the logics: two complete fuzzy systems joining Łukasiewicz and product logics

Arch. Math. Logic (2001) 40: 39–67

c© Springer-Verlag 2001

The LΠ and LΠ 12 logics:

two complete fuzzy systemsjoining L ukasiewicz and Product Logics

Francesc Esteva1, Llu ıs Godo1, Franco Montagna2

1 Institut d’Investigacio en Intelligencia Artificial (IIIA), Consejo Superior de Investiga-ciones Cientificas (CSIC), Campus Universitat Autonoma de Barcelona, s/n,08193 Bellaterra, Spain (e-mail:{esteva; godo}@iiia.csic.es)

2 Dipartimento di Matematica, Universita degli Studi di Siena, Via del Capitano 15,53100 Siena, Italy (e-mail: [email protected])

Received: 28 December 1998 / revised version: 11 May 1999

Abstract. In this paper we provide a finite axiomatization (using two fini-tary rules only) for the propositional logic (calledLΠ) resulting from thecombination of Lukasiewicz and Product Logics, together with the logicobtained by fromLΠ by the adding of a constant symbol and of a defin-ing axiom for 1

2 , calledLΠ 12 . We show thatLΠ 1

2 contains all the mostimportant propositional fuzzy logics: Lukasiewicz Logic, Product Logic,Godel’s Fuzzy Logic, Takeuti and Titani’s Propositional Logic, Pavelka’sRational Logic, Pavelka’s Rational Product Logic, the Lukasiewicz Logicwith ∆, and the Product and Godel’s Logics with∆ and involution. Stan-dard completeness results are proved by means of investigating the algebrascorresponding toLΠ andLΠ 1

2 . For these algebras, we prove a theorem ofsubdirect representation and we show that linearly ordered algebras can berepresented as algebras on the unit interval of either a linearly ordered field,or of the ordered ring of integers,Z.

1. Introduction

As shown in [H98], in the literature of fuzzy logic, three systems emerge: theLogic of Lukasiewicz (cf. e.g. [COM95] or [CM97]), Godel’s Fuzzy Logic(cf. [Go33]), and the Product Logic (cf. [HGE96]). These logics correspondto the main three continuous t-norms over the unit real interval, namely theLukasiewicz conjunction⊗, defined byx ⊗ y = max(x + y − 1, 0), theoperation∧, defined byx ∧ y = min(x, y), and the product�, defined by

40 F. Esteva et al.

x�y = x ·y. To every t-norm∗ one can associate an implication→, definedas its residuum, that is, defined byx → y = sup {z ∈ [0, 1] : z ∗ x ≤ y},and a negation¬, defined as¬x = x → 0. Thus, we have three implicationsand three negations, corresponding to the t-norms⊗, ∧ and�. The firstones, the Lukasiewicz implication and negation, are defined by

x →L y = min {y + (1− x), 1}¬Lx = 1− x

The second ones, Godel’s implication and negation, are defined by

x →G y ={

1 if x ≤ yy otherwise

¬Gx ={

1 if x = 00 otherwise

Finally, the implication and negation corresponding to� are defined by

x →P y ={

1 if x ≤ yyx otherwise

¬Px = ¬Gx

The Logic of Lukasiewicz is probably the most important of these logics.Usually, in the Logic of Lukasiewicz, one assumes as basic the connectives¬L and⊕, whose interpretation is as follows:

x⊕ y = min {x + y, 1}It is well-known that, if the constant0 is present in the language, then thepairs of connectives{⊕, ¬L} and{⊗,→L} are interdefinable. The spe-cialists of this field have discovered deep connections between this logicand various fields of mathematics, like algebra, algebraic geometry, andfunctional analysis (Cf. e.g. [CM97], [Mu86], [Mu94], [Pa95]). However,the research about other fuzzy logics is also growing very quickly, and awhole book in this subject has recently appeared [H98] and another one isto appear very soon [Got99]. In [H98], the author fully characterizes thecommon fragment of the Logic of Lukasiewicz, of Godel’s Fuzzy Logic,and of the Product Logic. This common fragment, sketched in Sect. 2.1, isnamedBasic Logic. Very related logical systems are also Hohle’s monoidallogics [Hoe95].

On the other direction, one might try to combine the above logics. Manyattempts have been done already. In [B96] the author studies the logics re-sulting from Godel’s Fuzzy Logic by adding an operator∆ which is a com-bination of the Lukasiewicz negation and of Godel’s negation (the sameas the negation of Product Logic). In [H98], the author extends Baaz’s re-sults to the logics resulting from the Logic of Lukasiewicz and from theProduct Logic respectively by also adding the operator∆ (see Sect. 2.2).

TheLΠ andLΠ 12 logics 41

In [EGHN98] the authors study the extension of Product and Godel’s Log-ics with an involutive negation. In [TT92], the authors extendGodel logic(not using this name) by L´ ukasiewicz and product conjunction, L´ ukasiewicznegation and the constant1

2 . In [H98], Hajek extends the logic of Takeutiand Titani by the adding of product implication, and simplifies their axiomsystem considerably. Both Takeuti-Titani and Hajek also investigate the cor-responding predicate logic. Their systems are sound and strongly completewrt the interpretation in the unit real interval, but they include infinitaryrules.

Starting from recent (separate) results of the authors in [EG98] and[M98], this paper provides a finite axiomatization (using two finitary rulesonly) for the propositional logic (calledLΠ) resulting from the combinationof Lukasiewicz and Product logics, together with the logic obtained by fromLΠ by the adding of a constant symbol and of a defining axiom for1

2 , calledLΠ 1

2 . We prove that Godel’s Fuzzy LogicG is faithfully intepretable inLΠ,there is an interpretation∗ of G in LΠ such that, for every formulaϕ of L,one has:G � ϕ iff LΠ � ϕ∗. Moreover, Takeuti and Titani’s PropositionalLogic, Pavelka’s Rational Logic and Rational Product Logic are faithfullyinterpretable inLΠ 1

2 . For the sake of precision, we must add that, due to thepresence of infinitary rules in Takeuti-Titani and in Rational Product logics,the interpretation does not preserve infinitary deduction, i.e., it is not truethat, for every possibly infinite setΓ of formulas, one hasL ∪ Γ � ϕ iffLΠ 1

2 ∪ {ψ∗ : ψ ∈ Γ} � ϕ∗.However, with the caution shown above, we can say thatLΠ 1

2 con-tains all the most important propositional fuzzy logics: Lukasiewicz Logic,Product Logic, Godel’s Fuzzy Logic, Takeuti and Titani’s PropositionalLogic, Pavelka’s Rational Logic, Pavelka’s Rational Product Logic, theLukasiewicz Logic with∆, the Product Logic with∆ and Godel’s logicwith involution.

We investigate the algebras corresponding toLΠ andLΠ 12 , calledLΠ

algebras, and, respectively,LΠ 12 algebras. For these algebras, we prove

a theorem of subdirect representation, i.e., we show that anyLΠ (LΠ 12 )

algebra is isomorphic to a subdirect product of a family of linearly orderedLΠ (LΠ 1

2 ) ordered algebras. Then, we prove that every linearly orderedLΠ 1

2 algebra can be represented as a suitably structured algebra on theunit interval of a linearly ordered filed, and that every linearly orderedLΠalgebra can be represented as an algebra on the unit interval of either alinearly ordered field, or of the ordered ring of integers,Z. These resultsallow us to prove a standard completeness theorem: a formula ofLΠ (LΠ 1

2 )is provable inLΠ (LΠ 1

2 ) iff it is true in theLΠ (LΠ 12 ) algebra arising from

the unit real interval under any evaluation. Of course, the Completeness

42 F. Esteva et al.

theorem can be restated in terms of algebras: theLΠ (LΠ 12 ) algebra arising

from the unit real interval is a functionally freeLΠ (LΠ 12 ) algebra.

The paper is organized as follows. After this introduction, Sect. 2 con-tains necessary background knowledge about the basic fuzzy logic BL andits corresponding algebras, together with a short survey on MV-algebras.Section 3 summarizes the direct antecedents of this paper. Sections 4, 5 and6 contain the main results of the paper, already commented in the previousparagraph. In Sect. 7 we investigate deduction from partially true proposi-tions inLΠ 1

2 in the Pavelka’s style, introducing the RationalLΠ logic andgetting strong completeness results. Finally, Sect. 8 deals withLΠ algebrason the real unit interval [0, 1] and their subalgebras. We end up with someconclusions and remarks about the interpretation of other fuzzy logics inLΠ andLΠ 1

2 .

2. Preliminaries

In this section we summarize the basic notions and results from severalsystems of propositional fuzzy logic that will be used throughout this paper.

2.1. The basic fuzzy logic BL and BL-algebras

The language of the basic logic BL [H98] is built in the usual way froma set of propositional variables, a conjunction∗, an implication→ and theconstant0. Further connectives 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).

The following formulas are theaxiomsof 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

Thededuction ruleof BL is modus ponens.


If one takes a continuous t-norm for the truth function of∗ and thecorresponding residuum for the truth function of→ (and evaluating0 by0) then all the axioms of BL become 1-tautologies (have identically thetruth value 1). And since modus ponens preserves 1-tautologies all formulasprovable in BL are 1-tautologies.

It is shown in [H98] that the well-known L´ ukasiewicz logic, denoted L,is the extension of BL by the axiom

(L) ¬¬ϕ → ϕ,and Godel logic, denoted G, is the extension of BL by the axiom

(G) ϕ → (ϕ ∗ ϕ).Finally, product logic, denotedΠ, is just the extension of BL by the followingtwo axioms:

(Π1) ¬¬χ → (((ϕ ∗ χ) → (ψ ∗ χ)) → (ϕ → ψ)),(Π2) ϕ ∧ ¬ϕ → 0

A BL-algebrais an algebraA = 〈A,∧,∨, ∗,→, 0, 1〉 with four binaryoperations (we shall take the same symbols of the logical connectives) andtwo constants such that

(i) 〈A,∧,∨, 0, 1〉 is a lattice with the largest element1 and the least element0 (with respect to the lattice ordering≤),

(ii) 〈A, ∗, 1〉 is a commutative semigroup with the unit element1, i.e.∗ iscommutative, associative and1 ∗ x = x for all x,

(iii) the following conditions hold:(1) z ≤ (x → y) iff x ∗ z ≤ y for all x, y, z.(2) x ∧ y = x ∗ (x → y)(3) (x → y) ∨ (y → x) = 1.

Thus, in other words, a BL-algebra is aresiduated latticesatisfying (2) and(3). The class of all BL-algebras is a variety. Moreover, each BL-algebracan be decomposed as a subdirect product of linearly ordered BL-algebras.

Defining¬x = x → 0, it turns out that MV-algebrasare BL-algebrassatisfying¬¬x = x, G-algebrasare BL-algebras satisfyingx ∗ x = x, andfinally, product algebrasare BL-algebras satisfying

x ∧ ¬x = 0¬¬z → ((x ∗ z → y ∗ z) → x → y) = 1.

The logic BL is sound with respect to BL-algebras: ifϕ is provable inBL thenϕ is anA-tautology for each BL-algebraA, i.e.ϕ has the value1Afor each evaluation of variables by elements ofA extended to all formulasusing operations ofA as truth functions.

Theorem 1. (cf. [H98]) BL is complete, i.e. for each formulaϕ the followingthree conditions are equivalent:

(i) ϕ is provable in BL,

44 F. Esteva et al.

(ii) for each BL-algebraA, ϕ is anA-tautology,(iii) for each linearly ordered BL-algebraA, ϕ is anA-tautology.

This completeness of BL easily extends to the three main many-valuedlogics Lukasiewicz, Godel and Product (see [H98]). But these three logicsenjoy besides standard completeness, that is, a formulaϕ is provable in anyof these three logics iff it is true in the corresponding algebraic structure ofthe real unit interval [0, 1] with truth functions as operations1.

2.2. MV and MV∆ algebras

Herewe introduceanequivalentbutmorecompactdefinitionofMV-algebrasthat we shall use in this paper.

Definition 1. A MV algebra is a systemA = 〈A,⊕,¬L, 0, 1〉 such that,letting x ⊗ y = ¬L(¬Lx ⊕ ¬Ly), andx � y = x ⊗ ¬Ly, the followingconditions are satisfied:

(i) 〈A,⊕, 0〉 is a commutative monoid.(ii) x⊕ 1 = 1(iii) ¬L¬Lx = x(iv) x⊕ ¬Lx = 1(v) ¬L0 = 1(vi) x⊕ (y � x) = y ⊕ (x� y)

The literature of MV algebras is very wide. The paper [CM97] contains anexcellent survey which is more than sufficient for the scope of this paper.

Notation 1. We define:x →L y = ¬Lx⊕ y, |x− y| = (x� y)⊕ (y � x), x ∨ y = x⊕ (y � x),x ∧ y = x� (x� y), x ↔L y = (x →L y)⊗ (y →L x).

Remark 1.Notice that, regarding the definition of MV-algebra in the pre-vious subsection, the operations∗, →, ¬ correspond here to the operations⊗, →L and¬L respectively.

Notation 2. LetL be any fuzzy logic whose logic connectives are among⊕,¬L,�,→P , and whose constants are among0, 1, 1

2 . ByIL we mean thestructure whose domain is the real interval[0, 1], and whose constants andoperations are the natural interpretations of the propositional constants andconnectives ofL as described in the Introduction.

1 Standard completeness for BL has been first investigated by Hajek in [H98b] but finallystated in a recent paper [CEGT99] where it is shown that a formula is provable in BL iff itis a tautology with respect to all continuous t-norms in the real unit interval [0, 1]


According to this notation, if⊕ is a connective ofL, then it is interpretedasx ⊕ y = min {x + y, 1}, etc. Moreover,IL is the structure on the unitreal interval corresponding to the Logic of Lukasiewicz,IΠ is the structurecorresponding to the Product Logic, etc.

Proposition 1. In every MV algebra, the following conditions hold:

(i) x� 0 = x, andx� x = 0(ii) 0� x = 0(iii) Let x ≤ y iff x � y = 0. Then≤ is a lattice order with top element

1 and bottom element0 wrt the lattice operations∨ and∧ defined inNotation 1. Moreover, one has:x ≤ y iff x →L y = 1

(iv) If x ≤ y, thenx⊕ z ≤ y ⊕ z, x� z ≤ y � z, andz � y ≤ z � x.(v) x = y iff |x− y| = 0(vi) |x− y| = (x ∨ y)� (x ∧ y)(vii) x⊕ y = 1� ((1� x)� y)(viii) If c ≤ b ≤ a, thena� (b� c) = (a� b)⊕ c(ix) In IL, ⊗ is a continuous t norm, and→L is its corresponding impli-

cation, i.e.,x →L y = sup {z ∈ I : x⊗ z ≤ y}Proof. Cf. [COM95]. ��In [H98] Hajek studies the extension of Lukasiewicz logic with a new unary(projection) connective∆ whose truth function (denoted also by∆) is de-fined as follows:

∆(x) ={

1, if x = 10, otherwise

Theaxiomsof the extended Lukasiewicz logic L∆ are those of Lukasiewiczlogic L plus:

(∆1) ∆ϕ ∨ ¬L∆ϕ(∆2) ∆(ϕ ∨ ψ) →L (∆ϕ ∨∆ψ)(∆3) ∆ϕ →L ϕ(∆4) ∆ϕ →L ∆∆ϕ(∆5) ∆(ϕ →L ψ) →L (∆ϕ → ∆ψ)

Deduction rulesof L∆ are modus ponens andnecessitationfor ∆: fromϕ derive∆ϕ.

Axioms∆1-∆5 were first formulated by Baaz in [B96]. The correspond-ing algebraic structures for L∆ are the so-called MV∆ algebras. A MV∆algebra is a structureA = 〈A,⊕,¬L, 0, 1, ∆〉 such that〈A,⊕,¬L, 0, 1, 〉 isa MV-algebra and the unary operation∆ satisfies the following conditions:

1. ∆x ∨ ¬L∆x = 12. ∆(x ∨ y) ≤ ∆x ∨∆y3. ∆x ≤ x

46 F. Esteva et al.

4. ∆x ≤ ∆∆x5. (∆x)⊗ (∆(x →L y)) ≤ ∆y6. ∆1 = 1

Clearly, the class of all MV∆ algebras is also a variety and, analogously toMV algebras, any MV∆ algebra can be decomposed as a subdiredt productof linearly ordered MV∆ algebras. Notice that in linearly ordered MV∆

algebras we have that∆1 = 1 and∆(a) = 0, for a �= 1. Moreover weshall show in Lemma 4 that linearly ordered MV∆ algebras are subdirectlyirreducible and conversely. The completeness theorem for Lukasiewicz logicextends to L∆ as follows.

Theorem 2. L∆ is complete, i.e. for each formulaϕ the following threethings are equivalent:

(i) ϕ is provable in L∆,(ii) ϕ∗ = 1 is true inIL∆

,(iii) ϕ∗ = 1 is true in each linearly ordered MV∆ algebra,(iiiv) ϕ∗ = 1 is true in each MV∆ algebra.

Moreover, each of the remaining distinguished fuzzy logics, productand Godel logics, can be added the∆ connective, together with its axioms(∆1) − (∆5), leading to the so denoted logicsΠ∆ and G∆, which arecomplete w.r.t. their corresponding extended algebrasΠ∆ and G∆ algebras.See [H98] for further details.

Finally, let us briefly comment on the logicΠ∼ andG∼ (see [EGHN98],extensions of product logic and Godel logics with an involutive negation∼.Axioms ofΠ∼ (G∼) are those of product (Godel) logic with∆ plus

(∼1) (∼∼ ϕ) ≡ ϕ(∼2) ¬ϕ →∼ ϕ(∼3) ∆(ϕ → ψ) → ∆(∼ ψ →∼ ϕ)

where∆ϕ is ¬ ∼ ϕ. Completeness results with respect to suitable exten-sions of product (G) algebras with an involutive operation, calledΠ∼ (G∼)algebras, are also obtained. However there is a distinction.G∼ has stan-dard completeness, that is,G∼ is complete w.r.t. the standardG∼-algebra([0, 1],min,→G, 1 − x). HoweverΠ∼ is complete only w.r.t. the class ofsemi-standardΠ∼ algebras, that is, the class of algebras in [0, 1] of the form([0, 1],�,→P , n), wheren is an involutive negation function. The differ-ence arises from the fact that, unlike semi-standardΠ∼ algebras, allG∼algebras over [0, 1] are isomorphic.

3. LΠ, LΠ 12 and the corresponding algebras

The logical and algebraic systems we present here have two clear and re-cent antecedents [EG98,M98] that deserve some comments. The first one is


basically a logical approach, whereas the second one is basically algebraic.Namely, in [EG98] the authors propose an axiomatic system that explicitlyextends the axiomatics of both systemsL andΠ, taking three primitive con-nectives (Lukasiewicz implication and Product conjuntion and implication)and the truth-constant0. They define the corresponding algebraic structures,and show that the theorems of the system are the tautologies of all linearlyordered algebras. It turns out that the axiomatic system and algebras of[EG98] are alternative equivalent representations of theLΠ logic andLΠalgebras we are going to introduce below. In [M98] the author gives an al-ternative axiomatization of the equational class ofLΠ andLΠ 1

2 algebrascorresponding to the logicsLΠ andLΠ 1

2 , and establishes an equivalence ofcategories with the class of commutative regular f-rings with explicit semi-inverse operation (called f-semifields there). Unlike [EG98] and the presentpaper, [M98] only deals with algebra and not with logic.

Now, in this section, we introduce an axiomatization ofLΠ logic andprove standard completeness results.

Notation 3. Hereafter we shall follow the notation already (somehow) in-troduced in the previous sections. We take, besides the truth constant0, threeprimitive connectives:→L as a connective for the Lukasiewicz implication,� as a connective corresponding to the product t-norm, and→P as a con-nective for the product implication. Besides we shall also use the followingdefinable connectives:

¬Lϕ is ϕ →L 0¬Pϕ is ϕ →P 0∆ϕ is ¬P¬Lϕϕ⊗ ψ is ¬L(ϕ →L ¬Lψ)ϕ⊕ ψ is ¬Lϕ →L ψϕ� ψ is ϕ⊗ ¬Lψϕ ∧ ψ is ϕ⊗ (ϕ →L ψ)ϕ ∨ ψ is ¬L(¬Lϕ ∧ ¬Lψ)ϕ →G ψ is ∆(ϕ →L ψ) ∨ ψ

The standard semantics for the primitive connectives is provided bymeans of evaluation mappingse assigning to each propositional variablep a value of the unit interval [0, 1], and extending to arbitrary formulas bymeans the Lukasiewicz and product truth-functions, that is:

e(ϕ →L ψ) = min(1, 1− e(ϕ) + e(ψ)),e(ϕ� ψ) = e(ϕ) · e(ψ), and

e(ϕ →P ψ) =

{1, if e(ϕ) ≤ e(ψ)e(ψ)/e(ϕ), otherwise

With these definitions, the usual truth functions for the above definable

48 F. Esteva et al.

connectives are recovered:

e(¬Lϕ) = 1− e(ϕ);e(¬Pϕ) = 1 if e(ϕ) = 0, e(¬Pϕ) = 0 otherwise (Godel negation);e(∆ϕ) = 1 if e(ϕ) = 1, e(∆ϕ) = 0 otherwise;e(ϕ⊗ ψ) = max(0, e(ϕ) + e(ψ)− 1);e(ϕ⊕ ψ) = min(1, e(ϕ) + e(ψ));e(ϕ� ψ) = max(0, e(ϕ)− e(ψ));e(ϕ ∧ ψ) = min(e(ϕ), e(ψ));e(ϕ ∨ ψ) = max(e(ϕ), e(ψ)); ande(ϕ →G ψ) = 1 if e(ϕ) ≤ e(ψ), e(ϕ →G ψ) = e(ψ)

otherwise (Godel implication).

Definition 2 (The logic LΠ). LΠ denotes the theory whose axioms andrules are as follows:

1. The axioms and rules of Lukasiewicz Logic plus∆ (Cf. [H98]) (thus,including the rule: A

∆(A) )

2. ∆(A ↔L B) ∧ ∆(C ↔L D) →L ((A ◦ C) ↔L (B ◦ D)), for◦ ∈ {�,→P }.

3. (A�B) →L (B �A)4. (A� (B � C)) ↔L ((A�B)� C)5. A ∧ ¬PA →L 06. A� (B � C) ↔L (A�B)� (A� C)7. ∆(A →L B) →L (A →P B)8. ∆(B →L A) →L (A� (A →P B) ↔L B).

Definition 3 (The logicLΠ 12 ). LΠ 1

2 denotes the logic obtained fromLΠby the adding of a propositional constant1

2 together with the axiom:12 ↔L

¬L12

Before going further, let us state the following theorem.

Theorem 3. LΠ (and thus,LΠ 12 as well) extends both Lukasiewicz and

Product logics.

Actually, axioms of Lukasiewicz logic are explicitly contained in thedefinition ofLΠ andLΠ 1

2 , but this is not the case for Product logic axioms.Nevertheless it can be shown that they are indeed provable inLΠ. The proofwill be done later by algebraic means, once we have introduced the algebraicstructures corresponding toLΠ, LΠ algebras, and showed the subdirectdecomposition of these algebras into linearly-ordered ones. Then it will beeasier to show that all the identities holding in linear Product algebras alsohold in linearLΠ algebras.


Lemma 1. The following are theorems ofLΠ:(i) (A� 1) ↔L A(ii) ∆(A →P B) ↔L ∆(A →L B).

Proof. (i) A →L A is a theorem of Lukasiewicz logic, thus∆(A →L A)too, and thus, by Axiom 7,A →P A is derivable. But then, using Axiom 8,A� (A →P A) ↔L A is derivable, that is, using Axiom 2,(A� 1) ↔L Ais derived.

(ii) One direction is a direct consequence of Axiom 7. For the otherdirection, first notice that in Lukasiewicz logicC →L (C ↔L 1) is provable,and thus∆C →L ∆(C ↔L 1) is also provable inLΠ. Now, takeC asA →P B. Using∆((A →P B) ↔L 1) and∆(A ↔L A) in Axiom 2, andtaking into account (i), we get∆(A →P B) →L (A� (A →P B) ↔L A),and thus∆(A →P B) →L ∆(A � (A →P B) ↔L A) as well. Now,using this last equivalence in Axiom 2, together withB ↔L B, we obtain∆(A →P B) →L ((A � (A →P B) →L B) ↔L (A →L B))). But thenwe can safely substituteA � (A →P B) →L B by A →L B in Axiom 8and prove

∆(A →P B) →L (∆(B →L A) →L (A →L B)).

Now, applying∆ and distributing it inside the implications, and noticingthat∆A ↔L ∆∆A is a theorem ofL∆, we get

∆(A →P B) →L (∆(B →L A) →L ∆(A →L B)).

But, since(A →L B)∨(B →L A) is provable in Lukasiewicz logic, we canprove∆(A →L B) ∨∆(B →L A), and thus by reasoning in Lukasiewiczlogic we finally prove

∆(A →P B) →L ∆(A →L B). ��Remark 2.As a consequence, we have thatA →P B is a theorem ofLΠiff A →L B is also a theorem. Therefore, provable equivalence with respectto both implications are the same.

We proceed with the definition of the classes of algebras correspondingto the logicsLΠ andLΠ 1

2 . By abuse of language, we use the same symbolto denote a connective ofLΠ and the corresponding algebraic operation.Clearly, operations→L,⊗, etc... are defined in terms of⊕,¬L,�and→P bytakingx →L y = ¬L⊕ y and the rest in the same way as the correspondingconnective.

Definition 4. AnLΠ algebra is a structureA = 〈A,⊕,¬L,�,→P , 0, 1〉such that:

1. 〈A,⊕,¬L, ∆, 0, 1〉 is aMV∆ algebra (cf.[H98]), .

50 F. Esteva et al.

2. 〈A,�, 1〉 is a commutative monoid.3. x� (y � z) = (x� y)� (x� z).4. ∆(x ↔L y) ∧∆(z ↔L u) ≤ ((x ◦ z) ↔L (y ◦ u)) for ◦ ∈ {�,→P }.5. x ∧ ¬Px = 06. ∆(x →L y) ≤ (x →P y)7. ∆(y →L x)) ≤ (x� (x →P y) ↔L y).

Definition 5. AnLΠ 12 algebra is anLΠ algebra equipped with a constant

12 satisfying the axiom1

2 = ¬L12 .

We show next a number of propertiesLΠ algebras enjoy.

Proposition 2. In anyLΠ algebra, the following conditions hold true:

(i) 0� x = 0.(ii) If x ≤ y, thenx� z ≤ y � z; in particular, x� z ≤ z.(iii) (x� y)⊕ (x� ¬Ly) = x.(iv) x⊗ y ≤ x� y ≤ x ∧ y.(v) If y ≤ ¬Lz, thenx� (y ⊕ z) = (x� y)⊕ (x� z).(vi) If y ≤ x, thenx� (x →P y) = y.(vii) If x ≤ y thenx →P y = 1.

Proof. (i) 0� x = (y � y)� x = y � x� y � x = 0.(ii) If x ≤ y, thenx�y = 0. Thus,x�z�y�z = (x�y)�z = 0�z = 0,

i.e.x�z ≤ y�z. Now, sincey ≤ 1 it follows thatx�z ≤ 1�z = z.(iii) By Definition 4 (3), x� (¬Ly) = x� (1� y) = x� (x� y). Thus,

(x � (¬Ly)) ⊕ (x � y) = (x � y) ⊕ (x � (x � y)) = (by definitionof ∨ in Notation 1)= x ∨ (x� y) = (by (ii)) = x.

(iv) Sincey ≤ 1, from (ii) it follows: x � y ≤ x � 1 = x. Similarly,x� y ≤ y. So,x� y ≤ x ∧ y.From (iii) it follows x� y = x� (x�¬L(y)) and sincex�¬L(y) ≤¬L(y), by (iv) of Proposition 1 we havex�(x�¬L(y)) ≥ x�¬Ly =x⊗ y.

(v) By Proposition 1 (vii), we have:

x� (y⊕ z) = x� (1� ((1� z))� y) = x� (x� (1� z)� x� y).

Sincex� y ≤ x� (1� z) ≤ x, we conclude:

x� (y ⊕ z) = (x� x� (1� z))⊕ x� y

= x� (1� (1� z))⊕ x� y

= x� z ⊕ x� y

(vi) It follows immediately from axiom (7) of Definition 4.(vii) It follows immediately from axiom (6) of Definition 4. ��


Definition 6. Let L be any ofLΠ or LΠ 12 , and letA be anL algebra.

An evaluationof L into A is a mape fromL formulas intoA such thatemaps every propositional constant into its realization inA and such that,for every connective ofL, and for any twoL formulasϕ andψ, one has:e(ϕ ∗ψ) = e(ϕ) A e(ψ), where A is the realization inA of the operationcorresponding to .AnL formulaϕ is said to be anA-tautologyiff for every evaluatione ofLintoA, one hase(ϕ) = 1.

Lemma 2. LetL denote any ofLΠ or LΠ 12 . Then:

(i) If L � A ↔L B andL � C ↔L D, then, for ∈ {�, →P ,⊕,¬L},L � (A C) ↔L (B D).

(ii) LetS be the set of equivalence classes of formulas ofLmodulo provableequivalence inL, and let, for every formulaA of L, [A] denote theequivalence class ofA. Define, for all [A], [B] ∈ S, and for ∈{�, →P ,⊕,¬L}, [A] [B] = [A B]. Then, is well-defined, and thestructureS = 〈S, ⊕, ¬L, �, →P , [0], [1]〉 is anLΠ algebra.Moreover, ifL = LΠ 1

2 , thenS ′ = 〈S, ⊕, ¬L, �, →P , [0], [1], [12 ]〉 isanLΠ 1

2 algebra.(iii) Let be the map fromL formulas into terms ofL algebras defined

by: (a): (pi) = xi (pi any propositional variable); (b):c = c, c anypropositional constant; (c): commutes with all connectives. Then, forevery formulaϕ ofL, one has:L � ϕ iff ϕ is true in everyL algebra iffϕ = 1 is true in everyL algebra iffϕ is true in the structureS definedin (ii) iff ϕ = 1 is true inS.

Proof. (i) The claim for ⊕ and ¬L is a well-known property of theLukasiewicz logic. Now suppose that is one of�, →P .If L � A ↔L B andL � C ↔L D, thenL � ∆(A ↔L B), andL � ∆(C ↔L D), (by the ruleA/∆A). By Definition 2, 2) and ModusPonens we get the claim.(ii) Almost trivial.(iii) Straightforward. ��

4. Subdirect decompositions

Lemma 3. LetA be aMV∆ algebra with additional operations, and letA′be the underlyingMV∆ algebra. Suppose that, for every n-ary operationf

52 F. Esteva et al.

and for everyx1 · · ·xn, y1 · · · yn ∈ A one has2:

(+)n∧

i=1

∆(xi ↔L yi) ≤ (f(x1 · · ·xn) ↔L f(y1 · · · yn))

Then,A andA′ have the same congruences. In particular,A is subdirectlyirreducible iffA′ is.

Proof. Clearly, every congruence ofA is a congruence ofA′. For the con-verse, observe that, for every congruenceθ ofAor ofA′ and for allx, y ∈ A,one has:xθy iff (x ↔L y)θ1. Now letθ be any congruence ofA′. We provethat, for everyn-ary function symbolf , and for allx1 · · ·xn, y1 · · · yn ∈ A,if xiθyi for i = 1 · · ·n, thenf(x1 · · ·xn)θf(y1 · · · yn). Now, fori = 1 · · ·n,(xi ↔L yi)θ1, hence∆(xi ↔L yi)θ∆(1) = 1, and thus

n∧i=1

∆(xi ↔L yi)θ1.

By Condition (+),(f(x1 · · ·xn) ↔L f(y1 · · · yn))θ1, and finally

f(x1 · · ·xn)θf(y1 · · · yn).

So,θ is a congruence ofA. It follows thatA andA′ have the same congru-ences. Since being subdirectly irreducible only depends on the congruencelattice, we also obtain thatA is subdirectly irreducible iffA′ is. ��Lemma 4. A MV∆ algebra is subdirectly irreducible iff it is linearly or-dered.

Proof. Due to the fact proved in [H98] that everyMV∆ algebra is a subdirectproduct of linearly orderedMV∆ algebras, it is clear that a subdirectlyirreducibleMV∆ algebra must be linearly ordered. Therefore the only thingto prove is that linearly orderedMV∆ algebras are subdirectly irreducible.Actually we will show more, we will show that they are not only subdirectlyirreducible but also simple. To this end, we prove that the only filters of alinearly orderedMV∆ algebraL are 1 andL itself. Indeed, filters ofMV∆

algebras are filters which are closed by∆, that is, verifying that ifa ∈ Fthen∆a ∈ F as well (cf. [H98]). We shall refer to them as∆-filters forshort. Now if a∆-filter F of a l.o.MV∆ algebra has an elementa �= 1, then,∆a = 0 ∈ F and thereforeF = L. Thus, linearly orderedMV∆ algebrasare simple, and thus subdirectly irreducible. ��

2 A similar condition was investigated by Pavelka in [Pa79, Part I] under the name ofadmissibleoperation.


Theorem 4. EveryLΠ algebra (LΠ 12 algebra respectively) is isomorphic

to a subdirect product of linearly orderedLΠ algebras (LΠ 12 algebras

respectively).

Proof. By Birkhoff’s Theorem (Cf. [MMT]), everyLΠ (LΠ 12 algebra is

isomorphic to a subdirect product of subdirectly irreducibleLΠ (LΠ 12 al-

gebras respectively). By Lemma 3, and Condition (4) in Definition 4 anLΠalgebra (LΠ 1

2 algebra respectively) is subdirectly irreducible iff the under-lying MV∆ algebra is. Now, by Lemma 4, aMV∆ algebra is subdirectlyirreducible iff it is linearly ordered. The claim follows. ��Corollary 1. Let Φ be any universal Horn first-order formula in the lan-guage ofLΠ algebras (LΠ 1

2 algebras respectively). IfΦ is true in everylinearly orderedLΠ (LΠ 1

2 ) algebra, thenΦ is true in everyLΠ (LΠ 12 )

algebra.

Corollary 2. Let the mapping defined in (iii) in Lemma 2. A formulaϕis provable inLΠ (LΠ 1

2 ) iff the corresponding identityϕ = 1 is true inevery linearLΠ (LΠ 1

2 ) algebra.

Proof. One direction is soundness. For the other direction, ifϕ = 1 is truein every linearLΠ (LΠ 1

2 ) algebra, then, by the above corollary, it is also truein everyLΠ (LΠ 1

2 ) algebra, in particular in the corresponding algebras ofprovably equivalent formulas (see (ii) of Lemma 2), but, obviously,ϕ = 1is true in these algebras iffLΠ (LΠ 1

2 ) provesϕ. ��Proposition 3. In any linearly orderedLΠ algebra, the following condi-tions hold:

(i) For everyx, ∆(x) ∈ {0, 1}.(ii) ∆(x) = 1 iff x = 1.(iii) The elements of the form∆(x) constitute a Boolean algebra wrt¬L,

∨,∧. Moreover, the following families of connectives collapse on suchBoolean algebra:{¬L, ¬P }, {∨, ⊕}, {∧, ⊗, �}.

(iv) x ≤ y iff x →P y = 1.(v) x� (x →P y) = x ∧ y.(vi) ¬P (x� x) = ¬Px(vii) If x� y = 0, then eitherx = 0 or y = 0(viii) ¬P (x) ∈ {0, 1}.(ix) ¬P (x) = 1 iff x = 0.(x) ¬P (x) ≤ ¬L(x) and∆(x) = ¬P¬L(x).(xi) (x →P y) ∨ (y →P x) = 1.(xii) if x� y ≤ x� z andx �= 0, theny ≤ z.(xiii) ¬P¬Px ≤ ((x� y →P x� z) →P (y →P z)).

54 F. Esteva et al.

(xiv) z ≤ x →P y iff z � x ≤ y.(xv) x →P y ≤ x →L y.

Proof. (i) It holds in every linearly orderedMV∆ algebra.(ii) It holds in every linearly orderedMV∆ algebra.(iii) Easily folllows from (i).(iv) One direction is (vii) of Proposition 2. Now, letx →P y = 1 and

supposey < x. Then by (vi) of Proposition 2,y = x� (x →P y) =x� 1 = x, contradiction.

(v) Direct consequence from (vi) and (vii) of Proposition 2.(vi) The inequality¬P (x�x) ≥ ¬Px is easy. For the converse inequality

notice that, due to Axiom 5 of Definition 6, it is0 = x∧(x →P 0) and,by (iv), ¬P 0 = 1. Then, using (v) we have the following identities:1 = (x ∧ (x →P 0)) →P 0

= x� (x →P (x →P 0)) →P 0= (x →P (x →P 0)) →P (x →P 0)= (x� x →P 0) →P (x →P 0)= ¬P (x� x) →P ¬Px

and thus, by (iv) again,¬P (x� x) ≤ ¬Px.(vii) Assumex ≤ y. Thenx � x ≤ x � y, and therefore,¬P (x � y) ≤

¬P (x� x), and by (vi),

¬P (x� y) ≤ ¬Px.

Analogously, if we now assume thaty ≤ x we get

¬P (x� y) ≤ ¬P y,

that is, we have proved that

¬P (x� y) ≤ max(¬Px,¬P y).

Therefore, ifx�y = 0,¬P (x�y) = 1 and hencemax(¬Px,¬P y) =1, hence eitherx = 0 or y = 0.

(viii) If x �= 0, then0 < x, therefore, by (vi),x � ¬P (x) = x � (x →P

0) = x∧0 = 0. By (vii), we conclude¬P (x) = 0. On the other hand,¬P (0) = 0 →P 0 = 1, by (iv).

(ix) It easily follows from (iv).(x) Obvious from (viii), (ix) and (ii).(xi) Obvious from (iv).(xii) If x � y ≤ x � z, then, if x > 0, by axiom (3) of Definition 4,

0 = (x � y) � (x � z) = x � (y � z), thus, by (vii),(y � z) = 0,thusy ≤ z.


(xiii) Notice that¬P¬Px = 1 iff x > 0. Then assumex > 0, and wehave to prove(x� y) →P (x� z) ≤ y →P z. Notice that, by (xii),{t | t � x � y ≤ x � z} ⊆ {t | t � y ≤ z}. Therefore, due to theresiduation property(x � y) →P (x � z) = max{t | t � x � y ≤x� z} ≤ max{t | t� y ≤ z} = y →P z.

(xiv) For one direction, ifz ≤ (x →P y) thenx � z ≤ x � (x →P

y) = x ∧ y ≤ y. For the other direction ifx� z ≤ y andx > 0 thenx�z ≤ x∧y = x�(x →P y) and now by (xii) we getz ≤ x →P y.

(xv) Since, by (iv) of Proposition 2,x� y ≥ x⊗ y, we havey ≥ x∧ y =x� (x →P y) ≥ x⊗ (x →P y) and thus, by the residuation propertyof the pair(⊗,→L), x →P y ≤ x →L y. ��

Corollary 3. Properties (ii), (iii), (iv), (v), (vi), (ix), (x), (xi), (xii), (xiii) and(xiv) of Proposition 3 can be expressed by universal Horn formulas, so theyare true in everyLΠ algebra.

Notation 4. By Proposition 3 (iii) and by Corollary 3, for everyLΠ algebraA, the set of all elements ofAof the form∆(x) constitutes a Boolean algebrawrt ∨,∧,¬L. This algebra will be denotedBA.

Now we are in position to give a proof for Theorem 3 of Sect. 3.

Theorem 5. If A = 〈A,⊕,¬L,�,→P , 0, 1〉 is aLΠ algebra then:(i) 〈A,⊕,¬L,∧,∨, 0, 1〉 is a MV algebra(ii) 〈A,�,→P ,∧,∨, 0, 1〉 is a Product algebra

Proof. (i) directly follows from the definition ofLΠ algebra (cf. Definiton4). For proving (ii), due to the decomposition theorems forLΠ andΠalgebras, it suffices to show it for linearly ordered algebras. But now, theconditions for〈A,�,→P ,∧,∨, 0, 1〉 being a Product algebra are proved in(2) of Definition 4, (ii) of Proposition 2, and (v), (xi), (xiii) and (xiv) ofProposition 3. ��

As a consequence, since Lukasiewicz and Product axioms are tautologiesover any linear MV and Product algebras, they are tautologies in any linearLΠ algebra, and thus, by Corollary 2, they are provable inLΠ logic. Thiscompletes the proof of Theorem 3. By completeness of Lukasiewicz andProduct logics, it also follows that all their theorems are also theorems ofLΠ andLΠ 1

2 .

5. Linearly ordered LΠ and LΠ 12 algebras

and linearly ordered fields

Definition 7. A linearly ordered commutative unitary ringR is said to bea discretely ordered ring iffR |= ¬∃x(0 < x ∧ x < 1).

56 F. Esteva et al.

Definition 8. Let F = 〈F,+,−,�,≤, 0, 1〉 be either a linearly orderedfield, or a discretely ordered ring. Letx−1 denote the multiplicative in-verse ofx if such a multiplicative inverse exists, and0 otherwise. LetA ={x ∈ F : 0 ≤ x ≤ 1}. Define, for allx, y ∈ A, x⊕ y = min {(x + y), 1}and¬Lx = 1−x. By abuse of language, let us still denote by� the restric-tion of the product operation toA.

x →P y ={

1 if x ≤ yy � x−1 otherwise

The algebra〈A,⊕,¬L,�,→P , 0, 1〉 is called the intervalLΠ algebra ofF . Moreover ifF is a linearly ordered field, defining12 = 2−1, the algebra〈A,⊕,¬L,�,→P , 0, 1, 1

2〉 is called the intervalLΠ 12 algebra ofF

Notice that the intervalLΠ algebra of a discretely ordered ring is thetrivial two-element algebra{0, 1}. By abuse of language, we will use theexpressioninterval algebra ofF to denote structures on a weaker language.In that case, it is meant that only the symbols of the language are interpreted,according to Definition 8. Thus, e.g. if we do not interpret1

2 , we speak ofthe intervalLΠ algebra ofF , etc.

Lemma 5. LetF be either a linearly ordered field, or a discretely orderedring. Then, the intervalLΠ algebra ofF is anLΠ algebra. IfF is a linearlyordered field, the intervalLΠ 1

2 algebra ofF is anLΠ 12 algebra.

Proof. Almost trivial. ��We wish to prove the converse of Lemma 5.

Theorem 6. A is a linearly orderedLΠ algebra if, and only if,A′, thestructure obtained fromA by omitting the interpretation of→P , is iso-morphic to the interval algebra of a linearly ordered commutative unitarydomain of integrity satisfying the following divisibility condition: for allx, y ∈ A′, x < y there existsz ∈ A′ such thatx = y � z.

Proof. We prove both directions.

(a) It is known (see for example [COM95]) that it is possible to embed alinearly ordered MV-algebra into a linearly ordered abelian group (l.o.a.g.).In particular, given a L´Π-algebraA = 〈A,⊕,¬L,�,→P , 0, 1〉, consider itsMV subalgebraAMV = 〈A,⊕,¬L, 0, 1〉 and the l.o.a.gGA = 〈GA,+,−,0G,≤G〉 whereGA = {(n, x) | n ∈ Z, x ∈ A, x �= 1}, 0G = (0, 0) and

(n, x) + (m, y) ={

(n + m,x⊕ y), if x⊕ y < 1(n + m + 1, x⊗ y), if x⊕ y = 1

−(n, x) ={

(−n, 0), if x = 0(−(n + 1),¬Lx), if 0 < x < 1

(n, x) ≤G (m, y) if n < m or n = m andx ≤ y.


It can be shown thatGA is a l.o.a.g. with neutral element (0,0) and strongunit (1,0) and that the MV-algebraAMV is isomorphic to the interval[(0, 0), (1, 0)] = {(n, x) ∈ GA | (0, 0) ≤G (n, x) ≤G (1, 0)}, identifying(0, x) with x and (1,0) with 1.

On the other hand it is possible to define a product operation× onGA,extension of the product� of the algebraA. The product is defined asfollows:

(n, x)× (m, y) = (nm, x� y) + m(0, x) + n(0, y),

wherem(0, x) means: the sum(0, x) + . . .+ (0, x), m times ifm > 0; thesum(−(0, x)) + . . .+ (−(0, x)), |m| times ifm < 0; and0(0, x) = (0, 0).It can be checked that

RA = 〈GA,+,−,×, (0, 0), (1, 0),≤G〉is a commutative, unitary and linearly ordered ring (see [EG98] for a detailedproof). Moreover,RA is a domain of integrity: from(n, x)×(m, y) = (0, 0),we first can easily deduce that eithern = 0 or m = 0. If n = m = 0, thenx�y = 0, and eitherx = 0 ory = 0, asA is a linearly orderedLΠ algebra.Now supposen = 0andm �= 0. Then(n, x)×(m, y) = m(0, x)+(0, x�y).Supposex �= 0. Let, for (n, x) ∈ RA, |(n, x)| denote(n, x) if (n, x) ≥G

(0, 0) and−(n, x) otherwise. We obtain:

|n(0, x) + (0, x� y)| ≥G |(0, x� (x� y))| = |(0, x� ¬Ly)|Now, y �= 1, therefore¬Ly �= 0, x � ¬Ly �= 0 (asA is a linearly orderedLΠ algebra), and a contradiction is reached. The case wheren = 0 andm �= 0 is symmetric.

It remains to prove the divisibility condition inRA, but this reduces toprove this condition for linearly orderedLΠ algebras. Now, ify ≤ x, bycondition (7) of Definition 4∆(y →L x) = 1 and thusx� (x →P y) = y.Hence,x →P y is the required element.

(b). Now, let(A,+,−,×,≤, , 0, 1) be a commutative linearly ordered do-main of integrity with unit 1 and satisfying the divisibility condition for×. Let [0, 1]A = {x ∈ A | 0 ≤ x ≤ 1}. Then it is easy to check that〈[0, 1]A,⊕,¬L,�,→P , 0, 1〉 is a LΠ algebra with the operations definedby:

x⊕ y = min(1, x + y)¬Lx = 1− xx� y = x× y

x →P y ={

1 if x ≤ yz otherwise

58 F. Esteva et al.

wherez, due to the divisibility condition, is the only element such thaty = x� z. ��Notation 5. x/y will also used to denotey →P x, as usual in rings satis-fying the divisibility condition.

Theorem 7.(a) Every linearly orderedLΠ algebra with more than twoelements is aLΠ 1

2 algebra.(b) The intervalLΠ algebra ofQ can be embedded in every non-trivial

LΠ 12 algebra.

(c) Every linearly orderedLΠ 12 algebra is isomorphic to the intervalLΠ 1

2algebra of a linearly ordered field.

(d) Every linearly orderedLΠ algebra is isomorphic either to the intervalLΠ algebra of a linearly ordered field or to the intervalLΠ algebra ofZ.

Proof.(a) LetA be any linearly orderedLΠ algebra with more than two elements.By Theorem 6, we can safely asume thatA is the interval algebra of a linearlyordered commutative unitary domain of integrityB. Leta ∈ A be such that0 < a < 1, and letb = min {a, 1− a}. Clearly,0 < b < b+b = b⊕b ≤ 1.Let c = b/(b ⊕ b) (remind thatx/y is an abbreviation fory →P x). Wehave:b = b ∧ (b⊕ b) = (b/(b⊕ b))� (b⊕ b) (by Proposition 3, (vi)). So,in B we have:b = c � b + c � b, andb � (1 − (c + c)) = 0. It follows:1− (c+ c) = 0, c = 1− c, and finallyc = ¬Lc. Thus,A is aLΠ 1

2 algebra.Clearly, there is a uniquec such thatc = ¬Lc, call it 1

2 .

(b) Let A be any non-trivial linearly orderedLΠ 12 algebra. Define, for

n ∈ N and fora ∈ A, na = a ⊕ · · · ⊕ a n times (0a = 0). We define foreveryn ∈ N, 1

n as follows. Let12 = 12 . We further define:10 = 1

1 = 1.Finally, forn ≥ 2, we define

1n + 1

=(

12� 1

n

)/

(n + 1

(12� 1

n

))

It is easily seen that the mapψ fromQ∩ [0, 1] intoA defined by:ψ(0) = 0;ψ( n

m) = n 1m if 0 < n ≤ m is a monomorphism from the intervalLΠ

algebra ofQ intoA (see [M98] for a detailed proof). Thanks to this result,in the sequel we identify any element of the formn 1

m with the correspondingrational numbernm .

(c) By the previous theorem, ifA is any linearly orderedLΠ 12 algebra, then

the algebraA′ obtained fromA by omitting the interpretations of→P and12 is the interval algebra of a linearly ordered integral domain,R say. LetF be the fraction field ofR. It suffices to prove that everyc ∈ F ∩ [0, 1] is


in A. Writing, for z ∈ Z and forα ∈ A, z + α instead of(z, α), we canrepresent anyc ∈ F ∩ [0, 1] asc = z+α

y+β , whereα, β ∈ A, α < 1, β < 1,z, y ∈ Z, z ≥ 0, y ≥ 0, if y = 0, thenβ �= 0, and eitherz < y, or z = yandα ≤ β. Now, if y = 0, thenz = 0, andc = α/β ∈ A. Otherwise, let

d =(

12� 1

y

)/

(12⊕

(12� 1

y� β

))

It is easy to check thatc = zd⊕ (α� d) ∈ A.

(d) LetA be anyLΠ algebra. IfA has more than two elements, then, by(b), it is aLΠ 1

2 algebra, and the claim follows from (c). Otherwise,A isobviously isomorphic to the intervalLΠ algebra ofZ. ��At this point, we are in a position to prove the Completeness Theorem forLΠ andLΠ 1

2 .

Theorem 8. (Completeness)Let ∗ be the map defined in Lemma 2, letLbe eitherLΠ or LΠ 1

2 , and letϕ be anyL formula. The following areequivalent:

(i) L � ϕ.(ii) ϕ∗ = 1 is true in everyL algebra.(iii) ϕ∗ = 1 is true inIL.

Proof. (i) and (ii) are equivalent by Lemma 2. In order to prove that (ii) and(iii) are in turn equivalent, it is sufficient to prove next Lemma 6. ��

Recall that byIL we mean the structure whose domain is the real interval[0, 1], and whose constants and operations are the natural interpretations ofthe propositional constants and connectives ofL.

Lemma 6. LetL again be eitherLΠ or LΠ 12 . ThenIL is functionally free

in the equational class ofL algebras. In other words: LetΦ be any equationin the language ofL algebras. Then,Φ is true in the intervalIL iff Φ is truein everyL algebra.

Proof. For the non-trivial direction, argue contrapositively. Suppose thatΦ is not valid in someL algebraA. We know thatA is isomorphic to asubdirect product of a family{Ai : i ∈ I} of linearly orderedL algebras.It follows thatΦ is not true in at least oneAi. Now,Ai is isomorphic to theinterval algebra of a linearly ordered integral domainRi. We can expandRi first to its fraction field and then to its real closure, call itK. Clearly,Ai is a substructure of the interval algebraB of K. Since equations arepreserved under taking subalgebras,Φ is not true inB. Since the operationsof the intervalL algebra of a linearly ordered field are definable in the field,andK, being real closed, is elementarily equivalent toR, B is elementarilyequivalent toIL. So,Φ is not true inIL. ��

60 F. Esteva et al.

Corollary 4. LΠ extends bothL andΠ conservatively

Proof. We prove thatLΠ extends Product Logic conservatively, i.e., ifϕis any formula of Product LogicΠ, thenLΠ � ϕ iff Π � ϕ; the proof forLukasiewicz logicL is analogous. By the Completeness theorem forLΠ,LΠ � ϕ iff e(ϕ) = 1 for every evaluatione of LΠ into ILΠ . Since everyevaluation ofΠ into IΠ has exactly one extension to an evaluation ofLΠinto ILΠ , LΠ � ϕ iff h(ϕ) = 1 for every evaluationh of Π into IΠ . Bythe Completeness ofΠ, this holds iffΠ � ϕ. ��

Theorem 9. (Finite Strong Completeness)If T is a finite theory over L´Π,thenT |= ψ impliesT � ψ.

Proof. AssumeT = {ϕ1, . . . , ϕn} and thate(ψ) = 1 for every evaluatione of LΠ into ILΠ such thate(ϕi) = 1 for i = 1, . . . , n. We want toprove thatT � ψ. Notice thate(ϕi) = 1 for i = 1, . . . , n iff e(ϕ1 ⊗. . . ⊗ ϕn) = 1. Therefore the hypothesis can be equivalently expressed as(∆(ϕ1⊗ . . .⊗ϕn) → ψ)∗ = 1 being true inILΠ . By completeness of L´Π,∆(ϕ1 ⊗ . . .⊗ ϕn) → ψ is provable in L´Π. But if ∆(ϕ1 ⊗ . . .⊗ ϕn) → ψis provable then, using necessitation for∆ and modus ponens, we have thatϕ1 ⊗ . . . ⊗ ϕn � ψ, and thus we haveT � ψ as well since, obviously,T � ϕ1 ⊗ . . .⊗ ϕn. ��

6. Rational LΠ logic

In this section we investigate Pavelka’s style completeness forLΠ, in asimilar way it was originally done by Pavelka for Lukasiewicz logic [Pa79],and later simplified and further elaborated by Hajek [H98], or for productlogic in [EGHN98]. The idea of Pavelka’s approach was to introduce inthe language a truth constantr for each rationalr ∈ [0, 1], together with aset of book-keeping axioms for them, allowing for deductions from and ofpartially true propositions.

If we start fromLΠ 12 , we don’t need to introduce the truth constants

since rationals are definable inside the logic, as it has been shown in The-orem 7 (b) forLΠ 1

2 algebras. Moreover, the corresponding book-keepingaxioms

(RLΠ1) ¬Lr ≡ 1− r(RLΠ2) r →L s ≡ r →L s(RLΠ3) r � s ≡ r � s(RLΠ4) r →P s ≡ r →P s

are provable inLΠ 12 logic, by completeness ofLΠ 1

2 . Notice that in the


above expressions,→L, � and→P in the right hand side of the equiv-alences denote the Lukasiewicz implication, Product t-norm and Productimplication in the unit interval [0, 1].

Analogously to the case of Rational Product logic, due to the disconti-nuity of Product implication on (0, 0), we need to consider the followinginfinitary inference rule:

(IR): fromϕ →P r, for eachr > 0, deriveϕ →P 0.

This is the only thing we need to add toLΠ 12 .

Definition 9. The language and axioms of RationalLΠ logic, RLΠ, arethe language of and axioms ofLΠ 1

2 . Inference rules ofRLΠ are modusponens, necessitation for∆ and IR.

Notice thatRLΠ, as defined, is an extension of both rational Pavelkalogic and rational Product logic. We introduce now some basic notions.

A theoryT overRLΠ is just a set of formulas. The setCnRLΠ(T )of all provable formulas inT is the smallestT ′ containingT as a subset,containing all axioms ofRLΠ and closed under all deduction rules. Forsimplicity we shall denoteϕ ∈ CnRLΠ(T ) byT � ϕ. By definition, a theoryT is consistentif T �� 0. Further, a theoryT is completeif T � (ϕ →L ψ)or T � (ψ →L ϕ) for each pairϕ,ψ. An evaluatione of LΠ 1

2 formulasover the real unit interval [0, 1] is a model of a theoryT if e(ϕ) = 1 forall ϕ ∈ T . The notions of provability and truth degree of a formulaϕ in atheoryT , denoted by|ϕ|T and||ϕ||T respectively, are as usual:

|ϕ|T = sup{r | T � r →L ϕ}‖ϕ‖T = inf{e(ϕ) | e is a [0, 1]-valued model ofT}

Notice that, we can equivalently use→Π instead of→L in the above def-initions since in any theoryT over LΠ we have thatT � ϕ →L ψ iffT � ϕ →Π ψ.

Our first aim is to show completeness forRLΠ in the Pavelka-style. Themain steps are the following.

Lemma 7. T � 0 iff T � r for somer < 1.

Proof. If T � r for somer < 1, thenT � rn for each naturaln, and thusT � r′ for anyr′ < 1. Then the infinitary rule does the job. ��

Next step is just to check that the following known three results forrational Pavelka logic easily extend toRLΠ (cf. [H98] 2.4.2, 3.3.7 and3.3.8 (1) respectively).

Lemma 8.

1. Each consistent theoryT can be extended to a consistent and completetheoryT ′.

62 F. Esteva et al.

2. If T does not prove(r →L ϕ) thenT ∪ {ϕ →L r} is consistent.3. If T is complete, thensup{r | T � r →L ϕ} = inf{r | T � ϕ →L r}.

Lemma 9. If T is complete, the provability degree commutes with connec-tives, this is, we have the following equalities:

|¬Lϕ|T = 1− |ϕ|T|ϕ →L ψ|T = |ϕ|T →L |ψ|T|ϕ� ψ|T = |ϕ|T � |ψ|T

|ϕ →P ψ|T = |ϕ|T →P |ψ|Twhere, again, in the left-hand side of the equalities→L,� and→P denoteconnectives whereas in the right-hand side they denote the correspondingtruth-functions in [0, 1].

Proof. The proof for the connectives whose truth functions are continuousin [0, 1], i.e.¬,→L and⊗, is not difficult and the reader may consult in [H98]the proof for Rational Pavelka logic. Moreover, the truth function of productimplication→Π is also continuous in every point(x, y) ∈ [0, 1]×[0, 1] suchthatx > 0. Therefore, analogously to rational Product logic, we have onlyto actually check that|ϕ →Π ψ|T = 1 when|ϕ|T = 0. We reproduce theproof of it here. Assume|ϕ|T = 0 = inf{r | T � ϕ →Π r}. This meansthatT � ϕ →Π r for everyr > 0, and using the infinitary rule,T provesϕ →Π 0. But 0 →Π ψ is provable in product logic, and thusT also provesϕ →Π ψ, and thus|ϕ → ψ|T = 1. ��

Theorem 10 (Pavelka style completeness).In RLΠ the provability andtruth degrees coincide, i.e. it holds||ϕ||T = |ϕ|T , for any theoryT and anyformulaϕ.

Proof. One inequality,|ϕ|T ≤ ||ϕ||T , is soundness. The other inequality,|ϕ|T ≥ ||ϕ||T , is just a consequence of the preceeding lemma, since, ifTis a complete extension ofT , the mappingeT defined ase(ϕ) = |ϕ|T isindeed a truth evaluation and evaluates all formulas ofT to 1. Therefore, bydefinition of truth degree it must be||ϕ||T ≤ eT (ϕ). ��

Notice that Pavelka’s style completeness is weaker than strong complete-ness. If|ϕ|T = 1 it only means thatT � r →L ϕ for all rationalr < 1.However due to the infinitary deduction rule we can get strong completenessin RLΠ with respect to the unit interval algebra [0, 1]3.

Corollary 5. (strong completeness)For any theoryT overRLΠ and anyformulaϕ, T � ϕ iff ϕ is 1-true in all [0, 1]-valued models ofT .

3 It is worth noticing that the infinitary inference rule (IR) is not necessarily sound in allLΠ 1

2 algebras.


Proof. It suffices to show that if|ϕ|T = 1 thenT � ϕ. So, supposeT �r →L ϕ for all rationalsr < 1. This is equivalent toT � r →Π ϕ forall rationalsr < 1. By reasoning in L´Π logic4, T proves¬ϕ →Π 1− rfor all r < 1, that is,T proves¬ϕ →Π r for all r > 0. Now, using theinfinitary inference rule,T proves¬ϕ →Π 0, and again by contraposition,T � ¬0 →Π ϕ, i.e.T � ϕ. ��

7. LΠ algebras in [0, 1]

To study theLΠ algebras and subalgebras in the real unit interval [0, 1], wefirst recall the following results.

Proposition 4. ([A85]) If S is a continuous t-conorm, T is a continuoust-norm and N is a strong negation, the general solution of the functionalequation

S(T (x, y), T (x,N(y))) = x

is given by the following expressions:

S(x, y) ={g−1(g(x) + g(y)), if g(x) + g(y) ≤ 11, otherwise

T (x, y) = g−1(g(x) · g(y))N(x) = g−1(1− g(x))

whereg is a continuous strictly increasing functiong : [0, 1] → [0, 1] suchthatg(0) = 0 andg(1) = 1. That is, up to an isomorphism,S is Lukasiewiczdisjunction,T is product conjunction andN is Lukasiewicz negation.

Proposition 5. ([Hoe95,BEG98]) Let∗ be a t-norm and let→ be itsresiduum. The pair(∗,→) satisfies the following conditions:

(1) Residuation:x ≤ y → z if and only ifx ∗ y ≤ z(2) x ∗ (x → y) = min(x, y)

if, and only if,∗ is continuous with respect to the usual topology in[0, 1].

As a consequence of these results and properties ofLΠ-algebras shownin this paper we obtain the following characterization ofLΠ-algebras in [0,1].

Proposition 6. An algebra A = ([0, 1],⊕,¬L,�,→P , 0, 1) is aLΠ-algebra if and only if⊕ and� are respectively a t-conorm and a t-norm given by the following expressions:

4 In RLΠ the following is a derived inference rule: fromϕ →Π ψ derive¬ψ →Π ¬ϕ.

64 F. Esteva et al.

x⊕ y ={g−1(g(x) + g(y)), if g(x) + g(y) ≤ 11, otherwise

x� y = g−1(g(x) · g(y))¬Lx = g−1(1− g(x))

whereg is a continuous strictly increasing functiong : [0, 1] → [0, 1] suchthat g(0) = 0 andg(1) = 1. That is, up to an isomorphism,⊕ and¬L areLukasiewicz t-conorm and negation respectively, and� is product t-norm.

Proof. SinceA is aLΠ-algebra,(⊗,→L) and(�,→P ) satisfy conditions(1) and (2) of Proposition 5 and thus� and⊕ are continuous t-norms. On theother hand, the identity(x� y)⊕ (x�¬Ly) = x holds in anyLΠ-algebra,thus the claim easily follows from Proposition 4. ��

This proposition tells us that, up to isomorphisms, the onlyLΠ algebra on[0, 1] is the standard one. Let us see now what happens with its subalgebras.

It is known (see for instance [COM95]) that the subalgebras of the stan-dard Lukasiewicz algebra can only be of two types:

– Finite subalgebras: algebras of n elements, denoted byLn, for n ∈ N ,and whose elements are the set{ i

n | 0 ≤ i ≤ n}– Infinite subalgebras: Infinite subsets of [0,1] closed by⊗ and¬L. It is

proved that these subsets are dense in [0,1].

Moreover, if we denote byTaut( ) the set of tautologies, then:

– Finite case:Taut(Lm) ⊆ Taut(Ln) if and only ifn− 1 dividesm− 1.– If S is an infinite subalgebra of [0,1], thenTaut(LS) = Taut(L[0,1]).

Now, regardingLΠ-algebras, we have shown that any l.o.LΠ-algebra mustcontain a subalgebra isomorphic toQ ∩ [0, 1]. Thus there not exist finitesubalgebras of the standardLΠ algebra. Moreover the tautologies withrespect toQ ∩ [0, 1], asLΠ-algebra, stricly contain the tautologies withrespect to[0, 1], as the following example shows.

Let φ be the formula[(p ∧ ¬Lp) ⊕ (p ∧ ¬Lp)] →P (p ∧ ¬Lp). It canbe easily shown that, for any assignmente to the variablep such that0 <e(p) < 1, the evaluation ofφ is always1/2. Now, letψ denote the formula

¬P¬P (p ∧ ¬Lp) →L (∆((p� p) ↔ φ) →P 0).

We show thatψ is a tautology with respect toQ∩ [0, 1] but not w.r.t.[0, 1].Namely, assigning the value

√1/2 to p, the evaluation ofψ is 0 and thusψ

is not a tautology in[0, 1]. However, an easy checking shows that, for anyother assignment, the evaluation ofψ is 1. Then, since

√1/2 �∈ Q, ψ is a

tautology with respect toQ ∩ [0, 1].


8. Conclusions and final remarks

In this paper we have been concerned with defining and axiomatizing twostrong systems of fuzzy logic,LΠ andLΠ 1

2 , which contain both connectivesof Lukasiewicz and Product logics. Completeness results have been obtainedby investigating the corresponding algebraic structures, in the same spiritas most of the recent developments of mathematical fuzzy logic (see e.g.[H98]).

It turns out that most of the existing systems of fuzzy logic are faithfullyinterpretable inLΠ and inLΠ 1

2 . In ILΠ , the Godel t norm∧ and the Godelimplication→G can be defined in terms of⊕, �, � and∆ as follows:

(✷) x ∧ y = x� (x� y) x →G y = y ⊕∆(x →L y)

Using(✷), we can translate every formulaϕ of Godel’s fuzzy LogicG into aformulaϕ+ of LΠ simply replacing every connective ofG by its definitionby means of(✷). Now we can argue as follows:Since (✷) defines the operations corresponding to Godel’s t norm andGodel’s implication inILΠ , if a formulaϕ is provable inG, thenILΠ |=(ϕ)∗ = 1. Thus, by the Completeness Theorem forLΠ, LΠ � ϕ+.

Conversely, ifLΠ � ϕ+, thenILΠ |= (ϕ+)∗ = 1. So,IG |= (ϕ)∗ = 1.By the Completeness ofG, G � ϕ. We conclude that

G � ϕ iff ILΠ |= (ϕ+)∗ iff LΠ � ϕ+.

So,+ is a faithful interpretation ofG intoLΠ.Due to Proposition 6, if we restrict ourselves to the product operations, the

standardLΠ algebra reduces to only one of the semi-standardΠ∼ algebras(the one such that the involutive negation is just1 − x). Therefore, unlikeG∼, Π∼ is not fully interpretable inLΠ.

As already noted in Sect. 7, the rational numbers of [0,1] are definable inLΠ 1

2 . From this, one easily obtains that both the Rational Pavelka LogicRPL and Rational Product logicRLΠ are faithfully intepretable inLΠ 1

2 .The interpretations are obtained by translating any connective into itself(LΠ 1

2 contains the connective of bothRPL andRLΠ), and, translating,for every rational numberr = n

m , the corresponding symbol inRPL (RLΠ)into n� 1

m .

Acknowledgements.The authors are deeply indebted to Petr Hajek, Daniele Mundici, andAntoni Torrens for many valuable suggestions and remarks. The third author wishes also tothank Giovanni Panti, who contributed to make him interested in this field.

66 F. Esteva et al.

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the logics: two complete fuzzy systems joining Łukasiewicz and product logics

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