (Subdirect representation property)

• Each theory in L can be extended to one whose Lindenbaum-Tarski matrix is linear.(Linear extension property)

• T, ϕ → ψ ` χ and T, ψ → ϕ ` χ entails T ` χ. (Prelinearity property)

There are objections that some vague notions have incomparable degrees, because theyhave more components (e.g., human intelligence) and that fuzzy logic should be able to dealwith this. It is necessary to observe that linear matrices are not the only matrices for ourfuzzy logics, a particular algebra for a real-life application can be non-linear. However, theSubdirect representation property tells us that each such an algebra can be decomposed intolinearly ordered ones (in the case of human intelligence we can decompose it to memory,concentration, creativity, etc.).

The second question, “Is there any order in the class of fuzzy logic?” is much easier.The answer is: “not much, so far”. Of course, there are numerous results about relationsamong some particular fuzzy logics (stronger/weaker, conservative extension, fragment etc.).There are also results relating fuzzy logic with some non-fuzzy ones (see [34]). However, webelieve that only the formal definition of the notion of fuzzy logic can lead to the systematicdevelopment of this class. The following theorem, together with the fact that the class offuzzy logics is defined formally, allows us to speak about the weakest fuzzy logic with someproperties:

• The intersection of an arbitrary system of fuzzy logics is a fuzzy logic.

This allows us to extend the known result: MTL (Esteva and Godo’s Monoidal T-normFuzzy Logic) is an extension of FLew (Ono’s Full Lambek Calculus with Exchange andWeakening) to the following one: MTL is the weakest fuzzy logic extending FLew. In fact, foreach logic we can assign the weakest fuzzy logic extending it. We present some examples:

Logic Weakest fuzzy logic ReferencesIntuitionistic Godel-Dummett [44]

FLw psMTLr [48]FLew MTL [33]

AMALL− IMTL [33]

This turns out to be an important methodological guideline. For example there are nofuzzy logics without weakening defined in the literature (although some have been recentlydeveloped by George Metcalfe and his colleagues). The weakest fuzzy logic over FLe seemsto be a good starting point for this research.

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4 CHAPTER 1. FROM FUZZY LOGIC . . .

We give another example demonstrating the power of our approach. It is quite commonto introduce for fuzzy logics their extensions by a unary connective 4 (usually called Baazdelta). These extensions have some very interesting properties. Some of them entail that 4is a “globalization” connective (as known from intuitionistic logic), and thus can be viewedas a kind of “exponential”. However, Baaz delta in fuzzy logics has also some additionalproperties. We observe the constituting property of these extensions and define:

• We say that L is a logic with Baaz delta iff for each formula in the language {→,⊥,>}and for substitution σ(v) = 4v it holds: `L σϕ iff ϕ is a theorem of classical logic.

For each fuzzy logic we define its “Delta companion” as the weakest fuzzy logic with Baazdelta extending it. We present a uniform way how to construct an axiomatic system forthe Delta companion of each logic. We show that this general definition fits well with theusual usage (i.e., the existing extensions of fuzzy logics by Baaz delta coincide with our Deltacompanions). Then we prove an interesting interplay between the standard completeness ofa fuzzy logic and its Delta companion.

In upcoming Chapter 2 we elaborate these ideas in more details and the author hopesthat Chapter 3, where the known results about existing fuzzy logics are presented in thenew formalism, demonstrates the power and usefulness of the formal definition of the class offuzzy logics and general theorems resulting from this definition. We hope that both classesof weakly implicative and weakly implicative fuzzy logics represent reasonable compromisebetween generality (there are far more general classes of logic) and usefulness of their resultsfor particular logics. This is true especially for fuzzy logic, where the more general approachesomit the characteristic properties of existing fuzzy logics.

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Chapter 2

Weakly implicative (fuzzy) logics

2.1 General theory for propositional logics

We start with some obligatory introductory definitions. However, we will use rather non-standard approach towards propositional logic. Of course, it turns out to be equivalent (ornearly equivalent) to the known ones, but here we put stress on some issues, concerningmainly the notion of proof and meta-rule. For the comprehensive survey (with the standardterminology) into the problematic of general approach towards logic, consequence relation,logical matrices, etc. see the survey to Abstract algebraic logic (AAL) [40].

In the first subsection we introduce weakly implicative logics. Then, in the second subsec-tion, we present the semantic for them (using the well-known notion of logical matrix). Thethird subsection is the core part of this work. There we introduce the class of fuzzy logic (orbetter: weakly implicative fuzzy logics).

2.1.1 Syntax

Definition 2.1.1 A propositional language L is a triple (VAR,C, a), where VAR is anon-empty set of the (propositional) variables, C is a nonempty set of the (propositional)connectives, and a is a function assigning to each element of C a natural number. A con-nective c for which a(c) = 0 is called a truth constant.

The set VAR is usually taken as fixed countable set, and so we usually define the propo-sitional language L is a pair (C, a). Later on we fix symbols for some basic connectives(→,∧,⊥, . . . ) together with their arities and then we define propositional language just as aset of these connectives. Sometimes we write (c, 2) ∈ L instead of c ∈ C and a(c) = 2.

Definition 2.1.2 (Formula) Let L be a propositional language. The set of (propositional)formulae FORL is the smallest set which contains the set VAR and is closed under connec-tives from C, i.e., for each c ∈ C, such that a(c) = n, and for each ϕ1, . . . , ϕn ∈ FORL wehave c(ϕ1, . . . , ϕn) ∈ FORL.

Definition 2.1.3 (Substitution) Let L be a propositional language. A substitution is amapping σ : FORL → FORL, such that σc(ϕ1, . . . , ϕn) = c(σ(ϕ1), . . . σ(ϕn)). The set ofall substitutions will be denoted as SUBL.

Of course a substitution is fully determined by its values on propositional variables. Letv ∈ VAR and ϕ,ψ ∈ FORL, by ψ[v :=ϕ] we understand the formula σ(ψ), where σ is thesubstitution mapping v to ϕ and mapping all the remaining propositional variables to itself.

The term consecution in the following definition is from the Restall’s book [78]. However,we use it in a very simplified version.

5

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6 CHAPTER 2. WEAKLY IMPLICATIVE (FUZZY) LOGICS

Definition 2.1.4 (Consecution) A consecution in the propositional language L is a pair< X,ϕ >, where X ⊆ FORL and ϕ ∈ FORL. The set of all consecutions will be denoted asCONL.

Of course we have P(FORL)× FORL=CONL.

Convenction 2.1.5 Let X ⊆ FORL, X ⊆ CONL and σ a substitution.

• By σ(X) we understand the set {σ(ϕ) | ϕ ∈ X}• By σ(X) we understand the set {<σ(X), σ(ϕ)> | <X, ϕ> ∈ X}.

• By SUBL(X) we denote the set⋃

σ∈SUBL

σ(X).

Definition 2.1.6 (Axiomatic system) Let L be a propositional language. The axiomaticsystem AS in language L is a non-empty set AS ⊆ CONL, which is closed under arbitrarysubstitution (i.e., SUBL(AS) = AS).

The elements of AS of the form <X, ϕ > ∈ AS are called axioms for X = ∅, n-arydeduction rules for |X| = n, and infinitary deduction rules for X being of infinite.

The axiomatic system is said to be finite if there is a finite set X ⊆ AS such thatSUBL(X) = AS. Furthermore, the axiomatic system is said to be finitary if all its deductionrules are finite. Finally, the axiomatic system is said to be strongly finite if it is finite andfinitary.

The usual way of presenting an axiomatic system is in form of schemata.

Definition 2.1.7 (Consequence) Let L be a propositional language and AS an axiomaticsystem in L. Theory T in L is a subset of FORL. The set CNSAS(T ) of all provableformulae in T is the smallest set of formulae, which contains T , axioms of AS and is closedunder all deduction rules of AS (i.e., X ⊆ CNSAS(T ) and <X, ϕ> ∈ AS entails thatϕ ∈ CNSAS(T )). We shall write T `AS ϕ to denote ϕ ∈ CNSAS(T ) and `AS ϕ to denoteϕ ∈ CNSAS(∅).

Notice that the relation `AS can be understood as a subset of CONL and AS ⊆ `AS.

Definition 2.1.8 (Logic) Let L be a propositional language. A non-empty set L ⊆ CONL

is called a logic in language L if it is closed under arbitrary substitution and `L = L.

Logic is a consequence relation in the usual sense. The elements of a logic are consecutionsand we write X `L ϕ instead of <X,ϕ > ∈ L. Sometimes when the logic L is clear from thecontext, we write just ` instead of `L. Observe that `AS is the smallest logic containing AS.

Definition 2.1.9 (Presentation) Let L be a propositional language, AS an axiomatic sys-tem in L, and L a logic in L. We say that AS is an axiomatic system for (a presentationof) the logic L iff L = `AS. We denote the set CNSAS(∅) as THM(L). Logic is said to befinite (finitary, strongly finite) if it has some finite (finitary, strongly finite) presentation.

Later on we show that our notion of finitary logic coincides with the usual one. Observethat each logic has at least one presentation.

Definition 2.1.10 (Proof) Let L be a propositional language and AS an axiomatic systemin L. A proof of the formula ϕ in theory T is a founded tree labelled by formulae; the root islabelled by ϕ and leaves by either axioms or elements of T; and if a node is labelled by ψ andits preceding nodes are labelled by ψ1, ψ2, . . . then <{ψ1, ψ2, . . . }, ψ> ∈ AS. We shall writeT `p

AS ϕ if there is a proof of ϕ in T .

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2.1. GENERAL THEORY FOR PROPOSITIONAL LOGICS 7

We understand the tree in an top-to-bottom fashion: the leaves are at the top and theroot is at the bottom of the tree, so the fact that tree is founded just means that there is noinfinitely long branch.

Theorem 2.1.11 Let L be a propositional language and AS an axiomatic system in L. ThenT `AS ϕ iff T `p

AS ϕ.

Proof: Let us define the set CNSp(T ) = {ϕ | T `pAS ϕ}. If we show that CNSAS(T ) =

CNSpAS(T ) for each T the proof is done. Obviously CNSp

AS(T ) contains T , axioms of AS andis closed under all deduction rules of AS, thus CNSAS(T ) ⊆ CNSp

AS(T ). Reverse directionis trivial using the induction over well-founded relation. QED

Lemma 2.1.12 (Finitary logic) Let L be a logic. Then L is finitary iff for each theory Tand formula ϕ we have: if T ` ϕ then there is finite T ′ ⊆ T such that T ′ ` ϕ.

Proof: Then L is finitary then there is its finitary presentation AS. Observe that for eachfinitary AS the proofs are always finite (because the tree has no infinite branch and becauseAS is finitary each node has finitely many preceding nodes and so we can use Konig’s Lemmato get that the tree is finite). The reverse direction is almost straightforward. QED

Observe that in finitary case we can linearize the tree, i.e., define the notion of the proofin the usual way. The notion of proof allows us to illustrate one important (and usuallyoverlooked) feature of our way of introducing logic. The deduction rule < {ψ1, ψ2, . . . }, ϕ>,or better written as ψ1, ψ2, · · · ` ϕ gives us a way how to construct proof of ϕ using proofs ofψ1, ψ2, . . . . However, the meta-rule: from ` ψ1, ` ψ2, . . . get ` ϕ only tells us that if thereare proof of ψ1, ψ2, . . . , then there is proof of ϕ, without any hint how to construct it. Whenwe introduce semantics, we will se that the former corresponds to the so-called local andthe latter to the global consequence. This distinction deserves more treatment. Also rulesare rules of inference between formulae, whereas meta-rules are rules of inference betweenconsecutions. Both our definitions hide (in some sense) the default meta-rules of consequence(thinning, permutation, contraction, and cut).

Definition 2.1.13 A logic L is an extension of a logic L′ iff L′ ⊆ L. The extension isaxiomatic if the logic L is axiomatized by logic L′ (understood as an axiomatic system) withsome additional axioms. The extension is conservative if for each theory T and formula ϕin the language of L′ we have: T `L ϕ entails T `L′ ϕ.

Notice that the above definition does not mention the language of the logic in question.However, it is obvious that if L is an extension of the logic L′ then the language of L has tobe larger than the language of L′. If the language is strictly larger we speak about expansionrather than about extension.

Now we define the crucial concept of this paper: the notion of weakly implicative logic.We assume that there is a binary connective → in the propositional language. There isan obvious generalization of this concept, if we drop the condition of the presence of theconnective→ in language, and following AAL tradition we would understand the term ϕ → ψas a set of formulae of two propositional variables, where the formula ϕ is substituted forthe first one and the formula ψ for the second one. Observe that some of theorems provenin this paper remain theorems under this interpretation. It is easy to see, that in this moregeneral approach (which we will not pursuit in this paper) the defined class of logics wouldbe a subclass of the so-called equivalential logics, and in the more specific approach (weuse in this paper) the class of logics defined by the following definition is a subclass of theso-called finitely equivalential logics. If fact weakly implicative logics are precisely the finitelyequivalential logics with the set of equivalence formulae E = {p → q, q → p}. The nameweakly implicative logics is inspired by the notion of implicative logics by Helena Rasiowa (see[77]). As we will see our notion is really a generalization of her notion (her only additionalassumption is ϕ ` ψ → ϕ).

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8 CHAPTER 2. WEAKLY IMPLICATIVE (FUZZY) LOGICS

Definition 2.1.14 (Weakly implicative logics) Let L be a propositional language, suchthat (→, 2) ∈ L and let L be a logic in L. We say that L is a weakly implicative logic iff thefollowing consecutions are elements of L:

(Ref) `L ϕ → ϕ,(MP) ϕ,ϕ → ψ `L ψ,(WT) ϕ → ψ, ψ → χ `L ϕ → χ,(Cngi

c) ϕ → ψ, ψ → ϕ `L c(χ1, . . . , χi−1, ϕ, . . . , χn) → c(χ1, . . . , χi−1, ψ, . . . , χn) foreach (c, n) ∈ L and each i ≤ n.

The (Ref) is for reflexivity, (MP) is for modus ponens, (WT) is for weak transitivity, and(Cng) is for congruence. Let (c, n) ∈ L), then by (Cngc) we understand the set of consecutions{(Cngi

c) | i ≤ n}. Furthermore, by (CngL) we understand the set of consecutions⋃

c∈L

(Cngc).

Let ϕ ↔ ψ be a shortcut for {ϕ → ψ,ψ → ϕ}. Observe that we assume neither Exchangenor Weakening nor Contraction as a rules for implication. However, we have all of them asmeta-rules, i.e., the connective → is by no means an internalization of `. This approach isusually called Hilbert’s style calculus, and it is defined in the same fashion as the Hilbertcalculi for particular substructural logics. This paper can be seen as a contribution to thegeneral (universal, abstract) theory of these calculi. For another general treatment of thistopic see Restall’s book [78]. Our approaches are rather incomparable in generality. He hasmuch wider notion of logic, but we develop our logics more.

Lemma 2.1.15 (Alternative definition) The set of consecutions (CngL) can be equiva-lently replaced by one of the following consecutions:

• ϕ1 ↔ ψ1, . . . ϕn ↔ ψn `L c(ϕ1, . . . , ϕn) → c(ψ1, . . . , ψn) for each (c, n) ∈ L.

• ϕ ↔ ψ `L χ → χ′ for each formula χ ∈ FORL, where χ′ is a result of replacing someoccurrence of subformula ϕ with formula ψ in formula χ.

Proof: We show only the first part, the second one is analogous. Assume (for simplicity)that c is a binary connective, then we have:

1. ϕ1 ↔ ψ1 `L c(ϕ1, ϕ2) → c(ψ1, ϕ2) ((Cng1c) for χ2 being ϕ2)

2. ϕ2 ↔ ψ2 `L c(ψ1, ϕ2) → c(ψ1, ψ2) ((Cng2c) for χ1 being ψ1)

3. ϕ1 ↔ ψ1, ϕ1 ↔ ψ2 `L c(ϕ1, ϕ2) → c(ψ1, ψ2) (1., 2., and (WT))

The other direction is trivial (just take ψ2 = ϕ2 and use (Ref)). QED

Observe that the second part of this lemma can be understood as a substitution rule andthus we will use it heavily in the formal proofs in this paper. Now we list several lemmatawith rather trivial proofs.

Lemma 2.1.16 Let L be a weakly implicative logic in language L and L′ a logic in languageL′, which is an extension of L. Then L′ is weakly implicative logic iff for each n-ary con-nective c in L′ \ L and each i ≤ n we have ϕ → ψ, ψ → ϕ `L′ c(χ1, . . . , χi−1, ϕ, . . . , χn) →c(χ1, . . . , χi−1, ψ, . . . , χn).

Corollary 2.1.17 (Extension) An arbitrary extension of an arbitrary weakly implicativelogic in the same language is a weakly implicative logic.

Lemma 2.1.18 (Intersection) The intersection of an arbitrary system of weakly implica-tive logics is a weakly implicative logic.

Definition 2.1.19 (Consistency) Let L be a weakly implicative logic in L, T a theory inL. A theory T is consistent if there is formula ϕ such that T 6` ϕ. A logic L is consistent iffthe theory ∅ is consistent.

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2.1. GENERAL THEORY FOR PROPOSITIONAL LOGICS 9

Observe that a logic L is consistent iff L = CONL, i.e., each consecution is provable.No we introduce the notion of linear theory. In the existing fuzzy logic literature the termcomplete theory is usually used. However, we think that complete theory is a different concept(also known as maximal consistent theory). However this notion is crucial in our paper andwe want to avoid any potential confusions we decided to pick new neutral name. The reasonfor choosing the name “linear” will be obvious after Lemma 2.1.32.

Definition 2.1.20 (Linear theory) Let L be a weakly implicative logic in L, T a theoryin L. A theory T is linear if T is consistent and for each formulae ϕ, ψ we have T ` ϕ → ψor T ` ψ → ϕ.

At the end of this subsection we introduce the notion of power of the implication, to writeϕ3 → ψ instead of ϕ → (ϕ → (ϕ → ψ)) (for example).

Definition 2.1.21 Let m be a natural number and ϕ and ψ formulae. Then the formulaϕm → ψ is defined inductively as: ϕ0 → ψ = ψ and ϕi+1 → ψ = ϕ → (ϕi → ψ).

2.1.2 Semantics

We start be recalling some well-known definitions. The completeness theorem for weaklyimplicative logics (which we prove in this section) is a consequence of some more generaltheorem known in AAL. However, our concern is not to reprove known facts, we concentrateon the notion of linearity of a logical matrix, which (as far as the author knows) was not sodeeply studied.

Definition 2.1.22 (Algebra and matrix) Let L be a propositional language. An algebraA = (A,C) with signature (C, a) is called L-algebra. Let us denote the realization of c inA as cA. A pair B = (AB, DB), where AB is L-algebra and DB is a subset of A is calledL-matrix.

The elements of the set D are called designated elements. Notice that substitution can beunderstand as a endomorphism of the absolutely free L-algebra. We shall write cB insteadof cAB

Definition 2.1.23 (Evaluation) Let L be a propositional language and A an L-algebra.Then the A-evaluation is a mapping e: FORL → A, such that for each (c, n) ∈ L and eachn-tuple of formulae ϕ1, . . . , ϕn we have: e(c(ϕ1, . . . , ϕn)) = cA(e(ϕ1), . . . e(ϕn).

Of course, each A-evaluation is fully determined by its values on propositional variables.We can understand the A-evaluation as a homomorphism from the absolutely free L-algebrato A. Again, we speak about B-evaluation instead of AB-evaluation.

Definition 2.1.24 (Models) Let L be a propositional language, T a theory in L, and B anL-matrix. We say that B-evaluation is a B-model of T if for each ϕ ∈ T holds e(ϕ) ∈ DB.We denote the class of B-models of T by MOD(T,B).

Definition 2.1.25 (Semantical consequence) Let L be a logic in L, T a theory in L,and K a class of L-matrices. We say that ϕ is a semantical consequence of the T w.r.t.class K if MOD(T,B) = MOD(T ∪ {ϕ},B) for each B ∈ K; we denote it by T |=K ϕ. ByTAUT(K) we understand the set {ϕ | ∅ |=K ϕ}.

Observe that |=K⊆ CONL and that it is a logic in language L.

Definition 2.1.26 (Soundness and completeness) We say that the logic L is sound withrespect to class K iff |=K⊇ L. We say that the logic L is complete w.r.t. class K iff |=K⊆ L.

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10 CHAPTER 2. WEAKLY IMPLICATIVE (FUZZY) LOGICS

Definition 2.1.27 (L-matrices) Let L be a logic in L, T a theory in L, and B an L-matrix.We say that B is an L-matrix if L ⊆ |={B}. We denote the class of L-matrices byMAT(L). Finally, we write T |=L ϕ instead of T |=MAT(L) ϕ and TAUT(L) instead ofTAUT(MAT(L)).

Observe that for each presentation AS of L holds: L ⊆ |={B} iff AS ⊆ |={B}. The proofof the following lemma is trivial.

Lemma 2.1.28 (Soundness) The logic L is sound w.r.t. class MAT(L) (i.e., T `L ϕ,then T |=L ϕ). Furthermore, MAT(L) is the greatest class w.r.t. which is the logic L sound.

The proof of the following lemma is almost straightforward. Observe that the congruencerelation defined in the following lemma is the so-called Leibnitz congruence (see [40]).

Lemma 2.1.29 Let L be a weakly implicative logic and B an L-matrix. Then relation ≤B

defined as x ≤B y iff x →B y ∈ DB is a preorder. Furthermore, its symmetrization x ∼B yiff x ≤B y and y ≤B x is a congruence on A. Finally, the set DB is a cone w.r.t. ≤B, i.e.,if x ∈ DB and x ≤B y then y ∈ DB.

Definition 2.1.30 (Matrix preorder) Let L be a weakly implicative logic and let B be anL-matrix. The relation ≤B defined in the previous lemma is called matrix preorder of B.

The matrix is said to be ordered iff the relation ≤B is order. We denote the class ofordered L-matrixes by o−MAT(L).

The matrix is said to be linearly ordered (or just linear) iff the relation ≤B is linear order.We denote the class of linearly ordered L-matrixes by l−MAT(L).

Matrices for weakly implicative logics coincide with the class of the so-called prestandardmatrices (see Dunn [30]), whereas the ordered matrices coincides with the so-called standardmatrices. Ordered matrices also coincides with the class of the so-called reduced matrices inAAL (see [40]). Obviously, l−MAT(L) ⊆ o−MAT(L) ⊆ MAT(L). An interesting questionis where the opposite of Lemma 2.1.28 holds, i.e., whether the logic L is complete w.r.t. classMAT(L). To answer this question we recall the well-known concept of a Lindenbaum-Tarskimatrix.

Definition 2.1.31 (Lindenbaum-Tarski matrix) Let L be a weakly implicative logic inL, T be a theory in L. We define [ϕ]T = {ψ | T ` ϕ ↔ ψ} and LT = {[ϕ]T | ϕ ∈FORL}. We define L-matrix LinT , where the L-algebra has the domain LT , operationscLinT

([ϕ1]T , . . . [ϕn]T ) = [c(ϕ1, . . . ϕn)]T and the designated set D = {[ϕ]T | T `L ϕ}.

It is obvious that the definition is sound. Now we prove the fundament lemma, its parts(1) and (4) are consequences of known facts about equational logics. However, for us iscrucial the part 5. which links the notion of linear theory with the notion of linearity of amatrix.

Lemma 2.1.32 Let L be a weakly implicative logic, T a theory, and e an LinT -evaluationdefined as e(ϕ) = [ϕ]T . Then:

(1) LinT ∈ MAT(L),

(2) e ∈ MOD(T,LinT),

(3) [ϕ] ≤LinT [ψ]T iff T ` ϕ → ψ,

(4) LinT ∈ o−MAT(L),

(5) LinT ∈ l−MAT(L) iff T is a linear theory.

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2.1. GENERAL THEORY FOR PROPOSITIONAL LOGICS 11

Proof:

(1) We show that if X `L ϕ then X |=LinTϕ, i.e., if X `L ϕ and f ∈ MOD(X,LinT)

then f(ϕ) ∈ DLinT. Recall that f(ϕ) = [ψ]T for some formula ψ.

Let us define substitution σ by setting σ(v) = ψ ∈ f(v) (arbitrarily). Next we showthat for each ϕ we get σ(ϕ) ∈ f(ϕ): let ϕ = c(ϕ1, . . . , ϕn), from σ(ϕi) ∈ f(ϕi) we getσ(c(ϕ1, . . . , ϕn)) = c(σ(ϕ1), . . . σ(ϕn)) ∈ cLinT

(f(ϕ1), . . . , f(ϕn)) = f(c(ϕ1, . . . , ϕn)).Thus we know that for each ϕ we get f(ϕ) = [σ(ϕ)]T

From f ∈ MOD(X,LinT) we get that f(ψ) ∈ DLinTfor each ψ ∈ X and thus

T ` σ(ψ). From X ` ϕ we get σ(X) ` σϕ. Thus together we have T ` σ(ϕ) and sof(ϕ) = [σ(ϕ)]T ∈ DLinT

.

(2) Trivial.

(3) [ϕ] ≤LinT[ψ]T iff [ϕ] →LinT

[ψ]T ∈ DLinTiff [ϕ → ψ]T ∈ DLinT

iff T ` ϕ → ψ.

(4) Since [ϕ] ≤LinT[ψ]T and [ψ] ≤LinT

[ϕ]T entails T ` ϕ ↔ ψ the proof is obvious.

(5) Straightforward. QED

Theorem 2.1.33 (Completeness) Let L be a weakly implicative logic in L. Then for eachtheory T and formula ϕ holds: T ` ϕ iff T |=L ϕ.

Proof: One direction is Lemma 2.1.28. Reverse direction: we get LinT ∈ MAT(L) (fromLemma 2.1.32) and for LinT -evaluation e defined as e(ψ) = [ψ]T holds e ∈ MOD(T,LinT).Thus from T |=L ϕ we get that [ϕ]T = e(ϕ) ∈ DLinT

and so T ` ϕ. QED

The proofs of the following two corollaries are trivial.

Corollary 2.1.34 Let L be a weakly implicative logic. Then THM(L) = TAUT(L).

Corollary 2.1.35 Let L be a weakly implicative logic in L, T a theory in L. Then for eachformula ϕ holds T ` ϕ iff T |=L ϕ iff T |=o−MAT(L) ϕ.

Now we recall the definition of a direct and subdirect product of matrices. These defi-nitions are obvious, if we understand matrix as a first-order structure with functions corre-sponding to the operations and one unary predicate, whose realization is the set of designatedelements.

Definition 2.1.36 Let L = (VAR,C, a) be a propositional language and I a (nonempty)class of L-matrices. The direct product of matrices from I (denoted as

∏B∈I

B) is a ma-

trix X = (X, (cX)c∈C, DX), where X is a cartesian product od domain of matrices from I,operations are defined pointwise, and (xB)B∈I ∈ DX iff xB ∈ DB for each B ∈ I.

Furthermore, we say that the matrix X is a subdirect product of matrices from I if thereis an embedding f : X → ∏

B∈I

B, such that for each B ∈ I holds πB(f(X)) = B.

By πB we mean the projection to the component B.

Lemma 2.1.37 Let L be a logic and I a class of L-matrices. Then each matrix X which isa subdirect product of matrices from I is an L-matrix.

Proof: Trivial. QED

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12 CHAPTER 2. WEAKLY IMPLICATIVE (FUZZY) LOGICS

2.1.3 Fuzzy logic

In the previous section we have seen that each weakly implicative logic is sound and completew.r.t. class of its ordered matrices. There is an obvious question, which of them are completew.r.t. class of its linearly ordered matrices. This will lead us to the second central definitionof this paper: the notion of weakly implicative fuzzy logics.

Definition 2.1.38 (Fuzzy logics) Weakly implicative logic L in language L is called fuzzylogic if L = |=l−MAT(L) (i.e., if the logic L is complete w.r.t. linearly ordered L-matrices).

The full proper name of the above defined class of logics if weakly implicative fuzzy logics,however since all the logics we encounter from now on are weakly implicative, we just saythat a logic is fuzzy. There is a joint paper by the author and Libor Behounek [8] givenphilosophical, methodological, and pragmatical reasons for using the term fuzzy, and for formaldelimitation of the existing informal class of fuzzy logic. Observe that in fuzzy logics we haveTHM(L) = TAUT(L) = TAUT(l−MAT(L)).

We are going to show the equivalent definitions the class of weakly implicative fuzzylogics.

Definition 2.1.39 (Linear extension) A weakly implicative logic L has the Linear Exten-sion Property (LEP) if for each theory T formula ϕ such that T 6` ϕ there is a linear theoryT ′, such that T ⊆ T ′ and T ′ 6` ϕ.

Definition 2.1.40 (Prelinearity) A weakly implicative logic L has the Prelinearity Prop-erty (PP) if for each theory T we get T ` χ whenever T, ϕ → ψ ` χ and T, ψ → ϕ ` χ.

Definition 2.1.41 (Subdirect Decomposition) A weakly implicative logic L has the Sub-direct Decomposition Property (SDP) if each ordered L-matrix is a subdirect product of linearL-matrices.

Theorem 2.1.42 (Characterization of fuzzy logics) A weakly implicative logic is fuzzyiff it has LEP.

Proof: Assume that L is fuzzy logic. If L and T 6` ϕ then there is linear L-matrix B andB-evaluation e such that e(ϕ) 6∈ DB. Let us define T ′ = T ∪ {ψ | e(ψ) ∈ DB}. ObviouslyT ⊆ T ′ and T ′ 6` ϕ. Since ≤B is linear order we get e(χ) ≤B e(δ) or e(δ) ≤B e(χ) for each χand δ. Thus either e(χ → δ) ∈ DB or e(δ → χ) ∈ DB.

Reverse direction: we need to prove that `L = |=l−MAT(L). One inclusion is trivialconsequence of Theorem 2.1.33, the reverse one we prove by contradiction: assume thatT 6` ϕ. Let us take a linear supertheory T ′ 6` ϕ. We know that LinT ′ ∈ l−MAT(L)(from Lemma 2.1.32), LinT ′ is linearly ordered and for the LinT ′ -evaluation e defined ase(ψ) = [ψ]T ′ holds e ∈ MOD(T ′,LinT′). This entails that also e ∈ MOD(T,LinT′). Therest of the proof is analogous to the one for weakly implicative logics. QED

Lemma 2.1.43 Let L be a weakly implicative logic with LEP. Then L has PP.

Proof: We argument contrapositively: let T 6` χ, then (using LEP) there is linear theoryT ′, such that T ′ 6` χ. Assume that T ′ ` ϕ → ψ, then obviously T, ϕ → ψ 6` χ. QED

To reverse this lemma it seems that we need one additional assumption: the logic hasto be finitary. However, it is obvious that for some infinitary rules the equivalence will holdas well, the question exactly for which subclass of weakly implicative logics the equivalenceholds seems to be interesting open problem.

Lemma 2.1.44 Let L be a finitary weakly implicative logic with PP, T a theory, and ϕ aformula, such that T 6` ϕ. Then there is a linear theory T ′, such that T ⊆ T ′ and T ′ 6` ϕ.

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2.1. GENERAL THEORY FOR PROPOSITIONAL LOGICS 13

Proof: Let ||L|| = κ. Let us enumerate all tuples of formula by ordinals < κ. Let T0 = T .We construct theories Tµ using transfinite induction. Let Tµ =

⋃ν<µ

Tν .

Observe that if Tν 6` ϕ for each ν < µ then Tµ 6` ϕ as well (the logic is finitary). Weknow that either Tµ ∪ {ϕµ → ψµ} 6` ϕ or Tµ ∪ {ψµ → ϕµ} 6` ϕ (otherwise using PP we getcontradiction with Tµ 6` ϕ), define Tµ accordingly. Finally, define T ′ = Tκ; obviously T ′ 6` ϕand T ′ is linear. QED

Theorem 2.1.45 (Equivalent characterization) Let L be a finitary weakly implicativelogic. Then the following are equivalent:

(1) L is a fuzzy logic,

(2) L has LEP,

(3) L has PP,

(4) L has SDP.

Proof: (1) → (2) : C.f. Theorem 2.1.42.(2) → (3) : C.f. Lemma 2.1.43.(3) → (4) : Let us denote the language of L as L = (VAR,C, a). Let B = (A, D) be

an ordered L-matrix and ≤ its matrix order. Let us take VAR′ = A, for clearness we willuse va ∈ VAR′ and a ∈ A. We define the propositional language L′ = (VAR′,C, a) andthe logic L′ as `SUBL′ (L). The logic L′ has obviously PP as well and thus it has LEP (usingLemma 2.1.44). Notice that here we use the assumption that L is finitary.

We define T = {c(va1 , . . . van) ↔ vcB(a1,...an) | c ∈ C,a(c) = n, and a1, . . . , an ∈ A} ∪{va | a ∈ D}. Observe that B is an L′-matrix. Let us define B-evaluation e(va) = a andobserve that {e} = MOD(T,B). Now we show T ` va → vb iff a ≤ b: one direction is simple(from e ∈ MOD(T,B) we get that e(va) →B e(vb) ∈ D and so a ≤ b); the other direction issimilar (if a ≤ b then a → b ∈ D thus T ` va→b and so T ` va → vb). Finally, we observethat for each formula ϕ there is a ∈ A, such that T ` ϕ ↔ va

Let us define the set I of all linear theories extending T . Next we define L-matrixX =

∏S∈I

LinS (direct product of Lindenbaum-Tarski matrices). Finally, we define f(a) =

([va]S)S∈I.Now we show that f is an embedding of B into X: since for each S ∈ I we have

[vcB(a1,...,an)]S = [c(va1 , . . . van)]S = cLinS([va1 ]S , . . . [van ]S) and so we get f(cB(a1, . . . , an))

= (cLinS([va1 ]S , . . . [van ]S))S∈I = cX(f(a1), . . . f(an)). Since obviously a ∈ D entails that

f(a) ∈ DX (from a ∈ D we have T ` va and so S ` va and thus [va]S ∈ DLinS for each S ∈ I)we know that f is a morphism. It remains to be shown that f is one-one: from a 6= b we getthat either a 6≤ b or b 6≤ a. Let us assume that a 6≤ b then T 6` va → vb, using Lemma 2.1.44we know that there is a linear theory S ∈ I such that S 6` va → vb thus [va]S 6≤LinS

[vb]S andso f(a) 6≤X f(b).

Finally, we observe that for each S we have πS(f(A)) = LS (just recall that for each ϕthere is a ∈ A, such that T ` ϕ ↔ va).

Since LinS is linearly ordered L-matrix for each S ∈ I and B can be embedded into directproduct of (LinS)S∈I in the way that πS(f(A)) = LS . We conclude that B is a subdirectproduct of linearly ordered L-matrices.

(4) → (1) : Trivial.QED

Lemma 2.1.46 Let L be a fuzzy logic. Then (ϕ → ψ)i → χ, (ψ → ϕ)j → χ `L χ for eachnaturals i and j.

Proof: Let T = {(ϕ → ψ)i → χ, (ψ → ϕ)j → χ}. Obviously T, ϕ → ψ ` χ andT, ψ → ϕ ` χ (using (MP)). Since each fuzzy logic has PP the proof is done. QED

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14 CHAPTER 2. WEAKLY IMPLICATIVE (FUZZY) LOGICS

Table 2.1: Structural rules

consecution symbol nameϕ ` ψ → ϕ W Weakening

ϕ → (ψ → χ) ` ψ → (ϕ → χ) E Exchangeϕ → (ϕ → ψ) ` ϕ → ψ C Contraction

Table 2.2: Addition rules

consecution symbol name(ϕ → ψ) → χ, (ψ → ϕ) → χ ` χ PL prelinearity

` ϕ → ((ϕ → ψ) → ψ) As assertionϕ → ψ ` (ψ → χ) → (ϕ → χ) Sf suffixingψ → χ ` (ϕ → ψ) → (ϕ → χ) Pf prefixing

` ϕ → (ϕ → ϕ) M mingle

The proofs of the following lemmata are obvious. The first one is especially important.It allows us to speak about weakest fuzzy logic with some property (eg. being stronger thatsome know logic).

Lemma 2.1.47 (Intersection) The intersection of an arbitrary system of fuzzy logics is afuzzy logic.

Lemma 2.1.48 (Axiomatic extension) An axiomatic extension of arbitrary fuzzy logicin the same language is a fuzzy logic.

The assumption of being in the same language can be omitted if we assume some addi-tional properties of the logic in question.

Lemma 2.1.49 (Conservative expansion) Let L′ be a conservative expansion of a fini-tary fuzzy logic L. Then L is fuzzy logic as well.

Proof: We know that L′ has PP, we show that L has PP as well and because L is finitarywe get that L is fuzzy. Let us take theory T and formulae ϕ,ψ, χ in language of L. Assumethat T, ϕ → ψ `L χ and T, ψ → ϕ `L χ, then also T, ϕ → ψ `L′ χ and T, ψ → ϕ `L′ χ.Using PP for L′ we get T `L′ χ. Conservativeness completes the proof. QED

2.2 Special propositional logics

In this section we introduce some additional rules (see Tables 2.1 and 2.2) and some additionalconnectives (see table 2.3) and prove some facts about these extensions. It is just a sketch ofa huge work to be done. The ultimate goal is to characterize known logics and put them intoour context with modularly designed Hilbert’s style calculi. Having this done, we identifywhich of them are fuzzy, or we find minimal fuzzy logics extending some known logics (eg.minimal fuzzy logic over intuitionistic logic is Godel logic, minimal fuzzy logic over FullLambek calculus with exchange and weakening is MTL logic, etc.) A lot of work in thisdirection can be found in Restall’s book [78] (of course for logic in general, he does not dealwith fuzzy logic in any form).

Having stronger logic/language can lead to simplified semantics, eg. having 1 we canreplace matrices with ordered structures (the designated set will be the upper cone of 1B),having one of the lattice connectives we can even work with algebras, thus being able to

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2.2. SPECIAL PROPOSITIONAL LOGICS 15

Table 2.3: Propositional connectives

Symbol Arity Name Alternative name> 0 verum additive truth1 0 one multiplicative truth0 0 zero multiplicative falsum⊥ 0 falsum additive falsum4 1 Baaz delta globalization∧ 2 min-conjunction additive conjunction∨ 2 max-disjunction additive disjunction& 2 strong conjunction fusion, multiplicative conjunctionà 2 c-implication reverse implication

use powerful methods of Abstract Algebraic Logics (which, of course, we can do anyway,but some theorems of ALL hold in algebraizable logics only). There are other possiblesimplification of the semantics allowed be adding some structural rules (weakening leads toalgebraic semantics, exchange to ordered structures, with ϕ being valid iff e(ϕ → ϕ) ≤ e(ϕ),etc.).

2.2.1 Adding rules

In this section we restrict ourselves to the propositional languages with implication only.The basic rules correspond to the structural rules are exchange, contraction and weakening(see Table 2.1), extended by some additional important rules summarized in Table 2.2. Forthe sake of simplicity we speak about rules and not the rule schemata. We formulate themas rules, however in some situation we can use their stronger forms—we formulate then asaxioms. To do this in a general way we present the following definition.

Definition 2.2.1 Let R be a unary deduction rule of the form ϕ ` ψ. By the correspondingaxiom we understand axiom ` ϕ → ψ, we will denote it as ax (R).

Of course if L is weakly implicative logic then `L ϕ → ψ entails ϕ `L ψ, i.e., if ax (R) ∈ Lthen R ∈ L. Recall that in the literature some of the axioms are known under differentnames.

Definition 2.2.2 (Adding rules) Let L be a weakly implicative logic in language {→} andQ be a set of consecutions. We say that L is an Q-implication fragment if the consecutionsfrom Q are elements of L.

We say that L is an fuzzy Q-implication fragment if L is fuzzy logic and L is Q-implicationfragment.

Definition 2.2.3 The weakest Q-implication fragment is denoted as MIN(Q). Let us denotethe weakest weakly implicative logic as WIL. Furthermore, the weakest fuzzy Q-implicationfragment is denoted as FUZZ(Q).

Both definition are sound thanks to the Lemma 2.1.18 and 2.1.47. Let us by WIL denotethe logic MIN(∅) (i.e., the weakest weakly implicative logic). Then obviously MIN(Q) =`WIL∪Q, i.e., the presentation of MIN(Q) results from the presentation of WIL by addingconsecutions from Q.

The following lemma shows the interplay between transitivity and exchange.

Lemma 2.2.4 The following logics are equivalent:

(1) BCI logic (implicational fragment of intuitionistic linear logic),

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16 CHAPTER 2. WEAKLY IMPLICATIVE (FUZZY) LOGICS

Table 2.4: Known implicational fragments

consecutions implicational fragment of∅ intuitionistic linear logicC relevance logic

C, M relevance logic with mingleW affine intuitionistic linear logic

W, C intuitionistic logic

(2) MIN({ax (Sf), As}),(3) MIN({ax (Sf), ax (E)}),(4) MIN({ax (Pf), ax (E)}),(5) MIN({Pf, ax (E)}),(6) MIN({ax (Sf), E}),(7) MIN({ax (Pf), E}).

Proof: We show that each logic is stronger than the next one (cyclicly):(1) ⊇ (2): Recall that BCI logic is axiomatized by ax (Sf), As, (Ref), and (MP). So all

we have to do is to show (Cng→) but this is almost straightforward.(2) ⊇ (3): All we need to show is `MIN({ax(Sf),As}) ax (E)

(i) ψ → ((ψ → χ) → χ) As(ii) (ψ → ((ψ → χ) → χ)) → ((((ψ → χ) → χ) → (ϕ → χ)) → (ψ → (ϕ → χ))) ax (Sf)(iii) (((ψ → χ) → χ) → (ϕ → χ)) → (ψ → (ϕ → χ)) (i), (ii), (MP)(iv) (ϕ → (ψ → χ)) → (((ψ → χ) → χ) → (ϕ → χ)) ax (Sf)(v) (ϕ → (ψ → χ)) → (ψ → (ϕ → χ)) (iv), (iii), (WT)

(3) ⊇ (4): Trivial.(4) ⊇ (5): Trivial.(5) ⊇ (6): All we need to show is `MIN({Pf,ax(E)}) ax (Sf):

(i) (ψ → χ) → (ψ → χ) (Ref)(ii) ψ → ((ψ → χ) → χ) (i), ax (E), (MP)(iii) (ϕ → ψ) → (ϕ → ((ψ → χ) → χ)) (ii),Pf(iv) (ϕ → ((ψ → χ) → χ)) → ((ψ → χ) → (ϕ → χ)) ax (E)(v) (ϕ → ψ) → ((ψ → χ) → (ϕ → χ)) (iii), (iv),Pf

(6) ⊇ (7): Trivial.(7) ⊇ (1): Trivial. QED

Notice two open problems: what about logics MIN({Sf, ax (E)}) and MIN({ax (Pf),As})?Table 2.4 puts some known logics into our context (we list axioms which has to be added

to MIN(ax (Sf),E)). We can add all of them as rules or axioms—in the presence of exchangethese two options are equivalent—as shown by the following observation:

1. MIN(ax (Sf), E) = MIN(ax (Sf), ax (E)),

2. MIN(ax (Sf), E, W) = MIN(ax (Sf), ax (E), ax (W)),

3. MIN(ax (Sf), E, C) = MIN(ax (Sf), ax (E), ax (C)).

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2.2. SPECIAL PROPOSITIONAL LOGICS 17

Proof: Part 1. was shown in Lemma 2.2.4. Part 2. is easy. To prove part 3. we only showthat `MIN(ax(Sf),E,C) ax (C):

(i) (ϕ → (ϕ → ψ)) → (ϕ → (ϕ → ψ)) (Ref)(ii) ϕ → ((ϕ → (ϕ → ψ)) → (ϕ → ψ)) (i) and E.(iii) ϕ → (ϕ → ((ϕ → (ϕ → ψ)) → ψ)) (ii), ax (E), ax (Sf)(iv) ϕ → ((ϕ → (ϕ → ψ)) → ψ) (iii) and C(v) (ϕ → (ϕ → ψ)) → (ϕ → ψ) (iv) and E

QED

Before we proceed further we observe some properties of BCI. The proofs are easy (byinduction).

Lemma 2.2.5 It holds:

1. `BCI (ϕn → (ψm → χ)) → (ψm → (ϕn → χ)),

2. `BCI (ϕn → ψ) → ((ψ → χ) → (ϕn → χ)).

Now we present an important definition—the deduction theorem—for rather wide classof weakly implicative logic. We present it in an unusual form. Our formulation allows us toshow exactly which logics have this deduction theorem.

Definition 2.2.6 Let L be a logic. We say that L has Implicational Deduction Theorem(DT→) if L has a presentation AX, where (MP) is the only deduction rule and for eachtheory T , formulae ϕ,ψ, and proof P of ψ in theory T, ϕ (in the presentation AX) we have:T, ϕ ` ψ iff T ` ϕn → ψ, where n is a number of occurrences of ϕ in the leaves of the proofP and there is a proof P′ of ϕn → ψ in T , such that each ψ ∈ T occurs in the leaves of P

same number of times as in the leaves of P′.

Observe that each logic with DT→ has also the “standard” form of the local deductiontheorem.

Corollary 2.2.7 Each logic with DT→ has so called Local Deduction Theorem (LDT): foreach theory T and formulae ϕ, ψ: T, ϕ ` ψ iff there is n such that T ` ϕn → ψ.

Now we present sufficient and necessary condition for L to have DT→.

Theorem 2.2.8 (Deduction theorem) Let L be a logic. Then L has DT→ iff L has apresentation, where (MP) is the only deduction rule and the implicational fragment of L isan extension of BCI.

Proof: First direction: All we have to do is to show that `L ax (Sf) and `L ax (E) (the rest isthe immediate consequence of the definition of DT→). Observe that ϕ,ψ, ϕ → (ψ → χ) `L χand each of the premises is used exactly once, applying DT→ three times we get `L ax (E).Observe that ϕ,ϕ → ψ, ψ → χ `L χ and each of the premises is used exactly once, applyingDT→ three times we get `L ax (Sf).

Reverse direction: we need to prove two directions. One is obvious. To prove the otherone we use the induction over the proof of ψ in T, ϕ (in AX). We show that it holds for eachχ in the proof P′:

• χ is a leaf of P, i.e., χ ∈ T , χ is an axiom, or χ = ϕ: trivial.

• χ has predecessors ψ2 = ψ1 → χ and ψ1, using the induction property we get thatT ` ϕn → (ψ1 → χ) and T ` ϕm → ψ1 (and the number of occurrences of formulaefrom T is the same), we distinguish two cases:

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18 CHAPTER 2. WEAKLY IMPLICATIVE (FUZZY) LOGICS

– ψ2 = ϕ, from Lemma 2.2.5 (2) we get ` (ϕm → ψ1) → ((ψ1 → χ) → (ϕm → χ)),thus we get T ` ϕ → (ϕm → χ) and so T ` ϕm+1 → χ.

– ψ2 6= ϕ, using Lemma 2.2.5 (1) we get T ` ψ1 → (ϕn → χ). From Lemma 2.2.5(2) we obtain T ` ϕm → (ϕn → χ). Thus T ` ϕm+n → χ.

In both cases the number of occurrences of formulae from T is not changed. QED

We have proved even more: given proof of ψ in T, ϕ, we give a constructive procedurewhich produces the proof of ϕn → ψ in T .

Corollary 2.2.9 Let L be a language and L a logic in L, such that L has a presentation,where (MP) is the only deduction rule and implicational fragment of L is an extension ofBCI. Then L has LDT.

In the presence of LDT we can prove the “converse” of Lemma 2.1.46. Thus having anequivalent definition of fuzzy logics in some class of logics.

Lemma 2.2.10 Let L be a finitary logic with LDT. Then `L (ϕ → ψ)i → χ, (ψ → ϕ)j →χ `L χ iff L is fuzzy.

Proof: One direction is just Lemma 2.1.46. To prove the other direction we only show thatL has PP. Assume that T, ϕ → ψ ` χ and T, ϕ → ψ ` χ. By LDT we get T ` (ϕ → ψ)i → χand T ` (ψ → ϕ)j → χ and so we have T ` χ. QED

At the end of this section we present an infinite axiomatic system for the minimal fuzzy{ax (Sf), E,W}-implication fragment. The question whether there is finite system seems tobe open. We decided to use classical names for axioms ax (Sf), ax (E), and ax (W).

Definition 2.2.11 The fuzzy BCK logic (FBCK) has the following presentation:

B ` (ϕ → ψ) → ((ψ → χ) → (ϕ → χ)),C ` (ϕ → (ψ → χ)) → (ψ → (ϕ → χ)),K ` ϕ → (ψ → ϕ),Fn ` ((ϕ → ψ)n → χ) → (((ψ → ϕ)n → χ) → χ) for all n,(MP) ϕ,ϕ → ψ ` ψ.

Observe that FBCK = MIN(ax (Sf),E, W, Fn). Using Lemma 2.2.4 we could write severaldifferent equivalent axiomatic systems. Also observe that ordered FBCK-matrices are exactlyBCK-algebras satisfying axioms Fn.

Theorem 2.2.12 FBCK = FUZZ({ax (Sf), E,W}).

Proof: First, we show that FBCK, is fuzzy logic. Using Theorem 2.1.45 it is enough toshow that FBCK has PP. Let T, ϕ → ψ ` χ and T, ψ → ϕ ` χ then using DT→ we get thatT ` (ϕ → ψ)m → χ and T ` (ψ → ϕ)n → χ for some n and m. Let us take t = max(m,n),using W we get the T ` (ϕ → ψ)t → χ and T ` (ψ → ϕ)t → χ, axiom Ft completes theproof.

Next, we have to show that each fuzzy logic extending BCK proves Fn. We recall thateach fuzzy logic has PP. Now observe that

(i) ((ϕ → ψ)n → χ) → ((ϕ → ψ)n → χ) (Ref)(ii) (ϕ → ψ)n → (((ϕ → ψ)n → χ) → χ) (i) and Lemma 2.2.5 1.(iii) ϕ → ψ ` (((ϕ → ψ)n → χ) → χ) (ii)(iv) ϕ → ψ ` ((ψ → ϕ)n → χ) → (((ϕ → ψ)n → χ) → χ) (iii) and K

(v) ϕ → ψ ` ((ϕ → ψ)n → χ) → (((ψ → ϕ)n → χ) → χ) (iv) and C

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2.2. SPECIAL PROPOSITIONAL LOGICS 19

Table 2.5: Rules

Consecution symbol Name match forψ&ϕ → χ a` ϕ → (ψ Ã χ) ÃR Ã-residuation Ãϕ → (ψ → χ) a` ϕ&ψ → χ R residuation &

ϕ → ψ a` ϕ Ã ψ Imp implications Ã` ϕ → > Tr veritas ex quolibet >` ⊥ → ϕ Fa ex-falso quodlibet ⊥

ϕ a` 1 → ϕ PP push and pop 1ϕ → χ, ψ → χ ` ϕ ∨ ψ → χ ∨1 supremum ∨

` ϕ → ϕ ∨ ψ ∨2 idempotency ∨` ϕ ∨ ψ → ψ ∨ ϕ ∨3 commutativity ∨

χ → ϕ, χ → ψ ` χ → ϕ ∧ ψ ∧1 infimum ∧` ϕ ∧ ψ → ϕ ∧2 idempotency ∧

` ϕ ∧ ψ → ψ ∧ ϕ ∧3 commutativity ∧

Table 2.6: Matching rules for 4

Consecution symbol Name` 4(ϕ → ψ) → (4ϕ →4ψ) 41 4-monotonicity

` 4ϕ → ϕ 42 4-reflexivity` 4ϕ →44ϕ 43 4-transitivity

` 4ϕ → (4ψ → ϕ) 4W 4-weakening` 4(4ϕ → (4ϕ → ψ)) → (4ϕ → ψ) 4C 4-contraction

` 4(4ϕ → (4ψ → χ)) → (4ψ → (4ϕ → χ)) 4E 4-exchangeϕ ` 4ϕ (NEC) necessitation

By replacing ϕ and ψ in (iv) we get:

(vi) ψ → ϕ ` ((ϕ → ψ)n → χ) → (((ψ → ϕ)n → χ) → χ)

Since we L has PP we get ` Fn QED

Since obviously BCK = MIN(ax (Sf), E,W) we get the following corollary.

Corollary 2.2.13 FBCK is the weakest fuzzy logic stronger than BCK.

We can alter the axioms Fn by using two different natural number m,n as “exponents”,we get axioms Fm,n. If we add axioms Fm,n to the BCI logic, we get fuzzy logic (by Lemma2.2.10). However, we are not able to the prove the converse statement, i.e., that this logic isminimal fuzzy logic over BCI.

2.2.2 Adding connectives

As mentioned before, we consider the connectives from Table 2.3. For the matching rulesfor the particular connective see Tables 2.5 and 2.6. There are many interesting interplaysbetween them and the matching consecutions (eg. in the presence of residuation rule we canprove that the residuation axiom is equivalent to the associativity axiom for &). Again, wepresent only the basic definitions, a lot of work is to be done yet.

Definition 2.2.14 (Adding connectives) A logic L is Q-weakly implicative logic in L ifits implication fragment is Q-implication fragment and if some of the connectives from the

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20 CHAPTER 2. WEAKLY IMPLICATIVE (FUZZY) LOGICS

Table 2.7: Known logics

L Q MINL(Q)⊥, & ∨ ax (Sf),E, W, C Intuitionistic logic

Ã, ⊥, 1, 0, &, ∧, ∨ Sf, Pf Full Lambek⊥, & ax (Sf), E, W, C, P� Godel logic⊥, &, ∧ ax (Sf), E, W, P� MTL logic

&, ∧ ax (Sf), E, W, P� MTLH logic

set {>, 0, 1,⊥,∧,∨,&, Ã,4} are in L, then their matching rules and instances of (Cngc)rules for the connectives c in question are elements of L. We say that L is an Q-fuzzy logicif L is fuzzy logic and L is Q-weakly implicative logic.

To simplify things we will write that L is Q-logic instead of L is Q-weakly implicativelogic.

Definition 2.2.15 Let L be a propositional language. We denote the weakest Q-logic in L

as MINL(Q) and the weakest fuzzy Q-logic is denoted as FUZZL(Q).

Table 2.7 puts some known logics into our context. The only thing we need to observe isthe fact that in presence of ax (Sf) and E we get from the residuation rules the residuationaxioms (having this it is easy to show that & is associative. We show one direction:

(i) (ϕ → (ψ → χ)) → (ϕ → (ψ → χ)) (Ref)(ii) ϕ → ((ϕ → (ψ → χ)) → (ψ → χ)) (i) and E(iii) ϕ → (ψ → ((ϕ → (ψ → χ)) → χ)) (ii), ax (E), (WT)(iv) ϕ&ψ → ((ϕ → (ψ → χ)) → χ) (iii) and residuation rule(v) (ϕ → (ψ → χ)) → (ϕ&ψ → χ) (iv),E.

The logic FUZZ{&}(ax (dT ), E,W) is the newest logic of Petr Hajek—the quasihoop logic(for details see [51]).

Definition 2.2.16 The quasihoop logic (QH) has the following presentation:

(1) ` (ϕ → ψ) → ((ψ → χ) → (ϕ → χ)),(2) ` ϕ&ψ → ψ&ϕ,(3) ` ϕ&ψ → ϕ,(4) ` (ϕ → (ψ → χ)) → (ϕ&ψ → χ),(5) ` (ϕ&ψ → χ) → (ϕ → (ψ → χ)),Fn ` ((ϕ → ψ)n → χ) → (((ψ → ϕ)n → χ) → χ),(MP) ϕ,ϕ → ψ ` ψ.

Observe that axiom (2) corresponds to E axiom (3) to W. Axioms (4) and (5) are ax-iomatic version of residuation rules R. The ordered QH-matrices are just BCK(RP)-algebras.

2.2.3 The connective ∨Having disjunction in the language, we can express several concepts of this paper in waysmore usual in the literature. In this subsection we assume that ∨ ∈ L for each logics weencounter here.

Lemma 2.2.17 Let L be a weakly implicative logic. Then ϕ ∨ ψ, ϕ → ψ ` ψ.

Proof: We give a formal proof:

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2.2. SPECIAL PROPOSITIONAL LOGICS 21

(i) ϕ → ψ, ψ → ψ ` ϕ ∨ ψ → ψ ∨1(ii) ϕ ∨ ψ, ϕ → ψ ` ϕ ∨ ψ → ψ (i)(iii) ϕ ∨ ψ, ϕ → ψ ` ψ (ii) and (MP)

QED

Now we define the notion of prime theory. This is more known and more widely usedconcept than the concept of linear theory. However, we will see that in fuzzy logics bothnotions coincide.

Definition 2.2.18 (Prime theory) Let L be a weakly implicative logic. A theory T isprime if from T ` ϕ ∨ ψ we get T ` ϕ or T ` ψ.

Definition 2.2.19 (Prime extension) A weakly implicative logic L has the Prime Exten-sion Property (PEP) if for each theory T formula ϕ such that T 6` ϕ there is a prime theoryT ′, such that T ⊆ T ′ and T ′ 6` ϕ.

Definition 2.2.20 (Proof by cases) A weakly implicative logic L has the Proof by CasesProperty (PCP) if for each theory T we get T, ϕ ∨ ψ ` χ whenever T, ϕ ` χ and T, ψ ` χ.

Observe that above defined principles PCP and PEP differ from seemingly analogousprinciples PP and LEP. For example Intuitionistic logic has both PCP and PEP but doesnot have the other two. Let us examine this in more details:

Lemma 2.2.21 Let L be a weakly implicative logic. Then:

1. each linear theory is prime;

2. if L has PP we have ` (ϕ → ψ) ∨ (ψ → ϕ);

3. if ` (ϕ → ψ) ∨ (ψ → ϕ) then each prime theory is linear;

4. if L has PP then L has PCP;

5. if ` (ϕ → ψ) ∨ (ψ → ϕ) and L has PCP then L has PP.

Proof:

1. Let T ` ϕ ∨ ψ. Since T is linear we know that T ` ϕ → ψ or T ` ψ → ϕ. Thus (usingLemma 2.2.17) we get T ` ψ or T ` ϕ.

2. Trivial.

3. Trivial.

4. From PP and ∨2 we easily get `L (ϕ → ψ) ∨ (ψ → ϕ). Now let T be a theory suchthat T, ϕ ` χ and T, ψ ` χ. Using Lemma 2.2.17 we know that T, ϕ ∨ ψ, ϕ → ψ ` ψand T, ϕ ∨ ψ,ψ → ϕ ` ϕ. Thus T, ϕ ∨ ψ, ϕ → ψ ` χ and T, ϕ ∨ ψ,ψ → ϕ ` χ. PPcompletes the proof.

5. just observe that from T, ϕ → ψ ` χ and T, ψ → ϕ ` χ we get T, (ϕ → ψ)∨(ψ → ϕ) ` χ.Knowing that `L (ϕ → ψ) ∨ (ψ → ϕ) we get T ` χ. QED

This lemma has three interesting corollaries.

Corollary 2.2.22 In fuzzy logic the theory T is prime iff T is linear.

Corollary 2.2.23 A logic L is fuzzy iff L has PEP and ` (ϕ → ψ) ∨ (ψ → ϕ).

Corollary 2.2.24 A finitary logic L is fuzzy iff L has PCP and ` (ϕ → ψ) ∨ (ψ → ϕ).

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22 CHAPTER 2. WEAKLY IMPLICATIVE (FUZZY) LOGICS

2.2.4 The connective 4The connective 4 is a special one, in intuitionistic logic it is known as globalization (withsome additional assumptions); in linear logic it is a kind of exponential; and in fuzzy logicsit is known as Baaz delta. Roughly speaking, this connective allows us controlled use ofstructural rules. In this subsection we assume that 4 is an element of all the propositionallanguages L (unless the opposite is explicitly mentioned).

Now we present the analog of Definition 2.2.6 and prove the analog of Theorem 2.2.8.However, the 4 connective allows us much simpler formulations (resembling the deductiontheorem of modal logic S4) . In the following definition and theorem we understand L asarbitrary weakly implicative logic (not necessarily with matching rules 41, 42, 43,4W,4C,4E, and (NEC)).

Definition 2.2.25 Let L a weakly implicative logic. We say that L has Delta DeductionTheorem (DT4) if for each theory T and formulae ϕ,ψ we have: T, ϕ ` ψ iff T ` 4ϕ → ψ.

Now we present sufficient and necessary condition for L to have DT4. And we also showthat in each weakly implicative logic with DT4 the matching rules for 4 hold.

Theorem 2.2.26 (Deduction theorem) Let L be a finitary logic. Then L has DT4 iff Lhas some presentation AX, where (MP) and (NEC) are the only deduction rules and all thematching axioms for 4 are provable in L.

Proof: First direction: Assume that L has DT4, then obviously L has presentation where(MP) and (NEC) are the only deduction rules (just replace each rule ϕ1, . . . ϕn ` ψ with thefollowing axiom ` 4ϕ1 → (· · · → (4ϕn → ψ) . . . )). All we have to do is to show that thematching rules for 4 hold:

• (NEC): From ` 4ϕ →4ϕ we get ϕ ` 4ϕ (using DT4).

• 41: From ϕ,ϕ → ψ ` ψ we get ϕ,ϕ → ψ ` 4ψ (using (NEC)). Applying DT4 twicecompletes the proof.

• 42: From ϕ ` ϕ we get ` 4ϕ → ϕ (using DT4).

• 43: From ϕ ` 4ϕ we get ` 4ϕ →44ϕ (using DT4).

• 4W: From ϕ,ψ ` ϕ we get ` 4ϕ → (4ψ → ϕ) (using DT4 twice).

• 4C: We know that ϕ,4ϕ → (4ϕ → ψ) ` ψ ((NEC) and (MP) twice). Applying DT4twice completes the proof.

• 4E: We know that ϕ,ψ,4ϕ → (4ψ → χ) ` χ ((NEC) twice and (MP) twice).Applying DT4 three times completes the proof.

Reverse direction: we need to prove two directions. One is obvious. To prove the otherone we use the induction over the proof of ψ in T, ϕ (in AX). We show that it holds for eachχ in the proof of ψ in T, ϕ:

• χ ∈ T , χ is an axiom - trivial using 4W.

• χ = ϕ - trivial using 42.

• χ = 4ψ1 is obtained from its predecessor ψ1 by (NEC). From the induction propertyfor ψ2 we know that T ` 4ϕ → ψ1. We apply (NEC), 41, and (MP) to get thatT ` 44ϕ →4ψ1. Using 43 and (WT) we get T ` 4ϕ →4ψ1.

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2.2. SPECIAL PROPOSITIONAL LOGICS 23

• χ is obtained from its predecessors ψ2 = ψ1 → χ and ψ1 by (MP). From the inductionproperty for ψ2 we know that T ` 4ϕ → (ψ1 → χ). We apply 41 twice to getT ` 44ϕ → (4ψ1 →4χ), using 43 and 4E we get T ` 4ψ1 → (4ϕ →4χ).

From the induction property for ψ1 we know that T ` 4ϕ → ψ1. We apply 41 and 43to get T ` 4ϕ → 4ψ1. Now using (WT) we obtain T ` 4ϕ → (4ϕ → 4χ). Axiom4C gets us T ` 4ϕ →4χ and 42 completes the proof. QED

We can easily prove analogy of Lemmata 2.1.46 and 2.2.10. Thus having an equivalentdefinition of fuzzy logics in some class of logics.

Lemma 2.2.27 Let L be a fuzzy logic. Then 4(ϕ → ψ) → χ,4(ψ → ϕ) → χ `L χ.

Lemma 2.2.28 Let L be a finitary logic with DT4. Then L is fuzzy iff in L we have:4(ϕ → ψ) → χ,4(ψ → ϕ) → χ `L χ.

Unlike in Lemma 2.2.10, in this case the rule appearing in the previous lemma is equivalentto the following axiom: `L 4(4(ϕ → ψ) → χ) → (4(4(ψ → ϕ) → χ) → χ). Having ∨ inthe language we can formulate even stronger claim.

Corollary 2.2.29 Let L be a language, ∨ ∈ L and L a finitary logic in L with DT4. ThenL is fuzzy iff `L 4(ϕ → ψ) ∨4(ψ → ϕ).

Now we observe that 4(ϕ → ϕ) can be used as definition of 1. Thus we may assume thatwhenever 4 ∈ L, then 1 ∈ L. We prove the matching rules for 1 together with the fact thatthe selection of the formula ϕ in the definition 4(ϕ → ϕ) = 1 does not matter.

Lemma 2.2.30 It holds:

• ` 4(ϕ → ϕ) →4(ψ → ψ),

• ψ ` 4(ϕ → ϕ) → ψ,

• 4(ϕ → ϕ) → ψ ` ψ,

• ` 4(ϕ → ϕ).

Recall the in presence of ⊥ we can define the derived connective negation as ¬ϕ = ϕ → ⊥.Now we show some rather trivial properties of logics with 4 and ⊥ in the language.

Lemma 2.2.31 Let L be a language such that ⊥ ∈ L and L a logic in L. Then

1. `L 4(1 → ⊥) → ⊥,

2. `L ⊥ ↔ 4⊥,

3. `L 1 ↔41,

4. `L (4ϕ → ¬4ϕ) → ¬4ϕ.

Proof: The only non-trivial is Part 1. We give a formal proof.

(i) 4(1 → ⊥) →4(1 → ⊥) (Ref)(ii) 4(1 → ⊥) → (41 →4⊥) (i), 41, and (WT)(iii) 41 → (4(1 → ⊥) →4⊥) (ii), (NEC), 4E, (WT)(iv) 4(1 → ⊥) →4⊥ (iii) and (MP)(v) 4(1 → ⊥) → ⊥ (iv), 42 and (WT)

QED

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24 CHAPTER 2. WEAKLY IMPLICATIVE (FUZZY) LOGICS

Let us assume from now on that ⊥ is an element of all the propositional languages L.The reader familiar with fuzzy logics with 4 can notice that matching rules are somehow“weak”, 4 has more properties in this case, which do not hold in general. In the rest of thissection we add some additional rules for 4 to get known fuzzy logics with 4. We start byshowing that fuzzy logic L, where `L (¬4ϕ →4ϕ) →4ϕ has some interesting properties.

Lemma 2.2.32 Let L be a fuzzy logic in L, such that `L (¬4ϕ →4ϕ) →4ϕ. Then:

1. For each linearly ordered L-matrix B holds: 4Bx = 1B if 1B ≤B x and 4Bx = ⊥B

otherwise.

2. L has DT4.

3. T,4ϕ ` χ and T,¬4ϕ ` χ entails T ` χ.

4. ` (1 → ⊥) → ⊥ iff ` ¬4ϕ →4¬4ϕ.

5. ` ¬4ϕ →4¬4ϕ iff ` ¬4(ϕ → ψ) →4(ψ → ϕ).

Proof: 1. We omit the subscripts B in this proof. Let 1 ≤ x, then obviously 1 ≤ 4x (using(NEC)). Using 4W we get 1 ≤ 41 ≤ 4x → 1, thus 4x ≤ 1.

If x < 1 then 4x < 1 as well (because (42) we know 4x ≤ x). Since ¬4x →4x ≤ 4xwe have to have ¬4x > 4x (otherwise 1 ≤ ¬4x → 4x and so 1 ≤ 4x—a contradiction).Observe that from 4C we get `L 4(4ϕ → (4ϕ → ⊥)) → (4ϕ → ⊥), which leads to4(4x → ¬4x) ≤ ¬4x. We know that 4x ≤ ¬4x so 4(4x → ¬4x) = 1 and so1 ≤ 4x → ⊥. Finally, 4x ≤ ⊥.

2. One direction is obvious. We show the reverse one contrapositively: if T 6` 4ϕ → ψ,then there is linearly ordered L-matrix B and B-model e of T and e(4ϕ → ψ) < 1 (again weomit the subscripts B). Then obviously 1 ≤ e(ϕ) (otherwise e(4ϕ) = ⊥ and so we get that1 ≤ e(4ϕ → ψ)—a contradiction), thus e(4ϕ) = 1 and so e(4ϕ → ψ) = 1 → e(ψ) = e(ψ).So we know that e is B-model e of T, ϕ and e(ψ) < 1. Thus T, ϕ 6` ψ.

3. We observe that since ` 4(¬4ψ → 4ψ) → 4ψ and ` 4(4ψ → ¬4ψ) → 4¬4ψ(from Lemma 2.2.31), whenever we prove T ` 4ψ → χ and T ` 4¬4ψ → χ then T ` χ(using Lemma 2.2.28 and the fact that L has DT4). Using DT4 once more completes theproof.

4. Assume that ` (1 → ⊥) → ⊥, then also ¬1 → χ. We use part 3 of this lemma: firstnotice that 4ϕ ` 1 ↔ 4ϕ and so 4ϕ ` ¬4ϕ → ¬1. Thus 4ϕ ` ¬4ϕ → 4¬4ϕ. Second,we also know that ¬4ϕ ` 1 ↔ ¬4ϕ and since ` 1 →41 we get ¬4ϕ ` ¬4ϕ →4¬4ϕ.

To prove the reverse direction just set ϕ = ⊥ and get ¬4⊥ → 4¬4⊥. Lemma 2.2.31completes the proof.

5. Observe that ψ → ϕ ` 4¬4(ϕ → ψ) → 4(ψ → ϕ) (using 4W, 43, and DT4)and so ψ → ϕ ` ¬4(ϕ → ψ) → 4(ψ → ϕ) (using ` ¬4ϕ → 4¬4ϕ). Recall thatϕ → ψ ` 4(ϕ → ψ) ↔ 1 and since we know ` (1 → ⊥) → χ (from part 4. of this lemma) weget ϕ → ψ ` ¬4(ϕ → ψ) →4(ψ → ϕ).

To prove the reverse direction just set ϕ = ⊥, ψ = 1, and get ` ¬4(⊥ → 1) →4(1 → ⊥).Observe that ` 4(⊥ → 1) ↔ 1 (using DT4) and so we have ` (1 → ⊥) → 4(1 → ⊥).Lemma 2.2.31 and Part 4. of this lemma complete the proof. QED

Observe that the part 1. holds even without assumption that L is fuzzy and that part 2.has an interesting corollary:

Corollary 2.2.33 Let L be a finitary fuzzy logic in L, such that `L (¬4ϕ → 4ϕ) → 4ϕ.Then L has a presentation, where (MP) and (NEC) are the only deduction rules.

Of course, we even know this presentation: just replace each rule ϕ1, . . . ϕn ` ψ with theaxiom ` 4ϕ1 → (· · · → (4ϕn → ψ) . . . ). Now we try to formulate the “essence” of theconnective 4, when used in fuzzy logics.

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2.2. SPECIAL PROPOSITIONAL LOGICS 25

Definition 2.2.34 Let L be a language such that ⊥,4 ∈ L and L a fuzzy logic in L. Wesay that L is logic with Baaz delta iff for each formula in language {→,⊥, 1}, if we definesubstitution σ(v) = 4v then `L σϕ iff ϕ is a theorem of classical logic.

Obviously, not all logics with 4 connective are logic with Baaz delta (take Intuitionisticlogic, and in all Heyting algebras interpret 4 as identity). Observe that if L is a logic withBaaz delta it is consistent. This definition can be viewed as rather peculiar, but we are goingto present more convenient alternative definition. The following lemma works for fuzzy logiconly: even with Intuitionistic logic fulfills the properties 1.–5. from the upcoming definitionit is not a logic with Baaz delta, analogously ÃLukasiewicz logic with globalization is not fuzzylogic and so it is not a logic with Baaz delta.

Lemma 2.2.35 Let L be a consistent fuzzy logic. Then L is a fuzzy logic with Baaz delta iffthe following holds:

1. ` (¬4ϕ →4ϕ) →4ϕ,

2. ` (1 → 1) → 1,

3. ` (⊥ → 1) → 1,

4. ` (⊥ → ⊥) → 1,

5. ` (1 → ⊥) → ⊥.

Proof: One direction is obvious (only non-trivial part is to show ` (¬4ϕ →4ϕ) →4ϕ—todo this just notice that (¬p → p) → p is a theorem of classical logic).

To prove converse direction we need to show two things: first, if ϕ is a theorem of theclassical logic, then ` σϕ. We prove this contrapositively: assume that 6` σϕ, then there islinear L-matrix B and B-evaluation e such that e(σϕ) < 1. Observe that form Lemma 2.2.32part 1. we know that e(4v) is either 1 or ⊥, from the form of the formula σϕ and fromtheorems 2.–5. we know that e(σϕ) = ⊥ and if we define evaluation f(v) = e(4v) then ϕ isa classical evaluation not satisfying ϕ.

Second, we need to show that if ` σϕ then ϕ is a theorem of the classical logic. We provethis by a contradiction: assume that there is a classical evaluation e, such that e(ϕ) = ⊥.Then there is substitution ρ(v) = e(v) (we identify constants 1 and ⊥ with two truth valuesof classical logic) and ρϕ → ⊥ is a theorem of the classical logic. Thus ` σ(ρϕ → ⊥).Because σ(ρϕ → ⊥) = ρϕ → ⊥ (there are no variables in ρϕ → ⊥) and theorems 2.–5.we get ` ρϕ → ⊥ (using the previous direction). Since we assume that ` σϕ we also have` ρσϕ and so finally ` ρϕ (because 4⊥ ↔ ⊥ and 41 ↔ 1). Thus together we have ` ⊥—acontradiction with consistency of L. QED

In the literature, it is common that for fuzzy logic L there is defined its conservativeexpansion by the connective 4 (usually denoted as L4), which is fuzzy as well. It is usualthat L4 is a logic with Baaz delta and has DT4 (even if L has not some variant of deductiontheorem). Now we introduce general way of expanding the logic L into the logic L4. Firstwe give an indirect definition and then we show how to find a presentation of L4 based onthe presentation of L4.

Definition 2.2.36 Let L be a consistent fuzzy logic in L, such that ⊥ ∈ L and 4 6∈ L ByL4 we denote the weakest fuzzy logic with Baaz delta in language L ∪ {4} expanding L.

In the next chapter, we will see other axiomatic systems for logics with Baaz delta thanthe one we are going to present (we prove their equivalences). However, this one has severaladvantages. First, it is formulated in the language with →, ⊥, and 4 only and second, itdoes not needs any structural rules of the starting logic.

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26 CHAPTER 2. WEAKLY IMPLICATIVE (FUZZY) LOGICS

Theorem 2.2.37 Let L be a consistent finitary fuzzy logic in L, such that ⊥ ∈ L and 4 6∈ L

and let AX be some finitary presentation of L. Then the following is a presentation of L4:

A axioms of AX,B ` 4ϕ1 → (. . . (4ϕn → ψ) for each n-ary deduction rule <ϕ1, . . . , ϕn, ψ > ∈ AX,C matching rules for 4,(MP) ϕ,ϕ → ψ ` ψ,44 ` (¬4ϕ →4ϕ) →4ϕ,45 ` ¬4(ϕ → ψ) →4(ψ → ϕ),46 ` (1 → 1) → 1.

Proof: Let L′ be the logic with the above presentation. We have to show that L′ is afuzzy logic with Baaz delta (it obviously extends L). Since L is fuzzy we get (ϕ → ψ) → χ,(ψ → ϕ) → χ `L χ and so (ϕ → ψ) → χ, (ψ → ϕ) → χ `L′ χ. From Lemma 2.2.31 and 44we know that `L′ (¬4ϕ → 4ϕ) → 4ϕ and `L′ (4ϕ → ¬4ϕ) → ¬4ϕ. Thus if we proveT `L′ 4ϕ → χ and T `L′ ¬4ϕ → χ we get T `L′ χ.

Let us denote T = {4(ϕ → ψ) → χ,4(ψ → ϕ) → χ}. Observe that T,4(ϕ → ψ) `L′ χand T,¬4(ϕ → ψ) `L′ χ (the first is obvious, to prove the second use 44). Thus T `L′ χ.From Theorem 2.2.26 we know that L′ has DT4 and from the fact T `L′ χ we know that L′

is fuzzy (using Lemma 2.2.28).Now we observe that L′ is a conservative expansion of L (we can extend any linear

L-matrix B, which is counterexample to T ` ϕ, into the linear L′-matrix B4 and is acounterexample as well—since L′ is fuzzy). So we know that L′ is consistent (because L isconsistent) .

Thus we can use Lemma 2.2.35 to prove that L′ is a logic with Baaz delta. Notice thatall we need to show is the following:

1. ` (⊥ → 1) → 1,

2. ` (⊥ → ⊥) → 1,

3. ` (1 → ⊥) → ⊥.

To prove 1. use theorem ` (¬4ϕ →4ϕ) →4ϕ for ϕ = 1, we get ` (¬41 →41) →41.Lemma 2.2.31 completes the proof. To prove 2. use theorem ` ¬4(ϕ → ψ) → 4(ψ → ϕ)for ϕ = 1 and ψ = ⊥, we get ` ¬4(1 → ⊥) → 4(⊥ → 1). Lemma 2.2.31 completes theproof. To prove 3. use theorem ` ¬4(ϕ → ψ) → 4(ψ → ϕ) for ϕ = ⊥ and ψ = 1, we get` ¬4(⊥ → 1) →4(1 → ⊥). Lemma 2.2.31 completes the proof.

To complete the proof of this theorem we need to show that each fuzzy logic L′ with Baazdelta proofs all formulae from our presentation. From Lemma 2.2.35 we know that `L′ 46,`L′ 44, and `L′ (1 → ⊥) → ⊥. Observe that in this case we can use Lemma 2.2.32 Parts4. and 5. to get `L′ 45. Logic L′ obviously proves all consecution from the group A and B.So all we have to show are the axioms from the group B: to do this just observe that L′ hasDT4 (because L′ is fuzzy and `L′ 44 we can use Lemma 2.2.32) and since L ⊆ L′ the proofis done. QED

The presence of the axiom ` (1 → 1) → 1 seems to be unavoidable, however under somerather weak additional assumptions we can omit it. One of them is of course the presence ofweakening, the other one is the presence of Sf (then we get ` (1 → 1) → (⊥ → 1) and usingthe know fact that ` (⊥ → 1) → 1 we would get our axiom). Now we show that L4 hassome promised nice properties:

Lemma 2.2.38 Let L be a consistent fuzzy logic in L, such that 4 6∈ L and ⊥ ∈ L. ThenL4 is a fuzzy logic with DT4 and Baaz delta, which is a conservative expansion of L.

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2.3. FIRST-ORDER LOGIC 27

2.3 First-order logic

In this second part of this paper, we move to the first-order logics. We present the very basictheorems only. The broader treatment of this topic will be the content of the subsequentpapers. Our approach is inspired by the classical first-order logic and by its modification(the axiomatic system, the notion of Henkin theory) for non-classical logics, the main sourceis Hajek’s treatment of basic predicate fuzzy logic (for details see [44]).

2.3.1 Basic definitions

In the following let L be a fixed weakly implicative logic in propositional language L.

Definition 2.3.1 (Predicate language) By multi-sorted predicate language Γ we under-stand a quintuple (S, ¹, P, F, A), where S is a non-empty set of sorts, ¹ is an orderingon S (indicating the subsumption of sorts), P is a non-empty set of predicate symbols, F is aset of function symbols, and A is a function assigning to each predicate and function symbola finite sequences of elements of S.

Let |A(P )| denote the length of the sequence A(P ). The number |A(P )| is called the arityof the predicate symbol P . The number |A(f)| − 1 is called the arity of the function symbolf . The functions f for which A(f) = <s> are called the individual constants of sort s. Ifs1 ¹ s2 holds for sorts s1, s2 we say that s2 subsumes s1.

The L-logical symbols are individual variables xs, ys, . . . for each sort s, the logical con-nectives of L, and the quantifiers ∀ and ∃.

Let us denote by Cs the set of constants of the sort s. In the following let Γ be a fixedmulti-sorted predicate language for logic L∀.

Definition 2.3.2 (Terms) Each individual variable of sort s is a Γ-term of sort s. Let fbe a function symbol, t1, . . . , tn terms of sorts s1, . . . , sn, and A(f) = <w1, . . . , wn, wn+1>such that si ¹ wi for i ≤ n. Then f(t1, . . . , tn) is a Γ-term of sort wn+1.

Notice that the set of terms depends on Γ only, whereas the set of formulae depends ofthe propositional language as well. So we should speak about Γ-terms and (Γ, L)-formulae(however, we speak about Γ-formulae if the propositional language is clear from the contextand we speak about terms and formulae if both propositional and the predicate language areclear from the context).

Definition 2.3.3 (formulae) Let t1, . . . tn be terms of sorts s1, . . . sn, and P be a predicatesymbol, A(P ) = <w1, . . . wn>, such that si ¹ wi for i ≤ n. Then P (t1, . . . tn) is an atomicΓ-formula. The nullary logical connectives of L are atomic Γ-formulae as well.

Let ϕ be Γ-formula and xs an object variable of the sort s. Then (∀xs)ϕ and (∃xs)ϕ areΓ-formulae. Furthermore, the class of Γ-formulae is closed under logical connectives of L.

Bounded and free variables in a formula are defined as usual. A formula is called asentence iff it contains no free variables. A set of Γ-sentences is called a Γ-theory.

Instead of ξ1, . . . , ξn (where ξi’s are terms or formulae and n is arbitrary or fixed by thecontext) we shall sometimes write just ~ξ.

Unless stated otherwise, the expression φ(x1, . . . , xn) means that all free variables of φare among x1, . . . , xn.

If φ(x1, . . . , xn, ~z ) is a formula and we substitute terms ti for all xi’s in φ, we denote theresulting formula in the context simply by φ(t1, . . . , tn, ~z ).

Definition 2.3.4 (Substitutability) A term t of sort w is substitutable for the individualvariable xs in a formula ϕ(xs, ~z ) iff w ¹ s and no occurrence of any variable y occurring int is bounded in ϕ(t, ~z ).

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28 CHAPTER 2. WEAKLY IMPLICATIVE (FUZZY) LOGICS

Let B be fixed ordered L-matrix in the following text.

Definition 2.3.5 (Structure) An B-structure M for Γ has the following form:M = ((Ms)s∈S, (PM)P∈P, (fM)f∈F), where Ms is a non-empty domain for each s ∈ S andMs ⊆ Mw iff s ¹ w; PM is an n-ary fuzzy relation

∏ni=1 Msi → L for each predicate symbol

P ∈ P such that A(P ) = <s1, . . . , sn>; fM is a function∏n

i=1 Msi→ Msn+1 for each

function symbol f ∈ F such that A(f) = <s1, . . . , sn, sn+1>, and an element of Ms if f isa constant of sort s.

Definition 2.3.6 (Evaluation) Let M be a B-structure for Γ. An M-evaluation of theobject variables is a mapping e which assigns to each variable of sort s an element from Ms

(for all sorts s ∈ S).Let e be an M-evaluation, x a variable of sort s, and a ∈ Ms. Then e[x → a] is an

M-evaluation such that e[x → a](x) = a and e[x → a](y) = e(y) for each individual variabley different from x.

Definition 2.3.7 (Truth definition) Let M be a B-structure for Γ and v an M-evaluation.Values of the terms and truth values of the formulae in M for an evaluation v are defined asfollows:

||xs||BM,v = v(x) ,

||f(t1, t2, . . . , tn)||BM,v = fM(||t1||BM,v, ||t2||BM,v, . . . , ||tn||BM,v) ,

||P (t1, t2, . . . , tn)||BM,v = PM(||t1||BM,v, ||t2||BM,v, . . . , ||tn||BM,v) ,

||c(ϕ1, ϕ2, . . . , ϕn)||BM,v = cB(||ϕ1||BM,v, ||ϕ2||BM,v, . . . , ||ϕn||BM,v) ,

||(∀xs)ϕ||BM,v = inf{||ϕ||BM,v[xs→a] | a ∈ Ms} ,

||(∃xs)ϕ||BM,v = sup{||ϕ||BM,v[xs→a] | a ∈ Ms} .

If the infimum or supremum does not exist, we take its value as undefined. We say that aB-structure M for Γ is safe iff ‖ϕ||BM,v is defined for each Γ-formula ϕ and each M-evaluationv.

Definition 2.3.8 (Value of formula) Let M be a safe B-structure for Γ, and ϕ a Γ for-mula. A truth value of the formula ϕ in M is defined as follows:

||ϕ||BM = inf{||ϕ||BM,v | v is an M-evaluation}.

We say that ϕ is an B-tautology if ||ϕ||AM ∈ DB for each safe B-structure M.

Definition 2.3.9 (Model) Let M be a B-structure for Γ, and T a Γ-theory. Then theB-structure M for Γ is called B-model of T if ||ϕ||AM ∈ DB for each ϕ ∈ T . We denote theset of A-models od T by MOD(T,A).

Definition 2.3.10 (Semantical consequence) Let K be a class of L-matrices. We saythat ϕ is a semantical consequence of the T w.r.t. class K if MOD(T,B) = MOD(T ∪ {ϕ},B)for each B ∈ K; we denote it by T |=K ϕ. By TAUT(K) we understand the set {ϕ | ∅ |=K ϕ}.

We write T |=L ϕ instead of T |=o−MAT(L) ϕ and we also write T |=lL ϕ instead of

T |=l−MAT(L) ϕ.For a fixed B-model M and an M-valuation e such that e(xi) = ai (for all i’s), instead of

‖ϕ(x1, . . . , xn)‖BM,e we write simply ‖ϕ(a1, . . . , an)‖ and speak of the value of ϕ(a1, . . . , an).Now we define the predicate logic to each weakly implicative logic (and stronger predicate

logic for each fuzzy logics). As in the propositional case we understand the predicate logic asan asymmetric consequence relation (following Dunn’s terminology). For simplicity of thisintroductory chapter we made two extra design choices. First, we assume that ∨ is the partof the language (there are ways how avoid the need for ∨ under some additional assumptionseg. having exchange, or having à in the language). Second, we formulate consecutions (∀2)

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2.3. FIRST-ORDER LOGIC 29

and (∃2) as axioms rather than rules, which would result into the weaker definition. However,it is obvious that under some rather weak assumptions these two notions would coincide (eg.under the presence of & in the language). These two topics will be more elaborated in somesubsequent paper.

Definition 2.3.11 Let L be a weakly implicative logic. The logic L∀− is given by the fol-lowing axioms and deduction rules:

(P) the formulae and deduction rules resulting from the axioms and deduction rulesof L by the substitution of the propositional variables by the formulae of Γ,

(∀1) `L∀− (∀x)ϕ(x) → ϕ(t), where t is substitutable for x in ϕ,(∃1) `L∀− ϕ(t) → (∃x)ϕ(x), where t is substitutable for x in ϕ,(∀2) `L∀− (∀x)(χ → ϕ) → (χ → (∀x)ϕ), where x is not free in χ,(∃2) `L∀− (∀x)(ϕ → χ) → ((∃x)ϕ → χ), where x is not free in χ,(Gen) ϕ `L∀− (∀x)ϕ.

Furthermore, if L is a fuzzy logic we define the logic L∀ as an extension of L∀− by axiom:

(∀3) `L∀ (∀x)(χ ∨ ϕ) → χ ∨ (∀x)ϕ, where x is not free in χ.

Logics L∀ and L∀− are sometimes the same (ÃLukasiewicz predicate logic) and sometimesthey are different (Godel predicate logic). Now we recall the concept of the PrelinearityProperty, we will need this property to prove the completeness theorem of L∀ w.r.t. linearlyordered matrices.

Definition 2.3.12 (Prelinearity) Fuzzy logic L∀ has the Prelinearity Property (PP) if foreach theory T and each sentences ϕ and ψ we get T ` χ whenever T, ϕ → ψ ` χ andT, ψ → ϕ ` χ.

Unluckily, we are not able to prove that each predicate fuzzy logic has PP. However, wecan give some simply checkable sufficient conditions. Before we do so we observe that we caneasily prove the both deduction theorems (we assume the same definition of Implicational(Delta) Deduction theorem are in propositional case, we only assume that ϕ is a sentence).

Theorem 2.3.13 (Deduction Theorems) Let L be a logic with DT→ (DT4respectively).Then both logics L∀− and L∀ have DT→ (DT4 resp.).

Corollary 2.3.14 Let L be a fuzzy logic, such that L has some presentation where (MP) isthe only deduction rule and implicational fragment of L is an extension of BCK. Then L∀has PP.

Corollary 2.3.15 Let L be a propositional language, 4 ∈ L and L a fuzzy logic in L, suchthat L has some presentation where (MP) and (NEC) are the only deduction rules. Then L∀has PP.

2.3.2 Henkin and witnessed theories

In this subsection we prepare some technical means to proof the completeness.

Definition 2.3.16 (Henkin and ϕ-witnessed theories) The theory is called Henkin the-ory if for each sentence ϕ = (∀x)ψ and T 6` ϕ, there is a constant c such that T 6` ψ(c).

Furthermore, let ϕ(x1, . . . xn, y) be a formula. Henkin theory is called ϕ-witnessed theoryif for each formula ψ(y) = ϕ(x1 : t1, . . . xn : tn, y), where ti are closed terms holds: ifT ` (∃y)ψ(y), then there is a constant c such that T ` ψ(c).

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30 CHAPTER 2. WEAKLY IMPLICATIVE (FUZZY) LOGICS

Definition 2.3.17 (Henkin and ϕ-witnessed logics) Let L be a weakly implicative logicand ϕ a formula. We say that the logic L∀− is Henkin (ϕ-witnessed ) for each theory T andeach sentence α, T 6` α there is a Henkin (ϕ-witnessed ) theory T ′ such that T ⊆ T ′ andT ′ 6` α.

Let L be a fuzzy logic and ϕ a formula. We say that the logic L∀ is Henkin (ϕ-witnessed )for each theory T and each sentence α, T 6` α there is a linear Henkin (ϕ-witnessed ) theoryT ′ such that T ⊆ T ′ and T ′ 6` α.

Definition 2.3.18 (proto-ϕ-witnessed logics) Let L be a weakly implicative logic and ϕa formula. We say that the logic L∀− (or the logic L∀) is proto-ϕ-witnessed if for eachtheory T and for each formula ψ(y) = ϕ(x1 : t1, . . . xn : tn, y), where ti are closed termsholds: T ∪ {ψ(c)} is a conservative extension of T ∪ {(∃y)ψ(y)}.

In fact, the logic is proto-ϕ-witnessed iff it supports introduction of Skolem constants. Weuse this property to show, that in that case we can introduce Skolem function of arbitraryarity.

Lemma 2.3.19 Let L be a weakly implicative logic and ϕ a formula. If the logic L∀− isϕ-witnessed then it is proto-ϕ-witnessed .

Furthermore, let L be a fuzzy logic and ϕ a formula. If the logic L∀ is ϕ-witnessed thenit is proto-ϕ-witnessed.

Proof: Let T be a theory and ψ(y) = ϕ(x1 : t1, . . . xn : tn, y), where ti are closed terms,such that theory T ∪{ψ(c)} is not a conservative extension of T ∪{(∃y)ψ(y)}, i.e., there is aformula α such that T ∪{ψ(c)} ` α and T ∪{(∃y)ψ(y)} 6` α. Let us take ϕ-witnessed theoryT ′, such that T ∪ {(∃y)ψ(y)} ⊆ T ′ and T ′ 6` α. Since T ′ is ϕ-witnessed and T ′ ` (∃y)ψ(y)there is a constant d such that T ′ ` ψ(d). Since T ∪ {ψ(c)} ` α we get T ∪ {ψ(d)} ` α andso T ′ ` α—a contradiction

The proof of the second part as the same. QED

Definition 2.3.20 (Directed set of formulae) Let Ψ be a set of formulae. We say thatΨ is a directed set if for each ψ, ϕ ∈ Ψ there is δ ∈ Ψ such that ` ψ → δ and ` ϕ → δ (wecall δ the upper bound of ϕ and ψ).

This is the crucial lemma of this chapter. We use it for the proof of completeness of ourfirst-order calculi.

Lemma 2.3.21 (ϕ-witnessed extension) Let L be a finitary fuzzy logic and L∀ has PP,and ϕ be a formula. If the logic L∀ is proto-ϕ-witnessed then it is ϕ-witnessed.

Proof: We construct our extension by a transfinite induction. Let T be a theory and α aformula T 6` α. If Ψ is a set of formulae by T 6` Ψ we mean T 6` ψ for each ψ ∈ Ψ.

Before we start we extend our predicate language by new constants {csν | ν ≤ ||Γ||}

for each sort s. Let T0 = T and Ψ0 = {α}. We enumerate all formulae with one freevariable x by ordinal numbers as χµ and all formulae with one free variable x of the formϕ(x1 : t1, . . . xn : tn, x) by ordinal numbers as σµ.

We construct directed sets Ψµ and theories Tµ so Tµ 6` Ψµ and Tµ ⊆ Tν and Ψµ ⊆ Ψν forµ ≤ ν. Thus we get Tµ 6` α. Observe that theory T0 and set Ψ0 fulfill these conditions. Theinduction step:

Let us define the sets: Tµ =⋃

ν<µTν and Ψµ =

⋃ν<µ

Ψν . Notice that from the induction

property we get that Tµ 6` Ψµ and Ψµ is directed set. Now we construct theory T ′µ and setΨµ. We distinguish two cases:

(H1) There is ψ ∈ Ψµ such that Tµ ` ψ ∨ χµ(c). Let T ′µ = Tµ ∪ {ψ → (∀x)χµ(x)} andΨµ = Ψµ.

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2.3. FIRST-ORDER LOGIC 31

(H2) Otherwise, let T ′µ = Tµ and Ψµ = Ψµ ∪ {ψ ∨ χµ(c) | ψ ∈ Ψµ}.

We show that T ′µ 6` Ψµ and Ψµ are directed. Let c be the first unused constant of theproper sort.

(H1) Let ϕ ∈ Ψµ and δ is the upper bound of ϕ and ψ. Recall that Tµ 6` δ. We knowthat Tµ ` ψ ∨ χµ(x) (just replace c by x everywhere in the proof of ψ ∨ χµ(c)). ThusTµ ` ψ ∨ (∀x)χµ(x) (by the generalization and axiom (∀3)).

Obviously, Tµ ∪ {ψ} ` ψ and so Tµ ∪ {ψ} ` δ. Thus Tµ ∪ {(∀x)χµ(x)} 6` δ (otherwiseusing PCP we get Tµ ` δ—a contradiction). Finally, if we have Tµ ∪ {(∀x)χµ(x)} ` ϕ

then we get Tµ ∪ {(∀x)χµ(x)} ` δ—a contradiction. We have shown that T ′µ 6` Ψµ theother conditions are in this case obvious.

(H2) The proof of T ′µ 6` Ψµ is trivial. We only have to show that Ψµ is directed. We shoulddistinguish three cases, however we show only one (the other are analogous) ϕ, α ∈ Ψµ

and ψ = α∨χµ(c). Let δ be the upper bound of ϕ and α then obviously δ∨χµ(c) ∈ Ψµ

is the upper bound ϕ and ψ.

Next, we construct theory Tµ. Let c be the first unused constant of the proper sort.Again, we distinguish two cases:

(W1) There is ψ ∈ Ψµ such that T ′µ ∪ {(∃x)σµ(x)} ` ψ. Let Tµ = T ′µ.

(W2) T ′µ ∪ {(∃x)σµ(x)} 6` Ψµ. Let Tµ = T ′µ ∪ {σµ(c)}.

We show that Tµ 6` Ψµ:

(W1) Trivial.

(W2) Since the logic L∀ is proto-ϕ-witnessed T ′µ ∪ {σ(c)} is a conservative extension of T ′µ ∪{(∃x)σ(x)}. Since T ′µ ∪ {(∃x)σ(x)} 6` Ψµ the proof is done.

Let us define theory T = T||Γ|| and set Ψ = Ψ||Γ||. Now we construct a complete theoryT ′ such that T ⊆ T ′ and T ′ 6` Ψ. We do it again by a transfinite induction. Let us enumeratepair of formulae by ordinals. T ′0 = T . We construct theories T ′µ such that T ′µ 6` Ψ andT ′µ ⊆ T ′ν for µ ≤ ν. Let us define theory T ′µ =

⋃ν<µ

T ′ν . Notice that from the induction

property we get that T ′µ 6` Ψ. The induction step: we show that T ′µ ∪ {ϕµ → ψµ} 6` Ψor T ′µ ∪ {ψµ → ϕµ} 6` Ψ. By contradiction: let there be formulae β, γ ∈ Ψ such thatT ′µ ∪ {ϕµ → ψµ} ` β and T ′µ ∪ {ψµ → ϕµ} ` γ. Let us take upper bound δ of β and γ andwe get T ′µ ∪ {ϕµ → ψµ} ` δ and T ′µ ∪ {ψµ → ϕµ} ` δ. Thus T ′µ ` δ—a contradiction.

Finally, we define T ′ = T ′||Γ||. Recall that T ′ 6` Ψ. If we show that T ′ is ϕ-witnessedtheory the proof is done (because T ′ is obviously complete and T ′ 6` α).

Is T ′ Henkin? Let ϕ(x) be processed in the step µ. If T ′ 6` (∀x)ϕ(x) then we usedthe case (H2) (otherwise Tµ ` ψ ∨ ϕ(c) which leads to Tµ ` ψ ∨ (∀x)χµ(x) and so we getTµ ∪ {ψ → (∀x)ϕ(x)} ` (∀x)ϕ(x) and so Tµ ` (∀x)ϕ(x)—a contradiction). If T ′ ` ϕ(c) thenT ′ ` ϕ(c) ∨ ψ for all ψ ∈ Ψµ. Since we used case (H2) we know that ϕ(c) ∨ ψ ∈ Ψµ—acontradiction with T ′ 6` Ψ.

Is T ′ ϕ-witnessed ? Let ψ(x) = ϕ(x1 : t1, . . . xn : tn, x) be processed in the step µ. IfT ′ ` (∃x)ϕ(x) then we used the case (W2) (since Tµ ∪ {(∃x)ϕ(x)} ` ψ for some ψ ∈ Ψ weget T ′ ` ψ—a contradiction). Thus Tµ ` ϕ(c) and so T ′ ` ϕ(c). QED

Corollary 2.3.22 (Henkin extension in fuzzy logics) Let L be finitary fuzzy logic. IfL∀ has PP then the logic L∀ is Henkin.

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32 CHAPTER 2. WEAKLY IMPLICATIVE (FUZZY) LOGICS

Proof: Just read the proof of the latter lemma without parts (W1) and (W2) and noticethat the assumption that L∀ is proto-ϕ-witnessed was used only in part (W2). QED

Theorem 2.3.23 (Henkin extension) Let L be finitary weakly implicative logic. Then thelogic L∀− is Henkin.

Proof: Just read the proof of Theorem 3.4 in [53]. QED

2.3.3 Completeness

We introduce the notion of a Lindenbaum-Tarski matrix in the same fashion as in the propo-sitional level. We define the canonical LinT -structure MT in the usual way—elements are theclosed terms, and functions and predicates are defined accordingly. We have the followingimportant lemma.

Lemma 2.3.24 Let T be a Henkin theory and ϕ a formula with only one free variable x ofthe sort s. Then

• [(∀xs)ϕ]T = infc∈Cs

[ϕ(c)]T.

• [(∃xs)ϕ]T = supc∈Cs

[ϕ(c)]T.

Proof: Recall that [ϕ]T ≤ [ψ]T iff T ` ϕ → ψ (cf. Lemma 2.1.32). We prove only the firstclaim, the proof of the second one is analogous.

We show that [(∀xs)ϕ]T is the greatest lower bound of all [ϕ(c)]T. The proof that[(∀xs)ϕ]T is the lower bound is simple: [(∀xs)ϕ]T ≤ [ϕ(c)]T for all constants c ∈ Cs (byaxiom (∀1)).

Now suppose there is [χ]T such that [χ]T ≤ [ϕ(c)]T for all c ∈ Cs and [χ]T 6≤ [(∀xs)ϕ]T.Thus T 6` χ → (∀xs)ϕ and so T 6` (χ → ϕ) (by rule (Gen∀)). Thus T 6` (∀xs)(χ → ϕ) (byaxiom (∀1)) By a Henkin property we get a constant d ∈ Cs such that T 6` χ → ϕ(d). Finally[χ]T 6≤ [ϕ(d)]T - a contradiction. QED

Obviously, for T being Henkin the canonical LinT -structure is safe and we have [ϕ]T =||ϕ||LinT

MTand thus MT is a LinT -model of T . Since each theory can be extended into Henkin

theory (and in the case of fuzzy logic into the linear Henkin theory the proof of the followingtheorem is straightforward.

Theorem 2.3.25 Let L be a fuzzy logic, Γ be a predicate language, T a theory and ϕ aformula. If L∀ is Henkin, then T `L∀ ϕ iff T |=l

L ϕ.

Corollary 2.3.26 Let L be a finitary fuzzy logic. Then L∀ has PP iff for each T and eachϕ we have: T `L∀ ϕ iff T |=l

L ϕ.

Corollary 2.3.27 Let L be logic, such that L has some presentation AX, where (MP) is theonly deduction rule and implicational fragment of L is an extension of FBCK. Then the logicL∀ is sound and complete w.r.t. the corresponding class of linear matrices.

Corollary 2.3.28 Let L be a propositional language, 4 ∈ L and L a fuzzy logic in L, suchthat L has some presentation where (MP) and (NEC) are the only deduction rules. Then thelogic L∀ is sound and complete w.r.t. corresponding class of linear matrices.

There are fuzzy logic, described in the literature not covered by this general approach(so far), namely the logics RPΠ, RPΠ∼([0, 1]Π∼), and R�Π (because their infinitary rule,however this problem can be solved, see Subsection 3.5.3) and the logic P�′ (because of theunavoidable rule ¬(ϕ¯ ϕ) ` ¬ϕ).

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2.3. FIRST-ORDER LOGIC 33

At the end of this section we formulate completeness theorem of for weakly implicativelogics. The proof is analogous to the one for fuzzy logics, we only use Corollary 2.3.23instead of Corollary 2.3.22. This gives us some kind of first-order calculus for a very wideclass of logics. There is an interesting research task to examine existing first-order calculifor particular logics (substructural, modal, intuitionistic, etc.) and compare them to ourapproach.

Theorem 2.3.29 Let L be a weakly implicative logic, Γ a predicate language, T a theory,and ϕ a formula. Then T `L∀− ϕ iff T |=L ϕ.

2.3.4 Skolem functions

Again, this section is only a short sketch of what “can be done”. Observe that the majorityof known fuzzy logic are ϕ-witnessed for each formula ϕ. Whereas, the logics with 4 arerather limited in this aspect, as shown by the following lemma:

Lemma 2.3.30 Let L be a propositional language, 4 ∈ L and L a logic in L with DT4.Then the logics L∀− and L∀ are ϕ-witnessed iff ` 4(∃y)ϕ(y) → (∃y)4ϕ(y).

Proof: First, let us suppose that the logic is proto-ϕ-witnessed then {ϕ(c)} is a conservativeextension of {(∃y)ϕ(y)}. Since {ϕ(c)} ` (∃x)4ϕ(x) we get {(∃y)ϕ(y)} ` (∃x)4ϕ(x). Thededuction theorem gives us ` 4(∃y)ϕ(y) → (∃y)4ϕ(y).

Other direction if similar. Let ψ(y) = ϕ(x1 : t1, . . . xn : tn, y), where ti are closed terms.We want show that T ∪ {ψ(c)} is a conservative extension of T ∪ {(∃y)ψ(y)}. For each ϕwithout c we want to get : T ∪ {ψ(c)} ` ϕ iff T ∪ {(∃y)ψ(y)} ` ϕ. By deduction theoremand some simple steps we get: T ` (∃y)4ψ(y) → ϕ iff T ` 4(∃y)ψ(y) → ϕ. Now just noticethat if ` 4(∃y)ϕ(y) → (∃y)4ϕ(y) then ` 4(∃y)ϕ(y) ≡ (∃y)4ϕ(y). QED

Corollary 2.3.31 Let L be a propositional language, 4 ∈ L, L a logic in L with DT4, andϕ a formula. Then the logics L∀− and L∀− are 4ϕ-witnessed.

Let us examine the behavior of the ϕ-witnessed logic w.r.t. Skolem functions introduction.We formulate the theorem for fuzzy logics only, its reformulation for weakly implicative logicsneeds an analogy of Lemma 2.3.21. This can be done in a rather straightforward way, but weskip this here. For the sake of simplicity we formulate the theorem for the unsorted language.

Theorem 2.3.32 Let L be a proto-ϕ-witnessed finitary fuzzy logic with PP, T a theory, andϕ(x1, . . . , xn, y) a formula. If T ` (∀x1) . . . (∀xn) (∃y)ϕ(x1, . . . xn, y). Then the theory T ′ inthe language of T extended by new function symbol fϕ resulting from the theory T by addingthe axiom ` (∀x1) . . . (∀xn)ϕ(x1, . . . xn, fϕ(x1, . . . , xn)) is a conservative extension of T .

Proof: Let T be a ϕ-witnessed supertheory of T . Then if T 6` χ there is a canonicalLinT -modelMT of T . Since for each vector t1, . . . , tn of closed terms if T ` (∃y)ϕ(t1, . . . tn, y)there is a constant ct1,...,tn such that T ` ϕ(t1, . . . tn, ct1,...,tn). Since ct1,...,tn is an elementof MT (together with all other closed terms) we define (fϕ)MT

(t1, . . . , tn) = ct1,...,tn . Thenobviously MT is a model of T ′ and since MT 6|= χ we get that T ′ 6` χ. QED

To prove the Skolem function elimination we need to extend our language with some sortof equality.

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Chapter 3

Known fuzzy logics

This chapter serves as preliminary for the rest of this work. However, it has importance ofits own, because we are not only reviewing known logics, but we are putting them into thecontext of our general approach from the last chapter. It turns out that many theorems,usually proved in the literature, are merely consequences of our general theorems. The aimof this chapter is not to “process” all existing fuzzy logics and apply our new formalism, werestrict ourselves to logics and properties of logics we need in the further text.

In the previous chapter we have seen a general definition of the class of fuzzy logics. It isa formal definition, which delimits the class of fuzzy logics inside a broader class of weaklyimplicative logics. However, there is another informal delimitation of fuzzy logic: fuzzylogic is a logic appearing in papers written by fuzzy logicians. This is sometimes more andsometimes less restrictive condition. In this section we make some additional assumption tomake these two notions closer to each other. Propositional fuzzy logics in the literature aremostly formulated with (MP) as the only deduction rule (and with 4-necessitation in thecase with 4 in in the language), they have usually connectives → and & and are expressiveenough to define lattice connectives (as basic or derived connectives). Hajek’s Quasihooplogic [51] is an exception—it does not define the lattice connectives.

Now we make some even more restrictive assumptions to delimit the logics we will en-counter in the rest of this work. We list these assumptions together with names of logics,without (!) these assumptions and with references to the relevant papers:

Exchange pseudo Basic Logic [48, 49, 47]Weakening Uninorm logic [65]⊥ ∈ L Hoop logic [53]⊥ ∈ L and Exchange Flea logic [50]

With all these assumption we get to the logic FUZZ{&,∧,⊥}(E, W, ax (Sf)). This logic isknown as monoidal t-norm logic MTL of Esteva and Godo (see [33, 31, 13, 46]). However,we make one more assumption: we add the so-called divisibility axiom.

(D) ` ϕ&(ϕ → ψ) → ψ&(ψ → ϕ).

We start by showing that FUZZ{&,∧,⊥}(E, W, ax (Sf),D) is Hajek’s Basic fuzzy logic (BL).This will be our starting logic, the rest of this work is devoted to the study of some propertiesof some of its extensions.

Before we do so we prepare (in the first section) some definitions concerning generalnotions of fuzzy logic, i.e., without just mentioned design choices. We concentrate on thenotion of standard completeness. Then, in the second and third section, we examine the classof existing fuzzy logic fulfilling our design choices. The fourth section is devoted to the studyof logics resulting by adding truth constants into the language of our fuzzy logics. Finally,in the last section, we deal with the first-order fuzzy logics.

35

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36 CHAPTER 3. KNOWN FUZZY LOGICS

3.1 Standard completeness and Core fuzzy logics

In the literature, fuzzy logics are usually considered as unary consequence relations (as aset of theorems/tautologies). If we recall our approach we notice that it is more general.This results in some clash in terminology. What we call completeness is known under theterm strong completeness, whereas the term completeness is understood as equality betweentheorems and tautologies. As we have seen each fuzzy logic enjoys both completeness andstrong completeness and so there is no need for two terms. Thus we stick with our denotation.There is another kind completeness playing an important role in fuzzy logic.

Definition 3.1.1 (Standard matrix) Let L be a weakly implicative logic and let B be anL-matrix. We say that B is a standard L-matrix iff B = [0, 1] and ≤B is the usual order ofreals. Let us denote the class of standard L-matrices as s−MAT(L).

Definition 3.1.2 (Standard Completeness) Let L be a weakly implicative logic. We saythat L has the standard completeness if L = |=s−MAT(L) (i.e., if the logic L is sound andcomplete w.r.t. standard L-matrices.)

Observe that because the usual order of reals is linear each logic with standard com-pleteness is fuzzy. Again, the concept we have just defined is usually called strong standardcompleteness. We omit the word “strong”. However, we have a problem here, there are veryfew known logics with the standard completeness (namely Godel , Godel 4, and Godel ∼),many of them enjoy its weaker form only:

Definition 3.1.3 (Finite Standard Completeness) Let L be a weakly implicative logic.We say that L has the finite standard completeness if T `L ϕ iff T |=s−MAT(L) ϕ for eachfinite theory T and formula ϕ.

Furthermore, we say that L has the weak standard completeness if `L ϕ iff |=s−MAT(L) ϕfor each formula ϕ.

Of course, standard completeness implies finite standard completeness and it implies weakstandard completeness. There is another equivalent definition of Finite Standard Complete-ness using the so-called finitary companions.

Definition 3.1.4 Let L be a logic. The logic FIN(L) (finitary companion of L) is definedas: X `FIN(L) ϕ iff there is finite X ′ ⊆ X and X ′ `L ϕ.

Observe that FIN(L) is the greatest finitary logic contained in L (of course if L is finitary,then FIN(L) = L).

Theorem 3.1.5 Let L be a weakly implicative logic. Then L is a finitary logic with the finitestandard completeness iff L = FIN(|=s−MAT(L)) (i.e., if L is a finitary companion of thelogic given by the standard semantics of L).

Proof: Let us denote FIN(|=s−MAT(L)) as L′. We know that X `L′ ϕ iff there is finiteX ′ ⊆ X and X ′ `|=s−MAT(L)

ϕ iff X ′ |=s−MAT(L) ϕ iff X ′ `L ϕ. Thus from X `L′ ϕ we getX `L ϕ and vice-versa. Reverse direction: just observe that for finite T we have T `L ϕ iffT `L′ ϕ QED

This is interesting, as we will see in the further text Hajek’s BL has the finite stan-dard completeness but not standard completeness and that the standard semantics of BL isgiven by continuous t-norms. Thus it is sometimes said, that BL is the “logic of continuoust-norms”. The previous theorem says that this is true only to some extent. It is correct if weunderstand logic as a set of theorems/tautologies, if we understand logic as a consequencerelation then BL is “just” a finitary companion of the logic of continuous t-norms.

Now we show that in the case of fuzzy logic with 4, such that `L (¬4ϕ → 4ϕ) → 4ϕwe can easily show the equivalence of the notion of finite standard completeness and weakstandard completeness. Later on we show analogy of this lemma for other classes of logic.

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3.1. STANDARD COMPLETENESS AND CORE FUZZY LOGICS 37

Lemma 3.1.6 Let L be a fuzzy logic in language with 4, `L (¬4ϕ →4ϕ) →4ϕ. If L hasweak standard completeness, then it has finite standard completeness.

Proof: We show that for each finite T we have T `L ϕ iff T |=s−MAT(L) ϕ, one directionis trivial. We show the other one. First we observe that |=s−MAT(L) is fuzzy logic and|=s−MAT(L) (¬4ϕ → 4ϕ) → 4ϕ and so by Lemma 2.2.32 it has the DT4. Suppose thatT |=s−MAT(L) ϕ then by DT4 we get |=s−MAT(L) 4ψ1 → (4ψ2 → (· · · → (4ψn → ϕ) . . . )).From our assumption we get that `L 4ψ1 → (4ψ2 → (· · · → (4ψn → ϕ) . . . )). By DT4 weget that T ′ `L ϕ and so T `L ϕ QED

Lemma 3.1.7 Let L be a fuzzy logic and suppose logic L4 has the (weak, finite) standardcompleteness. Then L has the (weak, finite) standard completeness.

Proof: Observe that in the following argument the cardinality of T does not matter: assumethat T 6`L ϕ. Since L4 is a conservative extension of L we get T 6`L4 ϕ and so there is astandard L4-matrix B4, such that T 6|=B4 ϕ. Thus for the corresponding standard L-matrixB we have T 6|=B ϕ. QED

Theorem 3.1.8 Let L be a fuzzy logic in a language with ∨. Then L4 has the finite standardcompleteness iff L has the finite standard completeness.

Proof: The proof of one direction is Lemma 3.1.7. We prove the reverse direction con-trapositively: assume that L4 has not the finite standard completeness. Thus there is afinite theory T and a formula ϕ, such that T |=s−MAT(L4) ϕ and T 6`L4 ϕ. Using thecompleteness of L4 we know that there is L4-matrix B4 and B4-model e of theory T ,such that e(ϕ) 6∈ DB4 . Let us define the set of formulae SUB as the set of all subfor-mulae of ϕ or formulae from T starting with 4. Then we define two subsets of SUB:SUB1 = {ψ | ψ ∈ SUB and e(4ψ) = 1} and SUB⊥ = SUB \ SUB1. Observe that4ψ ∈ SUB1 iff e(ψ) ≥ 1B4 and 4ψ ∈ SUB⊥ iff e(ψ) < 1B4 . For each formula χ let usdenote by χ′ the formula resulting from χ by replacing each subformula from SUB1 by 1and each subformula from SUB⊥ by ⊥.

Let us define theory T ′ = {χ′ | χ ∈ T} ∪ {ψ | 4ψ ∈ SUB1} and the formula δ =ϕ′ ∨ ∨

4ψ∈SUB⊥ψ. First, we show T ′ 6` δ: take L-matrix B and B-evaluation e and observe

that e is a model of the theory T ′ and e(δ) 6∈ DB4 . If we show that T |=s−MAT(L) δ theproof is done. Let C be an arbitrary standard L-matrix and e a C-evaluation: we have toshow that whenever e is a model of T ′ we have e(δ) ∈ DC. Observe that to prove this isnon-trivial only if e is a model of {ψ | 4ψ ∈ SUB1} and e(ψ) 6∈ DC for each 4ψ ∈ SUB⊥.To prove this notice that for each L4-matrix C4 end each C4-evaluation e if e is a model ofSUB1 and e(ψ) 6∈ DC4 for each ψ ∈ SUB⊥ we have e(χ) = e(χ′). The fact that T |=C4 ϕcompletes the proof. QED

We would like to extend Lemma 3.1.6 to a broader class of fuzzy logics. However, we arenot able to do so, we can only prove the equivalence of this claim with another open problem.

Theorem 3.1.9 Let L be a logic in a language with ∨. The following statements are equiv-alent:

1. If L has weak standard completeness then L has finite standard completeness.

2. If L has weak standard completeness then L4 has weak standard completeness.

Proof: (1) → (2): Assume that L has weak standard completeness, then L has finitestandard completeness and so by Theorem 3.1.8 we know that L4 has the finite standardcompleteness.

(2) → (1): Assume that L has weak standard completeness, then L4 has weak standardcompleteness and so by Lemma 3.1.6 we know that L4 has the finite standard completeness.Lemma 3.1.7 completes the proof. QED

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38 CHAPTER 3. KNOWN FUZZY LOGICS

Notice that in the second part of the proof we have not used the assumption that ∨ isin the language. It is rather common that s−MAT(L) contains only one matrix (up toisomorphism). In the following definition we introduce the notion of core fuzzy logic, this israther auxiliary notion for the needs of this work, many other authors would not agree withthis delimitation.

Definition 3.1.10 (Core fuzzy logic) Let L be a fuzzy logic. We say that L is core fuzzylogic if L is an extension of BL with finite standard completeness and s−MAT(L) containsonly one matrix (up to isomorphism).

For each core fuzzy logic, we encounter in this text we select one of its standard matricesand denote it as [0, 1]L. Because this selection is unique (up to isomorphism) the logic isespecially interesting. The reason is that in this case it allows us to introduce some classicalsemantical notions like satisfiability, compactness, functional representation, etc. In moredetails, since there is a tight correspondence between a logic and one particular matrix wehave for this logic a unique semantics, which allows us to speak about semantical propertiesof a logic without either being general (speaking about classes of “semantics”) or having toindex the property by particular matrix (we can speak about satisfiability and not only aboutB-satisfiability).

For example let us take ÃLukasiewicz logic; it has finite standard completeness w.r.t. onlyone standard matrix [0, 1]ÃL. We say that a formula of ÃLukasiewicz logic is satisfiable if thereis an [0, 1]ÃL-evaluation e, such that e(ϕ) = 1. On the other hand, in intuitionistic logic wehave general satisfiability (if there is Heyting algebra B and a B-evaluation e, such thate(ϕ) = 1) and we also have B-satisfiability for each particular Heyting algebra B. Evenbetter example is the notion of functional representation.

Definition 3.1.11 Let L be a core fuzzy logic. Let C be an arbitrary function from [0, 1]n

into [0, 1] and let ϕ be an arbitrary formula with variables {v1, . . . , vn}. We say the functionC is represented by the formula ϕ in logic L (ϕ is a representation of C ) if e(ϕ) = C ( e(v1),e(v2), . . . , e(vm)), where e is an arbitrary [0, 1]L-evaluation.

Definition 3.1.12 (Functional Representation) Let L be a core fuzzy logic. Let C beclass of functions from any power of [0, 1] into [0, 1]. We say the C is functional representationof logic L if for each C ∈ C there is a formula ϕ such that ϕ is a representation of C andvice-versa (i.e., for each ϕ there is C ∈ C, such that ϕ is a representation of C ).

If we slightly changed Definition 3.1.11 we could say that boolean functions are functionalrepresentation of classical logic. It is obvious that for the formulation of the notion of thefunctional representation we need one unique matrix.

In the next theorem we relate the conservativeness and completeness of particular logic.

Theorem 3.1.13 Let L be a fuzzy logic with (finite) standard completeness. Let L′ be a(finitary) expansion of L, such that each standard L-algebra is a reduct of some standardL′-algebra. Then L′ is a conservative expansion of L.

Proof: We deal with the “finite” case only, the proof of the second one is analogous. Letus denote by K the class of all standard L-algebras and by K′ the class of their expansions.

Let T be a theory and ϕ a formula in the language of L, such that T `L′ ϕ. Since L′ isfinitary, there is a finite T ⊆ T and T `L′ ϕ. Using soundness of L′ we know that T |=K′ ϕ.Thus obviously T |=K ϕ and by finite completeness of L we get T `L ϕ, the rest is trivial.

QED

3.2 Basic fuzzy logic and its expansions

Now we are going to review some particular well-known fuzzy logic meeting our design choices.We start with Hajek’s Basic Fuzzy Logic and its axiomatic extensions. Then we proceed withstudy of expansions of Basic Logic with some particular additional connectives.

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3.2. BASIC FUZZY LOGIC AND ITS EXPANSIONS 39

3.2.1 Basic fuzzy logic and its extensions

The language of BL contains a conjunction &, an implication →, and the constant 0. Furtherconnectives are defined as follows:

ϕ ∧ ψ is ϕ&(ϕ → ψ),ϕ ∨ ψ is ((ϕ → ψ) → ψ) ∧ ((ψ → ϕ) → ϕ),¬ϕ is ϕ → 0,

ϕ ≡ ψ is (ϕ → ψ)&(ψ → ϕ),ϕ⊕ ψ is ¬ϕ → ψ,ϕª ψ is ϕ&¬ψ,

1 is ¬0.

We set > = 1 and ⊥ = 0. The following formulae are the axioms of BL:(A1) (ϕ → ψ) → ((ψ → χ) → (ϕ → χ)) ,(A2) ϕ&ψ → ϕ ,(A3) ϕ&ψ → ψ&ϕ ,(A4) ϕ&(ϕ → ψ) → ψ&(ψ → ϕ) ,(A5a) (ϕ → (ψ → χ)) → (ϕ&ψ → χ) ,(A5b) (ϕ&ψ → χ) → (ϕ → (ψ → χ)) ,(A6) ((ϕ → ψ) → χ) → (((ψ → ϕ) → χ) → χ) ,(A7) 0 → ϕ .

The only deduction rule of BL is modus ponens.There is a new result showing that axiom (A3) is redundant (see [18]). Let us by BL−

denote the logic resulting results from BL by omitting axiom (A3).

Lemma 3.2.1 The following are theorems of BL−:

1. ϕ → ((ϕ → ψ) → ψ) ,

2. (ϕ → (ψ → χ)) → (ψ → (ϕ → χ)) ,

3. ϕ → ϕ ,

4. ϕ&ψ → ψ&ϕ .

Proof:

1. From (A4) and (A2) we get ϕ&(ϕ → ψ) → ψ. The proof is completed by axiom (A5b).

2. From (A1) in the form (ψ → ((ψ → χ) → χ)) → ((((ψ → χ) → χ) → (ϕ → χ)) →(ψ → (ϕ → χ))) and 1. we obtain (((ψ → χ) → χ) → (ϕ → χ)) → (ψ → (ϕ → χ)).Finally, axiom (A1) in the form (ϕ → (ψ → χ)) → (((ψ → χ) → χ) → (ϕ → χ))completes the proof.

3. From (A2) and (A5b) we get ϕ → (ψ → ϕ). Using 2. we get ψ → (ϕ → ϕ). Fixing ψto be an arbitrary theorem completes the proof.

4. We start with 3. in the form ψ&ϕ → ψ&ϕ. From (A5b) and 2. we get ϕ → (ψ → ψ&ϕ).Axiom (A5a) completes the proof. QED

Now we list some useful theorems of BL for their proofs see [44].

Lemma 3.2.2 The following are theorems of BL:(T1) ϕ → (ψ → ϕ)(T2) (ϕ → (ψ → χ)) → (ψ → (ϕ → χ))(T3) ϕ → ϕ

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40 CHAPTER 3. KNOWN FUZZY LOGICS

(T4a) (ϕ → ψ) → (ϕ&χ → ψ&χ))(T4b) (ϕ → ψ) → (ϕ ∧ χ → ψ ∧ χ))(T4c) (ϕ → ψ) → (ϕ ∨ χ → ψ ∨ χ))(T5) (ϕ → (ψ → ϕ&ψ)

(T6) ϕ&ψ → ϕ ∧ ψ(T7) ϕ ∧ ψ → ψ(T8) ((χ → ϕ) ∧ (χ → ψ)) → (χ → ϕ ∧ ψ)(T8’) ((χ → ϕ)&(χ → ψ)) → (χ → ϕ ∧ ψ)(T9) (ϕ → ψ) → (ϕ ≡ ϕ ∧ ψ)(T9’) (ϕ → ψ) ≡ (ϕ → ϕ ∧ ψ)

(T10) ϕ → ϕ ∨ ψ(T11) ((ϕ → χ) ∧ (ψ → χ)) → (ϕ ∨ ψ → χ)(T11’) ((ϕ → χ)&(ψ → χ)) → (ϕ ∨ ψ → χ)(T12) (ϕ → ψ) ∨ (ψ → ϕ)(T13) (ϕ → ψ)n ∨ (ψ → ϕ)n

(T14) (ϕ → ψ) → (ψ ≡ ϕ ∨ ψ)

(T15) ϕ → ¬¬ϕ(T16) ¬ϕ → (ϕ → ψ)(T17) (ϕ → ψ) → (¬ψ → ¬ϕ)(T17’) (ϕ ≡ ψ) → (¬ψ ≡ ¬ϕ)(T18a) ¬(ϕ ∧ ψ) ≡ (¬ϕ ∨ ¬ψ)(T18b) ¬(ϕ ∨ ψ) ≡ (¬ϕ ∧ ¬ψ)

(T19) ϕ&1 ≡ ϕ(T20) (1 → ϕ) ≡ ϕ

(T21) (ϕ ≡ ψ) → ((ϕ&χ) ≡ (ψ&χ))(T21’) (ϕ ≡ ψ)&(χ ≡ δ) → ((ϕ&χ) ≡ (ψ&δ))(T22) (ϕ ≡ ψ) → ((ϕ → χ) ≡ (ψ → χ))(T23) (ϕ ≡ ψ) → ((χ → ϕ) ≡ (χ → ψ))(T23’) (ϕ ≡ ψ)&(χ ≡ δ) → ((ϕ → χ) ≡ (ψ → δ))

(T24a) ϕ&(ψ ∧ χ) ≡ (ϕ&ψ) ∧ (ϕ&χ)(T24b) ϕ&(ψ ∨ χ) ≡ (ϕ&ψ) ∨ (ϕ&χ)(T25a) ϕ ∨ (ψ ∧ χ) ≡ (ϕ ∨ ψ) ∧ (ϕ ∨ χ)(T25b) ϕ ∧ (ψ ∨ χ) ≡ (ϕ ∧ ψ) ∨ (ϕ ∧ χ)(T26a) (ϕ ∧ ψ)&(ϕ ∧ ψ) ≡ (ϕ&ϕ) ∧ (ψ&ψ)(T26b) (ϕ ∨ ψ)&(ϕ ∨ ψ) ≡ (ϕ&ϕ) ∨ (ψ&ψ)(T27) (ϕ&ψ)&χ ≡ ϕ&(ψ&χ)(T28) (ϕ → ψ)&(χ → δ) → ((ψ → χ) → (ϕ → δ))(T29) (ϕ → ψ)&(χ → δ) → ((ϕ&χ) → (ψ&δ))

Lemma 3.2.3 Let I and J be finite sets. The following are theorems of BL:

(S1)∧i∈I

ϕi&∧

j∈J

ψj ≡∧

i∈I,j∈J

(ϕi&ψj)

(S2)∨i∈I

ϕi&∨

j∈J

ψj ≡∨

i∈I,j∈J

(ϕi&ψj)

(S3)∨i∈I

ϕi →∧

j∈J

ψj ≡∧

i∈I,j∈J

(ϕi → ψj)

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3.2. BASIC FUZZY LOGIC AND ITS EXPANSIONS 41

(S4)∧i∈I

ϕi →∨

j∈J

ψj ≡∨

i∈I,j∈J

(ϕi → ψj)

(S5)∧i∈I

ϕi ∨∧

j∈J

ψj ≡∧

i∈I,j∈J

(ϕi ∨ ψj)

(S6)∨i∈I

ϕi ∧∨

j∈J

ψj ≡∨

i∈I,j∈J

(ϕi ∧ ψj)

(T1) ϕ → (ϕ ∧ ψ ≡ ψ)

(T2) ϕ → (ϕ ∨ ψ ≡ 1)

(T3) (ϕ → ψ) → (ϕ ∧ ψ ≡ ϕ)

(T4) (ϕ → ψ) → (ϕ ∨ ψ ≡ ψ)

(T5) (ϕ → ψ) → (ϕ ∧ χ → ψ)

(T6) ((ϕ → ψ)&(ϕ → χ)) → (ϕ → ψ ∧ χ)

Lemma 3.2.4 BL is a {E, W, ax (Sf)}-logic in language {→, &, 0} with DT→.

Proof: Obviously, BL extends BCI and fulfills the conditions of Theorem 2.2.8. Also allthe matching rules for connectives are valid in BL. QED

BL also validates all the matching rules for ∧,∨, 1,⊥,>.

Lemma 3.2.5 BL = FUZZ{&,∧,0}(ax (Sf), E,W,D).

Proof: First, we show that BL is fuzzy. To show that BL has PP, just use DT→ andtheorems (T11) and (T13). To show that BL is the weakest fuzzy logic with E, W, ax (Sf),Dwe show that all its axioms are provable in FUZZ{&,∧,0}(E, W, ax (Sf), D).

• (A1): This is just ax (Sf).

• (A2): From W and E we get ϕ → (ψ → ϕ), residuation rule completes the proof.

• (A3): Using E and residuation rule we get ϕ → (ψ → ψ&ϕ), residuation rule completesthe proof.

• (A4): This is just D.

• (A5a): We give a formal proof

(i) (ϕ → (ψ → χ)) → (ϕ → (ψ → χ)) (Ref)(ii) ϕ → ((ϕ → (ψ → χ)) → (ψ → χ)) (i) and E(iii) ϕ → (ψ → ((ϕ → (ψ → χ)) → χ)) (ii), ax (E), (WT)(iv) ϕ&ψ → ((ϕ → (ψ → χ)) → χ) (iii) and residuation rule(v) (ϕ → (ψ → χ)) → (ϕ&ψ → χ) (iv), E.

• (A5b): Analogous to (A5a).

• (A6): Just read the proof of the second part of Theorem 2.2.12.

• (A7): This is a matching rule for 0. QED

The following observation holds in much broader context (in presence of weakening),however for our needs it is sufficient to formulate it like this:

Lemma 3.2.6 Let L be an extension of BL and B be an ordered L-matrix. Then DB ={1B}.

The BL-matrices tightly correspond to the so-called BL-algebras.

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42 CHAPTER 3. KNOWN FUZZY LOGICS

Definition 3.2.7 A BL-algebra is a structure B = (B,∪,∩, ∗,⇒,0,1) such that:

(1) (L,∪,∩,0,1) is a bounded latticem

(2) (L, ∗,1) is a commutative monoidm

(3) z ≤ (x ⇒ y) iff x ∗ z ≤ y for all x, y, z, (adjointness)

(4) x ∩ y = x ∗ (x ⇒ y), (divisibility)

(5) (x ⇒ y) ∪ (y ⇒ x) = 1. (prelinearity)

Theorem 3.2.8 Let B = ((B, ∗,⇒,0), {1B}) be an ordered BL-matrix. Then the structure(B,∧B,∨B, ∗,⇒,0,1) is a BL-algebra. If B = (B,∪,∩, ∗,⇒,0,1) is a BL-algebra, then((B, ∗,⇒,0), {1}) is an ordered BL-matrix.

Proof: To show the first part we have to show that all defining conditions of BL-algebrasare valid in arbitrary BL-matrix. This is simple because all theorems of BL are tautologiesin each BL-matrix (soundness) and all the conditions we want to prove are among theoremsfrom Lemma 3.2.2. To show the second part it is enough to observe that all axioms of BLare tautologies of arbitrary BL-algebra and that (MP) preserves validity. QED

This allows us to replace the validity condition e(ϕ) ∈ DB by far simpler conditione(ϕ) = 1B. As we restrict ourselves to extension of BL in this work, we will from now on towork with algebras and not with matrices (they are the “same”—the translation shown abovewill work for an arbitrary extension of BL as well). Thus we rephrase all semantical notions(validity, model, semantical consequence, etc.) to accommodate this change. For example,we are getting the completeness theorem:

Theorem 3.2.9 Let ϕ be a formula and T a theory. Then the following are equivalent:

• T `BL ϕ.

• e(ϕ) = 1 for each BL-algebra B and each B-model e of theory T.

• e(ϕ) = 1 for each linearly ordered BL-algebra B and each B-model e of theory T.

The interpretations of a fusion in standard BL-algebras correspond to the well-known classof continuous t-norms and the interpretations of a implication correspond to the so-calledresidua of these t-norms. For details about t-norms see Appendix A. Here we only recall theconnection between t-norms and standard BL-algebras.

Theorem 3.2.10 Let B be a standard BL-algebra. Then &B is a continuous t-norm and→B is its residuum. Conversely, if ∗ is a continuous t-norm and ⇒ is its residuum, then([0, 1],min, max, ∗,⇒, 0, 1) is a standard BL-algebra.

We will denote the standard BL-algebra induced by a continuous t-norm ∗ as BL∗.

Theorem 3.2.11 BL has the finite standard completeness but not standard completeness.BL is not core fuzzy logic.

Proof: The finite standard completeness of BL was proven in [14]. The fact that BL hasnot the standard completeness is part of the folklore. The last fact is trivial (Godel andÃLukasiewicz t-norms are not isomorphic). QED

We have seen that BL is not core fuzzy logic, however BL is complete w.r.t. one particularstandard BL-algebra (this algebra is based on t-norm which is an infinite ordinal sum of ÃLu-kasiewicz t-norms, see [1]). This raises a question whether our definition of core fuzzy logicis not too restrictive, whether we shouldn’t replace the condition that all standard algebrasshould be isomorphic with a weaker one, like elementary equivalent (or even weaker). We

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3.2. BASIC FUZZY LOGIC AND ITS EXPANSIONS 43

answer this question simply by recalling that the notion of core fuzzy logic is just an auxiliaryterm used in this work only.

Now we define the class of extensions of BL, which has finite standard completeness w.r.t.one particular t-norm.

Definition 3.2.12 Let ∗ be a continuous t-norm. Let us denote the finitary companion of|=BL∗ as PC(∗) (propositional calculus induced by ∗).

Obviously TAUT(PC(∗)) = TAUT(s−MAT(PC(∗))) = TAUT(BL∗). It is known (see[55]), that there are only countably many different sets TAUT(PC(∗)) and for each ∗ thereis a finite axiomatic extension of BL L∗ such that TAUT(PC(∗)) = THM(L∗) (see [37]). Weconjecture that PC(∗) = L∗ and:

Conjecture 3.2.13 Each PC(∗) is a finite axiomatic extension of BL. Thus there are count-ably many PC(∗).

This fact is supposed to hold (personal communication with Franco Montagna), howeverwe can not present the proof here so we leave is as a conjecture (it is sufficient for our needs).We end this section by defining one important extension of BL. In this extension we “fix”the interpretation of the negation:

Definition 3.2.14 The Strict Basic logic (SBL) is an axiomatic extension of BL by theaxiom ¬ϕ ∨ ¬¬ϕ. We denote this axiom as St.

The following theorem is a part of folklore:

Theorem 3.2.15 SBL has the finite standard completeness but not standard completeness.SBL is not core fuzzy logic.

The proof of the following lemma is trivial:

Lemma 3.2.16 Let L be an arbitrary extension of SBL and B a linear L-algebra. Then¬0 = 1 and ¬x = 0 for each x > 0.

3.2.2 Basic fuzzy logic with 4Basic logic can be extended to the Basic logic with delta connective. In fuzzy logics is theunary connective 4 known as Baaz delta or 0-1 projector. This connective is well knownand was used in many papers. The axiomatic system for the Godel logic equipped with thisconnective was published by Baaz in his paper [2]. The generalization of this system to thelogic BL is from [44]. We show the usual presentation of the logic BL4 (we call it BL4’) andshow its equivalence with the one from Theorem 2.2.37.

Definition 3.2.17 The logic BL′4 is an extension of BL by axioms:(A41) 4ϕ ∨ ¬4ϕ,

(A42) 4(ϕ ∨ ψ) → (4ϕ ∨4ψ),

(A43) 4ϕ → ϕ,

(A44) 4ϕ →44ϕ,

(A45) 4(ϕ → ψ) → (4ϕ →4ψ).The deduction rules of BL′4 are modus ponens and 4-necessitation.

Lemma 3.2.18 BL′4 = BL4.

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44 CHAPTER 3. KNOWN FUZZY LOGICS

Proof: To prove one direction we have to show that `BL4 A41 and `BL4 A42. The firstaxiom is get easily using Lemma 2.2.32 Part 3. The provability of the second axiom is justa consequence of the fact that BL4 is a fuzzy logic.

To prove the converse direction we have to go through the presentation of BL4 and showits containment in BL′4:

• A: Trivial.

• B: Trivial.

• C: The only non trivial matching rule to prove is 4C, we give the formal proof:

(i) (4ϕ → (4ϕ → ψ)) → (4ϕ → (4ϕ → ψ)) (Ref)(ii) 4ϕ → ((4ϕ → (4ϕ → ψ)) → (4ϕ → ψ)) (i) and E(iii) ¬4ϕ → (4ϕ → ψ) T16(iv) (4ϕ → (4ϕ → ψ)) → (¬4ϕ → (4ϕ → ψ)) (iii) and W(v) ¬4ϕ → ((4ϕ → (4ϕ → ψ)) → (4ϕ → ψ)) (iv) and E(vi) 4ϕ ∨ ¬4ϕ → ((4ϕ → (4ϕ → ψ)) → (4ϕ → ψ)) (ii), (v), and T11(vii) (4ϕ → (4ϕ → ψ)) → (4ϕ → ψ) (vi) and A41

• (MP): Trivial.

• 44: Just use theorem T7 and the definition of ∨.

• 45: Again we give a formal proof:

(i) 4(ψ → ϕ) → (¬4(ϕ → ψ) →4(ψ → ϕ)) ax (W)(ii) 4(ϕ → ψ) → (¬4(ϕ → ψ) →4(ψ → ϕ)) T16 and E(iii) 4(ϕ → ψ) ∨4(ψ → ϕ) →45 (i), (ii), and T11(iv) 4((ϕ → ψ) ∨ (ψ → ϕ)) T12 and (NEC)(v) 4(ϕ → ψ) ∨4(ψ → ϕ) (iv) and (A42)(vi) ¬4(ϕ → ψ) →4(ψ → ϕ) (iii) and (v)

• 46: Trivial. QED

This lemma allows us to apply the machinery prepared in the previous section and obtainknown results about BL4 as corollaries of our general theorems. For example Lemma 3.1.7and Theorem 3.1.8 give us:

Theorem 3.2.19 BL4 has the finite standard completeness but not standard completeness.BL4 is not a core fuzzy logic.

Of course, we also know that BL4 is a fuzzy logic (i.e., it is sound and complete w.r.t.linear BL4-algebras), BL4-algebras have the Subdirect Decomposition Property and BL4is a conservative extension of BL with DT4. It should be obvious how BL4-algebras look,however we present the definition:

Definition 3.2.20 A BL4-algebra is a structure B = (B,∪,∩, ∗,⇒, ∆,0,1) such that:(0) (B,∪,∩, ∗,⇒,0,1) is a BL-algebra,

(1) ∆x ∪ (∆x → 0) = 1,

(2) ∆(x ∪ y) ≤ (∆x ∪∆y),

(3) ∆x ≤ x,

(4) ∆x ≤ ∆∆x,

(5) ∆(x → y) ≤ ∆x → ∆y,

(6) ∆1 = 1.

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Observe that conditions (1)–(5) directly correspond to the axioms, and condition (6)corresponds to 4-necessitation. Of course, BL4-algebras and BL4-matrices are in the samecorrespondence as BL-algebras and BL-matrices. It is obvious, that all what was written inthis subsection remains valid if we replace the logic BL with its arbitrary extension, in moredetails:

Theorem 3.2.21 Let L be a fuzzy logic extending BL and AX its presentation. Then L4has the following presentation:

A axioms of AX,B ` 4ϕ1 → (. . . (4ϕn → ψ) whenever <ϕ1, . . . , ϕn, ψ > ∈ AX,(A41) 4ϕ ∨ ¬4ϕ,(A42) 4(ϕ ∨ ψ) → (4ϕ ∨4ψ),(A43) 4ϕ → ϕ,(A44) 4ϕ →44ϕ,(A45) 4(ϕ → ψ) → (4ϕ →4ψ),(MP) ϕ,ϕ → ψ ` ψ,(NEC) ϕ ` 4ϕ,

Again, in group B it is necessary to add the axioms for the rules different from (MP)only. Thus if the logic L has DT→ we can omit the group B completely.

3.2.3 Strict Basic fuzzy logic with involutive negation

Basic strict fuzzy logic was introduced as a schematic extension of BL by a single axiom. Ofcourse, we can introduce logic SBL4, with the expected properties. Since the negation inSBL is “strict”, a new negation ∼ (co-called involutive negation) can be meaningfully addedto the logic SBL4. This logic and its extension was introduced and examined by Esteva,Godo, Hajek and Navara in their paper [35].

Definition 3.2.22 Basic strict fuzzy logic with involutive negation SBL∼ is SBL4 plus newunary connective ∼ and the following axioms:

(∼1) ∼∼ϕ ≡ ϕ,

(∼2) 4(ϕ → ψ) → (∼ψ → ∼ϕ).

Since SBL∼ is an axiomatic extension of fuzzy logic SBL4 we use Lemma 2.1.48 to get:

Theorem 3.2.23 SBL∼ is a fuzzy logic, i.e., SBL∼ is sound and complete w.r.t. the classof linearly ordered SBL∼-algebras.

It is not necessary to present the definition of SBL∼-algebras. We only present oneinteresting lemma:

Lemma 3.2.24 Let B be an SBL∼-algebra. Then ∼B is a order-reversing involution, i.e.,∼B∼Bx = x and x ≤ y then ∼Bx ≥ ∼By. Furthermore, if C is a linear SBL4-algebraand f is a an order-reversing involution, the algebra C′, resulting from C by adding f as aninterpretation for ∼ is a linear SBL∼-algebra.

Proof: From (∼1) we get ∼B∼Bx = x). We also know that: x ≤ y iff x →B y = 1 iff4(x →B y) = 1 iff ∼By →B ∼Bx = 1 iff ∼By ≤ ∼Bx. The second part is simple. QED

This lemma has two obvious corollaries:

Corollary 3.2.25 SBL∼ is a conservative expansion of both SBL and SBL4.

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46 CHAPTER 3. KNOWN FUZZY LOGICS

Corollary 3.2.26 Let B be an SBL∼-algebra. Then ¬B∼Bx = 4x and ¬Bx ≤ ∼Bx.

We can use this corollary to present simpler axiomatic system for SBL∼.

Theorem 3.2.27 SBL∼ is also an extension of SBL with the unary connective ∼, the axioms(∼1), (∼2) and ¬ϕ → ∼ϕ and 4-necessitation, where 4ϕ is defined as ¬∼ϕ.

Proof: To prove one direction we only have to show that ¬ϕ → ∼ϕ and 4ϕ ≡ ¬∼ϕ aretheorems of SBL∼. It is just a simple consequence of the completeness theorem and previouscorollary. For the proof of the reverse direction see [21, Theorem 3.1]. QED

Now we generalize the above definitions and results:

Definition 3.2.28 Let L be an extension of SBL in the same language. Then the logic L∼is a extension of L4 by axioms (∼1) and (∼2).

Of course, we could easily prove an analogy of Theorem 3.2.27. Now we get to a smallterminological problem. The notion of standard matrix as defined in Definition 3.1.1 is notsatisfactory in logics with ∼, the intended (standard) interpretation of ∼ is 1 − x and wewant to reflect this in our notion of “standard” algebra, we also want to change the notionof standard completeness and core fuzzy logic (to be consonant with terminology from [35]).Thus we make a slightly non-standard step: we change the definition of a standard algebra(matrix). We do this for logic with ∼; for any other logics the former definition remainsvalid. Later we do it once more, when we introduce rational constants.

Definition 3.2.29 Let L be an extension of SBL∼ and B be a standard L-matrix in thesense of Definition 3.1.1. We say that B is a standard L-matrix iff ∼B = 1− x.

With the new definition of a standard matrix we can state the following theorem (itsproof is a part of folklore):

Theorem 3.2.30 SBL∼ has finite standard completeness but not standard completeness.SBL∼is not a core fuzzy logic.

3.3 Core fuzzy logics

3.3.1 ÃLukasiewicz logic and its expansions

ÃLukasiewicz logic (ÃL) was introduced by Jan ÃLukasiewicz in his paper [62] in 1920. Thislogic was originally three-valued, he also studied (with A. Tarski) an infinite-valued versionof his logic, but mainly from the philosophical point of view. The proof of the completenessof the infinite-valued ÃLukasiewicz logic was done by Rose and Rosser in [79]. Nowadays theÃLukasiewicz logic is the most developed and most widely used many-valued logic. It has veryinteresting properties. In this chapter we will understand this logic as an extension of thebasic logic BL by one axiom, which we denote (�).

Definition 3.3.1 ÃLukasiewicz logic (ÃL) is an extension of BL by ` ¬¬ϕ → ϕ.

Theorem 3.3.2 It holds:

1. ÃL is fuzzy logic with DT→.

2. ÃL has finite standard completeness but not standard completeness.

3. ÃL is core fuzzy logic.

4. ÃL4 has finite standard completeness but not standard completeness.

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3.3. CORE FUZZY LOGICS 47

5. ÃL4 is core fuzzy logic with DT4.

Proof:

1. Trivial.

2. Finite standard completeness can be proved in many ways. See for example [44, The-orem 3.2.13] or [12]. Original proof of the second part can be found in [84].

3. Folklore.

4. See Theorem 3.1.8 and Lemma 3.1.7.

5. Trivial. QED

In ÃLukasiewicz logic we will denote the multiplicative conjunction as ⊗ rather than &,the reasons for this lies in Chapter 4 where we add one more multiplicative conjunction,which we will denote ¯. The problem is that in the literature about ÃLukasiewicz logic themultiplicative conjunction is usually denoted as ¯. However, we think that our denotation isbetter rooted in the tradition of substructural logics and we also want to reserve the symbol¯ for already mentioned additional multiplicative conjunction, whose standard interpretationwill be the usual product of reals.

Definition 3.3.3 In ÃL we define two additional connectives:

• ϕ⊕ ψ = ¬ϕ → ψ,

• ϕª ψ = ¬(ϕ → ψ).

Since, ÃL is a core fuzzy logic, all its standard algebras are mutually isomorphic. We pickthe one we denote [0, 1]�, as the one where ⊗ is interpreted as ÃLukasiewicz t-norm and → asits residuum. Then the standard interpretation of the derived connectives is rather simple:

• x⊕[0,1]� y = min(1, x + y),

• xª[0,1]� y = max(0, x− y),

• ¬[0,1]�y = 1− x.

The algebras for ÃLukasiewicz logic are usually taken with the different signature (⊕,¬,0),they are the so-called MV-algebras:

Definition 3.3.4 An MV-algebra is a structure L = (L,⊕,¬,0) such that, letting xª y =¬(¬x⊕ y), and 1 = ¬0 the following conditions are satisfied:

(MV1) (L,⊕,0) is a commutative monoid,(MV2) x⊕ 1 = 1 ,(MV3) ¬¬x = x ,(MV4) (xª y)⊕ y = (y ª x)⊕ x .

In each MV-algebra, we define the additional connectives: x⊗ y = ¬(¬x⊕¬y), x → y =¬x ⊕ y, x ∨ y = (x ª y) ⊕ y, x ∧ y = ¬(¬x ∨ ¬y). It is easy to show that MV-algebrasand ÃL-algebras are termwise equivalent (there are even other termwise equivalent defini-tions: Wajsberg algebras and basic bounded Wajsberg hoops). For deep algebraic results inMV-algebras see [12]. There the proof of the following proposition can be find:

Proposition 3.3.5 In every MV-algebra, the following conditions hold:

1. xª 0 = x ,

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48 CHAPTER 3. KNOWN FUZZY LOGICS

2. xª x = 0 ,

3. 0ª x = 0 ,

4. the following conditions are equivalent: x ≤ y, xª y = 0, x → y = 1 ,

5. if x ≤ y, then x⊕ z ≤ y ⊕ z, xª z ≤ y ª z, and z ª y ≤ z ª x ,

6. (xª y) ∧ (y ª x) = 0 ,

7. x⊗ (y ∨ z) = (x⊗ y) ∨ (x⊗ z) ,

8. x⊕ (y ∧ z) = (x⊕ y) ∧ (x⊕ z) ,

9. 1ª x = ¬x ,

10. a = (a⊕ ¬a)ª ¬a .

We conclude this section with McNaughton’s famous theorem about functional represen-tation of the ÃLukasiewicz logic.

Theorem 3.3.6 The class of all piece-wise linear functions with integer coefficients is afunctional representation of ÃLukasiewicz logic.

3.3.2 Godel logic and its expansions

Here we deal with Godel logic (denoted as G) and its extensions G4 and G∼. We will callthem together Godel logics. Godel logic is a well-known logical system developed from theoriginal Godel ’s system [42] by Rose and Rosser in [79], Dummett in [29], and put in thecontext of other many-valued logics by Hajek in [44]. G4 logic (Godel logic with delta) resultsfrom Godel logic by adding unary connective 4. Finally, G∼ logic (Godel involutive logic)arises from Godel logic by adding unary connective ∼ interpreted as an involutive negation.

Godel logic is extension of BL by single axiom, which we denote (G).

Definition 3.3.7 Godel logic (G) is an extension of BL by ` ϕ → ϕ&ϕ.

We show that G is a logic with contraction and even more. Before we do so we preparea few auxiliary theorems (we advise the reader to recall the names and denotations of theimportant consecutions). This lemma also demonstrates the power of contraction.

Lemma 3.3.8 The following consecutions hold in MIN&(E, W, C):

1. ` ϕ → (ψ → ϕ) ax (W )

2. χ → ϕ, χ → ψ ` χ → ϕ&ψ ∧1

3. ` ϕ&ψ → ϕ ∧2

4. ` ϕ&ψ → ψ&ϕ ∧3

5. ` (ϕ&ψ)&χ → ϕ&(ψ&ϕ) Associativity

6. ` ϕ&(ϕ → ψ) → ψ

7. ϕ → ψ ` ϕ&χ → ψ&χ

8. ` (ϕ → ψ) → ((ψ → χ) → (ϕ → χ)) ax (Sf)

Proof:

1. Observe that ϕ → ϕ ` ψ → (ϕ → ϕ) (from W). E completes the proof.

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3.3. CORE FUZZY LOGICS 49

2. We give a formal proof:

(i) ` ϕ → (ψ → ϕ&ψ) (Ref), and R(ii) χ → ϕ, χ → ψ ` ϕ → (ϕ → ϕ&ψ) (i)(ii) χ → ϕ, χ → ψ ` χ → (χ → ϕ&ψ) (ii), (WT), and E(iii) χ → ϕ, χ → ψ ` χ → ϕ&ψ (ii) and C

3. Trivial.

4. From ` ϕ → (ψ → ϕ&ψ) ((Ref) and R) we get ` ψ → (ϕ → ϕ&ψ) (using E). R again,completes the proof.

5. First, observe ` ϕ&ψ → ϕ and so ` (ϕ&ψ)&χ → ϕ. Second ` (ϕ&ψ) → ψ and so` (ϕ&ψ)&χ → ψ, this together with ` (ϕ&ψ)&χ → χ gets us ` (ϕ&ψ)&χ → ψ&χ(using ∧1). If we use ∧1 once more the proof is done.

6. Trivial.

7. Trivial.

8. From ` ϕ&(ϕ → ψ) → ψ we get ` (ϕ&(ϕ → ψ))&(ψ → χ) → ψ&(ψ → χ). Since` ψ&(ψ → χ) → χ we get ` (ϕ&(ϕ → ψ))&(ψ → χ) → χ. Associativity and R twicecomplete the proof. QED

Theorem 3.3.9 G = FUZZ⊥,&(E, W, C).

Proof: One direction is simple, all we have to do is to show ϕ → (ϕ → ψ) `G ϕ → ψ, butthis is quite straightforward (just use (A5), (G) and (A1)).

The second direction is not so trivial. Let us denote FUZZ⊥,&(E,W, C) as L. Notice thatwe can use the previous lemma. As a consequence of ∧1 we immediately get `L ϕ → ϕ&ϕ.Since we know that `L ax (Sf) all we need to show is `L D and the proof is done (usingLemma 3.2.5).

(i) ` ϕ → (ψ → ϕ) ax (W )(ii) ` ϕ&(ϕ → ψ) → (ψ → ϕ) (i), ∧2, and (WT)(iii) ` (ϕ → ψ) → (ϕ → ψ) (Ref)(iv) ` ϕ&(ϕ → ψ) → ψ (iii) and R(v) ` ϕ&(ϕ → ψ) → ψ&(ψ → ϕ) (ii), (iv), and ∧1

QED

Corollary 3.3.10 G is an extension of Intuitionistic logic by axiom A6. G is the weakestfuzzy logic extending Intuitionistic logic.

Observe that we have `G ϕ&ψ ≡ ϕ∧ψ, i.e., both conjunctions (multiplicative and latticeone) are the same. Thus we use only one symbol from now on—we choose ∧.

Theorem 3.3.11 Godel logic has so called Classical Deduction Theorem (CDT): for eachtheory T and formulae ϕ,ψ: T, ϕ ` ψ iff T ` ϕ → ψ. Furthermore, Godel logic is the weakestfuzzy logic with CDT in language →,&,⊥.

Proof: The proof that G logic has CDT is just like in classical logic. To prove the secondpart all we need to show is that in arbitrary logic with CDT we have W,E, C. To get thisjust use CDT to the following three consecutions, which obviously hold in arbitrary logic:ϕ,ψ ` ϕ, ϕ → (ψ → χ), ψ, ϕ,` χ, and ϕ → (ϕ → ψ), ϕ ` ψ. QED

Recall that the logic L∼ is defined for extension of SBL only, we show that this is thecase of Godel logic.

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50 CHAPTER 3. KNOWN FUZZY LOGICS

Lemma 3.3.12 We have: `G ¬¬ϕ ∨ ¬ϕ, i.e., G is an extension of SBL.

Proof: From Theorem (T16) we get ϕ ∧ ¬ϕ → 0. Theorem (T18a) completes the proof.QED

Now we look at the semantical aspects of Godel logics. Observe that there is exactly onestandard G,G4, and G∼-algebra (recall that the meaning of the term “standard” is changedin logic with ∼, under the original definition G∼ would have standard completeness but itwould not be a core fuzzy logic). The proof of standard completeness of G can be found inmany places (the original proof is due to Dummett [29], see also [44, Theorem 4.2.17]). Therest of the following theorem is partially part of folklore and partially consequence of ourgeneral theorems. Observe that if we would like to prove everything, all we need to do is toshow that G∼ has the standard completeness, the rest would be a consequence of the factthat G∼ conservatively extends G and G4.

Theorem 3.3.13 It holds:

1. G is fuzzy logic with CDT.

2. G4, G∼ are fuzzy logics with DT4.

3. G, G4, G∼ have the standard completeness.

4. G, G4, G∼ are core fuzzy logics.

3.3.3 Product logic and its expansions

Product logic (Π) was introduced by Hajek, Godo, and Esteva in [52] as an extension of BLby axioms (St and (Π)).

Definition 3.3.14 Product logic (Π) is an extension of BL by axioms ` ¬ϕ ∨ ¬¬ϕ and` ¬¬ϕ → ((ϕ&ψ → ϕ&χ) → (ψ → χ)).

Obviously, Π is an extension of SBL by axiom (Π). There is an obvious question, whetherthere is some more elegant presentation of Π, preferably with one axiom with only onevariable. We answer this question is Section 5.1. In the standard Π-algebra the conjunction isinterpreted by a product t-norm (usual multiplication of reals) and implication is interpretedby its residuum.

Theorem 3.3.15 It holds:

1. Π is fuzzy logic with DT→.

2. Π4 and Π∼ are fuzzy logics with DT4.

3. Π and Π4 have the standard completeness.

4. Π and Π4 are core fuzzy logics.

5. Π∼ has not the weak standard completeness.

6. Π∼ is not core fuzzy logic.

The only non-trivial part is part 3., for its proof see [44, Theorem 4.1.13]. For thefunctional representation of Π see Section 5.3.

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3.3.4 The ÃLΠ and ÃLΠ12

logics

In this section we deal with logics ÃLΠ, ÃLΠ12 and R�Π. These logic were introduced by Esteva,

Godo, and Montagna in papers [36] and [32] and then were further elaborated by the author([24], [21], [17], [23]), Montagna ([67], [68]), Vetterlein ([81]) and others.

Definition 3.3.16 Logic ÃLΠ has the following basic connectives (they are listed togetherwith their standard semantics in [0, 1]; we use the same symbols for logical connectives andthe corresponding algebraic operations):

0 0 truth constant falsum

ϕ →� ψ x →� y = min(1, 1− x + y) ÃLukasiewicz implication

ϕ →Π ψ x →Π y = min(1, xy ) product implication

ϕ&Πψ x&Πy = x · y product conjunction

Logic ÃLΠ12 has one additional truth constant 1

2 with standard semantics 12 . We define the

following derived connectives:

¬�φ is φ →� 0 ¬�x = 1− x

¬Πφ φ →Π 0 ¬�x = 0x

1 ¬�0 1

4φ ¬Π¬�φ 41 = 1; 4x = 0 otherwise

φ&�ψ ¬�(φ →� ¬�ψ) x&�y = max(0, x + y − 1)

φ⊕ ψ ¬�φ →� ψ x⊕ y = min(1, x + y)

φª ψ φ&�¬�ψ xª y = max(0, x− y)

φ∧ ψ φ&�(φ →� ψ) x∧ y = min(x, y)

φ ∨ ψ (φ →� ψ) →� ψ x ∨ y = max(x, y)

φ →G ψ 4(φ →� ψ) ∨ ψ x →G y = 1 if x ≤ y, otherwise y

We assume the usual precedence of connectives. Occasionally we may write ¬G and &G

as synonyms for ¬Π and ∧, respectively. We further abbreviate (φ →∗ ψ)&∗(ψ →∗ φ) byφ ↔∗ ψ for ∗ ∈ {G,�, Π}.

There are different axiomatic systems of the logic ÃLΠ. We use the one from [24].

Definition 3.3.17 Logic ÃLΠ is given by the following axioms:(ÃL) Axioms of ÃLukasiewicz logic,(Π) Axioms of product logic,(ÃL4) 4(ϕ →� ψ) →� (ϕ →Π ψ),(Π4) 4(ϕ →Π ψ) →� (ϕ →� ψ),(Dist) ϕ&Π(χª ψ) ↔� (ϕ&Πχ)ª (ϕ&Πψ).

The deduction rules are modus ponens and 4-necessitation (from ϕ infer 4ϕ). The logicÃLΠ 1

2 results from the logic ÃLΠ by adding axiom 12 ↔ ¬� 1

2 .

Observe that ϕ →A ψ ` ϕ →B ψ is valid deduction rule for arbitrary A,B ∈ {G,�, Π}

Lemma 3.3.18 The following formulae are theorems of the ÃLΠ logic:

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52 CHAPTER 3. KNOWN FUZZY LOGICS

(�Π1) ¬�4ϕ →Π ¬Π4ϕ(�Π2) (4ϕ¯ ψ) ↔Π (4ϕ⊗ ψ)(�Π3) (4ϕ →Π ψ) →� (4ϕ →� ψ)(�Π4) 4(ϕ ↔� ψ) ↔� 4(ϕ ↔Π ψ)(�Π5) (ϕ →G ψ) →� (ϕ →� ψ)(�Π6) (ϕ →G ψ) →� (ϕ →Π ψ)(�Π7) 4(ϕ ↔� ψ) ∧4(χ ↔� δ) →� ((ϕ¯ χ) ↔� (ψ ¯ δ))(�Π8) 4(ϕ ↔� ψ) ∧4(χ ↔� δ) →� ((ϕ →Π χ) ↔� (ψ →Π δ))

The consequence of this lemma is that it is not always necessary to write index for eachconnective, eg. (�Π3) shows that whenever an antecedent of �-implication is a formulastarting by 4 the implication is provably equivalent to B-implication for arbitrary B. Thefollowing theorem can be proven by showing corresponding formal proofs (although some ofits claim have shorter “semantical” proof). For details see [23], [24] and the authors masterthesis [22].

Theorem 3.3.19 The logic ÃLΠ extends the following logics: ÃL, ÃL4, G, G4, G∼, Π, Π4,and Π∼.

By saying that ÃLΠ extends Π we mean that if we restrict the logic ÃLΠ to the “prod-uct” connectives we obtain an extension of product logic. Now we introduce the notionof ÃLΠ-algebra and observe that ÃLΠ-algebras are just ordered ÃLΠ-matrices. For alternativedefinition of ÃLΠ-algebras see [17].

Definition 3.3.20 An ÃLΠ-algebra is a structure L = (L,⊕,¬�,→Π,&Π, 0, 1) such that:

• (L,⊕,¬�, 0) is an MV-algebra,

• (L,∨,∧,→Π, &Π, 0, 1) is a Π-algebra,

• x&Π(y ª z) = (x&Πy)ª (x&Πz).

Furthermore, the structure L = (L,⊕,¬�,→Π, &Π, 0, 1, 12 ) where the reduct

L′ = (L,⊕,¬�,→Π,&Π, 0, 1) is an ÃLΠ-algebra and the identity 12 = ¬� 1

2 holds is called theÃLΠ1

2 -algebra.

Theorem 3.3.21 We have:

1. Both ÃLΠ and ÃLΠ 12 are weakly implicative logic (the role of the principal implication can

be played by any of →G,→Π, or →�).

2. Both ÃLΠ and ÃLΠ12 have DT4.

3. Both ÃLΠ and ÃLΠ12 are core fuzzy logics.

4. Both ÃLΠ and ÃLΠ12 extend the following logics conservatively: ÃL, ÃL4, G, G4, G∼, Π,

and Π4.

5. Both ÃLΠ and ÃLΠ12 do not have standard completeness.

Proof:

1. Consequence of Lemmata 3.3.18 and 2.1.16.

2. Consequence of Theorem 2.2.26 and the fact that ÃLΠ extends ÃL4.

3. The fact that they are fuzzy logic is a consequence of Lemma 2.2.28. For the proof offinite standard completeness see [36], the rest is obvious.

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3.3. CORE FUZZY LOGICS 53

4. Easy consequence of Theorem 3.1.13 and the fact that all those logic are core fuzzylogics.

5. If not, we would get a standard completeness of ÃL (using the previous part)—a contra-diction. QED

Let us by [0, 1]�Π denote the ÃLΠ-algebra with operations as stated in Definition 3.3.16.(analogously for the standard ÃLΠ1

2 -algebra). The two-valued ÃLΠ algebra is denoted by {0,1}.In [21] the author proves that ÃLΠ-logic can be viewed as an axiomatic extension of Π∼ (if

we understand the connectives of Π∼ as “product” connectives of ÃLΠ and define the othersas in the following theorem).

Theorem 3.3.22 The following is a presentation of ÃLΠ:

(Π) axioms and deduction rules of Π∼,

(A) (ϕ →� ψ) →� ((ψ →� χ) →� (ϕ →� χ)),

where ϕ →� ψ is defined as ∼(ϕ&∼(ϕ → ψ)).

This result can be improved using results from the upcoming paper of Thomas Vetterlein[81]:

Theorem 3.3.23 The following is a presentation of ÃLΠ:

(Π) axioms and deduction rules of Π∼,

(A) ϕ&�ψ → ψ&�ϕ,

where ϕ&�ψ is defined as ϕ&∼(ϕ → ∼ψ).

The following definitions and theorems demonstrate the expressive power of ÃLΠ and ÃLΠ12

logics. Particularly, Corollary 3.3.26 shows that each propositional logic based on an arbitraryt-norm of a certain simple form is contained in ÃLΠ1

2 logic.

Definition 3.3.24 Let f be a function f : [0, 1]n → [0, 1]. Function f is called a rationalÃLΠ-function iff there is a finite partition of [0, 1]n such that each block of the partition isa semi-algebraic set and f restricted to each block is a fraction of two polynomials withrational coefficients. Furthermore, a rational ÃLΠ-function f is integral iff all the coefficientsare integer and {f(a1, . . . an) | ai ∈ {0, 1}} ⊆ {0, 1}.

The following theorem was proved in [69]. It has an interesting corollary, which wasindependently proved in [23].

Theorem 3.3.25 (Functional representation) The class of integral (rational) ÃLΠ func-tions is the functional representation of the ÃLΠ logic ( ÃLΠ1

2 logic resp.).

Corollary 3.3.26 Let ∗ be a continuous t-norm which is a finite ordinal sum of the threebasic ones (i.e., of G, � and Π), and ⇒ be its residuum. Then there are derived connectives&∗ and →∗ of the ÃLΠ1

2 logic such that their standard semantics are ∗ and ⇒ respectively.The logic PC (∗) of the t-norm ∗ is contained in ÃLΠ 1

2 if & and → of PC (∗) are interpretedas &∗ and →∗. Furthermore, if φ is provable in PC (∗) then the formula φ∗ obtained from φby replacing the connectives & and → of PC (∗) by &∗ and →∗ is provable in ÃLΠ1

2 .

Convention 3.3.27 Further on, the signs ∗ and ¦ will be reserved for t-norms definable inÃLΠ 1

2 (incl. G, ÃL and Π), and the indexed connectives will always have the meaning introducedin the previous Corollary. However, we omit the indices of connectives whenever they do notmatter, i.e., whenever all formulae obtained by subscripting any ∗ to such a connective areprovably equivalent (for example, ¬¬Gφ, 4(φ → ψ), etc.), or equivalently provable (e.g., theprincipal implication in axioms and theorems).

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54 CHAPTER 3. KNOWN FUZZY LOGICS

Corollary 3.3.28 Let r ∈ [0, 1] be a rational number. Then there is a formula ϕ of ÃLΠ12

such that e(ϕ) = r for any [0,1]-evaluation e.

This corollary tells us that in ÃLΠ12 we have a truth constant r for each rational number

r ∈ [0, 1]. Using the completeness theorem we get the following corollary.

Corollary 3.3.29 The following are theorems of the ÃLΠ 12 logic:

r&Πs = r&Πs,

r →Π s = r →Π s,

r →� s = r →� s.

where the symbols &Π,→Π,→� on the left side are operations in [0, 1] and on the right sidethey are logical connectives.

3.4 Adding truth constants

In this section we explore the so-called Pavelka-style extension of a particular fuzzy logic. Bythe term “Pavelka-style extension” we understand the logic created from some fuzzy logicby adding truth constants for all rations from [0, 1] which fulfills some additional properties.We present a general methodology for defining Pavelka-style extensions and some generaltheorems.

The original source of this approach is Pavelka’s work [76], where he adds truth constantsfor each real number form [0, 1] into the language of the ÃLukasiewicz logic and develops a newlogic in this framework. The most recent work in this area is a book by Novak, Perfilieva, andMockor [75], where the authors develop the so-called logic with evaluated syntax. The authorsintroduce a new kind of completeness. It relates the so-called provability and truth degreesof formulae. Although it is a completely natural notion of completeness in their approach, itis incomparable in power with our usual notion of completeness and thus we decided to callit a Pavelka-style completeness.

Logic with evaluated syntax is not a logic in our sense (it has non-classical syntax) andHajek in his book [44] showed how to interpret this logic in our framework. However, thisis a formal interpretation, which omits the extra-logical (linguistic, philosophical) merits oflogic with evaluated syntax. We do not want to discuss the difference between these twoapproaches here, but Chapter 8 contains some more ideas about this problem.

It seems reasonable to restrict the set of additional constants to rationals from [0, 1] only(let us denote this set by IQ). The reason is that we want to keep the language countable,so we can speak about decidability and complexity of the resulting logics.

As mentioned above we want to present a general methodology for defining Pavelka-styleextensions. There are previous works in this area which are incomparable with our work ingenerality. The original work of Pavelka (and then the work of Novak) can be extended tomore logics expanding ÃLukasiewicz logic than our approach. On the other hand our approachworks for logics not extending ÃLukasiewicz logic. The work of Esteva, Godo, and Noguera[38] deals with logics extending (in the same language) so-called weak nilpotent minimumlogic (from the logics defined in this work only Godel logic belongs in this class) and they alsodeals with standard completeness only, whereas we deal with both standard and Pavelka-stylecompleteness.

3.4.1 Rational extension and basic definitions

In this subsection we introduce the so-called Rational extension of some fuzzy logic, it willbe a first step in the definition of Pavelka-style extension. We also introduce some conceptswe will need in the further text.

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3.4. ADDING TRUTH CONSTANTS 55

Definition 3.4.1 Let L be a fuzzy logic and B a standard L-algebra. We say that B is arational standard L-algebra if the set IQ is closed under operations of B. We say that corefuzzy logic is rational if [0, 1]L is rational.

Definition 3.4.2 Let L be a propositional language, L a fuzzy logic in L, and B a rationalstandard L-algebra. The rational extension of L based on algebra B (RL(B)) in languageL∪{(r, 0) | r ∈ IQ} results from logic L by adding axioms c(r1, . . . , rn) ↔ cB(r1, . . . , rn) foreach (c, n) ∈ L and each r1, . . . , rn ∈ IQ.

Lemma 3.4.3 �Π12 = R�Π([0, 1]�Π) = R�Π1

2 ([0, 1]�Π 12).

For the proof see Corollary 3.3.29. Obviously, the logic RL(B) is weakly implicative. Weshow that it is also a fuzzy logic. But first we prove one more general lemma.

Lemma 3.4.4 Let L be a propositional language, L a fuzzy logic in L complete w.r.t. a classK of L-algebras, and B a rational standard L-algebra. The logic RL(B) is complete w.r.t.the class of all RL(B)-algebras, whose reducts to the language L are from K.

Proof: Let us assume that T 6`RL(B) ϕ, we know that there is RL(B)-algebra C andC-model e of theory T such that e(ϕ) < 1C.

Let us take an infinite set VAR0 of propositional variables not occurring in T and ϕand index its elements by rations from IQ, i.e., VAR0 = {vr | r ∈ IQ}. Define theoryT0 = {c(vr1 , . . . , vrn) ↔ vcC(r1,...,rn) | (c, n) ∈ L and r1, . . . , rn ∈ IQ}. For each formula ψdefine formula ψ′ as a formula resulting from ψ by replacing each occurrence of truth constantr with variable vr. Define theory T ′ = {ψ′ | ψ ∈ T}. Now observe that T0∪T ′ 6|={C} ϕ′ (justtake C-evaluation e′ resulting C-evaluation e by setting e(vr) = rC). Let us by C′ denotethe restriction of C to the language L. By observing that T0 ∪ T ′ ∪ {ϕ′} ⊆ FORL we getT0 ∪ T ′ 6|={C′} ϕ′ and so T0 ∪ T ′ 6`L ϕ′. Using the fact that L is complete w.r.t. class K weknow that there is L-algebra A and A-model f of theory T0 ∪ T ′ such that f(ϕ′) < 1A. Ifwe enlarge the L-algebra A into the RL(B)-algebra A′ by setting rA′ = f(vr) we observethat f is A′-model of T and f(ϕ) < 1A. QED

Corollary 3.4.5 Let L be a propositional language, L a fuzzy logic in L, and B a rationalstandard L-algebra. Then the logic RL(B) is a fuzzy logic.

Here we have the same problem as in the Section 3.2.3 about involutive negation. Fromthe last lemma we would get that if L is a logic with standard completeness, then RL(B)has standard completeness as well. However, the following example shows the non-intuitiveconsequence of such a definition.

Example 3.4.6 Let us consider the algebra B, whose reduct is the standard G-algebra [0, 1]Gand rational constants are interpreted in the following way: rB = 1B for each r > 0. ThenB is a RG([0, 1]G)-algebra.

We can observe that the interpretation of the rational constants in this example is rather“non-intended” and so we define:

Definition 3.4.7 Let L be a fuzzy logic, B be a standard L-algebra, and C a standardRL(B)-algebra in the sense of Definitions 3.1.1 and 3.2.29. We say that C is a standardL-algebra iff rC = r.

Corollary 3.4.8 The logic RG([0, 1]G) has not the finite standard completeness.

Proof: Let A be a standard RG([0, 1]G)-algebra, then 12 |=A 0 (since there is no A-model

of theory {12}. On the other hand, let B be the RG([0, 1]G)-algebra form Example 3.4.6.

Then obviously 12 6|=B 0—a contradiction. QED

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56 CHAPTER 3. KNOWN FUZZY LOGICS

¦Convention 3.4.9 Let us define formula ϕ →+ ψ as ϕ → ψ and formula ϕ →− ψ asψ → ϕ. Furthermore, we understand x <+ y as x < y and x <− y as y < x.

Definition 3.4.10 We say that the fuzzy logic L in language L is argument-monotonous ifffor each (c, n) ∈ L there is sequence c ∈ {+,−}n such that for each i ≤ n:

ϕ →C(i) ψ `L c(χ1, . . . , χi−1, ϕ, . . . , χn) → c(χ1, . . . , χi−1, ψ, . . . , χn).

Let us observe that all the logics we have defined are argument-monotonous, and we have:

Example 3.4.11 Let L be a fuzzy logic, then:

• → = (−, +)

• & = ∨ = ∧ = (+, +)

• ¬ = ∼ = (−)

• 4 = (+)

The following lemma demonstrated why we decided to call the above mentioned propertyby the term argument monotonous.

Lemma 3.4.12 Let L be a fuzzy logic. Then the following are equivalent:

1. the logic L is argument-monotonous,

2. {ϕi →c(i) ψi | i ≤ n} `L c(ϕ1, . . . , ϕn) → c(ψ1, . . . , ψn) for each (c, n) ∈ L,

3. for each linear L-algebra B, the operation cB is non-decreasing in the i-th argumentwhenever c(i) = + and non-increasing otherwise.

By monotonicity we understand monotonicity w.r.t. the matrix order ≤B. The proof issimple yet rather technical, so we leave it for the reader. Observe that for core fuzzy logicsit is enough to speak about standard L-algebras in Part 3.

Now we introduce the set of non-continuity points in some standard algebra. We assumethe standard topology of [0,1].

Definition 3.4.13 Let L be a propositional language, L a fuzzy logic in L, and B a standardL-algebra. Let us define the set (of non-continuity points)NC(B) = {<c, x1, . . . , xn > | (c, n) ∈ L and function cB is non-continuous in (x1, . . . , xn)}.If L is core fuzzy logic, then by NC(L) we denote the set NC([0, 1]L).

For each element l ∈ NC(B) we define the set consecutions ConB(l).

Definition 3.4.14 Let L be an argument-monotonous fuzzy logic and B some rational stan-dard L-algebra. Then the following consecutions are elements of ConB(<c, x1, . . . , xn >):

• {r →c(i) ϕi | r <c(i) xi} ` r → c(ϕ1, . . . , ϕn) for each r ≤ cB(x1, . . . , xn),

• {ϕi →c(i) r | xi <c(i) r} ` c(ϕ1, . . . , ϕn) → r for each r ≥ cB(x1, . . . , xn).

In some cases the set ConB(<c, x1, . . . , xn >) can be greatly simplified, now we presentsome particular deduction rules and then we show their relation to the general definitionabove.

Definition 3.4.15 Let a ∈ [0, 1]. Then we define the infinitary deduction rules:

• IR→: {ϕ → r | r > 0} ` ϕ → 0,

• IRa: {ϕ → r, s → ψ | r > a > s} ` ϕ → ψ,

• IR4: {r → ϕ | r < 1} ` ϕ.

Observe that the rules IR0 and IR→ mutually replaceable.

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3.4. ADDING TRUTH CONSTANTS 57

3.4.2 Pavelka-style extension

Now we define Pavelka-style extension of particular fuzzy logic.

Definition 3.4.16 Let L be an argument-monotonous fuzzy logic and B a rational standardL-algebra. The Pavelka-style extension of L w.r.t. B (denoted as RPL(B)) is the logicRL(B) extended by consecutions ConB(l) for each l ∈ NC(L) and consecutions r ` 0 for eachr < 1.

Furthermore, if L is an argument-monotonous rational core fuzzy logic we define thePavelka-style extension of L (denoted as RPL) as the logic RPL([0, 1]L).

We say that the fuzzy logic L is Pavelka-style fuzzy logic if it is Pavelka-style extensionof some fuzzy logic w.r.t. some its standard algebra.

There is a reason for introducing this definition in this general form and not only forthe core fuzzy logics. Recall that Product involutive logic Π∼ is not a core fuzzy logic (ithas not finite standard completeness) although there is only one standard Π∼-algebra (upto isomorphism). It is the standard Π-algebra with ∼ interpreted as 1 − x, let us denotethis algebra as [0, 1]Π∼ . We will see that the logic RPΠ∼([0, 1]Π∼) shares some interestingproperties of Pavelka-style extensions of core fuzzy logics..

Remark 3.4.17 We list the particular case of the set NC(L) for known fuzzy logics:

• NC(ÃL) = ∅• NC(ÃL4) = {<4, 1 >}• NC(Π) = {<→, 0, 0 >}• NC(�Π) = NC(Π4) = NC([0, 1]Π∼) = {<→, 0, 0 >} ∪ {<4, 1 >}• NC(G) = {<→, a, a > | a ∈ [0, 1]}• NC(G4) = NC(G∼) = {<→, a, a > | a ∈ [0, 1]} ∪ {<4, 1 >}

Now we look at some particular Pavelka-style logics and observe that some of the addi-tional consecution are sometimes redundant. The logic RPÃL is usually denoted as RPL (see[44]). Furthermore, the logic RP�Π is usually denoted as R�Π (see [36]).

Theorem 3.4.18 Particular Pavelka-style extension of known fuzzy logics can be axioma-tized as:

• RPL = RPÃL = RÃL([0, 1]�)

• RPÃL4 = RÃL([0, 1]�) + IR4

• RPΠ = RΠ([0, 1]Π) + IR→

• RPΠ4 = RΠ4([0, 1]Π4) + IR→ + IR4

• RPΠ∼([0, 1]Π∼) = RΠ∼([0, 1]Π∼) + IR4

• RPG = RG([0, 1]G) + {IRa | a ∈ [0, 1]}• RPG4 = RG4([0, 1]G4) + {IRa | a ∈ [0, 1]}+ IR4

• RPG∼ = RG∼([0, 1]G∼) + {IRa | a ∈ [0, 1]}+ IR4

• R�Π = RP�Π = RP�Π 12 = �Π1

2 + IR4

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58 CHAPTER 3. KNOWN FUZZY LOGICS

The proof is almost straightforward. Observe that except for RPL we do not knowwhether any logic mentioned in the previous theorem is actually a fuzzy logic. It is causedby the presence of infinitary deduction rules.

Definition 3.4.19 Let L be a Pavelka-style fuzzy logic w.r.t. the algebra B, T a theory, andϕ be a formula. Then we define:

• the truth degree of ϕ over T is ||ϕ||T = inf{e(ϕ) | e ∈ MOD(T,B)},

• the provability degree of ϕ over T is |ϕ|T = sup{r | T ` r → ϕ}.

Lemma 3.4.20 Let L be a Pavelka-style fuzzy logic w.r.t. algebra B, T a theory, and ϕ, ψformulae. Then:

(1) If T 6` r → ϕ then T ∪ {ϕ → r} is consistent.

Furthermore, if T is linear we have:

(2) |ϕ|T = sup{r|T ` r → ϕ} = inf{r|T ` ϕ → r},

(3) the provability degree commutes with all connectives, i.e., we have: |c(ϕ1, . . . , ϕn)|T =cB(|ϕ1|T , . . . |ϕn|T ),

(4) and for e(ϕ) = |ϕ|T we get e ∈ MOD(T,B).

Proof: Part (1) is trivial use of PP.Part (2): from linearity of T we get {r|T ` r → ϕ} ∪ {r|T ` ϕ → r} = [0, 1]. Obviously,

{r|T ` r → ϕ} = [0, a) and {r|T ` ϕ → r} = (b, 1] for some a, b ∈ [0, 1] (the intervals can beclosed). Thus all we need to show is that is not the case that b < a. In that case we wouldhave r, s ∈ IQ, r > s, T ` r → ϕ, and T ` ϕ → s. Thus T ` r → s. From book-keepingaxioms we know that r → s < 1 and so T ` 0—a contradiction.

Part (3): We show this on example of implication, the proof for other connectives areanalogous. Recall that in our setting (over BL) we have → = (−, +). Let us denote →B as⇒ in this proof. Assume that |ϕ|T = a and |ψ|T = b and we want to show that a ⇒ b =|ϕ → ψ|T . We distinguish two cases:

Case 1. <→, a, b> 6∈ NC(L), i.e., ⇒ is continuous in point (a,b). We start with definitionthen by Part (2) and continuity we get the following chain: a ⇒ b = sup{s|T ` s → ϕ} ⇒sup{r|T ` r → ψ} = inf{s|T ` ϕ → s} ⇒ sup{r|T ` r → ψ} = sup{s ⇒ r | T ` ϕ →s and T ` r → ψ} ≤ sup{s ⇒ r | T ` (s → r) → (ϕ → ψ)} = |ϕ → ψ|T . We prove thesecond direction indirectly, assume that a ⇒ b < |ϕ → ψ|T . From continuity there has to berationals r, s and t such that T 6` ϕ → s, T 6` r → ψ, and a → b ≤ r ⇒ s < t < |ϕ → ψ|T .Using linearity of T we can rewrite that as: T ` s → ϕ, T ` ψ → r and using the Lemma3.4.12 we get T ` (ϕ → ψ) → (r → s). Since we know that T ` t → (ϕ → ψ) (fromt < |ϕ → ψ|T we get T ` t → (r → s) i.e., T ` t ⇒ (r ⇒ s) and because t ⇒ (r ⇒ s) < 1 wehave a contradiction with consistency of T .

Case 2. <→, a, b > ∈ NC(L). We know that a = sup{s|T ` s → ϕ} = inf{s|T ` ϕ → s}and b = sup{r|T ` r → ψ}. We use this knowledge twice:

First, we get that T ` ϕ → s for each s > a and T ` r → ψ for each b > r and by the useof the rules from ConB(<→, a, b>) we get that T ` r → (ϕ → ψ) for each r ≤ a ⇒ b, i.e.,a ⇒ b ≤ |ϕ → ψ|T .

Second usage is analogous: we get that T ` s → ϕ for each a > s and T ` ψ → r for eachr > b and by the use of second group of rules in ConB(<→, a, b >) we get T ` (ϕ → ψ) → rfor each a ⇒ b ≤ r, i.e., |ϕ → ψ|T ≤ a ⇒ b.

Part (4) is trivial. QED

Observe that we use the fact that L is a fuzzy logic in Part (1) only. Now we use theprevious lemma to show the so-called Pavelka-style completeness.

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3.4. ADDING TRUTH CONSTANTS 59

Theorem 3.4.21 (Pavelka-style completeness) Let L be a Pavelka-style fuzzy logic w.r.tthe algebra B, T a theory, and ϕ a formula. Then: |ϕ|T = ||ϕ||T .

Proof: One direction is easy: whenever T ` r → ϕ then r ≤ e(ϕ) for each model e of T .Thus |ϕ|T ≤ ||ϕ||T . To prove the second direction assume that |ϕ|T < r < ||ϕ||T . ThusT 6` r → ϕ. By part (1) of the previous lemma T ∪ {ϕ → r} is a consistent theory. Let usextend T ∪ {ϕ → r} into the linear theory T ′ (using LEP). We know that for e(ψ) = |ψ|T ′we have e ∈ MOD(T ′,B). Thus also e ∈ MOD(T,B) and e(ϕ → r) = 1. Finally, e(ϕ) ≤ rand so ||ϕ||T ≤ r—a contradiction. QED

Corollary 3.4.22 Let L be a Pavelka-style fuzzy logic with 4 in the language. Then L hasthe standard completeness. Furthermore, if L is a Pavelka-style extension of some core fuzzylogic K, then L is a core fuzzy logic and it extends K conservatively.

Proof: Assume that T |=L ϕ. Then ||ϕ||T = 1 and so |ϕ| = 1, which means that T ` r → ϕfor each r < 1. Using IR4 we get T ` ϕ. QED

Theorem 3.4.23 Let L be an argument-monotonous rational core fuzzy logic stronger thanÃLukasiewicz logic, such that RPL is a fuzzy logic. Then RPL is a core fuzzy logic and itextends L conservatively.

Proof: All we need to show is the finite standard completeness of RPL. Assume thatT 6`RPL ϕ. Let K be the set of all rationals occurring in formulae of T and in ϕ. For eachrational r ∈ K let us take propositional variable vr and define ϕ′ = ϕ[r := vr] (i.e., replaceall occurrences of r by vr). Then we define T ′ analogously. Let T0 = {r ≡ vr | r ∈ K}.Observe that obviously T0∪T ′ 6`RPL ϕ′. Let k be the least common multiple of denominatorsof rationals in K. Let us pick a propositional variable v not used in T0, T

′, ϕ′. and define atheory T1 = {v ≡ 1

k}∪{v ab≡ ak

b v | ab ∈ K} (by ak

b v we mean v⊕v . . . v, where the argumentv appear ak

b -times). Again, observe that T1 ∪ T ′ 6`RPL ϕ′. Finally, we define theory T2 as T1

with formula v ≡ 1k replaced by {kv,¬((k − 1)v)k}. Again, T2 ∪ T ′ 6`RPL ϕ′ (by the same

counterexample, we only set e(v) = 1k , this is obviously a model of T2 ∪ T ′ and e(ϕ) < 1).

Observe that there is no constant in T2, T ′, and ϕ′ and so by finite standard completenessof L we get that there is [0, 1]L-model e of T2 ∪ T ′, such that e(ϕ′) < 1. To complete theproof just notice that e(v) = 1

k (otherwise e would not be a model of {kv,¬((k− 1)v)k} andso obviously e(vr) = r for each r ∈ K. Thus also e is a model of T and e(ϕ) < 1. QED

Corollary 3.4.24 Let T be a theory over RPL and ϕ a formula. Then: |ϕ|T = ||ϕ||T .Furthermore, RPL is a core fuzzy logic.

To get the Pavelka-style completeness for other logics mentioned in Remark 3.4.18 we firstneed to show that they are fuzzy logics, i.e., that by adding the infinitary rule(s) they remainfuzzy. We have to be careful, the main characterization theorem for fuzzy logics (Theorem2.1.45) works for finitary logics only. Thus our main strategy will be to use Theorem 2.1.42and show that the logics in question have LEP.

Lemma 3.4.25 Let L be a Pavelka-style extension of some fuzzy logic with DT4 and L hasa presentation AX, where there are no infinitary deduction rules except IR4, and IR→. Then

1. L has DT4,

2. L has PP,

3. if T 6` r → ϕ, then T ∪ {ϕ → r} is consistent,

4. L is fuzzy.

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60 CHAPTER 3. KNOWN FUZZY LOGICS

Proof: Part 1. Let T be a theory and ϕ,ψ formulae. We have to show that T, ϕ ` ψ iffT ` 4ϕ → ψ. One direction is obvious, we show the second by induction over the proof of ψin T, ϕ in AX (recall that the proof is well founded relation). The initial step is trivial. Theinduction step has to be done for each deduction rule in AX. Without loss of generality wecan assume that the only deduction rules are (MP), (NEC) an possibly IR4 and IR→. Forthe proof of the induction step of (MP) and (NEC) see the proof of Theorem 2.2.26. Thestep for IR4: assume that T ` 4ϕ → (r → ψ) for each r < 1, thus T ` r → (4ϕ → ψ) andso by IR4 we get T ` 4ϕ → ψ. The step for IR→ is analogous.

Part 2. Trivial.Part 3. See the proof of Part (1) of Theorem 3.4.20Part 4. We show that L has LEP. First, assume that IR4 is the only infinitary deduction

rule. Let us enumerate all formulae, stating ϕ0 = ϕ. We define sequence of theories Ti

• if Ti ` ϕ then Ti+1 = T, ϕ,

• if Ti 6` ϕ. Then there is r < 1 such that Ti 6` r → ϕi. We define Ti+1 = T, ϕi → r.Using Part 3. we get that Ti+1 is consistent.

Observe that T1 = T, r → ϕ. Let us define T =⋃

Ti. Now we show that each proof inT can be replaced by a finite proof. Let χ be a formula. We replace each use of infinitarydeduction rule by a singe formula. We start by such members of the proof of χ in T ′ obtainedby using of an infinitary rule from formulae proved without using any infinitary rule. Then weuse well-foundedness of the proof to obtain the result. Let us denote the formula in questionas ψ and suppose that ψ = ϕi. If Ti ` ψ then ψ ∈ T thus we can replace the proof of ψjust by the formula itself. Now we show that the case Ti 6` ψ can not happen. Recall thatTi+1 = T, ψ → r. Since T ` r → ψ for each r < 1 without using infinitary rules, there has tobe t > r and j such that Tj ` t → ψ. Thus Tmax(i+1,j) ` t → r. Since t →B r < 1 we have acontradiction.

Now construct theory T ′ in the same way as in Lemma 2.1.44 (building sequence ofconsistent theories Ti by adding either ϕi → ψi or ψi → ϕi). This theory is obviously linear.We show its consistency in the same way as we did it for T . Thus T ′ 6` ϕ (since T ′ ` ϕ → rwe would get T ` r—a contradiction with consistency of T ′.)

To prove this lemma even if we have IR→ in the presentation AX just augment thedefinition of Ti in the following way: if ϕi = ¬ψ and Ti 6` ϕi then there is not only r < 1such that T 6` r → ϕi but also s > 0 such that T 6` ϕi → s. We can safely assume that s < rand define Ti+1 = T ∪ {ϕi → r, s → ϕi}. The rest of the proof is analogous. QED

Corollary 3.4.26 Let C be one of the RPÃL4, RPΠ4, RPΠ∼([0, 1]Π∼), or R�Π, and let Tbe a theory over and ϕ a formula. We have:

1. |ϕ|T = ||ϕ||T ,

2. C is standard complete,

3. C is core fuzzy logic,

4. RPΠ∼([0, 1]Π∼) = R�Π.

Notice that although the logics Π∼ and �Π are different, adding truth constants andinfinitary deduction rule makes them coincide.

3.5 First-order logics

In this section we apply results from Section 2.3. Again, we advise the reader to go throughthe definitions and theorems in that section.

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3.5.1 Basic facts

Recall that Definition 2.3.11 gives us two predicate logic for each fuzzy logic. We startby recalling some known theorems of our logics from [44], where they are proved in BL∀.However, it is easy to observe that in fact they are provable already in BL∀−.

Theorem 3.5.1 Assume that ν does not contain x freely. The following are theorems ofBL∀− :(T∀1) (∀x) (ν → ϕ) ≡ (ν → (∀x)ϕ)(T∀2) (∀x) (ϕ → ν) ≡ ((∃x)ϕ → ν)(T∀3) (∃x) (ν → ϕ) → (ν → (∃x) ϕ)(T∀4) (∃x) (ϕ → ν) → ((∀x) ϕ → ν)(T∀5) (∀x) (ϕ → ψ) ≡ ((∀x) ϕ → (∀x) ϕ)(T∀6) (∀x) (ϕ → ψ) ≡ ((∃x) ϕ → (∃x) ψ)(T∀7) (∀x)ϕ&(∃x)ψ → (∃x) (ϕ&ψ)(T∀8) (∀x)ϕ(x) ≡ (∀y) ϕ(y)(T∀8′) (∃x)ϕ(x) ≡ (∃y) ϕ(y)(T∀9) (∃x) (ϕ&ν) ≡ ((∃x)ϕ&ν)(T∀10) (∃x)ϕn ≡ ((∃x)ϕ)n for each n ≥ 1(T∀11) (∃x)ϕ → ¬ (∀y)¬ϕ(T∀12) ¬ (∃x)ϕ ≡ (∀y)¬ϕ(T∀13) (∃x) (ν ∧ ϕ) ≡ (ν ∧ (∃x)ϕ)(T∀14) (∃x) (ν ∨ ϕ) ≡ (ν ∨ (∃x)ϕ)(T∀15) (∀x) (ν ∧ ϕ) ≡ (ν ∧ (∀x)ϕ)(T∀16) (∃x) (ϕ ∨ ψ) ≡ ((∃x)ϕ ∨ (∃x)ϕ)(T∀17) (∀x) (ϕ ∧ ψ) ≡ ((∀x)ϕ ∧ (∀x)ψ)

Let L be logic extending ÃL. The following are theorems of L∀ :

(T ÃL∀1) (∃x)ϕ ≡ ¬ (∀y)¬ϕ(T ÃL∀2) (∃x) (ν → ϕ) ≡ (ν → (∃x)ϕ)(T ÃL∀3) (∃x) (ϕ → ν) ≡ ((∀x)ϕ → ν)(T ÃL∀4) (∀x) (ϕ ∨ ν) ≡ ((∀x) ϕ ∨ ν)(T ÃL∀5) (∀x) (ϕ&ν) ≡ ((∀x) ϕ&ν)(T ÃL∀6) (∃x)nϕ ≡ n((∃x) ϕ) for each n ≥ 1

Observe that in ÃL∀ are quantifiers mutually definable. In [44] it is proved that in case ofÃLukasiewicz logic both predicate logic ÃL∀ and ÃL∀−. This can obviously be extended into thefollowing theorem :

Theorem 3.5.2 Let L be a fuzzy logic extending ÃL. Then L∀− = L∀.

Observe that for all logics L defined in this chapter, we know that L∀ has PP (seeCorollary 2.3.27 and 2.3.28). Thus we have the completeness w.r.t. their linear algebras.

Theorem 3.5.3 Let L be one of the following logics: BL, SBL, SBL4, SBL∼ ÃL, ÃL4, Π,Π4, Π∼, G, G4, G∼, �Π, and �Π1

2 . Then the logic L∀ is sound and complete w.r.t.corresponding class of linear matrices.

However, the notion of the standard completeness seems to be too strong in predicatelogic. We present that the sets of standard tautologies for the most of our logics are rathercomplex (from the point of view of the arithmetical hierarchy).

Definition 3.5.4 Let L be a fuzzy logic. We denote the set of s−MAT(L)-tautologies byTAUT∀(L).

We say that L∀ has the (finite) standard completeness iff T `L∀ ϕ iff T |=s−MAT(L) ϕfor each (finite) theory T and formula ϕ.

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62 CHAPTER 3. KNOWN FUZZY LOGICS

Theorem 3.5.5 We know:

• let C be one of the following logics: BL, BL4, SBL, SBL4, SBL∼, Π, Π4, Π∼, ÃL4,�Π, or �Π 1

2 . Then the set TAUT∀(C) is not arithmetical.

• The set TAUT∀(ÃL) is Π2-complete.

• The sets TAUT∀(G), TAUT∀(G4), and TAUT∀(G∼) are Σ1-complete.

For proof of this theorem see [44] and [45].

Corollary 3.5.6 Let C be one of the following logics: BL, BL4, SBL, SBL4, SBL∼, Π,Π4, Π∼, ÃL4, ÃL, �Π, or �Π 1

2 . Then THM(C∀) 6= TAUT∀(C), i.e., the logic C∀ has not thefinite standard completeness.

Theorem 3.5.7 The logics G∀, G4∀, and G∼∀ have the standard completeness.

This theorem for G∀ is proven in [44] (and in some older works), for G4∀ it is a part offolklore, and for G∼∀ see [35]. There is no known example of a predicate fuzzy logic withfinite standard completeness, which is not standard complete. It turns out that all fuzzypropositional logic lacking standard completeness (having only the finite one) lose the finitestandard completeness by a transition to the first order. This can be just a coincidence, butis definitively worth some further study.

Theorem 3.5.8 Let L be a fuzzy logic with DT→. Then L∀ is ϕ-witnessed for each ϕ.

The proof of this statement is a simple use of DT→ and theorem (T∀10).

Theorem 3.5.9 Let C be a logic with 4 in the language. If C∀ is complete w.r.t. corre-sponding linear algebras then it is undecidable.

Proof: The formula ϕ′ is created from the formula ϕ by replacing each atomic formulaP (t1, . . . , tn) with the formula 4P (t1, . . . , tn). Notice that formula ψ of a classical predicatelanguage is provable iff formula ψ′ is provable in C∀ (this is obvious from the definition of4 and completeness of the C∀). Thus if the C∀ logic would be decidable, it would be acontradiction to the undecidability of the classical predicate logic. QED

At the end of this subsection we recall some theorems of �Π∀.Lemma 3.5.10 Assume that ν does not contain x freely. The following are theorems of�Π∀:

(∀x) (ϕ →∗ ψ) → [(∀x)ϕ →∗ (∀x)ψ] (3.1)(∀x) (ϕ →∗ ψ) → [(∃x)ϕ →∗ (∃x)ψ] (3.2)(∀x) (ϕ∧ ψ) → [(∀x)ϕ∧ (∀x) ψ] (3.3)(∃x) (ϕ ∨ ψ) → [(∃x) ϕ ∨ (∃x)ψ] (3.4)(∀x) (ϕ1&∗ . . . ϕk →∗ χ) → [(∀x)ϕ1&∗ . . . &∗ (∀x)ϕk →∗ (∀x)χ] (3.5)(∀x) (ϕ1&∗ . . . ϕk →∗ χ) → [(∀x)ϕ1&∗ . . . &∗ (∀x)ϕk−1&∗ (∃x) ϕk →∗ (∃x) χ] (3.6)

Proof: In the proof we use an easy generalization of Corollary 3.3.26 to the predicate case.(3.1)–(3.4) are provable in BL∀ (see [44]). (3.5) is proved by a trivial inductive generalizationof the following proof in BL∀:

(∀x) (ϕ&ψ → χ) ↔ (∀x) (ϕ → (ψ → χ)) → [(∀x) ϕ → (∀x) (ψ → χ)]→ [(∀x)ϕ → ((∀x)ψ → (∀x)χ)] ↔ [(∀x)ϕ& (∀x)ψ → (∀x)χ].

(3.6) is proved in the same way, only applying (3.2) instead of (3.1) when distributing (∀x)over (ϕk−1 → ϕk). QED

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3.5. FIRST-ORDER LOGICS 63

3.5.2 Crisp equality

Now we enhance our logic with (crisp) equality. Let us assume that symbol = is a binarypredicate symbol and assume that = ∈ Γ in the rest of this section. We alter the definitionof atomic formula with the line:

• If t1 and t2 are terms of arbitrary sorts, then t1 = t2 is also an atomic Γ-formula.

Then we alter the definition of a value of a formula the following line:

• ||t1 = t2||LM,v = 1B iff ||t1||LM,v = ||t2||LM,v; and 0B otherwise.

And finally we define logic L∀= as a logic L∀ with three additional axioms:

(=1) x = x,(=2) x = y → (ϕ(x, z1, . . . , zn ) ≡ ϕ(y, z1, . . . , zn )),(=3) (x = y) ∨ ¬(x = y).

Observe that in logic with 4 we can remove the last axiom if we replace the second onewith:

(=2′) x = y →4(ϕ(x, z1, . . . , zn ) ≡ ϕ(y, z1, . . . , zn )),

Lemma 3.5.11 The following are theorems of L∀=:

• x = y → y = x

• x = y&y = z → x = z

• x1 = y1& . . . &xn = yn → (ϕ(x1, . . . , xn, z ) ≡ ϕ(y1, . . . , yn, z )).

Obviously, we have completeness theorem w.r.t. all linear algebras and all models.

Theorem 3.5.12 Let L be a fuzzy logic and L∀ be Henkin. Then for each theory T andformula ϕ we have: T `L∀= ϕ iff T |=l

L ϕ.

Corollary 3.5.13 Let L be a finitary fuzzy logic and L∀ with PP. Then for each theory Tand formula ϕ we have: T `L∀= ϕ iff T |=l

L ϕ.

We can easily extend Theorem 2.3.32 into the following theorem (we use the usual deno-tation (∃!y)ϕ(y) for the formula (∃y) (ϕ(y)& (∀x) (ϕ(x) → x = y))).

Theorem 3.5.14 Let L∀ be a proto-ϕ-witnessed finitary fuzzy logic with PP in predicate lan-guage Γ, T a theory, ϕ(x1, . . . , xn, y) a formula, and T ` (∀x1) . . . (∀xn) (∃y) ϕ(x1, . . . xn, y).Then the theory T ′ in the language of T extended by a new function symbol fϕ resulting fromthe theory T by adding the axiom ` (∀x1) . . . (∀xn) ϕ(x1, . . . xn, fϕ(x1, . . . , xn)) is a conser-vative extension of T .

Furthermore, if T ` (∀xs11 ) . . . (∀xn) (∃!y)ϕ((x1), . . . , x1, y) then for each Γ∪{fϕ}-formula

ψ there is a Γ-formula ψ′ such that T ′ ` ψ ≡ ψ′.

3.5.3 Adding truth constants

We conclude this section by extending properties of predicate versions of Pavelka-style fuzzylogics. Let us understand the class MOD(T,B) as the class of all B models of T . We presentthe definition of provability and truth degree and observe that they are quite the same as inthe propositional case.

Definition 3.5.15 Let L be a Pavelka-style fuzzy logic w.r.t. algebra B, T a theory over L∀,and ϕ be a sentence. The we define:

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64 CHAPTER 3. KNOWN FUZZY LOGICS

• the truth degree of ϕ over T is ||ϕ||T = inf{e(ϕ) | e ∈ MOD(T,B)},

• the provability degree of ϕ over T is |ϕ|T = sup{r | T ` r → ϕ}.

Here we run into two similar problems. If we have a fuzzy logic we do not know if itsPavelka-style extension remains fuzzy logic and for particular fuzzy logics we do not knowwhether their first-order variants have PP (i.e., the completeness w.r.t. linearly orderedalgebras). Let us write some sufficient conditions for a fuzzy logic to have Pavelka-stylefirst-order extension with Pavelka-style completeness.

Definition 3.5.16 We say that a logic L∀ has a Pavelka-style completeness iff for eachtheory T and formula ϕ: ||ϕ||T = |ϕ|T .

We continue by proving the analogy of Lemma 3.4.20. Recall that if the logic L∀ is Henkinthen L∀ has PP (see Corollary 2.3.26).

Lemma 3.5.17 Let L be a Pavelka-style fuzzy logic w.r.t. the algebra B; let T be a theory,and ϕ, ψ formulae. Then:

(1) If L∀ has PP then we have: T 6` r → ϕ then T ∪ {ϕ → r} is consistent.

Furthermore, if T is Henkin we define B-structure M with domain of all closed termsin language of T , the functions are defined as usual, and for each n-ary predicate we set:PM(x1, . . . , xn) = |P (x1, . . . , xn)|T . (the xi on the left-side are elements of M and on theright side they are the corresponding closed terms). Then we have:

(2) |ϕ|T = sup{r|T ` r → ϕ} = inf{r|T ` ϕ → r},

(3) the provability degree commutes with all connectives, i.e., we have |c(ϕ1, . . . , ϕn)|T =cB(|ϕ1|T , . . . |ϕn|T ),

(4) |(∀x)ϕ(x)|T = inf{|ϕ(x)|T | x ∈M},

(5) M ∈ MOD(T,B).

Proof: The proof of the first three parts is a direct analogy of the proof of Lemma 3.4.20.The proof of the part (4) is an analogy of the proof of Lemma 2.3.24: let T ` ϕ → ψ thenobviously T ` r → ψ whenever T ` r → ϕ and so |ϕ|T ≤ |ψ|T . Since T ` (∀x)ϕ(x) → ϕ(a)we get |(∀x)ϕ(x)|T ≤ |ϕ(a)|T for each a. Thus |(∀x)ϕ(x)|T ≤ inf{|ϕ(x)|T | x ∈M}

Now assume that |(∀x)ϕ(x)|T < inf{|ϕ(x)|T | x ∈ M}. Thus there is r such that|(∀x)ϕ(x)|T < r < inf{|ϕ(x)|T | x ∈ M}. So we know that T 6` r → (∀x)ϕ(x) andso T 6` (∀x)(r → ϕ(x)) (using axiom (∀2)). Since T is Henkin, there is c such thatT 6` (r → ϕ(c)) and so |ϕ(c)|T ≤ r—a contradiction.

The last part is trivial. QED

This gives us the proof of Pavelka-style completeness.

Theorem 3.5.18 Let L be a Pavelka-style fuzzy logic w.r.t. the algebra B and L∀ is Henkin.Then L∀ has Pavelka-style completeness.

Theorem 3.5.19 Let L be a fuzzy logic and B a standard algebra. The following are suffi-cient conditions for the logic RPL(B)∀ to have Pavelka-style completeness.

• RPL(B) has DT→.

• L has DT4 and L has a presentation AX, where there are no infinitary deduction rulesexcept IR4, and IR→.

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3.5. FIRST-ORDER LOGICS 65

Proof: The first part is easy: since RPL(B) has DT→, it is a finitary fuzzy logic (it is anextension of a fuzzy logic with DT→) and also RPL(B)∀ has PP. The rest is a consequenceof a Corollary 2.3.22.

The second part is more complicated. From Lemma 3.4.25 we know that RPL(B) is afuzzy logic with DT4. Thus RPL(B)∀ has DT4 and PP (Theorem 2.3.13 and Corollary2.3.15).

To show that RPL(B)∀ is Henkin we have to go through the proofs of Lemmata 2.3.21and 3.4.25. Observe that in the proof of Lemma 3.4.25 we have define a for each theory asupertheory in which each proof can be replaced by a finite proof. We do the same trick hereand then we continue with the proof of Lemma 2.3.21, which now can be carried in the sameway. QED

Corollary 3.5.20 The logics RPL∀, RPÃL4∀, RPΠ4∀, and R�Π∀ have Pavelka-style com-pleteness.

It turns out, that although predicate fuzzy logic with the connective 4 do not havestandard completeness, their Pavelka-style extensions fulfill an analogy of Corollary 3.4.22,i.e., they have standard completeness.

Lemma 3.5.21 Let L be a fuzzy logic with 4 in the language and RPL(B)∀ has Pavelka-stylecompleteness. Then for each theory T and formula ϕ we have T `RPL(B)∀ ϕ iff T |=B ϕ.

Corollary 3.5.22 Let L be a core fuzzy logic with 4 in the language and the logic RPL∀has Pavelka-style completeness. Then RPL∀ has standard completeness.

Also recall that the logic R�Π∀ coincides with logic TT from [44].

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Chapter 4

Product ÃLukasiewicz logic

As we have seen, the ÃLukasiewicz logic [62, 44] is one of the most important logics in thebroad family of many-valued logics. Its corresponding algebraic structures of truth values(MV-algebras) are well-known and deeply studied. Mundici’s famous result [12] establishedan important correspondence between MV-algebras and Abelian l-groups with strong unit.There is an obvious question if there is a logic, whose corresponding algebras of truth valuesare in the analogous correspondence with l-rings.

There are several papers dealing with the so-called product MV-algebras. Montagna’spapers [66, 67, 69] are fundamental to our aims. There is also a paper by Di Nola andDvurecenskij [73]. A product MV-algebra (PMV-algebra for short) is an MV-algebra en-riched by a product operation in such a way that the resulting structures correspond to thef -rings with strong unit. In [66], Montagna proved the subdirect representation theoremfor PMV-algebras and established a correspondence between linearly ordered f -rings withstrong unit and linearly ordered PMV-algebras. Later in [67], he introduced PMV4-algebras(PMV-algebras enriched by the 0-1 projector 4) and proved the categorical equivalence be-tween PMV4-algebras and certain extension of f -rings (the so-called δ-f -rings). Finally in[69], it was shown by Montagna and Panti that the variety of PMV4-algebras is generatedby the standard PMV4-algebra (over the real unit interval).

In the forthcoming paper [68], Montagna introduced a quasi-variety PMV+ containingonly the PMV-algebras without non-trivial zero-divisors and showed that PMV+ is gener-ated by the standard PMV-algebra (over the real unit interval).

However, so far there is no logic corresponding to the all above-mentioned algebras. Themain aim of this chapter is to define and develop such a logic. Our logic, which correspondsto PMV-algebras, is called P� logic. Further, we introduce P�′ logic corresponding to thealgebras from PMV+. We also study extensions of P� and P�′ logics by Baaz’s 4 (P�4 andP�′4 logics). The algebras of truth values of P�′4 logic correspond to PMV4-algebras. This,together with the fact that there are also several other different algebraic structures calledPMV-algebras, is the reason why we call P�-algebras the algebras of truth values correspond-ing to P� logic. Analogously, we introduce P�′-algebras, P�4-algebras, and P�′4-algebras.

We use the above-mentioned algebraic results to obtain completeness of all of these logicsand standard completeness for P�′ and P�′4 logic. Further, we show an example of theP�-algebra which demonstrates that P� logic is not standard complete. Then we show arelation of our logics to the �Π logic. Roughly speaking, the logic �Π is the extension of P�′by the product residuum.

Furthermore, we extend these logics by rational constants in the style of Section 3.4.Among others we obtain RPP�′ and RPP�′4 logics. Then we investigate the predicate ver-sions of all logics mentioned above. Later we deal with the arithmetical complexity of the setof tautologies of these logics which entails that they do not have the standard completeness.Finally, we show a relation of these logics to the predicate versions of �Π and R�Π logicsand the well-known logic of Takeuti and Titani [80].

67

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68 CHAPTER 4. PRODUCT ÃLUKASIEWICZ LOGIC

ÃL

ÃL4 P� RPL

P�4 RPP�P�′

P�′4 RPP�′

�Π RPP�4 = RPP�′4

R�Π

?

³³³³³)PPPPPq

HHj HHj ©©¼©©¼

?

?

?

?

?

©©¼ HHj

HHj ©©¼

HHHHj

Figure 4.1: Relations between logics of this chapter.

All logics considered in this chapter lie between ÃLukasiewicz logic and RÃLΠ logic [23, 36].Their mutual relations are depicted in Figure 4.1.

4.1 P� and P�′ logics

4.1.1 Syntax

In this section, we introduce the P� logic (P� for short), an extension of ÃLukasiewicz logicby a new binary connective ¯. This connective plays the role of multiplication. Thus thebasic connectives are ⊗,→,¯, 0. Additional connectives ⊕,ª,¬,∧,∨,≡, 1 are defined as inÃLukasiewicz logic. We also introduce the P�′ logic (P�′ for short), an extension of P� byone additional deduction rule. The reason for this extension is that the P� logic does notpossess the standard completeness property.

Definition 4.1.1 The axioms of P� logic are the axioms of ÃLukasiewicz logic and the fol-lowing axioms:

(P1) (χ¯ ϕ)ª (χ¯ ψ) ≡ χ¯ (ϕª ψ) ,(P2) ϕ¯ (ψ ¯ χ) ≡ (ϕ¯ ψ)¯ χ ,(P3) ϕ → ϕ¯ 1 ,(P4) ϕ¯ ψ → ϕ ,(P5) ϕ¯ ψ → ψ ¯ ϕ .

The only deduction rule is modus ponens. The P�′ logic is obtained from P� by adding anew deduction rule (ZD): from ¬(ϕ¯ ϕ) infer ¬ϕ.

It is obvious that all theorems of ÃLukasiewicz logic are also theorems of P� and alltheorems of P� are also theorems of P�′. Further, we show several useful theorems of P�logic. The most important one is theorem (TP4) stating that the connective ≡ is a congruencew.r.t. the product ¯, which entails that P� and P�′ are weakly implicative logics.

Lemma 4.1.2 The following are theorems of P� logic:(TP1) (ϕ → ψ) → (ϕ¯ χ → ψ ¯ χ) ,(TP2) (ϕ ≡ ψ) → (ϕ¯ χ ≡ ψ ¯ χ) ,(TP3) (ϕ1 → ψ1)⊗ (ϕ2 → ψ2) → (ϕ1 ¯ ϕ2 → ψ1 ¯ ψ2) ,(TP4) (ϕ1 ≡ ψ1)⊗ (ϕ2 ≡ ψ2) → (ϕ1 ¯ ϕ2 ≡ ψ1 ¯ ψ2) ,(TP5) ϕ⊗ ψ → ϕ¯ ψ ,(TP6) (ϕ ∧ ψ)¯ χ ≡ (ϕ¯ χ) ∧ (ψ ¯ χ) .

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4.1. P� AND P�′ LOGICS 69

Proof:(TP1 ): We start with one direction of equivalence (P1) (χ¯ϕ)ª (χ¯ψ) → χ¯ (ϕªψ).

By (P4) and (A1) we get (χ ¯ ϕ)ª (χ¯ ψ) → (ϕª ψ). Using (H7) we obtain ¬(ϕ ª ψ) →¬((χ¯ ϕ)ª (χ¯ ψ)). This is what we want to prove (because ¬(ϕª ψ) ≡ (ϕ → ψ)).

(TP2 ): We use (TP1) and (TP1) with ϕ,ψ exchanged. From theorem (H3) we get(ϕ → ψ)⊗ (ψ → ϕ) → (ϕ¯ χ → ψ ¯ χ)⊗ (ψ ¯ χ → ϕ¯ χ).

(TP3 ): Let us start with (TP1) (ϕ1 → ψ1) → (ϕ1 ¯ ϕ2 → ψ1 ¯ ϕ2) and (TP1) again(ϕ2 → ψ2) → (ψ1 ¯ ϕ2 → ψ1 ¯ ψ2). Now using (H3) we get (ϕ1 → ψ1) ⊗ (ϕ2 → ψ2) →(ϕ1 ¯ ϕ2 → ψ1 ¯ ϕ2) ⊗ (ψ1 ¯ ϕ2 → ψ1 ¯ ψ2). By axiom (A1) (after applying axiom (A5))we get (ϕ1 ¯ ϕ2 → ψ1 ¯ ϕ2) ⊗ (ψ1 ¯ ϕ2 → ψ1 ¯ ψ2) → (ϕ1 ¯ ϕ2 → ψ1 ¯ ψ2). Axiom (A1)completes the proof.

(TP4 ): This is an analogy of (TP2), except that we use (TP3) instead of (TP1).

(TP5 ): We start with (TP1) in the form (1 → ϕ) → (1¯ψ → ϕ¯ψ). By (H8) and (A5)we obtain ϕ⊗ (1¯ ψ) → ϕ¯ ψ. The rest is obvious.

(TP6 ): We start with the first direction: by (H9) and (TP1) we obtain (ϕ → ψ) →(ϕ¯χ → (ϕ∧ψ)¯χ). By (H5) and (A1) we get (ϕ → ψ) → ((ϕ¯χ)∧(ψ¯χ) → (ϕ∧ψ)¯χ).Analogously we get (ψ → ϕ) → ((ϕ ¯ χ) ∧ (ψ ¯ χ) → (ϕ ∧ ψ) ¯ χ). Axiom (A6) completesthe proof of this direction.

Reverse direction: by (T1) and (H9) we get (ϕ → ψ) → (ϕ¯χ → (ϕ¯χ)∧ (ψ¯χ)). By(H5) and (TP1) we get (ϕ ∧ ψ) ¯ χ → ϕ ¯ χ. By (A1) we get (ϕ → ψ) → ((ϕ ∧ ψ) ¯ ϕ →(ϕ¯ χ) ∧ (ψ ¯ χ)). The rest of the proof is analogous to the first part. QED

As in immediate corollary of this lemma and Lemma 2.1.16 we get:

Theorem 4.1.3 The logics P� and P�′ are weakly implicative logics.

Theorem 4.1.4 The logic P� is a weakly implicative fuzzy logic with DT→.

The proof that P� hat DT→ is a trivial application of Theorem 2.2.8 (in P�′ the situationis quite different, after we introduce the semantics we show that P� does not posses evenLDT). The rest of the theorem is a consequence of Lemma 2.2.10.

4.1.2 Semantics and completeness

Now we define the algebras corresponding to P�—P�-algebras. They coincide with theso-called PMV-algebras which were introduced in Montagna’s paper [66]. Furthermore,P�-algebras are also a subclass of more general algebras introduced by Dvurecenskij andDi Nola in paper [73] (they do not require 1 to be a neutral element for the product ¯and commutativity of ¯). However, we decided to use the name P�-algebras, because someauthors use the name PMV-algebras for the different structures (e.g. pseudo MV-algebras).

The P�′-algebras coincide with PMV+-algebras introduced in the forthcoming Mon-tagna’s paper [68]. These are exactly the subreducts of ÃLΠ-algebras.

Definition 4.1.5 A P�-algebra is a structure L = (L,⊕,¬,¯,0,1), where the reduct L∗ =(L,⊕,¬,0,1) is an MV-algebra and the following identities hold:

(1) (a¯ b)ª (a¯ c) = a¯ (bª c) ,(2) a¯ (b¯ c) = (a¯ b)¯ c ,(3) a¯ 1 = a ,(4) a¯ b = b¯ a ,

where a ª b = ¬(¬a ⊕ b) = a ⊗ ¬b and a ⊗ b = ¬(¬a ⊕ ¬b) . Moreover, we say that L is aP�′-algebra if it fulfills the following quasi-identity:

(5) if a¯ a = 0 then a = 0 .

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70 CHAPTER 4. PRODUCT ÃLUKASIEWICZ LOGIC

Observe that the P�-algebras form a variety and the P�′-algebras form a quasi-variety.

Example 4.1.6 Let us by [0, 1]S denote the algebra ([0, 1],⊕,¬,¯, 0, 1), where the reduct([0, 1],⊕,¬, 0, 1) is [0, 1]ÃL and ¯ is the usual algebraic product of reals. Notice that [0, 1]S isboth standard P�-algebra and standard P�′-algebra and that there is only standard P�- andP�′-algebra and (up to isomorphism)

Later, when we show that P�′ is core fuzzy logic, we set [0, 1]S = [0, 1]P�′ . Notice that alinearly ordered P�-algebra is P�′-algebra iff it has only trivial zero-divisors.

Theorem 4.1.7 P�-algebras coincides with ordered P�-matrices and P�′-algebras coincideswith ordered P�′-matrices.

Proof: The proof is an analogy of the proof of analogous theorem for BL-algebras (Theorem3.2.8). The only non trivial task is to show that all axioms and rules of P� and P�′ holds inthe corresponding algebras.

Let L be a P�-algebra. Since L∗ is an MV-algebra, we know that the axioms of ÃLuka-siewicz logic hold in L and modus ponens is a sound deduction rule. Axioms (P1)–(P3) and(P5) are obviously L-tautologies (cf. conditions (1)–(4) in the definition of P�-algebra).

We check L-tautologicity of (P4). By Proposition 3.3.5,4, we know that a¯ b → a = 1 iffa¯ b ≤ a = a¯ 1. Now using [66, Lemma 2.9(ii)], the proof of the first statement is done.

To prove the statement for P�′-algebras just observe that the rule (ZD) is obviously soundin each P�′-algebra. QED

This theorem has three important corollaries. The first states the connection of our logicsand ÃLukasiewicz logic and the second is the promised proof that the deduction theorem doesnot hold in P�′. This together gives us the third corollary that the logic P�′ is strictlystronger than the logic P�.

Corollary 4.1.8 P� and P�′ are conservative extensions of ÃLukasiewicz logic.

Proof: Just use Theorem 3.1.13. QED

Corollary 4.1.9 P�′does not have LDT. The rule (ZD) cannot be replaced by axioms.

Proof: Since obviously {¬(v ¯ v)} ` ¬v, the deduction theorem would give us that forsome n the formula (¬(v ¯ v))n → ¬v is a theorem of P�′. Hence (¬(v ¯ v))n → ¬v is[0, 1]S-tautology (by the latter theorem), i.e., there is n such that (¬(x¯ x))n ≤ ¬x for eachx ∈ [0, 1]. Notice that the derivatives of (¬(x¯x))n and ¬x at the point 0 are equal to 0 and−1, respectively. Thus for each n, there is x such that (¬(x¯ x))n > ¬x, a contradiction.

The second part is a consequence of Theorem 2.2.8. QED

Corollary 4.1.10 The logic P�′ is strictly stronger than the logic P�.

Now we recall several known results about P� and P�′-algebras. In [66, Lemma 4.3 (a)],Montagna showed that there is a correspondence between linearly ordered P�-algebras ando-rings. The analogous result for P�′-algebras is a consequence of [68, Corollary 4.3].

Theorem 4.1.11 An algebra L is a linearly ordered P�-algebra if and only if L is isomorphicto the interval algebra of some o-ring RL. Furthermore, L is P�′-algebra iff RL is a domainof integrity.

The following fact is a corollary of the previous theorem and was originally proved in [68,Corollary 4.4].

Theorem 4.1.12 The quasi-variety of P�′-algebras is generated by [0, 1]S.

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4.1. P� AND P�′ LOGICS 71

Theorem 4.1.13 Both P� and P�′ logics has SDP.

Proof: The claim for P�-algebras is trivial (it was first proved in [66, Theorem 5.1]). Theclaim for P�′-algebras is a trivial consequence of the previous theorem. QED

Corollary 4.1.14 The logic P�′ is a weakly implicative fuzzy logic.

Since we have shown that both P� and P�′ logics are fuzzy and that the P�-algebras(P�′-algebras) corresponds to the P�-matrices (P�′-matrices), we can state the completenesstheorem in the following form:

Theorem 4.1.15 (Completeness) Let C be either P� or P�′, T a theory over C, and ϕ aformula. Then the following are equivalent:

1. T `C ϕ,

2. e(ϕ) = 1L for each linearly ordered C-algebra L and each L-model e of theory T ,

3. e(ϕ) = 1L for each C-algebra L and each L-model e of theory T .

The question whether our logics possess the standard completeness is answered by thefollowing theorem.

Theorem 4.1.16 (Finite Standard Completeness) The logic P�′ has the finite stan-dard completeness, i.e., for each finite theory T and formula ϕ we have:

• T `P�′ ϕ iff e(ϕ) = 1 for each [0, 1]S-model e of theory T .

The logic P�′ is a core fuzzy logic without standard completeness.

The proof of the first part is a consequence of the completeness theorem and Theorem4.1.12. The second part is a consequence of Corollary 4.1.8 and Theorem 3.3.2. This theoremtogether with Corollary 4.1.10 gives us:

Corollary 4.1.17 The logic P� has not the finite standard completeness.

4.1.3 More on P� and P�′-algebras

In this section we examine the classes of P�-algebras and P�′-algebras. Let PÃL′, PÃL, and[0,1]S respectively denote the quasi-variety of P�′-algebra, the variety of P�-algebras, andthe variety generated by [0, 1]S resp. The results in the previous section allows us to statethe following theorem:

Theorem 4.1.18 The following holds:

1. PÃL′ is not a variety.

2. PÃL′ $ [0,1]S $ PÃL.

Proof:

1. Assume that PÃL′ is a variety. Then we get a contradiction with Corollary 4.1.9.

2. Both inequalities are obvious. The strictness of the first inequality is a consequence ofPart 1. and the strictness of the second one is a consequence of Corollary 4.1.17.

QED

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72 CHAPTER 4. PRODUCT ÃLUKASIEWICZ LOGIC

The direct algebraic proof of the first part can be found in [68, Theorem 3.1]. The factthat the variety of P�-algebras is not generated by [0, 1]S is already mentioned in Montagna’spaper [66, Problem 1], but there is no proof, only a reference to Isbell’s paper [59]. Inthat paper, Isbell proved that the equational theory of formally real f -rings (lattice-orderedrings satisfying all lattice-ring identities that are true in a totally-ordered field) does nothave a finite base, or even a base with a finite number of variables. Thus it seems to usthat the connection between the second part of Theorem 4.1.18 and Isbell’s paper is not sostraightforward. We have seen that it can be proven very simply as a natural consequence ofour previous results.

In the rest of this section we give one more alternative proof of the previous theorem.We construct two algebras directly demonstrating the strictness of both inequalities. First,Example 4.1.22 is an algebra from [0,1]S, which is not P�′-algebra (this also shows that PÃL′

is not closed under HSP , so it is not a variety). Second, Example 4.1.23 is a P�-algebrawhich is not in [0,1]S. Before we present our examples, we study the relation betweennontrivial zero-divisors and infinitesimal elements of P�-algebras. We recall the definition ofan infinitesimal element and continue with the lemma showing the distributivity of ¯ w.r.t.⊕.

Definition 4.1.19 An element a in a P�-algebra is said to be infinitesimal iff a > 0 andna ≤ ¬a for each n ∈ N, where na = a⊕ · · · ⊕ a.

Lemma 4.1.20 In each P�-algebra the following inequality holds:

b¯ (x⊕ y) ≤ (b¯ x)⊕ (b¯ y) .

Proof: The inequality is equivalent to (b ¯ (x ⊕ y)) ª ((b ¯ x) ⊕ (b ¯ y)) = 0. Now(b¯(x⊕y))ª((b¯x)⊕(b¯y)) = (b¯(x⊕y))⊗¬(b¯x)⊗¬(b¯y) = [(b¯(x⊕y))ª(b¯x)]ª(b¯y) =b¯ [((x⊕ y)ª x)ª y] = b¯ [(x⊕ y)⊗ ¬x⊗ ¬y] = b¯ [(x⊕ y)ª (x⊕ y)] = 0. QED

Proposition 4.1.21 Let L be a linearly ordered P�-algebra, and a ∈ L, a > 0. If a is azero-divisor then a is an infinitesimal.

Proof: Let us suppose that a is a zero-divisor which is not infinitesimal. Then there existsn ∈ N such that na = 1. By Lemma 4.1.20 a¯ na ≤ (a¯ a)⊕ · · · ⊕ (a¯ a) = 0 because a isa zero-divisor. Thus a = a¯ 1 = a¯ na = 0, a contradiction. QED

Example 4.1.22 Let us take the following set:

L1,∞ = {0, 1, 2, . . . ,∞,∞− 1,∞− 2, . . .} ,

where we identify ∞ with ∞− 0. The operations are defined as follows:

n ∈ N: ¬n = ∞− n,¬(∞− n) = n.

k, n ∈ N: k ⊕ n = k + n,(∞− k)⊕ (∞− n) = ∞,

k ⊕ (∞− n) =

{(∞− n + k) if k ≤ n,

∞ otherwise.k, n ∈ N: k ¯ n = 0,

k ¯ (∞− n) = k,(∞− k)¯ (∞− n) = (∞− k − n).

The structure (L1,∞,⊕,¯,¬, 0,∞) is a P�-algebra. Observe also that this algebra possessesnontrivial zero-divisors, thus it is not a P�′-algebra. Notice that the MV-reduct of this algebrais the well-known Chang algebra, thus the elements of L1,∞ are ordered as follows: 0 < 1 <2 < · · · < ∞− 2 < ∞− 1 < ∞.

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4.1. P� AND P�′ LOGICS 73

We show how to generate this P�-algebra with nontrivial zeros-divisors from the standardP�-algebra. It is the well-known fact that each algebra in a variety generated by [0, 1]S canbe obtained as A ∈ HSP ([0, 1]S), where P means the direct product, S a subalgebra, and Ha homomorphic image. So we will construct the example in the following three steps.

1. Step P : Take the algebra of all functions L = [0, 1][0,1].

2. Step S: Restrict to the subalgebra S ⊆ L of all continuous piecewise polynomial func-tions with integer coefficients such that either f(0) = 0 or f(0) = 1.

3. Step H: Factorise by the equivalence ∼, where f ∼ g iff f(0) = g(0) and f ′(0) = g′(0).By f ′(0), we denote the right-derivative of f in 0.

Example 4.1.23 Now we will show an example of a P�-algebra which cannot be generatedfrom the standard P�-algebra. Firstly, we construct an o-monoid and then we construct thealgebra of polynomials over this monoid. In this way, we obtain a ring; its interval algebrais the desired P�-algebra (cf. Theorem 4.1.11).

The following example of an o-monoid can be found in [39]. For any a, b, c, d ∈ N, 〈a, b, c〉will denote the sub-o-monoid of N generated by a, b, c, and 〈a, b, c〉/d will denote the o-monoidobtained by identifying with infinity all elements of 〈a, b, c〉 that are greater than or equal tod.

Let S = {32∗}∪ 〈9, 12, 16〉/30 denote the o-monoid obtained from 〈9, 12, 16〉/30 by addingone additional element, denoted by 32∗. This element satisfies 16 + 16 = 32∗, 32∗ + z = ∞,and the whole monoid is to be ordered as follows:

0 < 9 < 12 < 16 < 18 < 21 < 24 < 25 < 27 < 28 < 32∗ < ∞ .

All the relations that do not involve 32∗ are as in N, so we have only to check that x ≤ yimplies x + z ≤ y + z if x, y, or z is equal to 32∗, but it is easy to see.

Let R be the o-ring of integers. Then the monoid ring R[S] is the set of all finite formalsums r1X

s1 + · · ·+rnXsn , where X is an indeterminate, ri ∈ R and si ∈ S. Multiplication isdefined by XsXt = Xs+t and by distributivity (so X0 = 1). An element r1X

s1 + · · ·+ rnXsn

is said to be in normal form if ri 6= 0 and s1 < · · · < sn.We identify X∞ with 0 and denote the resulting quotient of R[S] by R[S]h. An element

r1Xs1 + · · ·+rnXsn in normal form is positive iff r1 > 0. Thus for all a, b ∈ S, a < b implies

Xa > Xb.It can be checked that R[S]h is an o-ring. Finally, we get the desired P�-algebra L as the

interval algebra of R[S]h. In L the following identity, which is valid in [0, 1]S, does not hold:

(x1 ¯ z1 ª y1 ¯ z2) ∧ (x2 ¯ z2 ª y2 ¯ z1) ∧ (y1 ¯ y2 ª x1 ¯ x2) = 0 (4.1)

Indeed, let us evaluate the variables as follows:

x1 = X16 y1 = X18 z1 = X16

x2 = X12 y2 = X9 z2 = X12

Then the terms in brackets in (4.1) attain the following values:

x1 ¯ z1 ª y1 ¯ z2 = X16 ¯X16 ªX18 ¯X12 = X32∗ > 0 ,

x2 ¯ z2 ª y2 ¯ z1 = X12 ¯X12 ªX9 ¯X16 = X24 ªX25 > 0 ,

y1 ¯ y2 ª x1 ¯ x2 = X18 ¯X9 ªX16 ¯X12 = X27 ªX28 > 0 .

To see that (4.1) is valid in the standard P�-algebra [0, 1]S, just observe that wheneverone of the variables xi, yi, zi is 0, then the equality trivially holds. Further, if x1z1 > y1z2

and x2z2 > y2z1, then∏

xi

∏zj >

∏yi

∏zj and this implies

∏xi >

∏yi.

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74 CHAPTER 4. PRODUCT ÃLUKASIEWICZ LOGIC

4.2 P�4 and P�′4 logics

In this section, we extend the language of P� logic by 4 connective and introduce P�4 andP�′4 logics. Later we show that these logic coincide with logics P�4 and P�′4.

Definition 4.2.1 Let C be either P� or P�′. The C4 logic results from C by adding axioms(A41) – (A45) and the deduction rule of necessitation.

Both just defined logics obviously extends ÃL4 logic.

Lemma 4.2.2 The formulae 4ϕ⊗ ψ ≡ 4ϕ ∧ ψ and 4ϕ¯ ψ ≡ 4ϕ ∧ ψ are theorems of theP� logic.

Proof: The first formula is the known theorem of ÃL4. For the proof of the second one, usethe first formula and theorem (TP5). QED

Theorem 4.2.3 We have: P�4 = P�4 and P�′4 = P�′4 (in the sense of Theorem 2.2.37).

Proof: The first claim is trivial (see Theorem 3.2.21). Let us denote the axiom4¬(ϕ¯ϕ) →¬ϕ from the group B in presentation of P�′4 as (P4). To prove the second claim we firstly,we show that the deduction rule (ZD) can be derived in P�4 extended by axiom (P4). Let¬(ϕ ¯ ϕ) be a theorem. Then 4¬(ϕ ¯ ϕ) is a theorem as well and so ¬ϕ is provable (bymodus ponens and axiom (P4)).

Converse direction: It is sufficient to prove ¬(¬(P4)¯¬(P4)) (let us denote this formulaby F ). Then proof is done by the use of (ZD). Observe that ¬(P4) ≡ 4¬(ϕ¯ϕ)⊗ϕ. Thus byLemma 4.2.2 we get ¬F ≡ (4¬(ϕ¯ϕ)∧ϕ)¯(4¬(ϕ¯ϕ)∧ϕ). After repeated use of theorem(TP6) we get ¬F ≡ (4¬(ϕ¯ϕ)¯4¬(ϕ¯ϕ))∧ (4¬(ϕ¯ϕ)¯ϕ)∧ (ϕ¯4¬(ϕ¯ϕ))∧ (ϕ¯ϕ).

By use of Lemma 4.2.2 we may write ¬F ≡ 4¬(ϕ ¯ ϕ) ∧ ϕ ∧ (ϕ ¯ ϕ). Finally, by theobvious fact that ϕ∧ (ϕ¯ϕ) ≡ ϕ¯ϕ is a theorem (use (H9), (H5)) and using Lemma 4.2.2,we get F ≡ ¬(4¬(ϕ¯ϕ)⊗ (ϕ¯ϕ)) ≡ 4¬(ϕ¯ϕ) → ¬(ϕ¯ϕ). Since 4¬(ϕ¯ϕ) → ¬(ϕ¯ϕ)is an instance of axiom (ÃL43), the formula F is a theorem. QED

Corollary 4.2.4 We have:

• P�4 and P�′4 are fuzzy logics with DT4,

• P�′4 is a core fuzzy logic without standard completeness,

• P�′4 is strictly stronger than P�4logic,

• P�4 is has not the finite standard completeness.

Proof: Everything in this corollary is just a consequence of the previous theorem. We onlypresent an alternative proof of the one but last statement, where the connective 4 allowsus to find a simple proof. According to Example 4.1.22, we have a P�4-algebra L withan element c > 0 such that c ¯ c = 0. Then ¬c = (∞ − c) < ∞, thus 4¬c = 0. Since¬(c¯ c) = ∞ and so 4(¬(c¯ c)) = ∞, we know that the axiom (P4) is not an L-tautology.Thus (P4) is not a theorem of P�4. QED

Now we summarize known facts about connections between our logics and ÃLukasiewiczand ÃLΠ logic.

Theorem 4.2.5

(1) P�, P�′, P�4, and P�′4 logics are conservative extensions of ÃLukasiewicz logic.

(2) P�4 and P�′4 logics are conservative extensions of ÃL4.

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4.3. PAVELKA STYLE EXTENSIONS 75

(3) P�4 logic is a conservative extension of P� logic and P�′4 logic is a conservativeextension of P�′ logic.

(4) P�′4 logic is not a conservative extension of P� logic.

(5) �Π logic is a conservative extension of P�′and P�′4 logics.

(6) �Π logic is not a conservative extension of P� and P�4 logics.

Proof: (1): For P� it is already proven in Theorem 4.1.8, the proof for the remaining logicsis analogous.

(2) Analogous to (1).(3) It follows from the fact that we can extend each linearly ordered P�-algebra by 4

and from the completeness theorems for both logics.(4) Since P�′4 is a conservative extension of P�′and P�′is strictly stronger than P�, the

proof easily follows.(5) A consequence of Theorems 3.3.21, 4.1.16, and Corollary 4.2.4.(6) A consequence of (5) and the fact that P�4 is strictly weaker than P�′4 (and that

P� is strictly weaker than P�′). QED

4.3 Pavelka style extensions

In this section we add rational constants into our language together with the book-keepingaxioms in the way of Section 4.3. This section is rather straightforward application of theresults of that section. The only not expected result is that if we add book-keeping axiomsand corresponding infinitary deduction rule into the logic P�4 and P�′4 (which are not thesame) we get one logic.

Theorem 4.3.1 Logics P�′ and P�′4 are argument-monotonous rational core fuzzy logic.NC(P�′) = ∅ and NC(P�′4) = {<4, 1 >}.

Observe that although P� is not a core fuzzy logic, it has only one standard algebra[0, 1]P�′ up to isomorphism and analogously for the logic P�4. Thus we define:

Theorem 4.3.2 Particular Pavelka style extension of fuzzy logics defined in this chapter canbe axiomatized as:

• RPP�′ = RP�′([0, 1]P�′)

• RPP�′4 = RP�′([0, 1]P�′4) + IR4

• RPP�([0, 1]P�′) = RP�([0, 1]P�′)

• RPP�4([0, 1]P�′4) = RP�4([0, 1]P�′4) + IR4

Let us denote the logic RPP�([0, 1]P�′) (or RPP�4([0, 1]P�′4) respectively) as RPP�(resp. RPP�4), even if this contradicts our definitions in Section 4.3.

Theorem 4.3.3 Let L be one of the P�, P�′, P�4, P�′4 and T a theory over RPL and ϕbe a formula. Then:

1. |ϕ|T = ||ϕ||T ,

2. RPP�4 and RPP�′4 have the standard completeness,

3. RPP�′ and RPP�′4 are core fuzzy logics,

4. RPP� has not the standard completeness.

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76 CHAPTER 4. PRODUCT ÃLUKASIEWICZ LOGIC

Proof: We know that both RPP� and RPP�′ are fuzzy logic from Corollary 3.4.5, weknow that the remaining two logics are fuzzy from Lemma 3.4.25. Thus the part 1. is justa consequence of Theorem 3.4.21. Part 2. is a consequence of Corollary 3.4.22. Part 3. isconsequence of Theorem 3.4.23 and Corollary 3.4.22. Part 4. is obvious (P� has not thestandards completeness). QED

Remark 4.3.4 We can also obtain Pavelka style completeness for RPP� from [44, Section3.3], where the author defines the logic RPL(¯). It is ÃLukasiewicz logic plus book-keepingaxioms of RPP� and the axioms:

(ϕ → ψ) → (ϕ¯ χ → ψ ¯ χ) ,

(ϕ → ψ) → (χ¯ ϕ → χ¯ ψ) .

This logic enjoys the same Pavelka’s style completeness as RPP� (see [44, Theorem 3.3.19]or our Theorem 3.4.21 ).

Thus we have three different logics, namely RPP�, RPP�′, RPL(¯). All of them enjoythe same Pavelka’s style completeness. However, the question whether these logics (as setsof theorems) coincide seems to be open for us.

Corollary 4.3.5

1. The RPP�4 and RPP�′4 logics coincide.

2. RPP�′ is a conservative extension of P�′and RPP�′4 is a conservative extension ofP�′4.

3. RPP� and RPP�′ are a conservative extensions of RPL.

4. R�Π is a conservative extension of RPP�4.

Proof:

1. This is an obvious consequence of the latter theorem and the fact that standard algebrasfor these logics are the same.

2. See Theorem 3.4.23 and Corollary 3.4.22.

3. Obvious using Corollary 3.4.24.

4. It is the consequence of Corollary 3.4.26.

QED

4.4 The predicate logics

Using our knowledge of predicate logic we can define for each of our four logics C predicateversion C∀− and get completeness w.r.t. all corresponding algebras. We can also definelogics C∀. However there is a problem with logic P�′∀. All other have clearly PP and we getcompleteness w.r.t. corresponding linear algebras. Since the logic P�′ does not have DT→we have no way to prove completeness and it seems to be an interesting open problem.

The questions whether P�∀ is a conservative extension of the ÃLukasiewicz predicate logic,and whether �Π∀ is a conservative extension of P�∀ logic seems to be open as well (andmany analogous one involving other predicate logics defined in this section). The standardcompleteness of P�∀, P�4∀, and P�′4∀ logics is a related problem. But this question can beanswered. Here we assume that the reader is familiar with the basic concepts of undecidabilityand arithmetical hierarchy ([44, Section 6.1] is satisfactory for our needs). In the followingwe assume that our predicate language is at most countable. Before we proceed we look atthe predicate version of Pavelka style extension of our logic. The following theorem is just acorollary of Theorem 3.5.19.

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4.4. THE PREDICATE LOGICS 77

Theorem 4.4.1 Let C be one of the P�, P�4, or P�′4. Then the logic RPC∀ has Pavelkastyle completeness. Furthermore, the logics RPP�4∀ and RPP�′4∀ have the standard com-pleteness.

Corollary 4.4.2 The logics RPP�4∀ and RPP�′4∀ coincide.

Corollary 4.4.3 R�Π∀ is a conservative extension of RPP�4∀.

Proof: It is the consequence of Theorems 3.5.20 and 4.4.1. QED

Observe that TAUT∀(P�∀) = TAUT∀(P�′∀) and TAUT∀(P�4∀) = TAUT∀(P�′4∀).

Theorem 4.4.4 The set TAUT∀(P�∀) is Π2-complete and the set TAUT∀(P�4∀) is notarithmetical.

Proof: The Π2-hardness is an obvious corollary of Theorem 3.5.5. The fact that the setTAUT∀(P�∀) is in Π2 is a corollary of Theorem 4.4.1 (we know that ϕ ∈ TAUT∀(P�∀) iff(∀r ∈ IQ))(∃ proof ω)(ω is the proof of r → ϕ)).

The proof of the second part is an obvious modification of the analogous proof for productpredicate logic (see [45, Corollary 2]). QED

Corollary 4.4.5 Let C be either P�, P�4 or P�′4. Then the logic C∀ has not the finitestandard completeness property.

At the end of this section we prove another consequence of the standard completeness ofRPP�∀. We make a small modification of the language of RPP�∀ and show that the resultinglogic TT coincides with the famous logic of Takeuti and Titani. The logic TT results fromRPP�∀ by omitting truth constants r such that r can not be expressed in the form k

2n . Inother words, we may say that the logic TT has only one additional truth constant, 1

2 , and theother truth constants are defined by using the connectives of P�4∀. Now we can easily provethe analogy of Theorem 4.4.1 for TT . Just go through the proofs leading to this theoremand notice that the set of truth constants in TT is dense. Thus all the proofs will be soundfor TT as well. This gives us the following corollary:

Corollary 4.4.6 R�Π∀ is a conservative extension of TT .

Takeuti and Titani’s logic was introduced by Takeuti and Titani in their work [80]. Itis a predicate fuzzy logic based on Gentzen’s system of intuitionistic predicate logic. Theconnectives used by this logic are just the connectives of RPP�4∀ logic. This logic has twoadditional deduction rules (named R1, R2) and 46 axioms (namely F1 – F46). We willnot present the axiomatic system and we only recall that this logic is sound and completew.r.t. the standard P�-algebra (cf. [80, Theorem 1.4.3]). All this leads us to the followingconclusion:

Theorem 4.4.7 Takeuti and Titani logic coincides with the logic TT . Furthermore, thelogic RPP�4∀ is a conservative extension of Takeuti and Titani logic.

This theorem allows to translate the results from the Takeuti and Titani’s logic into ourmuch more simpler (in syntactical sense) logical system of the TT or RPP�4∀ logic. Aninteresting corollary of this theorem and the previous corollary is a very simple proof of onepart of [24, Theorem 10]:

Theorem 4.4.8 Takeuti and Titani’s logic is contained in R�Π∀ logic.

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78 CHAPTER 4. PRODUCT ÃLUKASIEWICZ LOGIC

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Chapter 5

New results in Product logic

This chapter is devoted to the study of product logic. Recall that Product logic has beendefined in [52] in order to describe the logic of the product t-norm and its correspondingresiduum and it turns out to be an axiomatic extension of basic fuzzy logic corresponding toproduct t-norm.

In the first section we present an alternative axiomatic system for product logic and showthat it is the simplest possible (it the sense we specify later).

Then we continue with the study of normal forms formulae of product logic. We show thatevery time we fix a set of variables and we consider evaluations that positively interpretedsuch variables, then we can give a representation of a formula as conjunction of disjunctions(disjunction of conjunctions, resp.). This will be called a conjunctive (disjunctive, resp.)semi-normal form or CsNF (DsNF). Each disjunct (conjunct) will be called literal and hasthe form

vk11 & . . . &vkl

l → vkl+1l+1 & . . . &vkn

n .

Then we show how to join suitably these forms in order to obtain the conjunctive (disjunctive)normal form or CNF (DNF).

We also describe the steps that can be used for simplifying formulae expressed in CsNF(DsNF). Such simplification are based on the observation that each literal can be identifiedwith the n-tuple of powers of the variables occurring in it.

We conclude this section with simple algorithm for checking tautologies that is basedon the conjunctive normal form and on the simplification rules. This algorithm reduces theproblem of checking tautologies to a problem of integer linear programming. Note that TAUTproblem for product logic is co-NP complete (see [44]). We did not address the complexityof our algorithm.

In the last section we give a functional representation of formulae of product logic. Weprove the analogous of McNaughton theorem for ÃLukasiewicz logic (Theorem 3.3.25), statingthat truth tables of product formulae are such that in each region of positivity they are con-tinuously composed by several monomials (see also [41]). Each of this monomial correspondsto a basic literal.

5.1 On axioms of Product logic

This section has two goals. The first is to show a new axiomatic system of product fuzzylogic with only one non-BL axiom which has only two variables. The second goal is to provethat there cannot be any axiomatic system of the product fuzzy logic with single non-BLaxiom with only one variable.

79

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80 CHAPTER 5. NEW RESULTS IN PRODUCT LOGIC

5.1.1 New axiomatic system

Definition 5.1.1 Let NΠ be the schematic extension of BL with the followingnon-BL axiom:

(AΠ) ¬¬ϕ → ((ϕ → ϕ&ψ) → ψ&¬¬ψ).

Lemma 5.1.2 Product logic proves the following theorems:(Aux1) ϕ → ϕ&¬¬ϕ,(AΠ) ¬¬ϕ → ((ϕ → ϕ&ψ) → ψ&¬¬ψ).

Proof: (Aux1): ` ¬¬ϕ → (Aux1) (BL1)` ¬ϕ → (Aux1) (BL2)` ¬ϕ ∨ ¬¬ϕ → (Aux1) (BL3)` (Aux1) (TΠ1)

(AΠ): ` ¬¬ϕ → ((ϕ&1 → ϕ&ψ) → (1 → ψ)) (Π1)` ¬¬ϕ → ((ϕ → ϕ&ψ) → ψ)` ¬¬ϕ → ((ϕ → ϕ&ψ) → ψ&¬¬ψ) (Aux1)` (AΠ)

QED

Lemma 5.1.3 The following theorems are provable in NΠ:(Aux2) ¬ϕ&¬ϕ → ¬(¬ϕ → ϕ),(Aux3) ¬ϕ → ¬ϕ&¬ϕ,(Π2) ϕ ∧ ¬ϕ → 0,(Aux4) ¬¬ϕ → ((ϕ → ϕ&ϕ) → ψ),(Π1) ¬¬ϕ → ((ϕ&χ → ϕ&ψ) → (χ → ψ)).

Proof: (Aux2): ` (¬ϕ → ϕ) → (¬ϕ → ϕ) (BL4)` ¬ϕ → ((¬ϕ → ϕ) → ϕ)) (BL5)` ¬ϕ → (¬ϕ → ¬(¬ϕ → ϕ)) (BL6)` ¬ϕ&¬ϕ → ¬(¬ϕ → ϕ) (A5)

(Aux3): ` ¬¬1 → ((1 → 1&¬ϕ) → ¬ϕ&¬¬¬ϕ), this is the axiom (AΠ) for ϕ being 1 andψ being ¬ϕ` (Aux4)

(Π2): ` ¬ϕ → ((¬ϕ → ϕ) → 0) (Aux2), (Aux3)` ¬ϕ&(¬ϕ → ϕ) → 0 (A5)` (Π2)

(Aux4): ` ¬¬ϕ → ((ϕ → ϕ&ψ) → ψ&¬¬ψ) (AΠ)` ψ&¬¬ψ → ψ (A2)` (Aux4)

(Π1): ` ¬¬ϕ → ((ϕ → ϕ&ψ) → ψ) (Aux4)` ((ϕ → ϕ&ψ) → ψ) → ((χ → (ϕ → ϕ&ψ)) → (χ → ψ)) (A1)` (Π1) (A1) and (A5)

QED

Corollary 5.1.4 The schematic extension NΠ is equivalent to product logic (in other words:NΠ is an axiomatic system of product logic).

The proof is obvious.

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5.1. ON AXIOMS OF PRODUCT LOGIC 81

5.1.2 More about possible axiomatic systems

Now we prove that there is no single non-BL axiom with single variable, which would axiom-atize product logic. Let ψ be the formula with only one variable provable in Π. Let Π′ be theschematic extension of BL by the axiom ψ. Thus each Π-algebra is a Π′-algebra. To provethat Π is strictly stronger than Π′ it suffices to find a Π′-algebra which is not a Π-algebra.We offer the following algebra:

Definition 5.1.5 Let A = ([0, 1]\{12}, ∗,⇒,≤) be a BL-algebra. ≤ is a usual order of reals.

The operations are defined as follows:

• x ∗ y =

2xy for x, y < 12 ,

2xy − x− y + 1 for x, y > 12 ,

min(x, y) otherwise .

• x ⇒ y =

1 for x ≤ yy2x for x, y < 1

2 and x > y,x+y−12x−1 for x, y > 1

2 and x > y,

y otherwise .

Remark 5.1.6 The reader familiar with ordinal sums of BL-algebras will notice one fact.The algebra A from the latter definition is obtained from the ordinal sum of two standardΠ-algebras by the deleting of the central idempotent point.

Lemma 5.1.7 Let A be the algebra from the last definition. Then:

• A is NOT a Π-algebra,

• A is a Π′-algebra.

Proof: Just take the axiom (Π1). For x being 14 , y being 1 and z being 3

4 we get: 1 →(( 1

4 ⇒ 14 ∗ 3

4 ) ⇒ 34 ), but because 1

4 ∗ 34 = 1

4 it obviously does not hold. So A is NOT theΠ-algebra.

We want to prove that ψ holds in the algebra A. ψ has only one variable x. So we have twopossible cases:

• x ∈ [0, 12 ), we take the algebra A− = ([0, 1

2 ) ∪ {1}, ∗′,⇒′), where the operations ∗′,⇒′

are the restrictions of those from the algebra A to the new domain (the new domainis indeed closed to these restricted operations). It is obvious that the algebra A− is aΠ-algebra (it is isomorphic to the standard Π-algebra.) And the value of the formulaψ is the same in the A− as in the A itself. And because ψ is the tautology in eachΠ-algebra ψ holds in A−. So if x ∈ [0, 1

2 ) then ψ holds in A.

• x ∈ ( 12 , 1], we take the algebra A+ = ({0} ∪ ( 1

2 , 1], ∗′,⇒′). The arguments are similarto the previous ones.

QED

Remark 5.1.8 Notice one thing: we say ψ has only one variable, but there is another one”hidden” - the truth constant 0 in the definition of ¬. Which is the reason we may notuse these arguments for the ÃLukasiewicz logic (A+ would not be isomorphic to the standardMV-algebra, because the negation does not behave ”as is expected”).

Corollary 5.1.9 There is no schematic extension of BL with single axiom with single vari-able equivalent to product logic.

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82 CHAPTER 5. NEW RESULTS IN PRODUCT LOGIC

5.2 Semi-Normal Forms and Normal Forms

5.2.1 Literals

In this subsection we define the notion of literal and prove some of its basic properties.Before we do so, we prepare few technical lemmata. By V ARϕ we denote the set of variablesoccurring in ϕ.

Definition 5.2.1 Let ϕ be a formula and let V ⊆ V ARϕ. Then:

• an evaluation e is called (V, ϕ)-positive if for each v ∈ V ARϕ holds: e(v) > 0 iff v ∈ V ;

• an evaluation is called ϕ-positive if it is (V ARϕ, ϕ)-positive;

• the theory TV AR is defined as {¬¬p | p ∈ V AR};

• the theory T (V, ϕ) is defined as {¬¬p | p ∈ V } ∪ {¬p | p ∈ V ARϕ − V };

• the theory Tϕ is defined as T (V ARϕ,ϕ).

Corollary 5.2.2 An evaluation e is a model of the theory TV AR iff e(v) > 0 for everyv ∈ V AR.

An evaluation e is a model of the theory T (V,ϕ) iff e is (V, ϕ)-positive.An evaluation e is a model of the theory Tϕ iff e is ϕ-positive.

Lemma 5.2.3 For any formula ϕ only containing connectives & and → (thus ϕ not con-taining 0 and negation)

Tϕ ` ¬¬ϕ.

Proof: Note that for any evaluation e,

e(¬¬ϕ) ={

0 if e(ϕ) = 0,1 if e(ϕ) > 0.

The proof proceeds by induction on the complexity of ϕ.

• If ϕ is a variable v then T v = {¬¬v} and trivially T v ` ¬¬v.

• If ϕ = ψ1&ψ2 then ψ1 and ψ2 do not contain 0 and so by induction hypothesis Tψ1 `¬¬ψ1 and Tψ2 ` ¬¬ψ2. Note that Tψ1 , Tψ2 ⊆ Tϕ. If e is a model of Tϕ thene(¬¬ψ1) = 1 and e(¬¬ψ2) = 1, hence e(ψ1) > 0 and e(ψ2) > 0 and since

e(¬¬ϕ) = e(¬¬(ψ1&ψ2)) ={

0 if e(ψ1) · e(ψ2) = 0,1 if e(ψ1) · e(ψ2) > 0,

then e satisfies ϕ.

• If ϕ = ψ1 → ψ2 we can repeat the same argument as before, by noting that e(¬¬ϕ) < 1if and only if e(ψ2) = 0.

QED

Note that since TV AR contains Tϕ, we also get that for every formula ϕ not containing 0

TV AR ` ¬¬ϕ.

Notation. For a conjunction with n equal arguments ϕ, we use the abbreviation ϕn. Aconjunction of zero formulae (also written ϕ0) is considered as equal to 1. The set {1, . . . , n}will be denoted by n.

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5.2. SEMI-NORMAL FORMS AND NORMAL FORMS 83

Definition 5.2.4 A literal is a formula in form:

vk11 vk2

2 . . . vkl

l → vkl+1l+1 v

kl+2l+2 . . . vkm

m ,

where ki are natural numbers and vi arbitrary propositional variables.

Remark 5.2.5 Note that literals are always not equivalent to 0. Thus the only formula inCsNF (DsNF) equivalent to the truth constant 0 is truth constants 0 itself. Furthermore, theonly formula in CsNF (DsNF) containing 0 is again 0 itself. Thus we can use Lemma 5.2.3and show that if ϕ is a formula in CsNF (DsNF) then either ϕ = 0 or TV AR ` ¬¬ϕ.

Remark 5.2.6 Notice that we do not assume that variables vi are pairwise distinct. Itis obvious that we may assume that variables vi for i ≤ l are pairwise distinct as well asvariables vi for i > l.

If v1, v2 and v3 are variables, then in product logic holds the following:

T v1 ` ((v1&v2) → (v1&v3)) ≡ (v2 → v3)

(cancellation property, see [44]). Then if α is a literal, there is a literal α′, such that variablesoccurring in α′ are pairwise distinct and Tα ` α ≡ α′. It is obvious that also TV AR ` α ≡ α′.

Lemma 5.2.7 Let α, β, γ, δ be literals. Then the following formula is provable in productlogic:

(Aux1) (α → β)&(γ → δ) ≡ (α → β) ∧ (γ → δ) ∧ (α&γ → β&δ).

The following formula is provable in TV AR:

(Aux2) ((α → β) → (γ → δ)) ≡ (γ → δ) ∨ (β&γ → α&δ).

Proof: Let us use a denotation for formulae as the denotation for their evaluation. Recallthat (α → β) = 1 ∧ β

α for α 6= 0. Thus we may write, for α 6= 0 and δ 6= 0:

(Aux1): (1 ∧ β

α) · (1 ∧ δ

γ) = 1 ∧ β

α∧ δ

γ∧ βδ

αγ= (1 ∧ β

α) ∧ (1 ∧ δ

γ) ∧ (1 ∧ βδ

αγ),

that is what we want to prove. The case of α = 0 or γ = 0 is trivial.(Aux2): Let us suppose that α 6= 0, γ 6= 0, δ 6= 0. Then

(1 ∧ β

α) → (1 ∧ δ

γ) = (1 ∧

1 ∧ δγ

1 ∧ βα

) = (1 ∧ (1

1 ∧ βα

∧δγ

1 ∧ βα

) =

= (1 ∧ (δγ

1∨

δγ

βα

)) = 1 ∧ (δ

γ∨ αδ

βγ) = (1 ∧ δ

γ) ∨ (1 ∧ αδ

βγ).

Thus we proved that theory T = {¬¬α,¬¬β,¬¬γ,¬¬δ} proves formula (Aux2) (i.e. T `(Aux2)) by the strong completeness theorem.

Recall Lemma 5.2.3, which states that TV AR ` ¬¬α, TV AR ` ¬¬β, TV AR ` ¬¬γ andTV AR ` ¬¬δ (since truth constant 0 does not appear in any literal). Thus TV AR ` (Aux2).

QED

Lemma 5.2.8 Let αi, βi, γi and δi be literals. The following formulae are provable in productlogic:

(Aux3)∨i∈I

(αi → βi)&∨

j∈J

(γj → δj) ≡≡ ∨

i∈I

(αi → βi) ∧∨

j∈J

(γj → δj) ∧∨

i∈I,j∈J

(αi&γj → βi&δj);

(Aux4)∧i∈I

(αi → βi)&∧

j∈J

(γj → δj) ≡≡ ∧

i∈I

(αi → βi) ∧∧

j∈J

(γj → δj) ∧∧

i∈I,j∈J

(αi&γj → βi&δj).

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84 CHAPTER 5. NEW RESULTS IN PRODUCT LOGIC

The following formulae are provable in TV AR:

(Aux5)∨i∈I

(αi → βi) →∧

j∈J

(γj → δj) ≡≡ ∧

j∈J

(γj → δj) ∨∧

i∈I,j∈J

(βi&γj → αi&δj);

(Aux6)∧i∈I

(αi → βi) →∨

j∈J

(γj → δj) ≡≡ ∨

j∈J

(γj → δj) ∨∨

i∈I,j∈J

(βi&γj → αi&δj).

Proof: (Aux3): This is a direct analogy of the proof of theorem (Aux2). For αi 6= 0 andγj 6= 0,

i∈I

(1 ∧ βi

αi)&

j∈J

(1 ∧ δj

γj) = (1 ∧

i∈I

βi

αi)&(1 ∧

j∈J

δj

γj) =

1 ∧∨

i∈I

βi

αi∧

j∈J

δj

γj∧

i∈I,j∈J

βiδj

αiγj=

i∈I

(1 ∧ βi

αi)&

j∈J

(1 ∧ δj

γj) ∧

i∈I,j∈J

(1 ∧ βiδj

αiγj),

that is what we want to prove. The case of αi = 0 or γj = 0 is trivial.(Aux4): Just use theorems (S1) and (Aux1).(Aux5): An analogy of (Aux2) and (Aux3).(Aux6): Just use theorems (S4) and (Aux2). QED

Definition 5.2.9 We shall write (ϕ → ψ)&(γ → δ) for (ϕ&γ → ψ&δ) and (ϕ → ψ)→(γ →δ) for (ψ&γ → ϕ&δ).

Remark 5.2.10 Notice that if α and β are literals then α&β is a literal as well. Furthermoreif α and β are literals then α&β = α ∧ β ∧ α&β (thanks to (Aux1)), which is a conjunctionof literals.

We may strengthen this using theorems (Aux3) and (Aux4): Multiplication of two con-junctions of literals is a conjunction of literals. Multiplication of two disjunctions of literalsis a conjunction of some disjunctions of literals.

We can do the same for implication. Implication of two literals is a disjunction of someliterals (thanks to theorem (Aux2)). This can be also strengthen using theorems (Aux5) and(Aux6).

5.2.2 Semi-Normal Forms and Normal Forms

In this subsection we define a Conjunctive and Disjunctive semi-normal forms (CsNF, DsNF).We start with the definition of a literal, which is more complex than in the classical logic, andthen we build a CsNF (DsNF) just like in classical logic. We will continue with the proof ofthe partial equivalence of the formulae of product logic with formulae in CsNF (DsNF). Thereason we will prove only partial equivalences lies in the fact that semantics of the productimplication is not continuous in the point (0,0). We will develop a machinery to help usovercome this problem.

In the second part of this subsection we define Conjunctive and Disjunctive Normal Forms(CNF, DNF). Furthermore, we prove that each formula of can be equivalently written in CNF(DNF).

Definition 5.2.11 Let I and Ji for i ∈ I be finite sets and for every i ∈ I and j ∈ Ji letαi,j be literals. The formula ϕ is said to be in a Conjunctive semi-normal form (CsNF) if

ϕ =∧

i∈I

j=∈Ji

αi,j .

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5.2. SEMI-NORMAL FORMS AND NORMAL FORMS 85

The formula ϕ is said to be in a Disjunctive semi-normal form (DsNF) if

ϕ =∨

i∈I

j∈Ji

αi,j .

Furthermore, we define that truth constant 0 is in both CsNF and DsNF

Now we prove a crucial lemma of this subsection.

Lemma 5.2.12 Let ϕ be a formula. Then:

• there is formula ϕD in DsNF such that TV AR ` ϕ ≡ ϕD;

• there is formula ϕC in CsNF such that TV AR ` ϕ ≡ ϕC .

Proof: Notice that if some formula ϕ is in CsNF (DsNF) then by Remark 5.2.5 either ϕ = 0or TV AR ` ¬¬ϕ

Now we prove both our claims by induction on the complexity of formulae. We startwith the propositional variables and constants (1) and then we show the induction steps for& (2a), ¬ (2b) and → (2c). We prove the step for ¬, although it is a defined connective,because of simplicity.

1−(ϕ = v): Any propositional variable or truth constant is a formula in both CsNF andDsNF.

2a−(ϕ = γ&δ): By induction hypothesis, γ and δ are equivalent to some formula in CsNF(DsNF). We want to prove that the same holds for γ&δ. If γ or δ is the truth constant 0 theproof is obvious, otherwise:

TV AR ` γ ≡ γC γC =∧

i∈I

j∈Ji

αi,j ,

TV AR ` δ ≡ δC δC =∧

i′∈I′

j′∈J′i′

βi′,j′ ,

TV AR ` γ&δ ≡ γC&δC γC&δC = (∧

i∈I

j∈Ji

αi,j)&(∧

i′∈I′

j′∈J′i′

βi′,j′).

By theorem (S1) we get:

γC&δC ≡∧

i∈I,i′∈I′(

j∈Ji

αi,j&∨

j′∈J′i′

βi′,j′).

By theorem (Aux3) we get:

γC&δC ≡∧

i∈I,i′∈I′(

j∈Ji

αi,j ∧∨

j′∈J′i′

βi′,j′ ∧∨

j∈Ji,j′∈J ′i′

(αi,j&βi′,j′)).

Recall that if αi,j and βi′,j′ are literals then αi,j&βi′,j′ is a literal. Then:

γC&δC ≡∧

i∈I,i′∈I′(

j∈Ji

αi,j ∧∨

j′∈J′i′

βi′,j′ ∧∨

j∈Ji,j′∈J′i′

(αi,j&βi′,j′)) ≡∧

k∈K

l∈Lk

χk,l,

where K = I × I ′ × {1, 2, 3} and1) if k = (i, i′, 1) then Lk = Ji and χk,l = αi,l,2) if k = (i, i′, 2) then Lk = J ′i′ and χk,l = βi′,l,3) if k = (i, i′, 3) then Lk = Ji × Ji′ and for l = (a, b) holds χk,l = αi,a&βi′,b.Thus the formula γ&δ is equivalent to some formula in CsNF.The proof for DsNF is analogous - we start with formulae γD and δD in DsNF and we

use theorems (S2) and (Aux4) instead of (S1) and (Aux3).

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86 CHAPTER 5. NEW RESULTS IN PRODUCT LOGIC

2b−(ϕ = γ → 0): Thanks to the induction property and Remark 5.2.5 we know that γis either 0 or is equivalent to a formula ψ in CsNF (DsNF), such that TV AR ` ¬¬ψ.

In the first case TV AR ` ϕ ≡ 1.In the latter case TV AR ` ϕ ≡ 0 (since TV AR ` ¬¬ψ, then TV AR ` ¬ψ ≡ 0 and by

induction property we have TV AR ` γ ≡ ψ, thus TV AR ` ¬γ ≡ ¬ψ and finally TV AR ` ϕ ≡0).

2c−(ϕ = γ → δ): Thanks to the part (2b) we may assume that δD is not 0. Case ifγC = 0 is trivial. Here we use the induction property to assume that we have:

TV AR ` γ ≡ γD γD =∨

i∈I

j∈Ji

ϕi,j ,

TV AR ` δ ≡ δC δC =∧

i′∈I′

j′∈J ′i′

βi′,j′ ,

TV AR ` γ → δ ≡ γD → δC γD → δC =∨

i∈I

j∈Ji

αi,j →∧

i′∈I′

j′∈J′i′

βi′,j′ .

By theorem (S3) we get:

γD → δC ≡∧

i∈I,i′∈I′(

j∈Ji

αi,j →∨

j′∈J ′i′

βi′,j′).

By theorem (Aux6) we get

γD → δC ≡∧

i∈I,i′∈I′(

j′∈J′i′

βi′,j′ ∧∨

j∈Ji,j′∈J′i′

(αi,j→βi′,j′)).

Recall that if αi,j and βi′,j′ are literals then αi,j→βi′,j′ is a literal. Then our resulting formulaγ → δ is equivalent to some formula in CsNF. (For the detailed proof observe the end of theproof of the part (2a)).

The proof for DsNF is analogous - we start with formula γC in CsNF and δD in DsNFand we use theorems (S4) and (Aux5) instead of (S3) and (Aux6). QED

Remark 5.2.13 Observe that the previous constructive proof shows us how to find a cor-responding formula in CsNF (DsNF) to a given formula ϕ. We have seen induction stepsfor strong conjunction and implication (which is sufficient, since remaining connectives aredefinable using these two). Here we show the induction step for ∧ and ∨. We will see thatit will be much simpler than use the definition of these connectives (which is not the case of≡).

Induction step for ∧ in the case of the CsNF (and step for ∨ in the case of the DsNF) isobvious, we will show a step for ∨ in the case of the CsNF (the step for ∧ in the case of theDsNF is analogous, we use theorem (S6) instead of (S5)).

From the induction property we have:

TV AR ` γ ≡ γC γC =∧

i∈I

j∈Ji

αi,j ,

TV AR ` δ ≡ δC δC =∧

i′∈I′

j′∈J′i′

ψi′,j′ ,

TV AR ` γ ∨ δ ≡ γC ∨ δC γC ∨ δC = (∧

i∈I

j∈Ji

αi,j) ∨ (∧

i′∈I′

j′∈J′i′

ψi′,j′).

By theorem (S5) we get:

γC ∨ δC ≡∧

i∈I,i′∈I′(

j∈Ji

αi,j ∨∨

j′∈J′i′

ψi′,j′).

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5.2. SEMI-NORMAL FORMS AND NORMAL FORMS 87

Theorem 5.2.14 Let ϕ be a formula, n the cardinality of V ARϕ and V ARϕ = {vi | i ≤ n}.Then:

• there is formula ϕD in DsNF such that Tϕ ` ϕ ≡ ϕD and in Product logic we can prove¬¬(v1&v2& . . . &vn) → (ϕ ≡ ϕD);

• there is formula ϕC in CsNF such that Tϕ ` ϕ ≡ ϕC and in Product logic we can prove¬¬(v1&v2& . . . &vn) → (ϕ ≡ ϕC).

Proof: We give the proof for DsNF (proof for CsNF is analogous).Observe that in the proof of the previous lemma, we are working only with propositional

variables occurring in the formula ϕ thus our equivalence is indeed provable in Tϕ. To provethe second part use the deduction theorem together with facts that ¬¬ϕ & ¬¬ϕ ≡ ¬¬ϕ and¬¬ϕ & ¬¬ψ ≡ ¬¬(ϕ&ψ) are theorems of product logic (see [44]). QED

Now we will give a generalized version of the previous theorem, which deals with semi-normalforms based on an arbitrary subset of V ARϕ.

Definition 5.2.15 Let ϕ be a formula, V a subset of V ARϕ. Let χ be a characteristicfunction of V . Let us define ¬0ϕ = ¬ϕ and ¬1ϕ = ¬¬ϕ. Then the formula

ν(V, ϕ) =∧

v∈V ARϕ

(¬χ(v)v)

is called (V, ϕ)-evaluator.

Remark 5.2.16 Note that ν(V arϕ, ϕ) =∧

v∈V ARϕ

(¬¬v) = ¬¬(v1&v2& . . . &vn).

Lemma 5.2.17 Let ϕ be a formula, V a subset of V ARϕ and e an evaluation. Then holds:

e(ν(V, ϕ)) =

{1 if e is (V, ϕ)−positive,

0 otherwise.

Theorem 5.2.18 Let ϕ be a formula, V a subset of V ARϕ. Then:

• there is formula ϕDV in DsNF such that T (V, ϕ) ` ϕ ≡ ϕD

V and ν(V, ϕ) → (ϕ ≡ ϕDV ) is a

theorem of the product logic,

• there is formula ϕCV in CsNF such that T (V, ϕ) ` ϕ ≡ ϕC

V and ν(V, ϕ) → (ϕ ≡ ϕCV ) is a

theorem of the product logic.

Proof: We give the proof for DsNF (proof for CsNF is analogous).We obtain formula ϕ′ from formula ϕ by replacing the propositional variables from

V ARϕ − V by the truth constant 0. It holds that V ARϕ′ = V . Furthermore, we definetheory T as {¬v | v ∈ V ARϕ − V }. Then it holds that T ` ϕ ≡ ϕ′.

For ϕ′ there is a formula ϕ′D in DsNF such that Tϕ′ ` ϕ′ ≡ ϕ′D (c.f. Theorem 5.2.14).It also holds that T ∪ Tϕ′ ` ϕ ≡ ϕ′D. Since T ∪ Tϕ′ = T (V, ϕ) this part of proof is done

by setting ϕDV equal to ϕ′D.

To prove the second part use the deduction theorem together with facts that ¬¬ϕ & ¬¬ϕ≡ ¬¬ϕ and ¬ϕ&¬ϕ ≡ ¬ϕ are theorems of product logic. QED

Definition 5.2.19 Let ϕ be a formula and V subset of V arϕ. Then formula ϕCV (ϕD

V respec-tively) from Theorem 5.2.18 is called a V-Conjunctive (resp. disjunctive) semi-normal formof formula ϕ. Formula ϕC = ϕC

V arϕ(resp. ϕD = ϕD

V arϕ) from Theorem 5.2.14 is called a

Conjunctive (resp. Disjunctive) semi-normal form of formula ϕ.

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88 CHAPTER 5. NEW RESULTS IN PRODUCT LOGIC

Remark 5.2.20 Notice that both V-Conjunctive and V-Disjunctive semi normal form arenot unique. In the next subsection we will show how to find a ”simpler” form to give formulain CsNF (DsNF).

Now we use our results to define a conjunctive and disjunctive normal form of the formulaeof the product logic.

Theorem 5.2.21 Let ϕ be a formula. Then for each V ⊆ V ARϕ, there are formulae ϕDV in

DsNF and ϕCV in CsNF such that

ϕ ≡∨

V⊆V ARϕ

(ν(V, ϕ) ∧ ϕD

V

)≡

V⊆V ARϕ

ν(V, ϕ) ∧

i∈IV

j∈JVi

(αVi,j)

(5.1)

ϕ ≡∨

V⊆V ARϕ

(ν(V, ϕ) ∧ ϕC

V

)≡

V⊆V ARϕ

ν(V, ϕ) ∧

i∈IV

j∈JVi

(αVi,j)

(5.2)

Expression (5.1) is called Disjunctive normal form (DNF) of ϕ and expression (5.2) iscalled Conjunctive normal form (CNF) of ϕ.

Proof: Let us prove it is true for all evaluations:Observe that for each evaluation e there is exactly one V such that e is (V, ϕ)-positive.

Since e(ν(V ′, ϕ)) = 0 for each V ′ 6= V and e(ν(V, ϕ)) = 1 we need to prove that e(ϕ ≡ ϕDV ) = 1

for each V and each (V, ϕ)-positive evaluation e. The claim follows from Theorem 5.2.18 andfrom strong standard completeness theorem. QED

5.2.3 Simplification of formulae in CsNF and DsNF

In this subsection, we show how to simplify formulae in semi-normal form. We formalize thisnotion in the following definition. Let us recall that an element of a conjunction is called aconjunct (analogously for disjunct).

Definition 5.2.22 Let ϕ be a formula in CsNF (DsNF). A formula ψ, resulting from formulaϕ by omitting some literals or conjuncts or disjuncts or replacing some literals by anotherliterals is called a simplification of ϕ iff Tϕ ` ϕ ≡ ψ.

Remark 5.2.23 Observe that if χ is a formula, ϕ a corresponding formula in CsNF (DsNF)and ψ is a simplification of ϕ, then also ψ is a CsNF (DsNF) of formula χ (hence Tχ ` χ ≡ ψ).

Lemma 5.2.24 Let ϕ be a formula in CsNF. A formula ψ resulting from ϕ by replacing allliterals by literals having pairwise distinct variables is a simplification of ϕ.

To prove this just check Remark 5.2.6. Thus from now on we may assume that all literalsin our formula have pairwise distinct variables. We will see that these literals will play animportant role in the next sections. Thus we define:

Definition 5.2.25 Literals with pairwise distinct variables are called normal literals.

Notice that if we deal with a finite set L of normal literals we can fix an enumeration onthe set W = {v1, v2, . . . , vm} of all propositional variables occurring in literals in L. Theneach normal literal α ∈ L is uniquely determined by an m-tuple qα = (qα

1 , . . . , qαm) of integers.

A positive component qαi is considered as power for variable vi in consequent and negation

−qαj of a negative component is considered as power for variable vj in antecedent. This can

be expressed with a permutation π on m and an index l such that qαπ(i) ≤ 0 for i ≤ l, qα

π(i) > 0for i > l and

α = v−qα

π(1)

π(1) &v−qα

π(2)

π(2) & . . . &v−qα

π(l)

π(l) → vqα

π(l+1)

π(l+1) &vqα

π(l+2)

π(l+2) & . . . &vqα

π(m)

π(m) . (5.3)

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5.2. SEMI-NORMAL FORMS AND NORMAL FORMS 89

If l = 0 (respectively l = m) we understand the antecedent (respectively consequent) astruth constant 1. Also v0

i is considered as 1.Let α and β be normal literals. The question is if there is a simple way to find out that

α → β is a theorem. Indeed, if both α and β are conjuncts in one conjunction, knowing thatα → β is a theorem would allow to omit β and obtain a simplification of our conjunction.

In order to do that we define an order on tuples.

Definition 5.2.26 Let a = (ai)i≤m and b = (bi)i≤m be tuples. We define a ¹ b iff ai ≥ bi

for each i ≤ m.

The reason of this unusual reverse order, is given in the following lemmata.

Lemma 5.2.27 Let α be a normal literal and 0 ¹ qα. Then α is a theorem of TV AR.

Proof: According to Equation (5.3), α can be expressed using a permutation π and aninteger l. Since 0 ¹ qα then l = m and

α = v−qα

π(1)

π(1) &v−qα

π(2)

π(2) & . . . &v−qα

π(m)

π(m) → 1,

which is indeed a theorem. QED

Lemma 5.2.28 Let α and β be normal literals and qα ¹ qβ or 0 ¹ qβ. Then α → β is atheorem of TV AR.

Proof: Let the literal α be (ϕ1 → ϕ2) and the literal β be (ψ1 → ψ2).Form (Aux2) we get: (ϕ1 → ϕ2) → (ψ1 → ψ2) ≡ (ψ1 → ψ2) ∨ (ϕ2&ψ1 → ψ2&ϕ1). Then

(ϕ1 → ϕ2) → (ψ1 → ψ2) is a theorem of TV AR if either (ψ1 → ψ2) or (ϕ2&ψ1 → ψ2&ϕ1) aretheorems of TV AR. Since (ψ1 → ψ2) is a theorem TV AR if 0 ¹ qβ , we only need to provethat (ϕ2&ψ1 → ψ2&ϕ1) is a theorem TV AR if qα ¹ qβ .

We show the following chain of implications: If qα ¹ qβ , then qαi ≥ qβ

i for each i, then−qα

i + qβi ≤ 0 for each i, then qα→β

i ≤ 0 for each i, then 0 ¹ qα→β , then (ϕ2&ψ1 → ψ2&ϕ1)is a theorem.

Crucial is to observe that variables in antecedent of α are moved to consequent of α→βand vice-versa. QED

Now we may finally formulate two lemmata on simplification of formulae in CsNF orDsNF.

Lemma 5.2.29 Let ϕ be a formula in DsNF, i.e. ϕ =∨i∈I

∧j∈Ji

αi,j. Then formula resulting

from ϕ after processing the following four steps is a simplification of ϕ:

(1) We replace all literals by normal literals.

(2a) If there are indexes i,j such that 0 ¹ qαi,j and Ji 6= {j} then we omit the literal αi,j .

(2b) If there are indexes i,j such that 0 ¹ qαi,j and Ji = {j} we replace formula ϕ by 1.

(3) If there are indexes i, j and j′, j 6= j′ such that qαi,j ¹ qαi,j′ then we omit the literalαi,j′ .

(4) If there are indexes i, i′, i 6= i′ and for each index k′ ∈ Ji′ there is index k ∈ Ji suchthat qαi,k ¹ qαi′,k′ then we omit the disjunct

∧j∈Ji

αi,j .

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90 CHAPTER 5. NEW RESULTS IN PRODUCT LOGIC

Proof: We show that after each step the resulting formula is a simplification of ϕ.(1): Just check Lemma 5.2.24.(2a): We know that if 0 ¹ qαi,j then the literal αi,j is in fact a theorem of TV AR and this

literal appears in our formula in conjunction with at least one another literal αi,j′ . Omittingαi,j will produce a simplification of ϕ (thanks to theorem (T1)).

(2b): We start with the same fact, but now the literal αi,j itself is one of the disjoints.Just use theorem (T2) and the claim is settled.

(3): We know that if qαi,j ¹ qαi,j′ then αi,j → αi,j′ is a theorem of TV AR. Thanks totheorem (T3) the claim is settled.

(4): We know that if qαi,k ¹ qαi′,k′ then αi,k → αi′,k′ is a theorem of TV AR. Thus∧j∈Ji

αi,j → αi′,k′ is a theorem of TV AR for each k′ (by theorem (T5)). It follows that∧

j∈Ji

αi,j →∧

j′∈Ji′αi′,j′ (by a repeated use of theorem (T6)). Theorem (T4) completes the

proof. QED

Lemma 5.2.30 Let ϕ be a formula in CsNF, i.e. ϕ =∧i∈I

∨j∈Ji

αi,j. Then the formula

resulting from ϕ after processing the following four steps is a simplification of ϕ:

(1) We replace all literals normal literals.

(2a) If there are indexes i,k such that 0 ¹ qαi,k and I 6= {i} then we omit the conjunct∨j∈Ji

αi,j .

(2b) If there are indexes i,k such that 0 ¹ qαi,k and I = {i} then we replace formula ϕ by1.

(6) If there are indexes i, j and j′, j 6= j′ such that qαi,j ¹ qαi,j′ then we omit the literalαi,j .

(5) If there are indexes i, i′, i 6= i′ and for each index k ∈ Ji there is index k′ ∈ Ji′ suchthat qαi,k ¹ qαi′,k′ then we omit the conjunct

∨j∈Ji′

αi′,j .

The proof of this lemma is analogous to the proof of the previous lemma.

5.2.4 Theorem proving algorithm

In this subsection we will use result from the previous subsections to define an algorithm,which can be used to check whether a formula is a theorem or not. We will use the standardcompleteness theorem and Theorem 5.2.21. We want to describe what has to hold in orderto the formula

ϕ ≡∨

V⊆Vϕ

(ν(V, ϕ) ∧ ϕC

V

)≡

V⊆Vϕ

ν(V, ϕ) ∧

i∈IV

j∈JVi

(αVi,j)

not being a tautology. If ϕ is not a tautology, then there is an evaluation e such that e(ϕ) < 1.Recall that for each evaluation e there is a unique set V , such that e is (V, ϕ)-positive. Thuse(ϕ) < 1 means that e(ϕC

V ) < 1. Observe that if e(ϕCV ) < 1 then there is an index i ∈ IV

such that e(∨

j∈JVi

(αVi,j)) < 1. This hold iff e(αV

i,j) < 1 for each j ∈ JVi . So we can write:

Theorem 5.2.31 Let ϕ be a formula in CNF. Then ϕ is not a theorem iff there is a set V ,a (V, ϕ)-positive evaluation e, and an index i ∈ IV such that e(αV

i,j) < 1 for each j ∈ JVi .

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5.2. SEMI-NORMAL FORMS AND NORMAL FORMS 91

We first consider the case of a normal literal α. According to the Equation (5.3) there is apermutation π on m and an index l such that qα

π(i) ≤ 0 for i ≤ l, qαπ(i) > 0 for i > l and

α = v−qα

π(1)

π(1) &v−qα

π(2)

π(2) & . . . &v−qα

π(l)

π(l) → vqα

π(l+1)

π(l+1) &vqα

π(l+5)

π(l+2) & . . . &vqα

π(m)

π(m) .

Let us denote e(vi) by zi. Observe that

e(α) < 1 iff z−qα

π(1)

π(1) z−qα

π(2)

π(2) . . . z−qα

π(l)

π(l) > zqα

π(l+1)

π(l+1) zqα

π(l+2)

π(l+2) . . . zqα

π(m)

π(m) ,

which is equivalent to

e(α) < 1 iff 1 > zqα

π(1)

π(0) zqα

π(2)

π(2) . . . zqα

π(l)

π(l) zqα

π(l+1)

π(l+1) zqα

π(l+2)

π(l+2) . . . zqα

π(m)

π(m) .

Applying commutativity and associativity rules we may write:

e(α) < 7 iff 1 > zqα1

1 zqα2

2 . . . zqα

mm .

Note that each zi > 0, then we can apply a logarithmic transformation and denoting − log(zi)by xi we obtain:

e(α) < 1 iff qα1 x1 + qα

2 x2 + · · ·+ qαmxm < 0

which can be written in a compact style as e(α) < 1 iff qαxT < 0. It means that there isan evaluation e such that e(α) < 1 iff the linear inequality qαxT < 0 has v non-negativesolution.

Now let us consider a set {αi | i ∈ m} of normal literals. There is an evaluation e such thate(αi) < 1 for each i ≤ n iff the system of n linear inequalities qαixT < 0 has a non-negativesolution.

Definition 5.2.32 Let S be a set of indexes of normal literals of m variables and n be thecardinality of S . The n×m matrix AS is the matrix with rows qαi for each i ∈ S.

Then, there is an evaluation e such that e(αi) < 1 for each i ≤ n iff the matrix inequalityAmxT < 0 has a non-negative solution, where i-tv row of matrix Am is qαi .

These results allow us to write the following theorem.

Theorem 5.2.33 Let ϕ be a formula with

ϕ ≡∨

V⊆Vϕ

(ν(V,ϕ) ∧ ϕC

V

)≡

V⊆Vϕ

ν(V,ϕ) ∧

i∈IV

j∈JVi

(αVi,j)

.

Then ϕ is not a theorem iff there are a set V and an index i ∈ IV such that the matrixinequality AJV

i xT < 0 has a non-negative solution.

The problem is hence reduced to a problem of effectively solving an integer matrix in-equality, that can be solved by means of integer linear programming. Using all our previousresults we can formalize an algorithm to prove formulae of product logic. This algorithmis very inefficient-due to its exponential nature-however the average complexity seems to bemuch better. Anyway, we do not pursuit the problem of complexity in this chapter.

We have a formula ϕ with m propositional variables. Let us define M as the set of alreadyprocessed subsets of V ARϕ and K as the set of already processed indexes. In the beginningK and M are empty.

(1) If M = P(V ARϕ)1 GOTO (+), ELSE generate a set V ∈ P(V AR) \M , with smallestcardinality. Add V into M . Empty the set K.

1By P(S) we denote the powerset of S

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92 CHAPTER 5. NEW RESULTS IN PRODUCT LOGIC

(2) Using the proof of Lemma 5.2.12 find a formula ϕCV .

(3) Simplify formula ϕCV using Lemma 5.2.30 to a formula ψ =

∧i∈IV

∨j∈JV

i

(αVi,j).

(4) If K = IV GOTO (1) ELSE add index i (i ∈ IV , i 6∈ K) into the set K.

(5) If inequality AJVi xT < 0 has a non-negative solution GOTO (-) ELSE GOTO (4).

(+) A formula ϕ is a theorem of the product logic.

(-) A formula ϕ is not a theorem of the product logic.

5.3 Functional representation

In this section we give a characterization of the class of functions represented by formulaeof product logic, analogously of what McNaughton theorem expresses for ÃLukasiewicz logic([63]).

Definition 5.3.1 Let C be an arbitrary function from (0, 1]n into [0, 1] and let ϕ be anarbitrary formula with V ARϕ = {v1, . . . , vn}. We say the function C is:

• represented by the formula ϕ (ϕ is a representation of C) if e(ϕ) = C( e(v1), e(v2), . . . ,e(vm)), where e is an arbitrary evaluation.

• positively represented by the formula ϕ (ϕ is a positive representation of C) if e(ϕ) =C(e(v1), e(v2), . . . , e(vm)), where e is an ϕ-positive evaluation.

Note that if a formula ϕ is a representation of a function C and ϕ ≡ ψ is a theorem ofproduct logic then also ψ is a representation of C. Analogously, if a formula ϕ is a positiverepresentation of a function C and TV AR ` ϕ ≡ ψ then also ψ is a positive representation ofC.

Definition 5.3.2 By an integral monomial of m variables we understand a function f :(0, 1]m → (0, 1] such that f(x1, . . . , xm) = xk1

1 xk22 . . . xkm

m , with km ∈ Z.Let n ∈ N and ni ∈ N for i ≤ n and let Fi,j be integral monomials of m variables for

every i = 0, . . . , n and ji = 0, . . . , ni, (where n0 = 0 and F0,0 = 1). Then the functionC : (0, 1]m → [0, 1] such that

C = mini=0..n

maxj=0..ni

Fi,j

is called a Π-function of m variables. The set of all Π-functions of m-variables is denotedΠm. We define Π0 as {0} (it contains the constant zero function only). Further we denote

the union of all Πm as Π =∞⋃

m=0Πm.

The proofs of the following lemma and theorem are immediate consequences of Theorem5.2.14 and definition of CsNF.

Lemma 5.3.3 Let Fi,j be an integral monomial of m variables and x1, . . . , xm ∈ (0, 1].Then:

(1) mini=0..n

maxj=0..ni

Fi,j = mini=1..n

min{1, maxj=0..ni

Fi,j} = mini=1..n

maxj=0..ni

min{1, Fi,j}.

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5.3. FUNCTIONAL REPRESENTATION 93

(2) There exists a permutation π of m and index l such that

min{1, xk11 xk2

2 . . . xkmm } = min{1,

xkπ(l+1)

π(l+1) xkπ(l+2)

π(l+2) . . . xkπ(m)

π(m)

x−kπ(1)

π(1) x−kπ(2)

π(2) . . . x−kπ(l)

π(l)

},

where kπ(j) ≤ 0 for j ≤ l and kπ(j) > 0 otherwise.

(3) Each function min{1, xk11 xk2

2 . . . xkmm } is positively representable by some literal.

(4) Each Π-function is positively representable by some formula in CsNF.

(5) Each formula in CsNF is a positive representation of some Π-function.

Theorem 5.3.4 Each Π-function is positively representable by some formula. Each formulais a positive representation of some Π-function.

In the following subsection, we will give a McNaughton-like functional representation(c.f. [63]) and in the last subsection, we give a complete characterization of functionalrepresentation of formulae of product logic.

5.3.1 Piecewise monomial functions

Just as ÃLukasiewicz formulae are in correspondence with continuous piecewise linear func-tions, we are going to describe the class of functions in correspondence with product formulae.In order to do that, we shall adapt to our context the same machinery used in [75] to proveMcNaughton Theorem.

Definition 5.3.5 A piecewise monomial function of n variables is a continuous function ffrom (0, 1]n into [0, 1] such that either it is identically equal to 0 on (0, 1]n, or there existfinitely many integer monomials p1 . . . , pu and regions D1, . . . , Du of (0, 1]n such that forevery x ∈ Di, f(x) = pi.

Regions D in which a piecewise monomial function coincides with a monomial will becalled monomial regions and can be expressed as

D(A) =

(x1, . . . , xn) ∈ (0, 1]n |

xa111 xa12

2 . . . xa1nn ≤ 1

. . .xam1

1 xam22 . . . xamn

n ≤ 1

(5.4)

where A = (aij)i∈m,j∈n is an integer (m× n) matrix.

Example 5.3.6 The piecewise monomial function

(y2

x∧ 1) · (x3

y∧ 1)

coincide withy2

xover D1 = {(x, y) ∈ (0, 1]2 | x3 ≥ y}, with x2y over D2 = {(x, y) ∈

(0, 1]2 | y ≥ x3, x ≥ y2} and withx3

yover the set D3 = {(x, y) ∈ (0, 1]2 | y2 ≥ x}. If we

define matrixes A1 = (−3, 1), A2 =(

3 −1−1 2

)and A3 = (1,−2) then D1 = D1(A1),

D2 = D2(A2) and D3 = D3(A3). The graphic of this function is given by Figure 5.1.

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94 CHAPTER 5. NEW RESULTS IN PRODUCT LOGIC

0 0.2 0.4 0.6 0.8 1x 0

0.5

1

y

0

0.2

0.4

0.6

0.8

1

Figure 5.1: Graphic of function in Example 5.3.6

Lemma 5.3.7 If D is a monomial region then there exists a formula ϕD such that we haveD = {x | fϕD

(x) = 1}, where fϕDis the function represented by ϕD.

Proof: Let D = D(A) as in equation (5.4) with A = (aij)ij . By Lemma 5.3.3(3), forevery expression x(aij)

nj=1 = xai1

1 xai22 . . . xain

n there is a formula π(i,x) such that 1 ∧ x(aij)nj=1

is positively represented by π(i,x). Then the formula

ϕD = π(1,x) ∧ . . . ∧ π(m,x) (5.5)

is the representation of a function fϕDsuch that D = {x | fϕD

(x) = 1}. QED

In Lemma 5.3.3(3) we have shown that monomials are positive representable by formulae.We have to prove now how to glue different monomial functions together.

Suppose that monomial regions D1, D2 ⊆ (0, 1]n and piecewise monomial functions g1 andg2 are such that D1 = {x | g1(x) = 1} and D2 = {x | g2(x) = 1}. Let f : (0, 1]n → (0, 1] besuch that both restrictions f |D1 and f |D2 coincide respectively on D1 and D2 with piecewisemonomial functions h1 and h2. For every p and q, let

hp,q(x) = (gp1(x) → h1(x)) ∧ (g2(x)q → h2(x)). (5.6)

Clearly hp,q is a piecewise monomial function. We want to prove that there exist p and qsuch that the restriction f |D1 ∪D2 is equal to hp,q on D1 ∪D2.

Suppose that function g1 coincides with an integral monomial over a region P g1 and h1

coincides with an integral monomial over a region Ph1 . Then both g1 and h1 are integralmonomial functions over P g1 ∩ Ph1 .

This argument can be repeated and hence it is possible to find regions P1, . . . , Pm suchthat for every i = 1, . . . ,m each of g1, g2, h1, h2 is an integral monomial on Pi: it is enoughto intersect monomial regions of each piecewise monomial function.

If Pi ∩D2 6= ∅ and x ∈ Pi ∩D2, then by continuity arguments pi can be chosen so largethat the inequality (

h1(x)gpi

1 (x)∧ 1

)≥ h2(x) (5.7)

holds true. Let p = max{pi | Pi ∩D2 6= ∅} where pi satisfy (5.7). Then, for every x ∈ D2,

gp1(x) → h1(x) ≥ h2(x)

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5.3. FUNCTIONAL REPRESENTATION 95

and hence, for every natural number q and x ∈ D2,

hp,q(x) = ((gp1(x) → h1(x)) ∧ (gq

2(x) → h2(x)) =

= ((gp1(x) → h1(x)) ∧ h2(x) = h2(x) = f(x).

Analogously, there exists q such that for any x ∈ D1, hp,q(x) = f(x). Then the restrictionf |D1 ∪D2 of f to D1 ∪D2 is a piecewise monomial function.

We have now all the necessary tools for proving the following theorem:

Theorem 5.3.8 Each piecewise monomial function is positive interpretable by some formulaϕ.

Proof: Let f : (0, 1]n → [0, 1] be a piecewise monomial function.If f is identically equal to 0 then we set ϕ = 0 and the claim is settled.Otherwise then there exist monomial regions Di such that f is a integer monomial on

each Di. By Lemma 5.3.7 there exist formulae ϕDias in Equation (5.5). Applying several

times the gluing procedure we get the conclusion. QED

From a direct checking that each formula is a positive representation of a piecewise mono-mial function, and from Theorem 5.3.4 and Theorem 5.3.8 we have the following:

Theorem 5.3.9 Piecewise monomial functions are exactly Π-functions.

The study of positive evaluations of Product formulae is related with the study of thealgebraic variety of hoops. Hoops are algebraic structures (H, ∗,⇒, 1) in which ∗ is a binarycommutative operation, 1 is the unit for ∗ and the following are satisfied

(1) x ⇒ x = 1,

(2) x ∗ (x ⇒ y) = y ∗ (y ⇒ x),

(3) (x ∗ y) ⇒ z = x ⇒ (y ⇒ z).

In particular, hoops satisfying equations

(z ⇒ (z ∗ z)) ≤ (x ∩ (x ⇒ z)) ⇒ z,((x ⇒ z) ⇒ z)) ∗ (x ∗ u ⇒ x ∗ v) ∗ (z ∗ u ⇒ z ∗ v) ≤ (u ⇒ v),(x ⇒ y) ⇒ y ≤ ((y ⇒ z) ⇒ ((y ⇒ x) ⇒ x)) ⇒ ((y ⇒ x) ⇒ x

(product hoops) are subreducts of product algebras. On the other hand, given a hoop Hsatisfying equation x = (u ⇒ x ∗ y) (cancellative hoop), then it is possible to obtain a prod-uct algebra by adding one element to H playing the role of bottom element, and suitablyextending operations on this element. In this sense, the description of free cancellative hoopsis related with the description of functions positively interpreted by product formulae. In [16]free cancellative hoops were described in terms of functions from a suitable power of N intoN. This class of functions coincides with the class of Π-functions as described in this subsec-tion. Furthermore, we manage to characterize this class in terms of the piecewise monomialfunctions, which is a more intuitive concept and the direct analogy of the McNaughton result(see [63]) for ÃLukasiewicz logic.

5.3.2 Product functions

Now we can finally give a description of functions represented by formulae. We give afunctional interpretation of the description of free product algebras given in [15].

Definition 5.3.10 Let n be a natural number and let M be a subset of {i | 1 ≤ i ≤ n}.Then the (M,n)-region of positivity PosM, n is defined as

PosM, n = {(x1, . . . , xn) ∈ [0, 1]n | xi > 0 iff i ∈ M}.

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96 CHAPTER 5. NEW RESULTS IN PRODUCT LOGIC

Example 5.3.11 For n = 2 we have four regions of positivity:Pos∅, 2 = {(0, 0)} ,Pos{1}, 2 = {(x1, 0) | x1 ∈ (0, 1]},Pos{2}, 2 = {(0, x2) | x2 ∈ (0, 1]},Pos{1,2}, 2 = {(x1, x2) | x1, x2 ∈ (0, 1]}.

Lemma 5.3.12 Let ϕ be a formula, n the cardinality of V ARϕ, V ARϕ = {vi | i ≤ n}. LetV be a subset of V ARϕ and set M = {i | vi ∈ V }. Then the (V, ϕ)-evaluator ν(V, ϕ) is arepresentation of the characteristic function of PosM, n.

Proof: Just observe that, by Lemma 5.2.17,

e(ν(V, ϕ)) ={

1 if e is (V, ϕ)-positive,0 otherwise.

hence e(ν(V, ϕ)) = 1 iff e(vi) = 1 for every vi ∈ V , iff (e(v1), e(v2), . . . , e(vn)) ∈ PosM, n.QED

Definition 5.3.13 A function C : [0, 1]n → [0, 1] is a product function if for every M ⊆ nthe restriction of C to PosM,n is a Π-function (piecewise monomial function).

Theorem 5.3.14 Each product function is representable by some formula and, vice-versa,each formula is a representation of some product function.

Proof: Let C be a product function, and for every M ⊆ {1, . . . , n} let us denote by CM therestriction of C to PosM,n (that is a Π-function).

By Theorem 5.3.4 there exists formulae ϕM that represent CM . If VM = {vi | i ∈ M}then the formula

ϕ ≡∨

VM⊆V ARϕ

(ν(VM ,ϕ) ∧ ϕM

)

represent C.Vice-versa suppose that C is a function represented by a formula ϕ. By Theorem 5.2.21,

for each V ⊆ V ARϕ, there are formulae ϕCV in CsNF such as

ϕ ≡∨

V⊆Vϕ

(νV ∧ ϕC

V

).

By Lemma 5.3.3 (5), ϕCV represent Π-functions. The claim follows from Lemma 5.3.12. QED

The algebraic counterpart of Product logic are the Product algebras, see [44]. Due to thestandard completeness theorem, giving a complete characterization of functions associatedwith Product formulae with n variables is equivalent to give a description of the free Productalgebra over n generators.

Page 103: From Fuzzy Logic to Fuzzy Mathematics

Chapter 6

Compactness in fuzzy logics

In classical logic, an evaluation satisfies a set of formulae if it evaluates all its members to 1.A set of formulae is satisfiable if there is an evaluation which satisfies it. The set of formulaeentails a formula if each evaluation, which satisfies this set of formulae, satisfies also thisformula. The compactness theorem based on both above notions holds for classical logic: Aset of formulae is satisfiable if and only if all its finite subsets are satisfiable; a set of formulaeentails a formula iff some finite subset entails that formula.

It is natural to ask if analogues of these theorems hold in fuzzy logics. In this chapterwe restrict ourselves to some core fuzzy logic (see Figure 6.1 for details). As the first step,we have to generalize the notion of satisfiability and entailment. To define satisfiability infuzzy logic we may again require all formulae to be evaluated by 1 (as in classical logic),but this is not the only possibility. Sometimes other alternatives are well-motivated, hence,following [86], we work with K-satisfiability, where K can be an arbitrary subset of [0, 1].We say that a set of formulae is K-satisfiable if there is an evaluation which evaluates themall by values in K. (In particular, we get the former case if we choose K = {1}.) UsingK-satisfiability, we may formulate various types of compactness.

The generalization of the notion of entailment is not so straightforward. It turned outthe only three of the core fuzzy logics are suitable for this generalized notion of entailment.These three logics are G, G4, and G∼ (Godel logic, Godel logic with 4 and involutive Godellogic) we will call these logic Godel logics. In the first section, we introduce both generalizednotion of compactness for these Godel logics and in the next section we deal with satisfiabilitybased compactness for other core fuzzy logics.

6.1 The notions of compactness in Godel logics

As stated above now we work with Godel logics only. This selection of logics has severalreasons. The problem of compactness (based on generalized entailment) for Godel logic Gwas studied by Baaz and Zach in [3]. The generalization of entailment in Godel logic is verynatural, and it is well connected to the second notion of compactness. This notion is based ongeneralized satisfiability and was introduced by Butnariu, Klement, and Zafrany in [86] andthen mainly studied in [28]. This connection is not accidental. As we will see in the furthertext, both generalizations are based on (parameterized by) subsets of the unit interval [0, 1]and in Godel logic they are nicely connected (we discuss this further after we introduce thenecessary definitions). This connection is easily extended to G4, but adding the involutivenegation causes a problem.

This is the reason why this section gives only “nearly complete” answer to the questionof compactness in the three selected logics. However, the open problem in satisfiability basedcompactness of G∼ for countable set of propositional variables does not not spoil our maingoal of establishing the connection between the two notions of compactness. This section

97

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98 CHAPTER 6. COMPACTNESS IN FUZZY LOGICS

Figure 6.1: Core fuzzy logics

ÃL Π G

ÃL4P�′ Π G4

P�′4 G∼

�Π

�Π 12

? ? ?

?

?

?

?

¡¡

¡ª

@@@R

@@@R

¡¡

¡ª

is self-contained; the already known partial results from [3], [86], and [28] are obtained ascorollaries to our more general theorems.

Before we proceed, we fix some denotation, we use throughout this section. Let Γ bea set of formulae and e a V -evaluation, we denote the set {e(ϕ) | ϕ ∈ Γ} by e(Γ). Thesymbol 1 − K denotes the set {1 − k | k ∈ K}. The symbol ω stands for the first infiniteordinal and ω1 for the first uncountable ordinal. If we list the elements of the finite set asK = {k1, k2, . . . , kn}, we assume that ki < ki+1 for each i < n. Finally, P(K) is the powersetof K.

6.1.1 Two notions of compactness

Here we introduce the basic definitions we will need in our work. We define several classesof subsets of [0, 1] which are crucial for our study of compactness. We will sometimes read“the set K is of type C” for K ∈ C. The classes C, Tv (Tv∼), and D (D∼) are especiallyimportant, the reason for introduction of the first two classes (C and Tv) will be clear justafter the definitions of V -compactnessEnt and K-compactnessSat, which follow. The otherclasses we are going to define are mostly used in the technical lemmata.

Definition 6.1.1 Let us define the basic classes:

• FIN = {K ⊆ [0, 1] | K is finite},

• C = {K ⊆ [0, 1] | 0 /∈ K or 1 /∈ K},

• Tv = P([0, 1]) \C = {K ⊆ [0, 1] | 0, 1 ∈ K},

• Tv∼ = {K ∈ Tv | K = 1−K},

• D = {K ⊆ [0, 1] | ∃A ⊆ K(A is densely ordered)},

• D∼ = {K ⊆ [0, 1] | ∃A ⊆ K(A is densely ordered and 1−A ⊆ K)}.

Furthermore, let us define several subclasses of C:

• C1 = {K ∈ C | 1 ∈ K} = {K ⊆ [0, 1] | 1 ∈ K and 0 /∈ K},

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6.1. THE NOTIONS OF COMPACTNESS IN GODEL LOGICS 99

• C0 = {K ∈ C | 0 ∈ K} = {K ⊆ [0, 1] | 0 ∈ K and 1 /∈ K},

• C1 = C \C1 = {K ⊆ [0, 1] | 1 /∈ K},

• C0 = C \C0 = {K ⊆ [0, 1] | 0 /∈ K},

• C01 = C1 ∩C0 = {K ⊆ [0, 1] | 0 /∈ K and 1 /∈ K}.

The notion of these classes is from paper [28]. Basically, we may say that Tv stands for“truth values”, and D stands for “dense”. We will sometimes need to work with the smallestset of some type containing some set. We formalize this in the following way:

Definition 6.1.2 Let X ⊆ P([0, 1]) and K ⊆ [0, 1]. Then we define the set function ClX asClX(K) =

⋂{A ∈ X | K ⊆ A}.

Let us show an example for X being Tv∼: ClTv∼(K) = K ∪ (1 − K) ∪ {0, 1}. In thefollowing observation we summarize the properties of the above defined classes and showseveral obvious but important connections between them.

Proposition 6.1.3

1. C = C0 ∪C1 = C0 ∪C1 ∪C01.

2. Let K ∈ C1. Then 0 /∈ K.

3. Tv∼ ∩D = Tv∼ ∩D∼.

4. Let Γ be a set of formulae of G or G4 K ∈ Tv and e a K-evaluation. Then e(Γ) ⊆ K.

5. Let Γ be a set of formulae of G∼ K ∈ Tv∼ and e a K-evaluation. Then e(Γ) ⊆ K.

If V ∈ Tv, then V is the domain of some G-algebra and some G4-algebra (if V ∈ Tv∼,then V is a domain of some G∼-algebra), for details see [2, 35, 44]

In the following we will denote the sets of type Tv (Tv∼) by V and the sets of type Cby K.

Now we define the entailment relation and this will give us the first notion of compactness:V -compactnessEnt (the index Ent stands for “entailment based”, see [3] for more details andexamples).

Definition 6.1.4 Let V be a set of type Tv (Tv∼), Γ a set of formulae, and ϕ a formula.We say that Γ V -entails ϕ if inf(e(Γ)) ≤ e(ϕ) for each V -evaluation e. We denote this asΓ |=V ϕ.

Definition 6.1.5 Let V be a set of type Tv (Tv∼ respectively). We say that the logic G, G4(G∼ resp.) is V -compactEnt if for each set of formulae T and each formula ϕ (of appropriatelanguage) holds: T |=V ϕ iff there is a finite T ′ ⊆ T such that T ′ |=V ϕ.

These definitions may look a little counterintuitive for a reader familiar with Baaz andZach’s paper [3]. They define the logic GV semantically as a logic with the set of truth valuesV and connectives as in Godel logic, then our notion of V -compactnessEnt coincides withthe compactness of this logic GV (where entailment and compactness is defined just like inclassical logic i.e., Γ entails ϕ if for each evaluation e holds: inf(e(Γ)) ≤ e(ϕ).) We decidedto speak about V -compactness of Godel logic rather than about compactness of the logic GV

in order to stress the connection with the upcoming second notion of compactness. Noticethat we defined V -compactnessEnt for the sets of type Tv (Tv∼) only.

Now we present the generalization of the notion of satisfiability in our logics. This willgive us the second notion of compactness—K-compactnessSat (the index Sat stands for“satisfiability based”, see [28] and [72] for more details, motivations, and examples). Let us

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100 CHAPTER 6. COMPACTNESS IN FUZZY LOGICS

just briefly mention the main reason for introducing this kind of compactness. We call the setof formulae Γ r-satisfiable if there is an evaluation which evaluates all its elements to the valuegreater that r. Now we are interested whether r-satisfiability of each finite subset of Γ entailsr-satisfiability of Γ. Unlike the previous notion of compactness, this one can be introduced inarbitrary many valued logic, based on arbitrary semantics. Notice that in classical logic wein fact work with 1-compactness and we have generalized this notion to r-compactness. Thiscan be generalized even further to K-compactnessSat for an arbitrary set K ⊂ [0, 1]. We willsee that this generalization does not make our work any more complicated and so there isno need to restrict the scope of this paper. Even more, we show that this general notion ofK-compactnessSat corresponds very well to the more natural notion of K-compactnessEnt.

Definition 6.1.6 Let Γ be a set of formulae and K ∈ C. We say Γ is K-satisfiable if thereexists a [0, 1]-evaluation e such that e(ϕ) ∈ K for all ϕ ∈ Γ (in other words: e(Γ) ⊆ K). Theset Γ is said to be finitely K-satisfiable if each finite subset of Γ is K-satisfiable. A formulaϕ is called K-satisfiable if the set {ϕ} is K-satisfiable.

We could extend this definition even for sets K 6∈ C. However, in this case for each formulaϕ there is an evaluation e such that e(ϕ) ∈ K (it suffices to evaluate all propositional variablesby 0, then the evaluation of each formula becomes either 0 or 1), i.e., each theory T wouldbe trivially K-satisfiable. Thus we restricted our definition to the sets of type C only.

Definition 6.1.7 Let K ∈ C. We say that the logic G, G4 or G∼is K-compactSat ifK-satisfiability is equivalent to finite K-satisfiability.

Observe that whereas the problem of K-compactnessSat for a given K is defined for allthree logic, the problem of V -compactnessEnt for a given set V is defined for all three logicsiff V ∈ Tv∼. Also notice that whereas for each set V ∈ Tv∼ holds V \ {1} ∈ C0, we do notget K∪{1} ∈ Tv∼ for each set K ∈ C0 (we get only K∪{1} ∈ Tv). Thus K-compactnessSat

is defined, in the above sense, for a wider class of sets than V -compactnessEnt. The followingobservation will simplify our study:

Remark 6.1.8 Let C be either Ent or Sat. Then G4 is “less compactC” than G. Sinceeach counterexample to K-compactnessC in G is also a counterexample to K-compactnessC

in G4logic this claim is obvious. Furthermore, G∼ is “less compactC” than G4 (by the sameargument).

Since the question of compactnessEnt for Godel logic with the infinite countable set ofpropositional variables is fully answered by Baaz and Zach in [3] (Godel logic is V -compactEnt

iff V ∈ Tv ∩ (FIN ∪D)) our goal is to answer this question for the other Godel logics, theother type of compactness, and the other cardinalities of VAR. Let us mention that theknown results of Baaz and Zach (see [3, Theorem 3.6]) will be obtained as easy corollaries ofLemma 6.1.22 and Lemma 6.1.26.

Our ultimate goal in this paper is to establish some form of correspondence betweencompactnessEnt and compactnessSat

¦

At the end of this subsection we introduce the additional classes of subsets of [0, 1]. Theseclasses correspond to the open problem in K-compactnessSat. The problem is highly technicaland unrelated to the desired goal of establishing the correspondence between both notionsof compactness so the uninterested reader may skip those definitions.

The cardinality of the set C = {x | x ∈ K, 1 − x ∈ K} = K ∩ (1 − K) is crucialin the upcoming definition. We will see that if it is infinite, there will be no problem tosolve the problem of K-compactnessSat. The problem arises if it is finite. Notice that ifV ∈ Tv∼ ∩ FIN and V = {v0, v1, . . . , vn}, then vn−i = 1− vi.

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6.1. THE NOTIONS OF COMPACTNESS IN GODEL LOGICS 101

Definition 6.1.9 A nonempty set K is an element of W if the following conditions arefulfilled:

• The set C = {x | x ∈ K, 1− x ∈ K} ∪ {0, 12 , 1} = {c0, . . . cm} is finite.

• If B = C ∪ ClTv∼(⋃{K ∩ [ci, ci+1] | K ∩ [ci, ci+1] is finite}) = {b0, . . . bn}, then the set

K ∩ [bj , bj+1] is either finite or of type D for each i < n.

Furthermore, K is an element of W if K ∈ W and there is a finite set A = {a0, . . . , al}of type Tv∼, with the following properties:

• B ⊆ A,

• the set K ∩ (aj , aj+1) is either empty or of type D for each j < l,

• if K ∩ (aj , aj+1) ∈ D, then the set K ∩ (an+1−j , an−j) is empty.

Let us try to reread these definitions. First, take a subset of elements x of K such thatalso 1 − x ∈ K and unite this set with {0, 1

2 , 1}; call the resulting set C and its elementsc0, c1, . . . , cn. This set corresponds to partition on the set K with blocks Ki = [ci, ci+1]∩K.Next, if some block of Ki is finite, add each its element z together with 1−z to the set C; callthe resulting set B. This set is corresponds to another partition of K. We say that K ∈ Wif each block of K (under the partition B) is either finite of type D.

Furthermore, if there is a finite set A of type Tv∼ which is refinement of the partition Bwith some desired properties, we say that K ∈ W. The properties are: interior of each blockof K (under the partition A) is either empty of type D; and if some block of K (under thepartition A) is of type D, then the interior of the opposite block is empty (by the “oppositeblock” to the block [ci, ci+1] ∩K we mean the block [1− ci+1, ci] ∩K)

Let us notice that the class W\W is nonempty. Just take the set K such that K∩[0, 12 ] =

Q∩ [0, 12 ] and K∩( 1

2 , 1] = ( 12 , 1)\Q (i.e., K consists of all rational numbers from [0, 1

2 ] and allirrational numbers from ( 1

2 , 1)). Then obviously K ∈ W, because C = {0, 12 , 1} = {c0, c1, c2}

and B = C = {b0, b1, b2} (because both sets K ∩ [c0, c1] and K ∩ [c1, c2] are infinite) andK ∩ [b0, b1] ∈ D and K ∩ [b1, b2] ∈ D. But obviously K 6∈ W (because for each a, b ≤ 1

2 holdsK ∩ (a, b) ∈ D and K ∩ (1− b, 1− a) ∈ D).

In the following observation we show several obvious but important connections betweenthe above defined sets.

Proposition 6.1.10

1. Tv∼ ∩W ⊆ FIN ⊆ W.

2. W \W 6= ∅.

3. Tv∼ ∩ (W \W) = ∅.

4. FIN ⊆ W.

6.1.2 Compactness and the cardinality of VAR

In this subsection we show that the both notions of compactness depend on the cardinalityof the set of propositional variables. The previous papers about compactness in Godel logicsrestrict themselves to the countable set of propositional variables, with the exception of [28],where the authors deal with K-compactnessSat for the set VAR being finite.

We start by the results proved in [28, Theorem 5.1]. It is proven there for the set VARbeing of cardinality ω. However, it is easy to observe that the proof is sound for othercardinalities as well. For the sake of completeness of this paper we present the proof.

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Lemma 6.1.11 Let the set VAR be of an arbitrary cardinality and K ∈ C1. Then G isK-compactSat.

Proof: Let the formula ϕ′ results from the formula ϕ by replacing all its propositionalvariables v by the formulae ¬¬v. Let us recall the evaluation of the double negation:

e(¬¬ϕ) =

{1 if e(ϕ) > 0 ,

0 if e(ϕ) = 0 .

Now we prove by induction that e(¬¬ϕ) = e(ϕ′). If ϕ is a propositional variable the claimis obvious.

a) If ϕ = ψ∧χ, then e(ϕ) = e(ψ)∧e(χ). We may write the following chain of equivalences:e(¬¬ϕ) = 1 iff e(ψ) > 0 and e(χ) > 0 iff e(¬¬ψ) = 1 and e(¬¬χ) = 1 iff e(ψ′) = 1 ande(χ′) = 1 iff e(ϕ′) = e(ψ′ ∧ χ′) = 1.

b) If ϕ = ψ → χ, we may write the following chain of equivalences: e(¬¬ϕ) = 0 iffe(ψ → χ) = 0 iff e(ψ) > 0 and e(χ) = 0 iff e(¬¬ψ) = 1 and e(¬¬χ) = 0 iff e(ψ′) = 1 ande(χ′) = 0 iff e(ϕ′) = e(ψ′ → χ′) = 0.

Now observe that a formula is satisfiable in classical logic iff it is K-satisfiable in G. Thefirst implication is obvious (since K ∈ C1). To prove the reverse implication, let us assumethat the formula ϕ is K-satisfiable. Then the formula ¬¬ϕ is K-satisfiable (since 0 /∈ K and1 ∈ K) and, by the fact proven above, the formula ϕ′ is K-satisfiable, as well. Thus there isan evaluation which evaluates propositional variables by either 0 or 1 and the formula ϕ by1. This is a classical evaluation satisfying the formula ϕ.

The above argument can be easily extended to a set of formulae and our claim is obtainedas an easy consequence. QED

We turn our attention to cardinalities bigger than ω. Observe that in the following wedo not assume the Continuum Hypothesis (however we assume the Axiom of Choice). SinceVAR is uncountable it contains a subset VAR′ of cardinality ω1. Let us enumerate theelements of VAR′ by ordinals ν < ω1.

Lemma 6.1.12 Let VAR be an uncountable set and K ∈ C.

1. G4 is not K-compactSat for K ∈ C1 and K = {0}.

2. G is not K-compactSat for {0} 6= K 6∈ C1.

Proof: First suppose that K ∈ C1. We define the set of formulae Γ = {¬4(vν → vµ) | µ <ν < ω1}. Observe that e(¬4(vν → vµ)) = 1 iff e(vν) > e(vµ), i.e., e K-satisfies ¬4(vν → vµ)iff e(vν) > e(vµ). To complete the proof just notice that Γ is finitely K-satisfiable (justevaluate variables occurring in that finite subset by properly ordered elements from [0, 1]) andnot K-satisfiable–this would mean e(VAR′) is a chain of type ω1, and since e(VAR′) ⊆ [0, 1]we have a contradiction (by simple set-theoretical arguments). The proof for K = {0} isanalogous, we would only work with formulae 4(vν → vµ) instead of ¬4(vν → vµ).

Now we prove the second claim: if K is not of type C1 and K 6= {0} there is an elementk ∈ K, 0 < k < 1. Again, we use the same idea as in the proof of the first claim. Let us fixa propositional variable k 6∈ VAR′ and define the set of formulae Γ = {(vν → vµ) ∨ k | µ <ν < ω1}.

Observe that e((vν → vµ) ∨ k) = 1 iff e(vν) ≤ e(vµ) or e(k) = 1. Again notice thatΓ is finitely K-satisfiable (just put e(k) = k and evaluate variables occurring in that finitesubset by properly ordered elements from [0, k]) and not K-satisfiable (this would mean thate(VAR′) is chain of type ω1, and since e(VAR′) ⊆ [0, 1] we have a contradiction). QED

Now we prove two lemmata concerning V -compactnessEnt. Both are interesting by thefact that they hold for an arbitrary infinite set VAR and so we can use them in the nextsubsections, where we will fix the cardinality of VAR to ω.

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6.1. THE NOTIONS OF COMPACTNESS IN GODEL LOGICS 103

Lemma 6.1.13 Let VAR be an infinite set and V ∈ Tv \FIN. If G logic is V -compactEnt,then it is (V \ {0, 1})-compactSat.

Proof: Let Γ be a set of formulae. We assume that each finite Γ′ ⊆ Γ is (V \{0, 1})-satisfiableand we want to prove that Γ is (V \ {0, 1})-satisfiable.

Since V is infinite, we can find V -evaluation eΓ′ which (V \ {0, 1})-satisfies Γ′ and thereis an element k ∈ V , such that sup(eΓ′(Γ′)) = k < 1. Let t be a variable not occurring inany formula of Γ. We define the set of formulae Γ′ = {¬¬ψ,ψ → t | ψ ∈ Γ′}. Let us take theV -evaluation eΓ′ resulting from the V -evaluation eΓ′ by setting eΓ′(t) = k. We notice thateΓ′(Γ′) = {1}. Since k < 1, we get Γ′ 6|=V t.

By the V -compactnessEnt (in the form: if Γ′ 6|=V ϕ for each finite Γ′ ⊆ Γ, then Γ 6|=V ϕ)we get {¬¬ψ,ψ → t | ψ ∈ Γ} 6|=V t, i.e., there is a V -evaluation e such that inf(e(Γ)) > 0and sup(e(Γ)) ≤ e(t) < 1, thus obviously e(Γ) ⊆ V \ {0, 1}. QED

Lemma 6.1.14 Let VAR be an arbitrary set. Then:

1. G and G4 are V -compactEnt for each V ∈ FIN ∩Tv.

2. G∼ is V -compactEnt for each V ∈ FIN ∩Tv∼.

Proof: We show the proof for G∼, the proofs for other logics are analogous. We will reducethe problem of V -compactEnt to the problem of compactness of the classical first-order logic(FOL). In the following text we assume that all the formulae of G∼ are written using thebasic connectives only and the connectives ¬, ⇔, and ⇒ are the connectives of the classicallogic.

The language LV , consists of the constants tvi for each i ∈ V ; two binary function symbols→ and ∧; one unary function symbol ∼; and one binary predicate ≤. The theory T0 consistsof the following sentences (we omit the quantifiers):

(1a) x ∧ y = y ∧ x,

(1b) x ∧ (y ∧ z) = (x ∧ y) ∧ z,

(1c) x ∧ x = x,

(1d) x ∧ y = x ⇔ x ≤ y,

(2a) ∼∼x = x,

(2b) x ≤ y ⇔ ∼y ≤ ∼x,

(3a) x → y = tv1 ⇔ x ≤ y,

(3b) ¬(x ≤ y) ⇒ x → y = y,

(4a) ¬(tvi = tvj) for each i 6= j,

(4b) tvi ≤ tvj for each i ≤ j,

(4c) tvi = ∼tv1−i for each i,

(4d)∨

i∈V

(x = tvi).

It is obvious that some function and predicate symbols and some axioms of T0 are redun-dant, however in this way we can very easily notice that all LV -models of T0 are finite andmutually isomorphic and the structure V = (V,∧,→,∼,≤, (i)i∈V ), where the operations arethose of G∼, is one of them (this is also an obvious consequence of the fact that a theory ofa finite structure is categorical).

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104 CHAPTER 6. COMPACTNESS IN FUZZY LOGICS

Let us extend the language LV to the language LVARV by adding a constant cv for each

v ∈ VAR. If ϕ is a formula of G∼ logic then the term ϕ# of the classical FOL resultsfrom ϕ, by replacing the connectives (including the truth constant 0) by the correspondingfunctional symbols of LVAR

V and replacing each propositional variable v by the constant cv.Analogously, if t is a term of LVAR

V , such that if tvi appears in t, then i = 0 or i = 1, we definethe formula t# in an obvious way. In the following text, when we work with an LVAR

V -modelwe assume that its LV -reduct is V.

Notice that for each LVARV -model N there is a V -evaluation eN defined as eN(v) =

cvN. Analogously, for each V -evaluation f we define an LVAR

V -model Mf as cvMf

= f(v).Obviously, MeN = N and eMf = f . We can easily prove that the defining relations can beextended to an arbitrary formula (term), i.e., for each LVAR

V -model N holds eN(t#) = tN,and for each V -evaluation f holds ϕ#

Mf= f(ϕ). In other words, we have established one-one

correspondence between LVARV -models and V -evaluations.

Now observe that for each set of formulae Γ and each formula ϕ the following holds:Γ 6|=V ϕ iff the first-order theory Γ# = {¬((ψ → ϕ)# = tv1) | ψ ∈ Γ} has an LVAR

V -model.Indeed, if Γ 6|=V ϕ, then e(ψ) > e(f) for each ψ ∈ Γ. Thus e(ψ → ϕ) 6= 1 and so we get(ψ → ϕ)#Me

6= tv1 and so Me |= ¬((ψ → ϕ)# = tv1) for each ψ ∈ Γ. Thus Me is anLVAR

V -model Γ#. The reverse direction is analogous (we again use the finiteness of V and byproving that eM(ψ) > eM(f) for each ψ ∈ Γ we conclude that inf eM(Γ) > eM(f)).

The compactness property of the classical FOL completes the proof. QED

Now we put everything together and obtain the main theorem of this subsection. It givesa complete answer to the problem of V -compactnessEnt and K-compactnessSat of all threelogics in the case of V AR not being an infinite countable set.

Theorem 6.1.15 Let K ∈ C, V ∈ Tv, and V ∈ Tv∼. For VAR being an uncountable setwe get:

1. Godel logic is K-compactSat iff K is of type C1 or K = {0},

2. Godel logic with 4 and Godel involutive logic are not K-compactSat,

3. Godel logic and Godel logic with 4 are V -compactEnt iff V ∈ FIN,

4. Godel logic with 4 and Godel involutive logic is V -compactEnt iff V ∈ FIN.

For VAR being a finite set we get:

5. All three Godel logics are K-compactSat,

6. Godel logic and Godel logic with 4 are V -compactEnt,

7. Godel involutive logic is V -compactEnt.

Proof:

1. For K being of type C1 we use Lemma 6.1.11. For K being {0} just observe that aformula ϕ is {0}-satisfiable iff the formula ¬ϕ is {1}-satisfiable. For other sets K thesecond part of Lemma 6.1.12 completes the proof.

2. For K ∈ C1 and K = {0} we use the first part of Lemma 6.1.12, for other sets we usethe previous claim and Remark 6.1.8.

3. For K ∈ FIN we use Lemma 6.1.14; for other sets K we use the claim (1) andLemma 6.1.13 to get that G is not V -compactEnt. The claim for G4 is just a conse-quence of Remark 6.1.8.

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6.1. THE NOTIONS OF COMPACTNESS IN GODEL LOGICS 105

4. Obvious.

To prove the remaining claims for the set VAR being finite just observe that in this casethere are only finitely many non-equivalent formulae. This well-known property of Godellogic can be easily extended to other logics. However, for the sake of completeness of thispaper we present the proof of this fact for G∼ (and thus for the other two as well) in thenext subsection after we develop certain formal techniques. QED

Based on the result of this subsection we consider our set of propositional variables beingan infinite countable set from now on.

6.1.3 K-compactnessSat

In this subsection we deal with K-compactnessSat of our logics. We develop a machinery ofthe so-called evaluation prototypes, which helps us understand the structure of an evaluation.We need to introduce the truth constant r for each rational r ∈ [0, 1]. We need these constantsjust as a technical tool to understand the structure of an evaluation, thus we do not add themto the language (i.e., we do not change the set of formulae), we only extend the definition ofan evaluation by the condition e(r) = r (thus from now on we understand the evaluation asa function VAR ∪ {r | r ∈ Q ∩ [0, 1]} → [0, 1], such that e(r) = r).

Definition 6.1.16 Let P = {0, p1, p2, . . . , pn, 1} be a finite set of type Tv∼. The sequenceof pairs ((0, <), (p1, <), (p2, <), . . . (pn, <), (1, =)) is called an evaluation P -prototype oforder 0. If the sequence S is an evaluation P -prototype of order l then the sequence S′ is anevaluation P -prototype of order l+1 iff S′ results from S by inserting (at arbitrary positions)pairs (vl+1, ∗) and (∼vl+1, ∗), where ∗ ∈ {<, =}.

If S is an evaluation P -prototype let Si denote its i-th element, S1i the first member of

the pair Si and S2i the second one. Finally, by |S| we denote the length of S.

Notice that S1i is either a propositional variable, its negation, or a truth constant; and

S2i is either < or =.

Definition 6.1.17 Let P ∈ Tv∼ ∩ FIN, e an evaluation and S an evaluation P -prototype.Then we say that e respects S if e(S1

i ) S2i e(S1

i+1) for each i < |S|.We say that S is acceptable if there is an evaluation, which respects S. We denote the set

of acceptable evaluation K-prototypes of order n by EPPn .

Let P = {0, 12 , 1}. The sequences ((0, <), (v, <), (1

2 , <), (∼v, <), (1, =)) and ((0, <), (v,=), (∼v, =), (1

2 , <), (1, =)) are examples of acceptable evaluation P -prototypes of order 1 (theevaluation e(v) = 1

4 respects the first one and the evaluation e(v) = 12 respects the second

one).The sequence ((0, <), (v, =), ( 1

2 , <), (∼v, =), (1,=)) is an example of an unacceptable one(using the definition for i = 2 we get e(v) = e(1

2 ) = 12 and for i = 4 we get e(∼v) = e(1) = 1, a

contradiction). Of course, we could derive the syntactical rules for an evaluation P -prototypeto be an acceptable one, but our “semantical” definition is quite simpler.

In the following text we omit the word acceptable and assume that we work with accept-able prototypes only. Observe that EPP

n is indeed finite, and EPP0 is a singleton, so let us

use the same denotation for this set and its element. Notice that if e respects S and i ≤ j,then e(S1

i ) ≤ e(S1j ).

Lemma 6.1.18 Let ϕ be a formula such that VAR(ϕ) ⊆ VARn, P ∈ Tv∼ ∩ FIN andS ∈ EPP

n . Then there is an index i such that e(ϕ) = e(S1i ) for each evaluation e respecting

S.

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106 CHAPTER 6. COMPACTNESS IN FUZZY LOGICS

Proof: We prove this by the induction on the complexity of the formula ϕ. Let v be apropositional variable, v ∈ VAR(ϕ), thus v occurs somewhere in S (since VAR(ϕ) ⊆ VARn),i.e., there is an index i such that v = S1

i .Let ϕ = ∼ψ. By the induction hypothesis there is an index i such that for each evaluation

e respecting S holds e(ψ) = e(S1i ). Since for each index i there is an index j such that

∼e(S1i ) = e(S1

j ) (from the construction of S) we get e(ϕ) = ∼e(S1i ) = e(S1

j ).Let ϕ = ψ ∧ χ. By the induction hypothesis there are indices i and j such that for each

evaluation e respecting S holds e(ψ) = e(S1i ) and e(χ) = e(S1

j ). Let us assume that i ≤ j(for j ≤ i the proof is analogous). We know that for each evaluation e respecting S holdse(ψ) = e(S1

i ) and e(χ) = e(S1j ). Since i ≤ j then e(S1

i ) ≤ e(S1j ) thus e(ϕ) = e(S1

i ).Let ϕ = ψ → χ. By the induction hypothesis there are indices i and j such that for each

evaluation e respecting S holds e(ψ) = e(S1i ) and e(χ) = e(S1

j ). Let us assume that i ≤ j,then e(S1

i ) ≤ e(S1j ). Thus e(ϕ) = 1 = e(S|S|). Now let us assume that i > j. There are two

possibilities: for each index k, such that j ≤ k < i holds S2k is =. Since e respects S we get

e(S1i ) = e(S1

j ) so e(ϕ) = 1. The second one is that there is an index k, j ≤ k < i, such thatS2

k is <. Since e respects S we get e(S1i ) > e(S1

j ) and so e(ϕ) = e(S1j ) QED

We show an example: we take P = {0, 12 , 1}, S = ((0, <), (v, <), (1

2 , <), (∼v,<), (1, =)),and ϕ = ∼v → v. Then obviously e(ϕ) = e(v) = e(S1

2) for each e respecting S. ForS = ((0, <), (v, =), (∼v, =), (1

2 , <), (1, =)) we obtain e(ϕ) = 1 = e(S15).

Corollary 6.1.19 Let P ∈ Tv∼ ∩ FIN, p ∈ P , S ∈ EPPn , e be an evaluation respecting S,

and ϕ a formula such that VAR(ϕ) ⊆ VARn.

1. If e(ϕ) = p, then f(ϕ) = p for each evaluation f respecting S.

2. If e(ϕ) < 1, then f(ϕ) < 1 for each evaluation f respecting S.

3. If e(ϕ) > 0, then f(ϕ) > 0 for each evaluation f respecting S.

Proposition 6.1.20 Let P ∈ Tv∼ ∩ FIN, n a natural number, and e an evaluation. Thenthere is a S ∈ EPP

n such that e respects S.

Before we prove the fundamental lemma of the paper we use the above results to give thepromised proof of the crucial fact used in the proof of the second part of Theorem 6.1.15.

Lemma 6.1.21 Let us define the set Formn as {ϕ | VAR(ϕ) ⊆ VARn} and the equivalence' on Formn as ϕ ' ψ iff e(ϕ) = e(ψ) for each evaluation e. Then the quotient of Formn

by ' is finite.

Proof: Let f be an arbitrary evaluation. Let us define the equivalence 'f on Formn asϕ 'f ψ iff f(ϕ) = f(ψ). Due to the latter lemma there is an evaluation prototype S ∈ EPn

such that f respects S. Using Lemma 6.1.18 we get for each formula ϕ an index iϕ such thatfor each evaluation e respecting S holds e(ϕ) = e(S1

iϕ).

This has two consequences. First: since the sequence S is finite and for each formula ϕholds f(ϕ) = f(S1

iϕ) we conclude that the quotient of Formn by 'f is finite. Second: if e

and f both respect S then 'e = 'f .To complete the proof just notice that'=

⋂e 'e =

⋂S∈EPn

'eS, where eS is an arbitrary

evaluation respecting S. Since we have expressed ' as finite intersection of equivalences forwhich the quotient of Formn is finite, the proof is done. QED

Now we present the crucial lemma of this paper. The proof is inspired by the proofof [3, Theorem 3.4] (especially the construction of the tree). However, that proof is forcompactnessEnt and for Godel logic only and cannot be generalized to the stronger logics (atleast not in a straightforward way). Our proof has two parts—one for sets of type D∼ andthe other for sets of type W. Whereas the first part is quite easy to follow, the second one if

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6.1. THE NOTIONS OF COMPACTNESS IN GODEL LOGICS 107

rather technical. For this reason we advise the reader to skip the parts related to the sets ofthe type W (i.e., the Case (2) in the introductory definitions and in Sublemma 6.1.23) by thefirst reading and concentrate to the tree construction, its pruning, and the construction ofrelevant evaluation in the last part. The second part illustrates the reasons for introductionof type W, the reader can notice that the proof (especially the proof of Sublemma 6.1.23)would not work for arbitrary set of type W.

Lemma 6.1.22 Let K ∈ C ∩ (D∼ ∪W). Then G∼ is K-compactSat.

Proof: We prove both cases (K ∈ C ∩ D∼ and K ∈ C ∩W) together. We only give adifferent definitions of two sets P and V .

Case (1): If K ∈ D∼, it contains a dense subsets A and (1 − A), we define set V =ClTv∼(A) = A ∪ (1−A) ∪ {0, 1} and set P as {0,1}. Notice that V ⊆ K ∪ {0, 1}.

Case (2): If K ∈ W, there is set A = {a1, . . . an} from the definition of W. Let us denoteKi = K ∩ [ai−1, ai]. Recall that Ki either finite (Ki ⊆ {ai−1, ai}) or of type D, in this casewe will denote its dense subset by Ai. We define the sets Vi:

Vi =

{[ai−1, ai] if both Ki and Kn+1−i are finite,

Ai ∪ {ai−1, ai} if Ki is of type D and Kn+1−i is finite.

Finally, we define sets V =⋃

Vi ∪ (1−⋃Vi) and P = A.

Notice that in both cases we have P ∈ Tv∼ ∩ FIN and V ∈ Tv∼ ∩ D∼. Now wecontinue our proof: we have a set of formulae Γ and assume that there are infinitely manypropositional variables in the formulae of Γ (otherwise the proof is trivial). We prove thateither Γ is K-satisfiable or there is finite subset which is not K-satisfiable. We enumeratethe formulae from Γ and then variables in the usual way (first we enumerate variables fromϕ1, then from ϕ2, and so on). Let o be a function which to each i assigns the number of thefirst formula in which vi+1 occurs (thus only variables from {v1, . . . , vi} occur in all previousformulae). We define set of formulae: Γi = {ϕ1, ϕ2 . . . , ϕo(i)−1}. Notice that Γ =

⋃Γi. (Γi

is an initial segment of the enumerating sequence, where the formulae have variables from{v1, . . . , vi})

We construct a finitary tree T . The nodes in the i-th level are labelled by the evaluationP -prototypes of order i. The root is labelled by EPP

0 . If the node n in the i-th level islabelled by Sn we add a successive node for each H ∈ EPP

i+1 resulting from Sn (in the senseof Definition 6.1.16) and label this node by H.

We construct a subtree T ′ by pruning the tree T . We cut the subtree in node n in leveli iff no evaluation e respecting Sn K-satisfies Γi (in other words the node n is the node oftree T ′ if there is an evaluation e respecting Sn which K-satisfies Γi). It is easy to show thatour three T ′ is finite or has an infinite branch (Konig’s lemma). Before we finish our proofwe prove one important sublemma.

Sublemma 6.1.23 If T is not cut in node n, then each V -evaluation e which respects Sn

K-satisfies Γi.

Proof: Case (1): K is of type C, so K is either of type C1 or C0 or C01. Let us supposethe last case (the other ones are analogous). If T is not cut in node n there is an evaluatione respecting Sn which K-satisfies Γi. Thus 0 < e(ψ) < 1 for each ψ ∈ Γi (because 0, 1 6∈ K).Let f be an arbitrary V -evaluation, we know that f(Γi) ⊆ V ⊆ K ∪ {0, 1}. Furthermore, iff respects Sn, then 0 < f(ψ) < 1 for each ψ ∈ Γi (due to Corollary 6.1.19 parts 2. and 3.).

Case (2): Again we assume that there is an evaluation e respecting Sn which K-satisfiesΓi. Let f be an arbitrary V -evaluation, we know that f(Γi) ⊆ V . We finish the proof bya contradiction, let us assume that there is a formula ψ ∈ Γi such that f(ψ) 6∈ K. UsingLemma 6.1.18 we know that there is index s such that f(ψ) = f(S1

s ). Let us find indicesk ≤ s ≤ l such that S1

k = aj−1 and S1l = aj , so we know f(ψ) ∈ [aj−1, aj ]. Since e also

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108 CHAPTER 6. COMPACTNESS IN FUZZY LOGICS

respects Sn and K-satisfies Γi then also e(ψ) = e(S1s ) ∈ K ∩ [aj−1, aj ] = Kj . We distinguish

two subcases:Subcase I: Both Kj and Kn+1−j are finite: then Kj ⊆ P so e(S1

s ) = a ∈ K ∩P . Then byCorollary 6.1.19 part 1. also f(S1

s ) = a ∈ K ∩ P—a contradiction.Subcase II: Kj ∈ D and Kn+1−j is finite: because f(ψ) 6∈ K we know that f(ψ) ∈ V \K

so f(ψ) ∈ (V ∩ [aj−1, aj ])\K and because V ∩ [aj−1, aj ] = Aj ∪{aj−1, aj} we get f(ψ) = a ∈{aj−1, aj} \K ⊆ P . Since also e(ψ) = a (by Corollary 6.1.19 part 1.) and e(ψ) = a ∈ K—acontradiction. QED

(The rest of the proof of Lemma 6.1.22) If the resulting tree T ′ contains an infinite branch,we construct a V -evaluation e, which K-satisfies Γ. Notice that if n is a node in level i, n′

its successor, and e a V -evaluation respecting Sn, then there is a V -evaluation e′ respectingSn′ such that e(vj) = e′(vj) for j ≤ i (this is due to the fact that V is of type D∼). Westart with the empty V -evaluation and follow our infinite branch, in each node we use theprevious observation. The resulting V -evaluation e respects Sn for each node n on our infinitebranch, thus V -evaluation e K-satisfies Γi for each i. Thus e K-satisfies Γ. Here we haveused Sublemma 6.1.23 which assures that if T is not cut in node n, then each (especially theone we have constructed) V -evaluation e respecting Sn K-satisfies Γi.

Now let us suppose that T ′ is finite and i is its depth. We show that Γi is not K-satisfiable.By contradiction: we suppose that there is an evaluation e K-satisfying Γi. There has to bea node n (of the tree T ) on the level i such that e respects Sn (cf. Proposition 6.1.20). Letus take the node n′ without successor in the tree T ′, such that there is a path from n′ to n(in the tree T ). Let us denote the level of n′ by i′. Notice that e K-satisfies Γi′ (becauseΓi′ ⊆ Γi). The evaluation e also respects Sn′ . Since the tree T is cut in node n′, no evaluationrespecting Sn′ K-satisfies Γi′—a contradiction. QED

The proof of this lemma gives us two useful corollaries:

Corollary 6.1.24 Let K be set of type C ∩D∼, Γ a set of formulae. If Γ is K-satisfiable,then Γ is K-satisfiable for some (K ∪ {0, 1})-evaluation.

Corollary 6.1.25 Let K ∈ C ∩D. Then G4 logic is K-compactSat.

Proof: We start with Remark 6.1.8 and the previous lemma to get that G4 is K-compactSat

for K being [0, 1), (0, 1), and (0, 1].Let K ∈ C0 (i.e., 0 ∈ K and 1 6∈ K), the cases K ∈ C1 and K ∈ C01 are analogous,

and let Γ be a set of formulae. We suppose that each finite subset of Γ is K-satisfiable.Since K ⊆ [0, 1), each finite subset of Γ is [0,1)-satisfiable and so Γ is [0,1)-satisfiable (bythe [0,1)-compactnessSat of G4). So there is an evaluation e such that e(Γ) ⊆ [0, 1). It iswell known that there is an order-preserving embedding f : e(VAR) ∪ {0, 1} → K ∪ {1}(because any countable order can be embedded into the dense countable order), thus f ◦ e isan evaluation. Finally notice that (f ◦ e)(Γ) ⊆ K (obviously (f ◦ e)(Γ) ⊆ K ∪ {1}, but if forsome ϕ ∈ Γ is (f ◦ e)(ϕ) = 1 then e(ϕ) = 1—a contradiction). QED

Lemma 6.1.26 Let K ∈ C \ FIN.

1. If K 6∈ C1 ∪D then Godel logic is not K-compactSat.

2. If K ∈ C1 \D then G4 is not K-compactSat.

3. If K 6∈ D∼ ∪W, then G∼ is not K-compactSat.

Proof: This proof is a modification of the proof of [28, Theorem 7.4]. We define a ternaryconnective sh(ϕ,ψ, δ) = ψ ∨ (ψ → ϕ) ∨ (δ → ψ). Observe:

e(sh(ϕ,ψ, δ)) =

{e(ψ) if e(ϕ) < e(ψ) < e(δ) ,

1 otherwise .

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6.1. THE NOTIONS OF COMPACTNESS IN GODEL LOGICS 109

The value of a formula sh(ϕ,ψ, δ) is either the value of its second argument (if the values ofarguments are strictly increasingly ordered) or 1 (otherwise).

Let us take a collection (va,b)a>0, b>1 of mutually distinct propositional variables, anddefine v0,1 = 0 and v1,1 = 1. We define the initial finite sequence V 1 = (v0,1, v1,2, v1,1). Foreach i, we define V i+1 as the sequence which results from sequence V i by inserting elementva+a′,b+b′ between each pair of its subsequent elements va,b and va′,b′ (in fact, V i becomesthe ith Farey partition if va,b is understood as the rational a

b ).We define the initial set Γ1 = {sh(v0,1, v1,2, v1,1)}. For each i, set Γi+1 is defined as

Γi+1 = {sh(va,b, va+a′,b+b′ , va′,b′) : va′,b′ is the successor of va,b in V i} .

Finally, we define Γ =⋃

i Γi.Observe that the K-satisfiability of the set Γi is equivalent to the existence of an evaluation

e which maps the elements of V i onto a strictly increasing sequence in K ∪ {0,1} (becauseK 6∈ C1 gives us 1 6∈ K). The same argument works for

⋃j≤i Γj instead of Γi.

Since K is an infinite set, it contains chains of any finite length. Thus each finite subsetof Γ is K-satisfiable.

Let us assume that Γ is K-satisfiable. This means that there is an evaluation e such thatfor each i the values of elements of V i form a strictly increasing sequence in K ∪{0,1}. Dueto the inductive construction of V i, the values of all elements of

⋃i V i form a subset of [0, 1],

which is dense—a contradiction.The proof of the second part can be found in [28, Theorem 7.4]—roughly speaking we

change the definition of sh: sh(ϕ, ψ, δ) = ψ ∧ ¬4(ψ → ϕ) ∧ ¬4(δ → ψ). So we get

e(sh(ϕ, ψ, δ)) =

{e(ψ) if e(ϕ) < e(ψ) < e(δ) ,

0 otherwise .

The proof of the last part is based on the previous parts of this proof. The set K 6∈ W, thiscan be caused by two possibilities:

The set C = {x ∈ K | 1 − x ∈ K} ∪ {0, 12 , 1} is infinite. In this case we define the

set of formulae Γ′ = Γ ∪ {∼ϕ | ϕ ∈ Γ} (where Γ is the set of formulae from the first partof this proof). This theory is not K-satisfiable, because K 6∈ D∼. However, it is finitelyK-satisfiable, because we can find an arbitrarily long increasing sequence (a)i together withsequence (1− a)i in K.

If the set C = {x ∈ K | 1− x ∈ K} ∪ {0, 12 , 1} = {c0, . . . cm} is finite, there has to be an

index l such that K ∩ [bl−1, bl] is infinite but not of type D (where B = C ∪ ClTv∼(⋃{K ∩

[ci−1, ci] | K ∩ [ci−1, ci] is finite}) = {b0, . . . bn}). To fulfill our goal, let us suppose thatK ∈ C0, the case of K ∈ C1 is analogous. In the following we will use the same symbols forthe propositional variables and the elements of K and sh for the connective defined in thefirst part of this proof.

We define the set of formulae

∆0 = {¬4(ci ≡ ∼cm−i)∨cm−i | i ≤ m}∪{sh(0, c1, c2), sh(c1, c2, c3), . . . sh(cm−2, cm−1,1)}.It is easy to observe that ∆0 is K-satisfied by e iff e(ci) = ci. Now if the set K ∩ [ci−1, ci] ={ki

0, . . . kin} is finite, we define ∆i = {sh(ki

j−1, kij , k

ij+1) | 0 < j < n}, otherwise we take ∆i

as an empty set. Again ∆i is K-satisfied by e iff e(kij) = ki

j for each j. Finally, let us defineΓ0 =

⋃ni≥0 ∆i. Now we notice that bl is either ki

j or 1 − kij or ci for some indices i and j

and the same holds for bl−1. Let us denote the corresponding propositional variables p andq. Thus we know that each evaluation e K-satisfying Γ0 holds e(p) = bl−1 and e(g) = bl.

Now we define the set of formulae Γ as in the proof for Godel logic, with the only changein the initial step: we set v0,1 = p and v1,1 = q. Finally we define Γ′ = Γ ∪ Γ0 and observethat Γ′ is finitely K-satisfiable (because in K ∩ [bl−1, bl] we have infinitely many elements)but not K-satisfiable (because it would mean that there is a dense subset in K ∩ [bl−1, bl]—acontradiction). QED

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110 CHAPTER 6. COMPACTNESS IN FUZZY LOGICS

Now we formulate the main theorem of this subsection by putting all our results together.We obtain the complete answer of the problem of K-compactnessSat in G and G4. However,in the case of G∼ we have only proved K-compactnessSat for K ∈ W and disproved it forK 6∈ W, the problem seems to be open for sets K ∈ W \W.

Theorem 6.1.27 Let K ∈ C.

1. Godel logic is K-compactSat iff K ∈ FIN ∪C1 ∪D.

2. Godel logic with 4 logic is K-compactSat iff K ∈ FIN ∪D.

3. Godel involutive logic is K-compactSat if K ∈ D∼ ∪W.

4. Godel involutive logic is not K-compactSat if K 6∈ D∼ ∪W.

Proof: We prove the claims of this theorem in the opposite order.

4. G∼ is not K-compactSat if K 6∈ D∼ ∪W (cf. Lemma 6.1.26).

3. G∼ is K-compactSat if K ∈ D∼ ∪W (cf, Lemma 6.1.22).

2. We know that G4 logic is K-compactSat for K ∈ D (cf. Corollary 6.1.25) and forK ∈ FIN (because FIN ⊆ W we get that G∼ logic is K-compactSat; Remark 6.1.8completes the proof). Furthermore, we know that it is not K-compactSat for K 6∈ D(if also K ∈ C1 we use the second part of Lemma 6.1.26, if K 6∈ C1 we use the firstpart of Lemma 6.1.26 and Remark 6.1.8).

1. We know that G is K-compactSat for K ∈ D ∪ FIN (because of part (3) of thistheorem and Remark 6.1.8) and for K ∈ C1 (cf. Lemma 6.1.11). Finally, G logic is notK-compactSat for K 6∈ (C1 ∪D) (cf. Lemma 6.1.26). QED

6.1.4 V -compactnessEnt

In this subsection we deal with V -compactnessEnt. We use our results from the previoussubsection to prove new results. We also obtain known results of Baaz and Zach as a corollaryof our new results. The proofs in this subsection are based on the facts that for certain sets weare able to reduce the problem of V -compactnessEnt to the problem of V \{0}-compactnessSat

(or prove different but similar reduction theorems).

Lemma 6.1.28 Let V ∈ Tv∼ ∩ D∼. Then G∼ logic is V -compactEnt. Furthermore, ifV ∈ Tv ∩D, then G and G4 are V -compactEnt.

Proof: We show the proof for G∼, the proofs for other logics are analogous. We want toprove that for set of formulae Γ: if Γ |=V ϕ, then there is finite Γ′ ⊆ Γ such that Γ′ |=V ϕ.Let us prove it indirectly:

For each finite Γ′ ⊆ Γ: Γ′ 6|=V ϕ, i.e., there is a V -evaluation such that inf(e(Γ′)) > e(ϕ).Let us take a variable t not occurring in Γ∪ {ϕ} and alter our K-evaluation e in such a waythat: inf(e(Γ′)) > e(t) > e(ϕ). Let us define Γ′ = {ψ → t | ψ ∈ Γ′}∪{t → ϕ}. Observe that Γ′is (V \{1})-satisfiable (by evaluation e). Since V \{1} ∈ C∩D∼, G∼ is (V \{1})-compactSat.Thus we know that Γ = {ψ → t | ψ ∈ Γ} ∪ {t → ϕ} is (V \ {1})-satisfiable by someV -evaluation e (cf. Corollary 6.1.24). Thus for each ψ ∈ Γ holds e(ψ) > e(t) > e(ϕ), thusinf(e(Γ)) ≥ e(t) > e(ϕ) and the proof is done. QED

Theorem 6.1.29

1. Godel logic and Godel logic with 4 are V -compactEnt iff V ∈ Tv ∩ (FIN ∪D).

2. Godel involutive logic is V -compactEnt iff V ∈ Tv∼ ∩ (FIN ∪D).

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6.2. COMPACTNESS IN OTHER CORE FUZZY LOGICS 111

Proof: Since V -compactnessEnt for G and G4 logics is defined for sets of type Tv and theV -compactnessEnt for G∼ logic is defined for sets of type Tv∼, we will assume that all setswe encounter in this proof are of the appropriate type.

1. We know that G and G4 logics are V -compactEnt for V ∈ FIN (cf. Lemma 6.1.14)and for sets V ∈ D (cf. Lemma 6.1.28). The fact that G is not V -compactEnt forV 6∈ FIN ∪ D follows from Theorem 6.1.27 and Lemma 6.1.13. Finally, G4 is notV -compactEnt for V 6∈ FIN ∪D due to Remark 6.1.8 and the same claim for G.

2. The proof for G∼ is completely analogous to the proof of the previous claim (just recallthat Tv∼ ∩D = Tv∼ ∩D∼). QED

6.1.5 Correspondence between two notions of compactness

Notice the open problem in Godel involutive logic where for the sets V ∈ W \ W thequestion of V -compactnessSat for G∼ seems to be open. However, in the definition of thecompactnessEnt for G∼ logic, we work with sets of type Tv∼ only and Tv∼ ∩ (W \W) = ∅.This observation allows us to fulfill our goal and establish the connection between bothnotions of compactness despite of our open problem.

The proof of the following theorem is just a corollary of the main theorems of the previoustwo subsections (namely Theorems 6.1.27 and 6.1.29).

Theorem 6.1.30 Let VAR be an infinite countable set. If V ∈ Tv, then the following areequivalent:

1. V ∈ FIN ∪D,

2. Godel logic is V -compactEnt,

3. Godel logic is (V \ {1})-compactSat,

4. Godel logic is (V \ {0, 1})-compactSat.

If V ∈ Tv (V ∈ Tv∼), then the following are equivalent:

1. V ∈ FIN ∪D

2. Godel logic with 4 (Godel involutive logic respectively) logic is V -compactEnt,

3. Godel logic with 4 (Godel involutive logic respectively) logic is (V \ {1})-compactSat,

4. Godel logic with 4 (Godel involutive logic respectively) logic is (V \{0, 1})-compactSat,

5. Godel logic with 4 (Godel involutive logic respectively) logic is (V \ {0})-compactSat.

6.2 Compactness in other core fuzzy logics

In this section we extend the notion of satisfiability and compactnessSat (Definitions 6.1.6and 6.1.7) to other core fuzzy logics. Since we work with satisfiability based compactnessonly we omit the index Sat.

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112 CHAPTER 6. COMPACTNESS IN FUZZY LOGICS

6.2.1 Satisfiability based compactness

Before we start we recall some definition from the previous section and generalized them toall core fuzzy logics. We also define two more types of subsets of [0,1]. By the term compactin the following definition me mean compact in the standard topology of [0,1].

Definition 6.2.1 We define:

COM = {K ⊆ C | K is compact},

Q = {K ⊆ [0, 1] | K ∩Q = [0, 1] ∩Q}.

Definition 6.2.2 For a set Γ of formulae in some core fuzzy logic and K ⊆ [0, 1], we saythat Γ is K-satisfiable if there exists an evaluation e such that e(ϕ) ∈ K for all ϕ ∈ Γ. Theset Γ is said to be finitely K-satisfiable if each finite subset of Γ is K-satisfiable. Formula ϕis called K-satisfiable if the set {ϕ} is K-satisfiable.

If we recall the standard semantics of the core fuzzy logics we notice that with exceptionof �Π 1

2 if K ∈ C, then for any formula ϕ there is an evaluation e such that e(ϕ) ∈ K(it suffices to evaluate all propositional variables by 0, then the evaluation of each formulabecomes either 0 or 1). Thus as in Godel logics we restrict ourselves to the set of type C.

Definition 6.2.3 Let K ∈ C. We say that logic is K-compact if K-satisfiability is equivalentto finite K-satisfiability.

We observe that Remark 6.1.8 remains valid in this more general setting.

Remark 6.2.4 If we extend the set of connectives of a logic, L, then the resulting logic, L′,becomes “less compact”, i.e., if L′ is K-compact (resp. has the compactness property) then Lis K-compact (resp. has the compactness property). This is obvious since each counterexampleto K-compactness in L is also a counterexample to K-compactness in L′.

6.2.2 ÃLukasiewicz logic and Product ÃLukasiewicz logic

As an introduction to further results, we give a short proof of compactness that works forÃLukasiewicz logic, as well as for some other fuzzy logics and for classical logic. See [86] forthe first application of this kind of proof in fuzzy logics (in rather restricted setting).

Theorem 6.2.5 Let VAR be of arbitrary cardinality and K ∈ COM. Then ÃL and P�′ logicsare K-compact.

Proof: Let Γ be a set of formulae. For each ϕ ∈ Γ, the mapping Hϕ : [0, 1]A → [0, 1] definedby Hϕ(e) = e(ϕ) is continuous (because all the standard interpretation of connectives ofthese logics are continuous). The preimages (H−1

ϕ (K))ϕ∈Γ are closed subsets of the set [0, 1]A

which is compact in the product (=weak) topology (Tichonoff theorem). Due to finite K-sat-isfiability, the collection (H−1

ϕ (K))ϕ∈Γ is centered (i.e., each finite subset has a nonemptyintersection). Hence the intersection

⋂ϕ∈Γ H−1

ϕ (K) is nonempty; each of its elements is anevaluation which verifies that Γ is K-satisfiable. QED

It was necessary to have the set K closed (compact); K-compactness does not hold inÃLukasiewicz logic (and thus in P�′ as well) if K is not closed. Let us prove this claim:

Lemma 6.2.6 Let a, b ∈ [0, 1]. Then there is a formula ψa,b with one atom v in ÃLukasiewiczlogic such that:

e(v) < a =⇒ e(ψa,b) = 0 ,

e(v) > b =⇒ e(ψa,b) = 1 .

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6.2. COMPACTNESS IN OTHER CORE FUZZY LOGICS 113

Proof: We use McNaughton theorem (Theorem 3.3.6). We can find rationals c, d ∈ (a, b),c < d, and a McNaughton function f such that

f(x) =

0 if x ≤ c ,x−cd−c if c ≤ x ≤ d ,

1 if x ≥ d .

Let ψa,b be the formula which is interpreted by function f . QED

Notice that the formula ψa,b in the latter lemma is not unique.

Corollary 6.2.7 Let a, a′, b′, b ∈ [0, 1], a < a′ < b′ < b. Then there is a formula ϕ with oneatom v in ÃLukasiewicz logic such that:

e(ϕ) =

0 if e(v) ≤ a ,

e(v) if a′ ≤ e(v) ≤ b′ ,

0 if e(v) ≥ b .

Proof: We use the derived connective ∧M

of the ÃLukasiewicz logic defined by α ∧M

β =

α&(α → β). Its interpretation is e(α ∧M

β) = min(e(α), e(β)). We take ϕ = v ∧M

ψa,a′ ∧M¬ψb′,b,

where ψa,a′ , ψb′,b are obtained from Lemma 6.2.6 (with v as the only atom). QED

Using these lemmata we can give full characterization of the K-compactness in ÃLukasie-wicz and P�′ logics.

Theorem 6.2.8 Let VAR be of arbitrary cardinality and K ∈ C. Then ÃL and P�′ logicsare K-compact iff K ∈ COM.

Proof: One direction is Theorem 6.2.5. We show the reverse direction for ÃLukasiewicz logicand we get it for P�′ using Remark 6.2.4.

As K is not closed, there is a sequence of points in K converging to b /∈ K. From thissequence, we select a monotonic subsequence, say (ai)i∈N. Let us assume that this sequenceis strictly increasing. We assume that b 6= 1. (The proof for b = 1 is just a simpler versionof the following procedure – just omit the term ¬ψb′,b in the proof of Corollary. 6.2.7 andmake the corresponding changes in the rest). We take a strictly decreasing sequence (bi)i∈Nconverging to b (here we do not care of its intersection with K).

First, let us assume that K is of type C0.We fix an propositional variable, v. Using Corollary. 6.2.7, we construct formula ϕi

(having v as its only propositional variable) such that:

e(ϕi) =

0 if e(v) ≤ ai ,

e(v) if ai+1 ≤ e(v) ≤ bi+1 ,

0 if e(v) ≥ bi .

We take Γ = {v} ∪ {ϕi : i ∈ N}. Then Γ is finitely K-satisfiable. Indeed, for a finite subsetΦ ⊆ Γ, we take the maximal index j such that ϕj ∈ Φ, and the evaluation e(v) = aj+1 mapsall elements of Φ onto aj+1 ∈ K.

Suppose that Γ is K-satisfiable. Notice that if ϕi is K-satisfiable, then e(v) ∈ (ai, bi)(otherwise e(ϕi) = 0 /∈ K). Thus the satisfiability of Γ implies that the evaluation of v mustbe in the intersection of all intervals (ai, bi), i ∈ N. This intersection contains only b whichis not in K—a contradiction. We have proven that Γ is not K-satisfiable.

The proofs for the sequence (ai)i∈N strictly decreasing or for K being of type C1 areanalogous. QED

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Notice that the proof of Th. 6.2.5 works also in classical logic; obviously, every function ona finite product of discrete spaces {0, 1} is continuous and, due to Tichonoff theorem, {0, 1}A

is compact. The compactness property can be proven similarly also in other fuzzy logicsin which all operations are interpreted by continuous functions. In particular, this appliesto the so-called S-fuzzy logics. These logic are not fuzzy in the sense of this work (theyare not even weakly implicative). These logics were introduced in [86] and investigated indetail in [71, 61, 56] S-fuzzy logics are given semantically, the basic connectives are negation¬, interpreted by the standard fuzzy negation NS, and conjunction ∧, interpreted by acontinuous triangular norm. Implication → in an S-fuzzy logic is a derived connective

ϕ → ψ = ¬(ϕ ∧ ¬ψ) .

The compactness property of S-fuzzy logics has been proved in [86, Th. 3.3].

6.2.3 Product logic

We start with observation that the proof of the Lemma 6.1.11 remains valid for Product logic(in fact it is valid for arbitrary fuzzy logic with strict negation).

Lemma 6.2.9 Let the set VAR be of an arbitrary cardinality and K ∈ C1 or K = {0}.Then Product logic is K-compact.

Proof: For K being of type C1 we use Lemma 6.1.11. For K being {0} just observe thatformula ϕ is {0}-satisfiable iff formula ¬ϕ is {1}-satisfiable. QED

To prove the reverse direction we need one additional assumption.

Theorem 6.2.10 Let the set VAR contain at least two elements and k ∈ C. Then productlogic is K-compact iff K ∈ C1 or K = {0}.

Proof: We prove the reverse direction contrapositively: if K is not of type C1 and K 6= {0}there is an element k ∈ K, 0 < k < 1. Let us define the set of formulae Γ = {(r → pn) →p | 0 < n} ∪ {¬r ∨ p}, where p and r are propositional variables.

We show that Γ is finitely K-satisfiable. Let Γ′ be a finite subset of Γ and let m be thebiggest index occurring in formulae of Γ′. Let us evaluate e(p) = k and e(r) = km. Obviouslye(Γ′) = {k}.

To complete the proof we show that is not Γ is not K-satisfiable. Let e(p) = x ande(r) = y. If y = 0, then e(¬r ∨ p) = 1 i.e., Γ is not K-satisfiable. Let us assume that y > 0.Notice that both x = 1 and x = 0 leads to non K-satisfiability. Let us assume that 0 < x < 1,then there is index n such that xn−1 ≤ y. So xn

y ≤ x and so e((r → pn) → p) = 1. QED

6.2.4 Logics with 4In this subsection we examine the compactness property for logics with the set of connectivesextended by an additional unary connective 4 (0-1 projector or Baaz delta)

Lemma 6.2.11 Let the set VAR contain at least two elements and K ∈ C0 ∪C1. Then ÃL4and Π4 logics are not K-compact for any subset K of [0, 1] of type C01.

Proof: We show the proofs for both logic at once. Let us assume that K is of type C1 (i.e.1 ∈ K and 0 /∈ K). Let Γ = {¬4¬r, ¬4p} ∪ {ϕi : i ∈ N}, where p and r are propositionalvariables and ϕi = 4(r → pi). Notice that for each evaluation e we have

e(¬4¬r) =

{1 if e(r) > 0 ,

0 otherwise .

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6.2. COMPACTNESS IN OTHER CORE FUZZY LOGICS 115

(In case of product logic with 4, it suffices to take ¬¬r instead of ¬4¬r.) Let Γ′ be a finitesubset of Γ. Let j be the greatest index i for which ϕi ∈ Γ′. Define e(r) = 1

2 and e(p) < 1such that 1

2 ≤ e(pj) (such a value e(p) indeed exists). Then e(¬4¬r) = 1, e(¬4p) = 1 ande(ϕi) = 1 for all i ≤ j. Thus Γ′ is K-satisfiable and Γ is finitely K-satisfiable.

On the other side, suppose that there is an evaluation e making Γ K-satisfiable. Thene(¬4p) > 0 which is possible only for e(¬4p) = 1 and this implies e(p) < 1. Analogously,e(¬4¬r) > 0 implies e(¬4¬r) = 1 and e(r) > 0. Then there is an integer j such thate(pj) < e(r) and e(ϕj) = 0 /∈ K, a contradiction. Thus Γ is not K-satisfiable.

Now let us assume that K is of type C0 (i.e. 0 ∈ K and 1 /∈ K). The proof is analogous,the contradictory set of formulae is Γ′ = {4¬r, 4p} ∪ {ϕi : i ∈ N}, where p and r arepropositional variables and ϕi = ¬4(r → pi). QED

Lemma 6.2.12 Let the set VAR contain at least two elements and K ∈ C01. Then ÃL4 logicis not K-compact.

Proof: This is just analogy of the previous lemma. We know that 1 /∈ K and 0 /∈ K), thusthere is k ∈ K, such that 0 < k < 1 Let Γ = {r, 4p ∨ r} ∪ {ϕi : i ∈ N}, where p and r arepropositional variables and ϕi = 4(r → pi) ∧ r.

Let Γ′ be a finite subset of Γ. Let j be the greatest index i for which ϕi ∈ Γ′. Definee(r) = k and e(p) < 1 such that k ≤ e(pj) (such a value e(p) indeed exists). Then e(r) = k,e(4p ∨ r) = k and e(ϕi) = k for all i ≤ j. Thus Γ′ is K-satisfiable and Γ is finitelyK-satisfiable.

On the other side, suppose that there is an evaluation e making Γ K-satisfiable. Thene(4p ∨ r) 6= 1 which is possible only for e(4p) 6= 1 and this implies e(p) < 1. Analogously,we get e(r) > 0. Then there is an integer j such that e(pj) < e(r) and e(ϕj) = 0 /∈ K, acontradiction. Thus Γ is not K-satisfiable. QED

The following theorem is just a corollary of the previous two lemmata and Remark 6.2.4.

Theorem 6.2.13 Let the set VAR contain at least two elements and K ∈ C. Then ÃL4,Π4, and P�′4 logics are not K-compact.

6.2.5 �Π and �Π12

logics

Here we investigate the compactness of �Π and �Π12 propositional logics. Using Remark 6.2.4

and the fact that both these logics contain the ÃLukasiewicz logic with delta we immediatelyget:

Corollary 6.2.14 Let the set VAR contain at least two elements and K ∈ C. Then �Π and�Π 1

2 logics are not K-compact.

However, in �Π12 logic, we may define arbitrary rational constants from [0, 1]. This

means that �Π12 logic may lack K-compactness for sets which are not of type C, because

the argument about trivial K-satisfiability of each set Γ does not hold (take Γ containing asingle constant).

Theorem 6.2.15 Let the set VAR contain at least two elements. Then the �Π 12 logic is

K-compact iff K ∈ Q.

Proof: One direction is obvious (just take arbitrary evaluation which evaluates to rationalsonly and observe that evaluation of arbitrary formula is rational as well)

The second direction: the only unproved case is if 0, 1 ∈ K (otherwise K would be ofbe of type C and we could use the previous corollary). Assume that rational q 6∈ K. Thetruth constant q, corresponding to the rational q, can be defined within �Π 1

2 logic. We mayobserve that a formula ϕ is {1}−satisfiable iff formula ϕ′ = 4ϕ ∨ q is K-satisfiable. Thusour logic cannot be K-compact (otherwise it would be {1}-compact, but the set {1} is oftype C). QED

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116 CHAPTER 6. COMPACTNESS IN FUZZY LOGICS

Table 6.1: K-compactnessSat

Logic Finite Countable UncountableÃL, P�′ COM COM COMΠ C1 ∪ {{0}} C1 ∪ {{0}} C1 ∪ {{0}}G All FIN ∪C1 ∪D C1 ∪ {{0}}G4 All FIN ∪D NoneG∼ All ? NoneÃL4, Π4, P�′4, �Π None None None�Π 1

2 Q Q Q

Table 6.2: K-compactnessEnt

Logic Finite Countable UncountableG All FIN ∪D FING4 All FIN ∪D FING∼ All FIN ∪D∼ FIN

6.3 Conclusion

Table 6.1 summarizes our results about satisfiability based compactness from this chapter.The first column indicates the type of logic, the other ones show for which sets K the logicenjoys the K-compactness for particular cardinalities of the set VAR. With exception of G∼for countable set VAR all of them are full characterizations. Recall that case we know onlythat Godel involutive logic is K-compact if K ∈ D∼ ∪W and Godel involutive logic is notK-compact if K 6∈ D∼ ∪W.

Also recall that if set K is not of type C than all our logics with exception of �Π12

are K-compact. Thus all but the last rows of the table should by completed by ∪Tv (forsimplicity we omit it). Also notice that for the results for the logics ÃL4, Π4, P�′4, �Π, and�Π1

2 in case of finite set VAR we assume that VAR contains at least two elements.Table 6.2 summarizes our results about entailment based compactness from this chapter.

The first column indicates the type of logic, the other ones show for which sets K the logicenjoys the K-compactness for particular cardinalities of the set VAR. Recall that entailmentbased compactness is defined for set of type Tv (Tv∼ in the case of G∼). Thus all rows ofthis table should by completed by ∩Tv (∩Tv∼ respectively), but for the sake of simplicitywe omit it.

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Chapter 7

Fuzzy class theory

In this chapter we give a formal grounding of the upcoming project of formalizing and ax-iomatizing of the fuzzy mathematics (as announced in [5], see also Chapter 8).

Although fuzzy mathematics is nowadays very broad, the notion of fuzzy set is still acentral concept. The obvious idea is to formalize fuzzy sets using already formalized fuzzylogic in analogy with classical set theory. Previous attempts to formalize this notion haveseveral drawbacks from the point of view of their suitability for the formalization of Zadeh’snotion of fuzzy set. The papers [54] and [80] are mainly interested by metamathematicalproperties of fuzzified Zermelo-Fraenkel set theory and the papers [82] and [4] restrict them-selves to logics given by one particular t-norm and so they lack a sufficient expressive powerto capture the general notion of fuzzy set. Inspecting these (and a few other) approaches wecame to two conclusions: first, we do not need a full-fledged set theory and second, we needan expressively rich fuzzy logic as a logical background.

By a full-fledged set theory we mean an analogy of classical set theory with all its con-cepts. We observed that real-world applications of fuzzy sets need only small portion ofset-theoretical concepts. The central notion in fuzzy sets is the membership of an elementof the universe into a fuzzy set (not the membership of a fuzzy set into another fuzzy set).Formalization of such a concept in the classical setting is called elementary set theory, or classtheory. Roughly speaking, class theory is a theory with two sorts of individuals—objects andclasses—and one binary predicate—the membership of objects into classes. In this chapterwe develop a fuzzy class theory. The classes in our theory correspond exactly to Zadeh’sfuzzy sets.

By an expressively rich logic we mean a logic of great expressive power, yet with a simpleaxiomatic system and good logical properties (deduction theorem, Skolem function introduc-tion and eliminability, etc.). ÃLΠ∀ seems to be the most suitable logic for our needs. In thischapter we developed fuzzy class theory over the predicate logic ÃLΠ, however if we examinethe definitions and theorems we notice that nearly all of them will work in other fuzzy log-ics as well. We think that fixing the underlying logic will make important class-theoreticalconcepts clearer. Fuzzy class theory for a wider class of fuzzy logics can be a topic of someupcoming paper.

We show that the proposed theory is a simple, yet powerful formalism for working withelementary relations and operations on fuzzy sets (normality, equality, subsethood, union,intersection, kernel, support, etc.). By a small enhancement of our theory (adding tools tomanage tuples of objects) we obtain a formalism powerful enough to capture the notion offuzzy relation. Thus we can formally introduce the notions of T -transitivity, T -similarity,fuzzy ordering, and many other concepts defined in the literature. Finally, we extend ourformalism to something which can be viewed as simple fuzzy type theory. Basically, weintroduce individuals for classes of classes, classes of classes of classes etc. This allows us toformalize other parts of fuzzy mathematics (e.g., fuzzy topology). Our theory thus aspiresto the status of foundations of fuzzy mathematics and a uniform formalism that can make

117

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118 CHAPTER 7. FUZZY CLASS THEORY

interaction of various disciplines of fuzzy mathematics possible.Of course, this chapter cannot cover all the topics mentioned above. For the majority of

them we only give the very basic definitions, and there is a lot of work to be done to show thatthe proposed formalism is suitable for them. We concentrate on the development of basicproperties of fuzzy sets. In this area our formalism proved itself worthy, as it allows us tostate several very general metatheorems that effectively reduce a wide range of theorems onfuzzy sets to fuzzy propositional calculus. This success is a promising sign for our formalismto be suitable for other parts of fuzzy mathematics as well.

As mentioned above, in this chapter we restrict ourselves to notions that can be definedwithout adding a structure (similarity, metrics, etc.) to the universe of objects. Nevertheless,our formalism possesses means for adding a structure to the universe (usually by fixing asuitable class which satisfies certain axioms), which is necessary for the development of moreadvanced parts of fuzzy set theory. Such extensions of our theory will be elaborated insubsequent papers, for some hints see Section 7.4.

All these ideas about formalization of fuzzy mathematics are summed up and elaboratedin much more details in Chapter 8. There we also deal with the relation of our approach andsome older attempts in this direction. Let us now just mention, that some of the roots of ourapproach (and some of our concepts, like graded properties of fuzzy relations) can be foundalready in Gotwalld’s book [43]. Another, rather surprising source of similar ideas is Hohle’spaper [58], where in Section 5 the author writes:

“It is the opinion of the author that from a mathematical viewpoint the impor-tant feature of fuzzy set theory is the replacement of the two-valued logic by amultiple-valued logic. [. . . I]t is now clear how we can find for every mathematicalnotion its ‘fuzzy counterpart’. Since every mathematical notion can be writtenas a formula in a formal language, we have only to internalize, i.e. to interpretthese expressions by the given multiple-valued logic.”

7.1 Class theory over ÃLΠ

7.1.1 Axioms

Fuzzy class theory FCT is a theory over ÃLΠ∀ with two sorts of variables: object variables, de-noted by lowercase letters x, y, . . ., and class variables, denoted by uppercase letters X, Y, . . .None of the sorts is subsumed by the other.

The only primitive symbol of FCT is the binary membership predicate ∈ between objectsand classes (i.e., the first argument must be an object and the second a class; class theorytakes into consideration neither the membership of classes in classes, nor of objects in objects).

The principal axioms of FCT are instances of the class comprehension scheme: for anyformula φ not containing X (it may, however, contain any other object or class parameters),

(∃X)4 (∀x) (x ∈ X ↔ φ(x))

is an axiom of FCT. The strange 4 is necessary for securing that the required class existsin the degree 1 (rather than being only approximated by classes satisfying the equivalencein degrees arbitrarily close to 1). The 4 is also necessary for the conservativeness of theintroduction of comprehension terms1 {x | φ(x)} with axioms

y ∈ {x | φ(x)} ↔ φ(y)

and their eliminability. In the standard recursive way one proves that φ in comprehensionterms may be allowed to contain other comprehension terms.

1I.e., the Skolem functions of comprehension axioms, see Theorem 3.5.14.

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7.1. CLASS THEORY OVER ÃLΠ 119

The consistency of FCT is proved by constructing a model. Let M be an arbitrary setand L be a complete linear ÃLΠ-algebra. The Zadeh model M over the universe M and thealgebra of truth-values L is constructed as follows:

The range of object variables is M , the range of class variables is the set of all functionsfrom M to L. For any evaluation e we define ‖x ∈ X‖LM,e as the value of the functione(X) on e(x). The value of the comprehension term {x | φ(x)} is defined as the functiontaking an object a to ‖φ(a)‖LM,e (in fact, the characteristic function of φ(x) where e fixesthe parameters). Then it is trivial that ‖y ∈ {x | φ(x)}‖LM,e = ‖φ(y)‖LM,e which proves thecomprehension axiom.

If L = [0,1], we call the described model standard.

Definition 7.1.1 Let M be a model and A a class in M. The characteristic function χx∈A

is denoted briefly by χA and also called the membership function of A. (Instead of χA(x) or‖x ∈ A‖ many papers use just Ax.)

It can be observed that the crisp formula (∀x)4(x ∈ X ↔ x ∈ Y ) expresses the identityof the membership functions of X and Y (as in all models ‖(∀x)4(x ∈ X ↔ x ∈ Y )‖ = 1 iffthe membership functions of X and Y are identical, otherwise 0). Since our intended notionof fuzzy class is extensional, i.e., that fuzzy classes are determined by their membershipfunctions, it is reasonable to require the axiom of extensionality which identifies classes withtheir membership functions:

(∀x)4(x ∈ X ↔ x ∈ Y ) → X = Y

(the converse implication follows from the axioms for identity). The consistency of this axiomis proved by its validity in Zadeh models.

The comprehension scheme of FCT still allows classical models, as the construction ofZadeh models works for the ÃLΠ-algebra {0,1}. Sometimes it may be desirable to excludeclassical models. This can be done either by taking ÃLΠ 1

2 instead of ÃLΠ as the underlying logic,or equivalently by adding two constants C, c and the axiom of fuzziness c ∈ C ↔ ¬�c ∈ Cwithout changing the underlying logic. In both cases there is a sentence with the value 1

2 inany model, and all rational truth constants are therefore definable. The consistency of thisextension follows from the fact that it holds in standard Zadeh models.

General models of FCT correspond in the obvious way to Henkin’s general models ofclassical second-order logic, while Zadeh models correspond to full second-order models.FCT with its axioms of comprehension and extensionality thus can be viewed as a notationalvariant of the second-order fuzzy logic ÃLΠ (monadic, in the form presented in this section; forhigher arities see Section 7.2). Second-order fuzzy logics will be more closely examined in [7].In this chapter we prefer FCT formulated as a two-sorted first-order theory, because it is closerto the usual formulation of classical set or class theory, and because of its axiomatizability.For even though (standard) Zadeh models are the intended models of FCT, the theory ofZadeh models is not arithmetically definable, let alone recursively axiomatizable. This followsfrom the obvious fact that classical full second-order logic (which itself is non-arithmetical)can be interpreted in the theory of Zadeh models by inscribing 4 (or ¬¬G) in front of everyatomic formula.

7.1.2 Elementary class operations

Elementary class operations are defined by means of propositional combination of atomicformulae of FCT.

Convention 7.1.2 Let φ(p1, . . . , pn) be a propositional formula and ψ1, . . . , ψn be any for-mulae. By φ(ψ1, . . . , ψn) we denote the formula φ in which all occurrences of pi are replacedby ψi (for all i ≤ n).

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120 CHAPTER 7. FUZZY CLASS THEORY

Table 7.1: Elementary class operations

φ Opφ(X1, . . . , Xn) Name0 ∅ empty class1 V universal class

4(α → p) Xα α-cut4(α ↔ p) X=α α-level

¬Gp \X strict complement¬�p −X involutive complement

¬G¬�p (or 4p) Ker(X) kernel¬¬Gp (or ¬4¬�p) Supp(X) support

p&∗q X ∩∗ Y ∗-intersectionp ∨ q X ∪ Y unionp⊕ q X ] Y strong union

p&∗¬Gq X \∗ Y strict ∗-differencep&∗¬�q X −∗ Y involutive ∗-difference

Definition 7.1.3 Let φ(p1, . . . , pn) be a propositional formula. We define the n-ary classoperation generated by φ as

Opφ(X1, . . . , Xn) =df {x | φ(x ∈ X1, . . . , x ∈ Xn)} .

Among elementary class operations we find the following important kinds:

• Class constants. We denote Op0 by ∅ and call it the empty class, and Op1 by V andcall it the universal class.

• α-Cuts. Let α be a truth-constant. Then we call the class Op4(α→p)(X), i.e., the class{x | 4(α → (x ∈ X))}, the α-cut of X and abbreviate it Xα. Similarly, Op4(α↔ p)(X)is called the α-level of X, denoted by X=α.

• Iterated complements, i.e., class operations Opφ where φ is p prefixed with a chain ofnegations. In ÃLΠ, there are only a few such formulae that are non-equivalent. Theyyield the following operations (their definitions are summarized in Table 7.1): involutiveand strict complements, the kernel and support, and the complement of the kernel.Except for the involutive complement, all of them are crisp.

• Simple binary operations. Some of the class operations Opp◦q where ◦ is a (primitive orderived) binary connective have their traditional names and notation, listed in Table 7.1(not exhaustively).

7.1.3 Elementary relations between classes

Most of important relations between classes have one of the two forms described in thefollowing definition:

Definition 7.1.4 Let φ(p1, . . . , pn) be a propositional formula. The n-ary uniform relationbetween X1, . . . , Xn generated by φ is defined as

Rel∀φ(X1, . . . , Xn) ≡df (∀x) φ(x ∈ X1, . . . , x ∈ Xn).

The n-ary supremal relation between X1, . . . , Xn generated by φ is defined as

Rel∃φ(X1, . . . , Xn) ≡df (∃x) φ(x ∈ X1, . . . , x ∈ Xn).

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7.1. CLASS THEORY OVER ÃLΠ 121

Table 7.2: Class properties and relations

Relation Notation NameRel∃p(X) Hgt(X) heightRel∃4p(X) Norm(X) normalityRel∀4(p∨¬p)(X) Crisp(X) crispnessRel∃¬4(p∨¬p)(X) Fuzzy(X) fuzzinessRel∀p→∗q(X,Y ) X ⊆∗ Y ∗-inclusionRel∀p↔∗ q(X,Y ) X =∗ Y ∗-equalityRel∃p&∗q(X, Y ) X‖∗Y ∗-compatibility

Among elementary class relations we find the following important kinds (they are sum-marized in Table 7.2):

• Equalities Rel∀p↔∗ q denoted =∗. The value of X =G Y is the maximal truth de-gree below which the membership functions of X and Y are identical. In standard[0,1]-models, 1−‖X = Y ‖ is the maximal difference of the (values of) the membershipfunctions of X and Y , and ‖X =Π Y ‖ is the infimum of their ratios. All =∗ get value 1iff the membership functions are identical. For crisp classes, these notions of equalitycoincide with classical equality.

• Inclusions Rel∀p→∗q, denoted ⊆∗. Their semantics is analogous to that of equalities.They get the value 1 iff the membership function of X is majorized by that of Y .

• Compatibilities Rel∃p&∗q. Their strict and involutive negations may respectively becalled strict and involutive ∗-disjointness.

• Unary properties of height, normality, fuzziness, and crispness.

Notice that due to the axiom of extensionality, the relation Rel∀4(p↔ q), which is obviouslyequivalent to 4(X =∗ Y ), coincides with the identity of classes. Thus it is 4(X =∗ Y ) thatguarantees intersubstitutivity salva veritate in all formulae (equalities generally do not).

It can be noticed that Godel equality =G is highly true only if the membership functionsare identical on low truth values; product equality =Π is also more restrictive on lower truthvalues. However, this does not conform with the intuition that the difference in the highvalues (on the “prototypes”) should matter more than a negligible difference on objects thatalmost do not belong to the classes under consideration. Equality of involutive complements,−X =∗−Y , is therefore a better measure of similarity of classes. Similarly, −Y ⊆∗−X maygive a better measure of containment of X in Y than X ⊆∗ Y .

7.1.4 Theorems on elementary class relations and operations

The following metatheorems show that a large part of elementary fuzzy set theory can bereduced to fuzzy propositional calculus.

Theorem 7.1.5 Let φ, ψ1, . . . , ψn be propositional formulae.Then ` φ(ψ1, . . . , ψn)

iff ` Rel∀φ(Opψ1(X1,1, . . . , X1,k1), . . . , Opψn

(Xn,1, . . . , Xn,kn)) (7.1)

iff ` Rel∃φ(Opψ1(X1,1, . . . , X1,k1), . . . , Opψn

(Xn,1, . . . , Xn,kn)) (7.2)

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122 CHAPTER 7. FUZZY CLASS THEORY

Proof: The substitution of the formulae x ∈ Xi,j for pi,j into ψi(pi,1, . . . , pi,ki) everywherein the (propositional) proof of φ(ψ1, . . . , ψn) transforms it into the proof of

φ(x ∈ Opψ1(X1,1, . . . , X1,k1), . . . , x ∈ Opψn

(Xn,1, . . . , Xn,kn)).

Then use generalization on x to get Rel∀φ and ∃-introduction to get Rel∃φ.Conversely, given an evaluation e that refutes φ(ψ1, . . . , ψn), we construct a Zadeh model

M refuting (7.1) and (7.2) by assigning to the class variables Xi,j the functions Ai,j suchthat Ai,j(a) = e(pi,j) for every a in the universe of M. Applying Theorems 3.3.21 and 3.5.3,the proof is done. QED

Corollary 7.1.6 Let φ and ψ be propositional formulae.If ` φ → ψ then ` Opφ(X1, . . . , Xn) ⊆ Opψ(X1, . . . , Xn).If ` φ ↔ ψ then ` Opφ(X1, . . . , Xn) = Opψ(X1, . . . , Xn).If ` φ ∨ ¬φ then ` Crisp(Opφ(X1, . . . , Xn)).

By virtue of Theorem 7.1.5, the properties of propositional connectives directly translateto the properties of class relations and operations. For example:

` 4p → p proves ` Ker(X) ⊆ X` p → p ∨ q ” ` X ⊆ X ∪ Y` 0 → p ” ` ∅ ⊆ X` p&q → p∧ q ” ` X ∩∗ Y ⊆ X ∩G Y` ¬Gp ∨ ¬¬Gp ” ` Crisp(\X)` 4(α → p) → 4(β → p) for α ≤ β ” ` Xα ⊆ Xβ for α ≤ β, etc.

In order to translate monotonicity and congruence properties of propositional connectivesto the same properties of class operations, we need another theorem:

Theorem 7.1.7 Let φi, φ′i, ψi,j , ψ

′i,j be propositional formulae. Then

`k

&∗i=1

φi(ψi,1, . . . , ψi,ni) →k′∧

i=1

φ′i(ψ′i,1, . . . , ψ

′i,n′i

) (7.3)

iff

`k

&∗i=1

Rel∀φi

(Opψi,1

( ~X), . . . , Opψi,ni( ~X)

)→

k′∧

i=1

Rel∀φ′i

(Opψ′i,1

( ~X), . . . , Opψ′i,n′

i

( ~X))

(7.4)

Proof: Without loss of generality, the principal implications of (7.3) and (7.4) can beassumed to be →∗. Replacing all propositional variables pj in the proof of (7.3) by theatomic formulae x ∈ Xj then yields the proof of

k

&∗i=1

φi

(Opψi,1

( ~X), . . . , Opψi,ni( ~X)

)→∗

k′∧

i=1

φ′i

(Opψ′i,1

( ~X), . . . , Opψ′i,n′

i

( ~X))

.

Generalization on x and distribution of ∀ over all conjuncts using (3.1), (3.5) and (3.3) ofLemma 3.5.10 proves (7.4). QED

Examples of direct corollaries of the theorem:

Provability in BL4 of Proves in FCT(p → q) → ((p&r) → (q&r)) X ⊆∗ Y → X ∩∗ Z ⊆∗ Y ∩∗ Z(p → q) → (p → (p∧ q)) X ⊆∗ Y → X ⊆∗X ∩G Y[(p → q)&(q → p)] → (p ↔ q) (X ⊆∗ Y &Y ⊆∗X) → X =∗ Y(p ↔ q) → [(p → q)∧(q → p)] X =∗ Y → (X ⊆∗ Y ∧ Y ⊆∗X)[(p → r)&(q → r)] → (p ∨ q → r) (X ⊆∗ Z&Y ⊆∗ Z) → X ∪ Y ⊆∗ Z4(p → q) → [4(α → p) → 4(α → q)] 4(X ⊆ Y ) → Xα ⊆ Yα

transitivity of →, ↔ transitivity of ⊆∗, =∗, etc.

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7.2. TUPLES OF OBJECTS 123

Similarly, � ` (¬p ↔ ¬q) ↔ (q ↔ p) proves −X ⊆�−Y ↔ Y ⊆� X, etc.To derive theorems about Rel∃, we slightly modify Theorem 7.1.7:

Theorem 7.1.8 Let φi, φ′i, ψi,j , ψ

′i,j be propositional formulae. Then

`k

&∗i=1

φi(ψi,1, . . . , ψi,ni) →k′∨

i=1

φ′i(ψ′i,1, . . . , ψ

′i,n′i

) (7.5)

iff

`k−1

&∗i=1

Rel∀φi

(Opψi,1

( ~X), . . . , Opψi,ni( ~X)

)&∗Rel∃φk

(Opψk,1

( ~X), . . . , Opψk,nk( ~X

)→

→k′∨

i=1

Rel∃φ′i

(Opψ′i,1

( ~X), . . . , Opψ′i,n′

i

( ~X))

(7.6)

Proof: Modify the proof of Theorem 7.1.7, using (3.6) of Lemma 3.5.10 instead of (3.5),and then (3.4) of the same Lemma to distribute ∃ over the disjuncts. QED

Examples of direct corollaries:

Provability in BL∆ of Proves in FCTp&(p → q) → q Hgt(X)&∗(X ⊆∗ Y ) → Hgt(Y )4(p ∨ q) → 4p ∨ 4q Norm(X ∪ Y ) → Norm(X) ∨ Norm(Y )(p → r)&(p&q) → (q&r) X ⊆∗ Z&∗X‖∗Y → Y ‖∗Z, etc.

7.2 Tuples of objects

In order to be able to deal with fuzzy relations, we will further assume that the languageof FCT contains an apparatus for forming tuples of objects and accessing their components.Such an extension can be achieved, e.g., by postulating variable sorts for any multiplicity oftuples (all of which are subsumed by the sort of objects), enriching the language with thefunctions for forming n-tuples of any combination of tuples and accessing its components, andadding axiom schemes expressing that tuples equal iff their respective constituents equal. Thedefinition of Zadeh model then must be adjusted by partitioning the range of object variablesand interpreting the tuples-handling functions. We omit elaborating this sort of syntacticsugar.

In what follows, instead of {z | (∃x1) . . . (∃xn) (z = 〈x1, . . . , xn〉&φ)} we write the formula{〈x1, . . . , xn〉 | φ}.

FCT equipped with tuples of objects contains common operations for dealing with rela-tions. We can define Cartesian products, domains, ranges and the relational operations asusual:2

X ×∗ Y =df {〈x, y〉 | x ∈ X&∗y ∈ Y }Dom(R) =df {x | 〈x, y〉 ∈ R}Rng(R) =df {y | 〈x, y〉 ∈ R}R ◦∗ S =df {〈x, y〉 | (∃z) (〈x, z〉 ∈ R&∗ 〈z, y〉 ∈ S)}

R−1 =df {〈x, y〉 | 〈y, x〉 ∈ R}Id =df {〈x, y〉 | x = y}

2Obviously for crisp arguments these operations yield crisp classes; X ×∗ Y is crisp iff both X and Y arecrisp. Unless X and Y are crisp, the property of being a relation from X to Y is double-indexed (a ∗′-subsetof the Cartesian product X ×∗ Y ). Also the definitions of usual properties (e.g., reflexivity, ∗-symmetry, etc.)of a relation on a non-crisp Cartesian product have to be defined with relativized quantifiers which bringanother index. It is doubtful that definitions combining various t-norms will have any real meaning. Thesituation is much easier if only relations on crisp classes are considered.

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124 CHAPTER 7. FUZZY CLASS THEORY

Table 7.3: Properties of relations

Notation Definition NameRefl(R) (∀x) (Rxx) reflexiveSym∗(R) (∀x, y) (Rxy →∗ Ryx) ∗-symmetricTrans∗(R) (∀x, y, z) (Rxy&∗Ryz →∗ Rxz) ∗-transitiveDich(R) (∀x, y) (Rxy ∨ Ryx) dichotomicQuord∗(R) Refl(R)&∗Trans∗(R) ∗-quasiorderingLinquord∗(R) Quord∗(R)&∗Dich(R) linear ∗-quasiorderingSim∗(R) Quord∗(R)&∗Sym∗(R) ∗-similarityEqu∗(R) Sim∗(R)&∗ (∀x, y) (4Rxy →∗ x = y) ∗-equality

The introduction of tuples of objects also allows an axiomatic investigation of various kindsof fuzzy relations (e.g., similarities) and fuzzy structures (fuzzy preorderings, graphs, etc.).We can define the usual properties of relations, as summarized in Table 7.3 (for brevity’ssake, we write just Rxy for 〈x, y〉 ∈ R).3

Classical definitions of some properties of relations (e.g., antisymmetry) make use of theidentity predicate on objects. One may be tempted to use the identity predicate = of ÃLΠ∀in the role of the classical identity in these definitions. However, since = is crisp, suchdefinitions do not yield useful and genuine fuzzy notions. A fuzzy analogue of the crispnotion of identity is that of similarity or equality (see Table 7.3). We can therefore definethese properties relative to a ∗-similarity or ∗-equality S. For details see the last sections.

In this way, the properties of being a ∗-antisymmetric relation, a ∗-ordering, a linear∗-ordering, a ∗-well-ordering, a ∗-function and a ∗-bijection (w.r.t. some fuzzy ∗-equality)can be introduced. By means of ∗-bijections, the notions of ∗-subvalence, ∗-equipotenceand ∗-finitude of classes (again w.r.t. some fuzzy ∗-equality) can be defined. A thoroughinvestigation of these notions, however, exceeds the scope of this chapter.

7.3 Higher types of classes

7.3.1 Second-level classes

Class theory does not contain an apparatus for dealing with families of classes. In many cases,a family of classes can be represented by a class of pairs or some other kind of ‘encoding’.For instance, a relation R may be understood as representing the family of classes Xi ={x | 〈i, x〉 ∈ R} for all i ∈ Dom(R).

In other cases, however, no suitable class of indices can be found and such an ‘encoding’is not possible. Then it is desirable to extend the apparatus of class theory by classes ofthe second level. This is done simply by repeating the same definitions one level higher. Weintroduce a new sort of variables for families of classes X,Y, . . . , a new membership predicatebetween classes and families of classes X ∈ X, and the comprehension scheme for families ofclasses

(∃X)4 (∀X) (X ∈ X ↔ φ(X))

for all formulae φ (where φ may contain any parameters except for X). The extensionalityaxiom for families of classes now reads

(∀X)4(X ∈ X ↔ X ∈ Y) → X = Y.

3Following the usual mathematical terminology, ∗-similarity may also be called ∗-equivalence; we respectthe established fuzzy set terminology here. Weak dichotomy (∀x, y) (Rxy ⊕ Ryx) could also be defined andweak versions of the properties that contain dichotomy, e.g. weakly linear ∗-ordering.

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7.3. HIGHER TYPES OF CLASSES 125

Again it is possible to introduce second-level comprehension terms {X | φ(X)}, which intro-duction is conservative and eliminable by Theorem 3.5.14.

The consistency of this extension is proved by a construction of second-level Zadeh modelsover a linear ÃLΠ-algebra L, in which the object variables range over a universe U , the classvariables over the set LU of all functions from U to L, and the second-level class variablesrange over the set LLU

of all functions from LU to L. The second-level class {X | φ(X)} isagain identified with the characteristic function of φ as in Section 7.1.1. Obviously, this con-struction makes both the second-level comprehension scheme and the axiom of extensionalitysatisfied in the model; the theory of second-level classes can thus be viewed as third-orderfuzzy logic (we omit details).

All definitions of elementary class relations and operations and all theorems can directlybe transferred from classes to second-level classes. Refining the language, axioms and Zadehmodels to tuples of classes is also straightforward.

It may be observed that the class operations and relations Opφ, Rel∀φ, and Rel∃φ, whichwere introduced in Sections 7.1.2 and 7.1.3 as defined functors and predicates, are nowindividuals of the theory, viz. second-level classes.

7.3.2 Simple fuzzy type theory

If there be need for families of families of classes, it is straightforward to repeat the wholeconstruction once again to get third-level classes. By iterating this process, we get a simpletype theory over ÃLΠ, for which the class theory described in Sections 7.1–7.2 is the inductionstep. The comprehension schemes and Zadeh models can easily be generalized to allowmembership of elements of any type less then n in classes of the n-th level.4

A type theory over a particular fuzzy logic (viz. IMTL4, extended also to ÃL4) hasalready been proposed by V. Novak in [74]. As mentioned in the Introduction, our theorycan be built over various fuzzy logics with 4; its variant over IMTL4 and Novak’s typetheory seem to be equivalent (though radically different in notation, as Novak uses λ-terms).

Since almost all classical applied mathematics can be formalized within the first few levelsof simple type theory, the formalism just described should be sufficient for all applicationsof fuzzy sets based on t-norms or other functions definable in ÃLΠ (see Theorem 3.3.25). Toillustrate this, we show the formalization of Zadeh’s extension principle.

Definition 7.3.1 A (fuzzy) binary relation5 R between objects is extended by Zadeh’s prin-ciple (based on a t-norm ∗) to a relation R∗ between (fuzzy) classes as follows:

R∗(X, Y ) ≡df (∃x, y) (Rxy&∗x ∈ X&∗x ∈ Y )

Since relations between classes are classes of the second level in our simple type theory,Zadeh’s extension principle in fact assigns to a first-level class R a second-level relation; suchan assignment itself is an individual of the third level. Thus we can define Zadeh’s principleas an individual of our theory—a special class Z∗ of the third level:

Definition 7.3.2 (Zadeh’s extension principle) Zadeh’s extension principle based on ∗is a third-level function Z∗ defined as follows (we adopt the usual functional notation forclasses which are functions):

Z∗(R) =df {〈X,Y 〉 | (∃x, y) (Rxy&∗x ∈ X&∗x ∈ Y )}Generally we can extend any fuzzy relation R(n+1) of type n + 1 to one of type n + 2

by Zadeh’s principle of type n + 3 (based on a t-norm ∗). All these ‘principles’ are in factindividuals of our theory, whose existence follows from the comprehension scheme.

4This is done simply by postulating that the n-th sort of variables is subsumed by the k-th sort if n < k.The sorts can further be refined to allow arbitrary tuples of individuals of lesser types with the appropriatetuple-forming, component-extracting and tuple-identity axioms added. The generalization of Zadeh modelsis again quite straightforward.

5Functions are a special kind of relations. The generalization to n-ary relations is trivial.

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126 CHAPTER 7. FUZZY CLASS THEORY

Definition 7.3.3 Zadeh’s extension principle for relations of type n + 1 (for n ≥ 0) basedon ∗ is the function of type n + 3 defined as follows:

Z∗(n+3)

(R(n+1)

)=df

{⟨X(n+1)

1, . . . , X(n+1)

k

⟩ ∣∣∣(∃W (n)

1, . . . ,W(n)

k

)(⟨W (n)

1, . . . , W(n)

k

⟩∈ R(n+1) &∗

k

&∗i=1

W(n)i ∈ X

(n+1)i

)}

7.4 Adding structure to the domain of discourse

As we have shown, in FCT we can define many properties of individuals of our theory(objects or classes). Since our theory contains classical class theory (for classes which arecrisp), we can introduce arbitrary relations and functions on the universe of objects whichare definable in classical class theory. As they can be described by formulae, their existenceis guaranteed by the comprehension axiom. So the only thing we need to add is a constant ofthe appropriate sort and the instance of the comprehension axiom. The following definitionis the formalization of this approach for the first-order theories.

Definition 7.4.1 Let Γ be a classical one-sorted predicate language and T be a Γ-theory.For each n-ary predicate symbol P of Γ let us introduce a new constant P for a class ofn-tuples, and for each n-ary function symbol F we take a new constant F for a class of(n+1)-tuples. We define the language FCT(Γ) as the language of FCT extended by thesymbols Q for each symbol Q ∈ Γ. The translation φ of a Γ-formula φ to FCT(Γ) is obtainedas the result of replacing all occurrences of all Γ-symbols Q in φ by Q.

We define the theory FCT(T ) in the language FCT(Γ) as the theory with the followingaxioms:

• The axioms of FCT

• The translations φ of all axioms φ of T

• Crisp(Q) for each symbol Q ∈ Γ (for the definition of Crisp, see Table 7.2)

• 〈x1, . . . xn, y〉 ∈ F ∧ 〈x1, . . . xn, z〉 ∈ F → y = z for each n-ary function symbol F ∈ Γ.

Lemma 7.4.2 Let Γ be a classical predicate language, T a Γ-theory, L an ÃLΠ-algebra. IfM is an L-model of FCT(T ), then Mc = (M, (QMc)Q∈Γ), where QMc = QM for eachQ ∈ Γ, is a model (in the sense of classical logic) of the theory T .

Vice versa, for each model M of T there is an L-model N of FCT(T ) such that Nc isisomorphic to M.

Therefore (in virtue of Theorem 3.5.3), T ` φ iff FCT(T ) ` φ, for any Γ-formula φ.

Proof: If M is an L-model of FCT, then for each Q ∈ Γ, QM is crisp due to the axiomCrisp(Q) of FCT(T ). Setting the universe of Mc to that of M, and for each symbol Q ∈ Γ,setting QMc to the set whose characteristic function is QM, we can see that Mc models T ,because the axioms of T , which contain only crisp predicates, are evaluated classically in Mc.

Conversely, we define M as the standard Zadeh model with the universe of N, in whichFM = FN for every function symbol F ∈ Γ, and for every predicate P ∈ Γ, PM is realizedas the characteristic function of PN. Then M obviously satisfies all axioms of FCT(T ); theaxioms of T are again evaluated classically in M, as the realizations of all predicates involvedare crisp. QED

Example 7.4.3 Let R be a constant for a class of pairs. Then in each L-model of the theoryCrisp(R), Refl(R), Trans(R), (∀x, y) (Rxy&Ryx → x = y), the constant R is representedby a crisp ordering on the universe of objects. (For the definitions of Refl and Trans, seeTable 7.3.)

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7.5. FUZZY MATHEMATICS 127

Example 7.4.4 If T is a classical theory of the real closed field, then in each L-model M ofthe theory FCT(T ), the universe of objects with ≤M, +M, −M, ·M, 0M, 1M is a real closed field.

In Lemma 7.4.2 we speak of first-order theories only. Nevertheless, it can be extendedto any theory formalizable in classical type theory. Here we present only one example; fordetails see [7].

Example 7.4.5 Let τ be a constant for a class of classes and T the theory with the axioms:

• Crisp(τ)

• (∀X) (X ∈ τ → Crisp(X))

• (∀X) (Crisp(X)&X ⊆ τ → {x | (∃X) ∈ X&x ∈ X)} ∈ τ

• (∀X1) . . . (∀Xn) (X1 ∈ τ& . . . &Xn ∈ τ → X1 ∩ . . . ∩Xn ∈ τ) for each n ∈ N

Then in each L-model of the theory T , the constant τ is represented by a classical topologyon the universe of objects.

7.5 Fuzzy mathematics

If we examine the above definitions we see the crucial role of the predicate Crisp. If weremove this predicate from the above definitions we get the “natural” fuzzification of theabove-mentioned concepts.

In order to illustrate the methodology of fuzzification, let us concentrate on the conceptof ordering. If we remove the predicate Crisp from the definition, then we have to distinguishwhich t-norm was used in the axioms of transitivity and antisymmetry. Thus we get theconcept of ∗-fuzzy ordering. This is the way this concept was introduced by Zadeh. However,some carefulness is due here not to overlook some “hidden” crispness. There is crisp identityused in the antisymmetry axiom, and also in the reflexivity axiom which can be written as(∀x, y) (x = y → Rxy). A more general definition is therefore parameterized also by a fuzzyequality in the following way:

Example 7.5.1 Let E and R be two constants of classes of tuples. The following axiomsdefine the concept of (∗, E)-ordering R:

• Equ∗(E)

• Trans∗(R)

• (∀x, y) (Exy → Rxy)

• (∀x, y) (Rxy&∗Ryx →∗ Exy)

Observe that E is a ∗-equality, and the last two conditions can be written as R∩∗R−1 ⊆E ⊆ R. We thus get the notion of fuzzy ordering as defined by Bodenhoffer in [9].

In contemporary fuzzy mathematics the methodology of fuzzification of concepts is some-what sketchy and non-consistent: usually only some features of a classical concept are fuzzifiedwhile other features are left crisp.

We would like to propose another “inductive” approach. We propose to follow the usual“inductive” development of mathematics (in some metamathematical setting—here in simpletype theory) and fuzzify “along the way”. In more words: develop a fuzzy generalization ofbasic classical concepts (the notion of class, relation, equality—as done in this chapter);then define compound fuzzy notions by taking their classical definitions and consistentlyreplacing classical sub-concepts in the definitions by their already fuzzified counterparts.The consistency of this approach promises that no crispness will be unintentionally “leftbehind”.

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128 CHAPTER 7. FUZZY CLASS THEORY

This approach is formal and sometimes may lead to too complex notions. In such cases,some features of the complex notion may intentionally be left crisp by retaining some of thecrispness axioms. The advantage of the proposed approach is that we always know whichfeatures are left crisp.

The framework presented in this chapter provides a unified formalism for various dis-ciplines of fuzzy mathematics. This may enable, i.a., an interchange of results and meth-ods between distant disciplines of fuzzy mathematics, till now separated by differences innotation and incompatibilities in definitions. It can also bring new (proof-theoretic andmodel-theoretic) methods to traditional fuzzy disciplines and enable their further develop-ment in both theory and applications. Finally, the axiomatization of the whole fuzzy mathe-matics, independent of particular [0, 1]-functions, can be an important step in understandingvague phenomena. Further elaboration of the proposed formalism and its application to var-ious disciplines of fuzzy mathematics is thus a possible direction towards firm foundations offuzzy mathematics.

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Chapter 8

. . . to Fuzzy Mathematics

This chapter is a full text of the talk From Fuzzy Logic to Fuzzy Mathematics: A Method-ological Manifesto, presented in Vienna 16.8.2004 at the conference Challenge of Semanticsby the author and Libor Behounek.

A Methodological Manifesto

One of the motives for theoretical studies in fuzzy mathematics is the pursuit of formalreconstruction of the methods commonly used in applied fuzzy mathematics. The greatestsuccess in such investigation is undoubtedly the area of formal fuzzy logic. By efforts of Hajek,Gottwald, Mundici, and others, this discipline has reached the point when it is reasonableto attempt using it as a ground theory for the formalization of other branches of fuzzymathematics.

This paper tries to provide certain guidelines for such a transition from formal fuzzylogic to formal fuzzy mathematics. The guidelines are based upon doctrines observed byPrague workgroup on fuzzy logic founded and led by Petr Hajek. We attempt to explicitlyformulate some distinct features of Petr Hajek’s approach, which we reconstruct from hisscattered remarks and the general direction of his papers, and implement them in the formof a research programme. We hope that Petr Hajek will find our reconstruction of his doctrinefaithful enough; or else that he will enter into a fruitful dispute with his own disciples overthe methodological foundations of our discipline. If the former is the case, then we deem thatthe best label for the enterprise would be Hajek’s Programme in the foundations of fuzzymathematics.

The cornerstone of Hajek’s approach to fuzzy mathematics is the doctrine of working in aformal axiomatic theory over a fuzzy logic, rather than investigating particular models. Forthe ease of reference, let us call it Hajek’s imperative. Good reasons for such an approachcan be found, both of philosophical and pragmatic nature.

A philosophical reason is found in the following argumentation. Fuzzy logic describes thelaws of truth preservation in reasoning under (a certain form of) vagueness. Its interpreta-tion in terms of truth degrees is just a model—a classical rendering of vague phenomena.Even though such models may have originally been employed for discovering the laws ofapproximate reasoning, they must be regarded as secondary—just because they are essen-tially classical, not genuinely vague. Only formal theories over fuzzy logic—assuming thatfuzzy logic faithfully approximates the laws of truth preservation in reasoning fraught withvagueness—are genuinely fuzzy.

The limited adequacy of such classical models of vague reasoning is seen, i.a., in the usualobjection against fuzzy logic which points at its practice of assigning to vague propositionslike ‘Charles is tall’ definite truth values, e.g., 0.7485 (using fuzzy numbers does not solve the

129

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130 CHAPTER 8. . . . TO FUZZY MATHEMATICS

problem, only shifts it a level higher). However, the [0, 1] truth values should only be seenas a model underlying vague inference, which helps us in understanding truth preservationunder vagueness. No relevant conclusion can be derived from the assumption that the truthvalue of ‘Charles is tall’ is 0.7485, because such an assumption is absurd. Only the lawsof inference that emerge from considering all possible models of approximate reasoning areepistemically valuable; and this means de facto the laws derivable in a formal theory or logic.

Admittedly, a formal theory over fuzzy logic is just a notational abbreviation of classicalreasoning about the class of all models. Nonetheless, the axiomatic method is the generalparadigm of mathematics. The appropriate choice of the language of the formal theoryscreens off irrelevant features of the models. An axiomatic system is thus not only the meansof generalization over all models, but rather an abstraction to their constitutive features.

Obviously, Hajek’s imperative applies mainly to the development of fuzzy logic and variousbranches of fuzzy mathematics, not to particular applications. In an application, we aremodeling particular phenomena and thus we naturally work with a particular model. Forinstance, some real-life problem (e.g., processing of a questionnaire with five grades betweenabsolute yes and absolute no) may invite a definite algebra of truth values. However, havinga general theory may of course help even in particular cases, since it will describe the generalfeatures of the problem. The programme of developing fuzzy mathematics in a theoreticalmanner stresses the priority of general theories over immediate applicational needs.

The idea that fuzzy inference cannot be reduced to a particular model that is able toexplicate its rules entails that in the investigation of fuzzy inference we should not limit our-selves to one particular fuzzy logic (e.g., ÃLukasiewicz). The model which underlies it—e.g.,a specific t-norm—is particular, while fuzzy reasoning in general is broader. There are ex-amples of fuzzy reasoning that follow variant inference rules, all of which are suitable fordifferent respective contexts of real-life situations and invite explanations in terms of variousindividual t-norms or other semantics. The multitude of existing fuzzy logics varying bothin expressive power and inference rules is not only explicable by the need of capturing of allaspects of fuzzy inference in diverse contexts, but even indispensable for this enterprise.

Similar considerations are related to Hajek’s preference of fuzzy logics without truthconstants in the language (except for those which are definable). First, the truth constantshave little support in natural language. Second, by incorporating the truth values into thesyntax, we force the logic to follow too closely a particular model of vague inference, viz. thatusing truth values. Of course, we cannot be too dogmatic about rejecting truth constants: itturns out that in sufficiently strong theories, at least rational truth constants are definable.Sometimes, the truth degrees are useful for a particular application. However, we should becautious of deliberately introducing them into logic and thus restricting the possible modelsof vague inference.

Thus, even though liberal in both the expressive power and inference rules, we—followingHajek—believe a certain style of logical systems to be a most suitable formalism for represent-ing fuzzy inference. For brevity’s sake, in what follows we shall call them Hajek-style fuzzylogics. Put in a nutshell, they are fuzzy logics retaining the syntax of classical logic (prefer-ably without truth constants), defined as axiomatic systems (rather than non-axiomatizablesets of tautologies). A prototypical example is Hajek’s Basic Logic BL, propositional orpredicate. In the following paragraphs we give some reasons for such preferences.

There is a pragmatic motive for retaining as much of classical syntax as possible. Theway of working in theories over Hajek-style fuzzy logics resembles closely the way of work-ing in classical logic: Hajek-style fuzzy logics are often just weaker variants of Booleanlogic—syntactically fully analogous, just lacking some of its laws. Therefore, many theoreti-cal and metatheoretical methods developed for classical logic can be mimicked and employed,resulting in a quick and sound development of the theory. This feature has already been uti-lized in metamathematics of fuzzy logic—the proofs of the completeness, deduction, andother metatheorems have often been obtained by adjustments of classical proofs.

To illustrate the utility of this guideline, we allege that an axiomatic theory of fuzzy setscan more easily be developed as a formal theory of binary membership predicate over some

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131

fuzzy logic than if the graded membership is rendered, e.g., as a ternary predicate betweenelements, sets, and truth values in the framework of classical logic. Many constructions andeven proofs of the classical theory will work in the former case and need not be rediscovered(nor even reformulated). Even though both theories may turn out equivalent, the resemblanceof fuzzy concepts to classical ones becomes more visible in Hajek’s approach: cf. the many‘breakthrough’ definitions of fuzzy set inclusion which, if put down in Hajek-style fuzzy logichave exactly the form of the classical definition of set inclusion. This is another reasonfor preferring the classical syntax in fuzzy logic, over non-standard logical systems, e.g., anevaluated syntax. (This does not mean that they cannot have their own merits; only theyare not preferred for the development of fuzzy mathematics by the Prague school.)

The imperative to work deductively in a formal theory explains also our preference ofaxiomatic systems over non-axiomatizable sets of tautologies. The infeasibility of algorithmi-cal recognition of valid inferences in the latter is a strong reason supporting the preference.Thus, e.g., predicate fuzzy logics are better to be understood as the systems of axioms andrules for quantifiers than the sets of valid [0, 1]-tautologies, even though the former usuallyadmits non-intended models.

The respect for the priority of formal theories to models can partly be seen as emphasizingthe syntax against the semantics of fuzzy logic. Hajek’s approach thus can be viewed as asyntactic turn in fuzzy logic. The accent on syntax is of course not meant to contest thefundamental role of semantics in logic, nor the heuristic value of the models. Nevertheless,playing up the importance of formal deduction in fuzzy logic corresponds to its motivationas a description of the rules of correct reasoning under vagueness.

Such, then, is a reconstruction of the methodological background we adopt. It has alreadyproved worthy in the area of metamathematics of fuzzy logic. Thus it seems reasonable toapply its doctrines to other branches of fuzzy mathematics as well.

The need for axiomatization of further areas of fuzzy mathematics besides fuzzy logic isbeyond doubt. Axiomatization has always aided the development of mathematical theories.There have been many—more or less successful—attempts to formalize or even axiomatizesome areas of fuzzy mathematics. However, these axiomatics are usually designed ad hoc:some concepts in a classical theory are turned fuzzy, however their selection is based onnon-systematic intuitions or intended applications; seldom is fuzzified all that could be. (Tofuzzify as much as possible is desirable for generality’s sake; if an application requires somefeatures to be crisp, they can be ‘defuzzified’ by an additional assumption of the crispness ofthese particular features.) Many of these axiomatics are in fact semi-classical, being foundedupon the notions of truth degrees and membership functions, which are merely a classicalrendering of fuzzy sets.

Further problems of contemporary fuzzy mathematics lie in its fragmentation. Even ifsome axiomatic theories of various parts of fuzzy mathematics exist, they use completelydifferent sets of primitive concepts and incompatible formalisms. This makes it virtuallyimpossible to combine any two of them into one broader theory. It would certainly be betterif fuzzy mathematics as a whole could employ a unified methodology in building its axiomatictheories, because it would facilitate the exchange of results between its branches. Applyingthe doctrines sketched above, we propose such a unified methodology for the axiomatizationof fuzzy mathematics. Obviously, in our approach it assumes the form of constructing formaltheories over Hajek-style fuzzy logics.

In the axiomatic construction of classical mathematic, a three-layer architecture provedworthy, with the layers of logic, foundations, and only then individual mathematical disci-plines. Individual disciplines are thus developed within the framework of a unifying formaltheory, be it some variant of set theory, type theory, category theory, or another sufficientlyrich and general kind of theory. In fuzzy mathematics, the level of logic seems to be de-veloped far enough so as to support sufficiently strong formal theories. The search for asuitable foundational theory is thus the task of the day. As hinted above, the close analogybetween Hajek-style fuzzy logics and classical logic gives rise to a hope that fuzzy analogues

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132 CHAPTER 8. . . . TO FUZZY MATHEMATICS

of classical foundational theories will be able to harbour all (or at least nearly all) parts ofexisting fuzzy mathematics.

As conceivable candidates for a foundational theory, several ZF-style fuzzy set theorieshas already arisen. Many of them are certainly capable of doing the job. Nevertheless, theaxiomatics of most ZF-style fuzzy set theories savour of a similar ad hoc axiom choices asother hitherto attempts at axiomatization of fuzzy mathematics. By large this is inducedby the fact that such theories have to deal with a specific set-theoretical agenda and takeinto the account the structure of the whole set universe (expressed, e.g., by the axiom ofwell-foundedness). Moreover, for many of them it is not clear whether they can straightfor-wardly be generalized to other fuzzy logics than the one in which they were developed; thusthey are only capable of providing the foundation for a limited part of fuzzy mathematics.Besides the repertoire of ZF-style set theories, fuzzy logic offers an alternative of set theoriesbased on naive comprehension. Although their axiomatic system is very elegant, their con-sistency is limited to (certain) fuzzy logics where no bivalent operator is definable (roughlyspeaking, to infinite-valued ÃLukasiewicz logic or weaker).

If nevertheless a universal foundational theory is successfully found, the development ofindividual concepts of fuzzy mathematics has to proceed in a systematic way, taking intothe account the dependencies between them as in classical mathematics. For example, thenotion of cardinality should only be defined after the introduction and investigation of thenotion of function, upon which it is based (and which in turn is based upon the concept offuzzy equality, i.e., similarity). Defined notions should also be checked against conformitywith category theory. For instance, a proposed definition of Cartesian product should accordwith that of mapping (one must, however, take into account that many natural notions ofmorphism become fuzzy under fuzzy logic).

Only this kind of systematic approach can avoid giving ad hoc definitions of fuzzy con-cepts, which often suffer from arbitrariness and hidden crispness, or even references to par-ticular crisp models of fuzziness (e.g. membership functions) which are not objects of theformal theory.

As a concrete implementation of the general programme sketched above we propose aspecific foundational theory described below. We do not claim it to be the only possibleway neither of doing the foundations of fuzzy mathematics, nor of fulfilling our foundationalprogramme. The methodology itself is independent of this particular solution we propose.Nevertheless, we think that our theory embodies its guidelines very well and is a viablefoundation for fuzzy mathematics of the present day. Moreover, because of the simplicity ofits apparatus, the work done within its framework can be of use for other possible systemsvia a formal interpretation.

Inspecting the existing approaches and having in mind the need for generality and sim-plicity, it becomes obvious that a full-fledged set theory is not necessary for the foundationsof fuzzy mathematics. What is necessary is only the ability to perform within the theorythe basic constructions of fuzzy mathematics. On the other hand, a great variability of thebackground fuzzy logic is required in order to encompass the whole of fuzzy mathematics.

Most notions of classical mathematics can be defined within the first few levels of a simpletype theory. The similarity between Hajek-style and classical theories hints that this couldbe true of fuzzy concepts defined in a fuzzified simple type theory as well. Indeed, manyimportant notions can be defined already at the first level, which is in fact second-orderpredicate fuzzy logic. Most notably, elementary fuzzy set theory, or the axiomatization ofZadeh’s notion of fuzzy set, is contained in second-order fuzzy logic (second-order models areexactly Zadeh’s universes of fuzzy sets). Some theories (e.g., topology), however, need morelevels of type hierarchy, thus we employ higher-order fuzzy logic (in the limit, logic of orderω).

Unfortunately, fuzzy higher-order logic is not recursively axiomatizable. Since we pre-fer axiomatic deductive theories over non-axiomatizable sets of tautologies, we choose itsHenkin-style variant, even though it admits non-intended models. We thus get a first-order

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133

theory, axiomatized very naturally by the extensionality and comprehension axioms for eachorder. Moreover, the construction works for virtually all imaginable fuzzy logics (and manynon-fuzzy logics as well). The bunch of foundational theories we propose thus can be calledHenkin-style higher-order fuzzy logic (for an individual fuzzy logic of one’s choice; expres-sively rich logics like ÃLΠ seem to be sufficient for all practical purposes; nevertheless, theinvestigation of the fragments over weaker logics has also its own importance). Equippingthe theory with the obvious axioms of tuples yields an apparatus which seems to be of enoughexpressive power for a great part of fuzzy mathematics, since a structure on the universe ofdiscourse (metric, measure, etc.) can then be introduced by means of relations. Furthermore,if the background logic is sufficiently strong, there is a general method of embedding anyclassical theory, and even of its natural fuzzification (as well as conscious and controlled ‘de-fuzzification’ of its concepts if some of their features are to be left crisp). The details of thisformalism can be found in [6].

As indicated above, elementary fuzzy set theory and some parts of the theory of fuzzyrelations are already formalized within our foundational theory. Several other parts of fuzzymathematics are currently (re-)developed in our formalism. However, the reconstruction (andexpected further advance) of the whole of fuzzy mathematics is an infinite task. Everybody istherefore invited to participate in this research programme of systematic formal developmentof fuzzy mathematics, as well as to continue the discussion of its best foundation.

Acknowledgements. As the reader could easily observe this methodological programme hasclose links to the works of many predecessors, and in fact only applies their accomplishmentsto the area of fuzzy mathematics.

Our formalistic approach to mathematics is close to that of Hilbert’s [57]. Our aspiration tolay down the logical foundations for fuzzy mathematics is only a derivative of the admirableenterprise of Russell and Whitehead [83]. Methodologically our programme is somewhatsimilar to that of Bourbaki [10], though we hope not only reconstruct and codify, but alsoadvance the field of our interest. The link to Vienna circle [11] which results from thecircumstances of the first presentation of this manifesto is rather incidental (though in someaspects one could perhaps find distant parallels).

We cannot mention all outstanding works of fuzzy logic upon which our contributionis based. However, a milestone in the development of fuzzy logic is certainly Hajek’s [44].Apparently the first monograph close in spirit to our programme was Gottwald’s [43]. And,needless to say, the whole field of fuzzy mathematics we try to formalize originated withZadeh’s [85].

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134 CHAPTER 8. . . . TO FUZZY MATHEMATICS

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Appendix A

T-norms

T-norms (triangular norms) form a special class of operations on [0, 1]. T-norms were intro-duced by Menger in his paper [64]. A very good survey to the t-norms and their applicationsas a recent book of Klement, Mesiar, and Pap [60]. We present a modern definition of at-norm.

Definition A.1 A t-norm is a commutative associative non-decreasing (in both arguments)binary operation on [0, 1] with unit element 1 and annihilator element 0, i.e., the followinghold for all x, y, z ∈ [0, 1] :

• x ∗ y = y ∗ x

• x ∗ (y ∗ z) = (x ∗ y) ∗ z

• x ≤ y implies x ∗ z ≤ y ∗ z

• x ≤ y implies z ∗ x ≤ z ∗ y

• x ∗ 1 = x

• x ∗ 0 = 0

A t-norm is continuous if it is continuous mapping in the usual sense.

There are three special continuous t-norms. These t-norms have many important prop-erties and we will see that they play a fundamental role among all t-norms.

• ÃLukasiewicz t-norm: x ∗ y = max(0, x + y − 1)

• Godel t-norm: x ∗ y = min(x, y)

• Product t-norm: x ∗ y = x.y (product of reals)

It is obvious that these t-norms are continuous. Each left-continuous t-norm has a specialcorresponding operation called residuum.

Definition A.2 Let ∗ be a left-continuous t-norm. Then we define a binary operation on[0, 1] called the residuum ⇒ of t-norm ∗ in the following way: x ⇒ y = max{z | x ∗ z ≤ y}

Now we are going to examine some basic properties of t-norms and their residua. Espe-cially notice property (2), which we will often use in our proofs.

135

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136 APPENDIX A. T-NORMS

Lemma A.3 Let ∗ be a continuous t-norm and ⇒ its residuum. Then the following hold:(1) ⇒ is the unique operation satisfying the condition from the latter

definition,

(2) x ≤ y iff (x ⇒ y) = 1,

(3) (1 ⇒ x) = x,

(4) max(x, y) = min((x ⇒ y) ⇒ y, (y ⇒ x) ⇒ x).

If ∗ is continuous t-norm we have also:(5) min(x, y) = x ∗ (x ⇒ y).

Having this we may compute residua of three the most important t-norms: (x ⇒ y) = 1for x ≤ y and for x > y:

• ÃLukasiewicz residuum: x ⇒ y = 1− x + y

• Godel residuum: x ⇒ y = y

• Product residuum: x ⇒ y = yx

Before we present the theorem, which states why our three t-norms are so important, weprepare some definitions. This theorem is a variant of the known Mostert-Shields theorem([70].

Definition A.4 Let ∗ be a continuous t-norm.

• An element x of [0, 1] is called an idempotent if x ∗ x = x.

• An idempotent x is non-trivial if x 6= 1 and x 6= 0.

• An non-trivial idempotent x is a cutpoint if there no closed interval of idempotentscontaining x as its inner point.

Theorem A.5 Let ∗ be a continuous t-norm and ⇒ its residuum. Then there is a countableset of pairs (finite or infinite) R = ([ri, si], Ai), where [ri, si] is a closed interval from [0, 1](borders are either cutpoints or 0 or 1) and Ai is an elements from {�, Π, G}. The union ofintervals [ri, si] is dense in [0, 1] and they overlap at most in their borders. Then there is at-norm ∗′ (and the corresponding residuum ⇒′) isomorphic to the t-norm ∗ (to the residuum⇒ respectively) so the following holds:

x ∗′ y =

{ri + (si − ri)( x−ri

si−ri∗A

y−ri

si−ri) if x, y ∈ [ri, si]

min(x, y) otherwise

x ⇒′ y =

1 if x ≤ y

ri + (si − ri)( x−ri

si−ri⇒A

y−ri

si−ri) if x, y ∈ [ri, si], x > y

y otherwise

A t-norm is said to be a finitely constructed is the set of covering intervals is finite.The converse of this claim also holds, i.e., each operation constructed in this way is a

continuous t-norm.

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