systematization of finite many-valued logics through the method of tableaux

Systematization of Finite Many-Valued Logics Through the Method of TableauxAuthor(s): Walter A. CarnielliSource: The Journal of Symbolic Logic, Vol. 52, No. 2 (Jun., 1987), pp. 473-493Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2274395 .

Accessed: 11/07/2014 13:48

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to TheJournal of Symbolic Logic.

http://www.jstor.org

This content downloaded from 151.227.68.60 on Fri, 11 Jul 2014 13:48:46 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=asl

http://www.jstor.org/stable/2274395?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


THE JOURNAL OF SYMBOLIC LOGIC

Volume 52, Number 2, June 1987

SYSTEMATIZATION OF FINITE MANY-VALUED LOGICS THROUGH THE METHOD OF TABLEAUX

WALTER A. CARNIELLI 1

Abstract. This paper presents a unified treatment of the propositional and first-order many- valued logics through the method of tableaux. It is shown that several important results on the proof theory and model theory of those logics can be obtained in a general way.

We obtain, in this direction, abstract versions of the completeness theorem, model existence theorem (using a generalization of the classical analytic consistency properties), compactness theorem and Ldwenheim-Skolem theorem.

The paper is completely self-contained and includes examples of application to particular many-valued formal systems.

?1. Introduction. The main purpose of this work is to show how the formalization of finite many-valued logics, usually presented in the most diverse ways, can be unified under the point of view of analytic tableaux.

Many-valued logics can be considered as natural generalizations of classical logic, due to the fact that, as in the classical case, the truth values of propositional complex formulas depend functionally on the truth value of elementary subformulas. To maintain a similar relationship for quantified formulas, we introduce the notion of distribution quantifiers: these are quantifiers which have associated with them an interpretation function which maintains a connection with the quantifier analogous to that between tables and logical connectives.

In this paper we show that it is possible to present a tableau-type proof theory for every many-valued logic with distribution quantifiers, provided that among the quantifiers there are at least two which could be viewed as having the minimal properties of the universal and existential quantifiers. Indeed, our method allows the construction of the proof theory of any of those logics as soon as we know the tables and the interpretation functions.

This proof theory is sound and complete with respect to the semantics of the calculus, and basic results in model theory for many-valued first-order logics can be obtained in general, as for example the compactness theorem, the Ldwenheim- Skolem theorem, and the model-existence theorem. This suggests that such a

Received December 16, 1984; revised July 6, 1986. 'This paper was written while the author was visiting the Department of Mathematics of the

University of California at Berkeley, and was partially supported by the Fundasao de Amparo a Pesquisa do Estado de Sao Paulo, grant 83/1030-3.

C 1987, Association for Symbolic Logic

0022-4812/87/5202-001 2/$03. 1 0

473



474 WALTER A. CARNIELLI

uniform treatment could be used in a systematic study of the model theory of many- valued logics, since some model-theoretical problems are closely related to the combinatorial relationships among the connectives and quantifiers of the language considered (as examples of this point of view, we recall the prenex theorems and the joint-consistency theorem).

Our treatment also introduces for the quantifiers a problem similar to that of the interdefinability of the connectives in propositional logic (see [6]): what are the quantifiers, in a given language, that can generate, while operating together with the connectives, all the quantifiers in the language? This seems to be a very difficult question; we have succeeded in solving only partially the three-valued case elsewhere (cf. [2], where we proved, in a constructive manner, that there exist exactly 360 three-valued quantifiers of a certain kind, called perfect quantifiers, which can generate all other quantifiers in three-valued logics).

Most of the results can be extended to some classes of higher-order many-valued logics (e.g., using the many-valued correspondents of Henkin structures).

?2. Basic notions. The methods of proof theory known as analytic tableaux [9], Beth tableaux, or Hintikka tableaux are closely related and have been extensively studied for the case of classical first-order logic, intuitionistic logic and modal logics. The only attempt to generalize this approach to many-valued logics as far as we know is the paper by S. Surma [10], where analogues of the analytic tableaux of Smullyan [9] are considered for presenting a method for finite axiomatization of propositional many-valued logics. His presentation of tableaux, however, besides dealing with a unique distinguished value, is a bit redundant in the sense that it produces tableaux systems with an excess of rules, making the related completeness theorems practically trivial in some cases. A Gentzen's version of Surma's presentation for propositional logics is given by P. Borowik [0], which is closer to our tableau systems.

Our treatment for the propositional cases will make use of some notions similar to the ones presented in [10]; we then show that this approach can be extended to all many-valued first-order logics with any number of distinguished truth values.

We shall make use of some definitions and results with respect to abstract algebras which can be found in [5]. We recall mainly the notions of abstract algebra of finite type, similar algebras, subalgebra, algebra generated by a set SO, and free algebra in a class V1 of algebras.

Let So = { pi: i E N } be a set of propositional variables and C = {F1,. . ., Fi,..., FrI} a (finite) set of mi-ary propositional connectives; we define the propositional language over the alphabet SO u C as the abstract algebra S, freely generated in the class vd of all algebras of the same similarity through the connectives in C (where similarity here is given by the number of connectives and their arities); i.e.,

SO = {pi: i E N},

Sk+1 = Sku J{F(Xl,.., Xm,): X,,..., Xm. E SKand I < i < r} S = U{Sk:keN}.

The formulas of Sk +I - Sk are called formulas of level k + 1, and the formulas of So are called atomic formulas.



FINITE MANY-VALUED LOGICS 475

Let N={0,1,2,...,(n-1)}, let DcN, and let A=<N,fi:?<i<r> be an abstract algebra similar to S (i.e., each fi: Nmi -* N has the same arity as the corresponding propositional connective); in this case we call the pair W = <A, D> a pre-model for S, and the functions fi are called interpretations of the correspondent connectives Fi. The set N is called the set of truth values of A, and D the set of distinguished truth values of 'ff.

A propositional valorization (from S into -/), or, for short, a p-valorization, is a homomorphism v from S into d. It is obvious that a propositional valorization is completely determined if we know its values on the set SO of all propositional variables.

If d = IDj, we call the pair L = <S,t> an n-valued propositional logic with d distinguished values.

We shall use the following notation: B, C, D (with or without indices) for sets of formulas, and X, Y, Z (with or without indices) for formulas.

?3. Propositional tableaux. We shall consider informally the notions of trees, branches, nodes, height of nodes, etc.

We consider the following symbols:

X andX Y1 o Y2 o o''m0Y n YY + Y2 + + YY

as denoting trees of the following forms, respectively:

X X

and ?Y2 < x t

* Y1 Y2 .m

and we shall abbreviate those symbols by the following expressions:

X and X

of : i < ml + { Yj: i < m}~

We shall consider a fixed language S and sets N and D of truth values and distinguished values. Also we shall consider a set of symbols ao, a, .. ., an_ 1 called signs (out of the language S). If X is a formula in S, the symbol ai(X) is called a signed formula, for any aj, 0 < i < n - 1. The set of all signed formulas is denoted by S*.

If L = <S'it> is a fixed propositional logic, we define the rule ltiF (for each 0 < i < n - 1 and each connective F) as follows:

Let X be the formula schema F(X1,. . ., Xn) where F is an m-ary connective; we define the rule hiF as a function which associates the following tree to the signed formula ai(X):

ai(F(X,..., Xm)) + {aj1(Xil) 0 o aj,(Xt):j1,. . . ,t < n, t < m and

the propositional condition Hi(F;j1,. .. ,j,) holds},




where Hi(F; j1,..., j]) means that there exists a homomorphism h: S -* v such that:

1) h(Xk)=j k for 1 < k < t; and 2) if f represents the connective F, then f(v1,. .., vi,..., vi2, , Vik, ..., Vm) = ifor

all values of the function f, where Vik = Ik and the other v's are arbitrary; and 3) no t' < t satisfies 1) and 2). The signed formula ai(F(X,... ., Xm)) is called the premise of the rule, and each set

of formulas {aj,(X),... , aj,(XJ)} is called a 7r-consequence of ai(F(X1,...,Xm)) by the rule tiF-

If there exists no propositional condition Hi(F;j1,...J,]a), the rule r F is not defined.

We denote by ni the set of all rules hiF for all connectives F. Now, having defined all rules riF, 0 < i < n - 1, we call a propositional tableau

(or p-tableau for short) for a formula ai(F(X,... ., Xm)) any tree whose first node is the formula ai(F(X,... ., Xm)) and whose next nodes are determined by the following procedure:

P1) After putting the first node, the next immediate nodes are given by an application of 7iEF to ai(F(X1,..., Xm)) (following the convention that formulas separated by "o" go in the same branch, and sets of formulas separated by "+ " go into different branches).

P2) We then continue, extending each branch of the tree by adding new nodes throughfthe application of appropriate rules 7i F to any formulas which occur in this branch (that is, applying rules hiF to formulas of the form aiF(X1,. .., XM

Note that the definition of p-tableaux imposes no order on the choice of formulas used to extend the tree. This nondeterministic procedure is a natural place for considerations of a heuristic nature which may permit shorter proofs to be given improving the efficiency of the method in terms of the number of steps in the proof (although we do not treat this problem here).

Tableaux are designed to be permissive, but not obligatory: the objective of the construction of a tableau is to obtain a closed tableau (to be defined below), which is a halting condition in a computation procedure. This halting condition can be obtained with more or less efficiency, or even not be obtained at all in the case of tableaux for first-order logics. However we need to define a systematic, deterministic procedure which will obtain the desired halting condition if any other procedure will.

We define then the systematic propositional tableau (sp-tableau, for short) for a formula aiF(X,... ., Xm) as a p-tableau endowed with an order of application of the rules, which will be determined by the following procedure.

SP1) The same as P1), but declaring the node 1 as used. SP2) Suppose that the node n has been used; if all nodes where nonatomic

formulas occur have been used, we stop. Otherwise, we take as the node n + 1 the leftmost nonused node with least height.

SP3) Suppose that aj(X) is the formula in the node n + 1; we then add to each branch which passes through n + 1 the 7c-consequences of aj(X), using the appropriate rule.

The notions of subformula and immediate subformula are defined as usual.




We say that a p-tableau for ak(X) is closed if for each branch, either there exist nodes with ai(Y) and aj(Y) for i # j, or the branch contains some nonatomic formula ai(Y) and there exists no defined rule 7riF, where F is the main connective of Y. Otherwise, the tableau is said to be open.

REMARKS. 1) It follows from the definition that if X is an atomic formula, all tableaux for ai(X) are open.

2) It is clear that an sp-tableau for a formula ak(X) is well-determined, so we can refer to it as the sp-tableau for ak(X). Also, if there exists a closed p-tableau for ak(X), then the sp-tableau for ak(X) is also closed (this fact follows from the completeness theorem to be proved, but it should be intuitively clear).

In intuitive terms, we can consider a p-tableau for some signed formula ai(X) as closed when either the assumption that there exists a p-valorization v such that v(x) = i leads to a contradiction such as there exists a formula Y (namely, a subformula of X) which assumes distinct values under some p-valuation, or when there exists no possibility a priori of X assuming value i.

The theorems of the logic L considered will be, then, those formulas which have no possibility of assuming nondistinguished truth values; that is, those formulas X for which all tableaux ai(X) are closed, if i is a nondistinguished value. This implies that, in order to prove X, we have to show that there exists a closed tableau for each ai(X), i being a nondistinguished truth-value.

EXAMPLE 1. As an example of application let us consider the three-valued propositional calculus P1 (see [7]), whose matrices for the connectives are the following:

m * 0 1 2 v 0 1 2 & 0 1 2

0 2 0 0 0 2 0 0 0 0 0 0 0 2 1 0 1 0 0 2 1 0 0 0 1 0 0 2 2 0 2 0 0 0 2 0 0 2 2 2 2 2

(here N = {0, 1, 2} and D = {0, l}). According to our definitions, the rules will be the following:

(R- 1) ao(n X)

a,(X) + a2(X)'

(R-2) a(2 X) ao(X)

(R-3) ao(X -* Y) a2(X) + ao(Y) + a,(Y)'

(R-4) a2(X -_* Y) {a0(X) a a2(Y)} + {a1(X) a2MY

(R-5) ao(X v Y) a((X) + a,(X) + ao(Y) + al(Y)'

a2(X V Y) (R-6) {2X




(R-7) aO(X & Y)

{aO(X) ao(Y)} + {ao(X)? al(Y)} + {al(X) ao(Y)} + {al(X) al(Y)}

(R-8) a2(X & Y)

a2(X) + a2(Y)

Rule (R-3), for example, is justified in the following ways: a) The propositional condition Ho(-*; 2) holds, since -* (2, v) = 0 for all values of v

in {O, 1, 2}; this justifies the presence of a2(X) in the conclusion of the rule. b) The propositional condition Ho(-+; 0) holds, since -+ (v, 0) = 0 for all values of v

in {0, 1, 2}; this justifies the presence of ao(Y) in the conclusion of the rule. c) Analogously, the propositional condition Ho(-?; 1) holds and justifies the

presence of a1 (Y). It is interesting to note that the clause 3) of the definition of the rule iiF implies

that the rules are defined with maximal syntactic economy in terms of the number of formulas which go into the same branch.

For example, our definition excludes a variant of (R-3) of the form

(R-3)' ao(X -3+ Y) {a2(X) o a2(Y)} + ao(Y) + a1(Y)

because of (a) above and clause 3). This syntactic economy simplifies the rules and the resulting systems, but collides

with the search for efficiency, the latter being desirable from the point of view of heuristics in automatic theorem proving (which we confessed not to be considering here).

In other words, (R-3)' would be a perfectly acceptable rule, but the completeness theorem will show that it is not necessary.

The following is an example of an sp-tableau; the nodes have numbers for reference and [i] means repetition of the node (i). Also, the symbol * indicates a closed branch.

In the construction of this (closed) sp-tableau we are making use of some of the rules (R-1)-(R-8).

(1) a2((P-+Q)( P v Q))

(2) ao(P Q) (3) a1(P Q)

(4) a2( P v Q) (5) a2( -P v Q)

(6) a2P (7) aoQ (8) a (Q)

(9) a2( P) (11) [9] (13) [9]

(10) a2(Q) (12) [10] (14) [10]

(15) aoP * *

*




Let B = {X1,X2. .X s ...} be a set of formulas (finite or denumerable); we define a p-tableau for the set {aj,(Xj),ai2(X2),... aj(Xs),...}, where ai1, ai2, a. are any signs, as follows:

(T-1) We construct a p-tableau for ai1(X1). This is step one. (T-2) After step s, the step s + 1 is completed by attaching to each open branch of

the tableau (if there is any) a p-tableau for aj, +,(Xs+ 1). (T-3) If there is no open branch, stop the procedure. We define, similarly, an sp-tableau for the set {aj1(Xj),ai2(X2),... aj(Xs),...}

exchanging p-tableaux for sp-tableaux in the clauses of the above definition. The notions of open and closed tableaux are extended in the obvious way for

tableaux for sets of formulas. REMARK. It is clear that a necessary (but not sufficient) condition for having an

open sp-tableau for a set is that there exist open sp-tableaux for every element of the set. Moreover, the order of the elements in B has no influence on whether the resulting tableau is open or closed.

Indeed, closed sp-tableaux are finite, and the formulas of B which cause the closure condition would appear in any new enumeration of B up to a certain point in the enumeration; hence the sp-tableau would be closed in the new ordering if it was in the old ordering.

In the case that B is infinite and there exists an open tableau for {aj1(X1),. . . aj(Xs),. .} this tableau is an infinite tree. Since it is a finitely generated tree (i.e., each node has a finite number of successors), we have:

LEMMA 1 (K6NIG's LEMMA). An infinite p-tableau has at least one infinite branch. PROOF. See [9]. D In what follows we shall make clear the relationship between open and closed

tableaux and the possible values for formulas under p-valorizations. THEOREM 2. Let X be a formula in S; if there exists a closed p-tableau for ai(X), then

there exists no p-valorization v such that v(X) = i. PROOF. Let us call a set S of signed formulas satisfiable if there exists a p-

valorization v such that v(X) = i if ai(X) e S. We call a tableau satisfiable if it has a satisfiable branch (that is, if the set of nodes of one of its branches is satisfiable).

It is easy to see that applications of propositional rules turn satisfiable tableaux into satisfiable tableaux.

Hence, if there exists a p-valorization v such that v(X) = i, the tableau consisting solely of {aj(X)} is satisfiable, and so are its extensions by the rules; if we reach a closed tableau starting from {ai(X)}, we get a contradiction, since closed tableaux are obviously not satisfiable. OI

We define a Hintikka set as a set H of signed formulas satisfying the conditions: (C-1) If ai(Y) e H then ak(Y) ? H, for any j =A k, j and k in {O1, .. .,- 1} and Y

a propositional variable. (C-2) If X = F(X1,.. ., Xm) is such that ai(X) e H, then there exists at least one t-

consequence of ai(X) by the rule ti F which is contained in H. A Hintikka set H is said to be saturated when it also satisfies: (C-3) For every propositional variable Y there exists j < n such that aj(Y) e H. (C-4) If X = F(X1,.. ., Xm), then ai(X) e H iff some 2t-consequence of ai(X) by

the rule 7TiF belongs to H.




LEMMA 3. Every Hintikka set H can be extended to a saturated Hintikka set H. PROOF. An immediate adaptation of the two-valued case. L THEOREM 4. Any saturated Hintikka set R determines a valorization u such that

u(Y) = i iff ai(Y) E R; in particular, if there exists an open sp-tableau for ai(X), then there exists a p-valorization v such that v(X) = i.

PROOF. The proof follows essentially the lines of the main theorem in [10]. Let u be the function defined as u(Z) = i if ai(Z) E R and Z E SO; since R is a saturated Hintikka set, u is well defined, and moreover, u can be extended to S (since S is freely generated by SO); hence, for any formula F(X1,..., Xn), we have u(F(X1,...,XJ) = f(u(X1),.. ., u(XJ), where f is the interpretation of the connective F. Thus u is a p-valorization.

It follows, by induction on the length of formulas, that u(Y) = i iff ai(Y) E R, for any formula of S. In particular, if there exists an open sp-tableau for ai(X), this sp- tableau has at least one open branch R. It is clear that open branches of sp-tableaux are Hintikka sets; then by Lemma 3, R can be extended to a saturated Hintikka set R such that aiO(X) E R. L

Given a formula X, our definition of rules makes it clear that we have just a finite number of possible p-tableaux for ai(X), for any given a', since our rules have the subformula property (i.e., in the ic-consequence of a formula by any rule we have only (signed) subformulas of this formula). Now we have only a finite number of truth values, and in consequence Theorem 2 and Lemma 3 allow us to decide, for a given X and i, if there exists a p-valorization v such that v(x) = i.

In the next section we shall extend this property to the case of finite sets of formulas as well, and we shall present the model existence theorem for propositional logic, which allows us to give a short proof of the completeness and compactness theorems.

?4. Model existence theorems for propositional calculi. As we have fixed a logic L - <S, 9>, given X E S, we say that X is a theorem in L if, for every i E N - D (i.e., for every nondistinguished truth value), there exists a closed p-tableau for ai(X). A set of closed p-tableaux for ai(X), one for each i E N - D, is said to be a proof for X. We write FL X to denote the fact that X is a theorem in L.

If B is a set of formulas and No is a subset of N, and f is a function from to B to No, we denote by aNon f(B) a set of signed formulas of the form {ai(X): i = f(X) e No } (that is, aNo f(B) means providing elements of B with signs a,, i E No, in a certain fixed way). When the particular function f does not matter we write aN0(B).

If B u {X} c S, we say that X is a syntactical consequence of B when the following condition holds: if there is no closed p-tableau for a set aD(B) then there exists a closed tableau for aD(B) u {ai(X)}, for each i which is not a distinguished truth-value.

We write B FL X to denote the fact that X is a syntactical consequence of B in L. Considering the formula X E S and L = <S, A>, we say that X is satisfiable in W if

there exists a p-valorization v such that v(X) E D. In this case we also say that v satisfies X. We say that a set of formulas B is satisfiable if all elements in B are satisfiable by the same p-valorization.

Given B u {X} c S, we say that X is a semantical consequence of B, denoted by B F-L X, if all p-valorizations which satisfy B also satisfy X.




From this point on we shall consider, for convenience, only unary and binary connectives in our formal languages, though the method is completely general. Moreover, in order to make the exposition simpler we will consider only binary connectives, viewing the unary ones as particular cases of the binaries.

Our model existence theorem will be a generalization of the unifying principle introduced by R. Smullyan [8] for the classical case.

Let F be a collection of sets of signed formulas such that F has finite character; we say that F is an analytic consistency property (abbreviated as ACP) if for any K E F we have

(ACP-1) K does not contain ai(X) and aj(X) if X is an atomic formula and i # j. (ACP-2) If Y = F(X1, X2) and ai( Y) E K, then there exists some n-consequence C

of ai(Y) by the rule hiF such that K u C belongs to F. The sets K which belong to F are said to have the property F, and if X E K e F

then X is said to be F-consistent. It is clear that K' c K E F implies K' E F, because of the finite character of F.

If B is a set of nonsigned formulas and F is an ACP, we say that B is F-consistent if all formulas of B occur signed in some K E F (i.e., there is K E F such that for each X E B there exists ai such that ai(X) E K).

We now pass to the model-existence theorem, which allows a unified treatment of the model-theoretic aspects of the propositional many-valued logics. Here we deal only with the general consequences of the model-existence theorem.

THEOREM 5 (Model Existence Theorem, propositional case). Let F be an ACP and B c S. If B is F-consistent, then there exists a valorization v such that v(X) = i if ai(X) occurs in some K E F, for each X E B. In particular, if aD ,(B) E F for some set

aDJf(B), then all elements of B are satisfiable. PROOF. Let Z1, Z2,. .. , Zn,. .. be an enumeration of all formulas in the set S* of all

signed formulas; we construct the following sets C, inductively:

C1 = K such that all elements of B occur signed in K (which exists due to the fact that B is F-consistent),

Cn+1 = Cn u {Zn } if Cn u {Zn } has the property F, or

Cn+ 1 = Cn otherwise.

It is clear that all Cn so constructed have the property F, and the C1 c C2 c c C, C .

Let M = U{Cj: i E N}; we claim that M is a saturated Hintikka set. Indeed, (i) M has the property F, since for each finite K c M there exists Cn such that K

c Cn. Since Cn has the property so has K, since F has finite character. Hence, as any finite subset K of M has the property F, M also has the property F, again using the fact that F has finite character.

(ii) If M u {Zi } has the property F, then Zi E M, for in this case, by construction,

Ci u {Zi } has the property F, and hence Zi e Ci +1 c M. Now, by Lemma 3, M can

be extended to a saturated Hintikka set M. We then define the following p- valorization v: v(X) = i iff ai(X) occurs in M and X is an atomic formula.

The condition (C-3) of the definition of a saturated Hintikka set guarantees that v is well defined on the whole set of generators of the algebra So, and then v can be uniquely extended to a homomorphism v from S to d. Using clause (C-4) of the




definition of a saturated Hintikka set, it is easy to see by induction on the level of formulas that this p-valorization v has the required properties. El

Before applying the model existence theorem for obtaining the soundness, completeness and compactness theorem, we shall prove some lemmas related to tableaux for sets of formulas.

LEMMA 6. Let B C S, and let i: S -+ N be any function; if there exists an open sp- tableau for the set {aj(x)(X): X E B} then there exists a p-valorization v such that v(X) - i(X) for every X E B.

PROOF. If there exists an open sp-tableau for the set {aj(x)(X): X E B}, this tableau has at least one open branch (either by definition if B is finite, or using Lemma 1 if B is denumerable; in the latter case, condition (T-3) of the definition of tableau for a set guarantees that infinite branches are open).

This open branch is a Hintikka set, and can be extended to a saturated Hintikka set. Then, using Theorem 4, we can construct the required p-valorization. Cl

COROLLARY 7 (Soundness of p-tableaux). Let B u {X} C S; if B FL X then Bk=LX.

PROOF. Immediate consequences of the preceding lemma and the definitions of semantic and syntactic consequences. LI

LEMMA 8 (Finite character of p-tableaux for sets). Let R be a finite or denumerable set of signed formulas (i.e., R c S*); then there exists an open sp-tableau for the set R if for every finite subset Ro of R there exists an open sp-tableau for Ro.

PROOF. On one hand, if there exists an open sp-tableau for R, then there exist open tableaux for all finite subsets of R.

For the converse, suppose R infinite. Let Z1, Z2,... , Zn,... be an enumeration of the elements of R, and consider the following finite subsets of R: Rn = {Z1,.. ., Zn }.

The following procedure defines an sp-tableau for R: 1) Construct an open sp-tableau for R1. 2) After we have constructed an open sp-tableau for Rn, we add to each open

branch of this tableau an open sp-tableau for Zn +1 such that the resulting tableau is open (which exists by hypothesis).

Since step 2) runs indefinitely, this is, by definition, an open sp-tableau for R. LI THEOREM 9 (Completeness of p-tableaux). Let B u {X } be a set of formulas in S;

then B ILL X implies B FL X.

PROOF. Suppose B F/L X; then there exists an open sp-tableau for a set aD(B) u {aj(X)}, where i is a nondistinguished truth value. Now, by Lemma 8, it is easy to see that the class F of all open sp-tableaux for sets of signed formulas is an ACP. By Theorem 5, since the set B u {X} is F-consistent, there exists a p- valorization v such that v(Y) E D, for every Ye B, and v(X) 0 D. Hence B # X. a

COROLLARY 10 (Compactness). Let B c S. Then B is satisfiable if all finite subsets Bo of B are satisfiable.

PROOF. A direct application of Lemma 8 and Theorem 9. L

?5. First-order n-valued logics (preliminaries). Let S be a propositional language; we define the first-order extension of S, S', as the language composed of the following symbols which are the alphabet of S':

1) the symbols of S which are not propositional variables;




2) a finite set QT = {Q1, Q2,... , Q } of quantifiers with at least two elements; 3) a denumerable set IV = {v1, v2,... , v,,. . .V } of individual variables; 4) a denumerable set IP -{c1,c2,... ,cn,...} of individual parameters, such that

IP r- IV =0; and 5) for each positive natural number m an (at most denumerable) set PRm of m-ary

predicates; we assume that the whole set of predicates is nonempty. Besides the metalinguistic variables used in the propositional part we shall use

x, y, x1, Yl, etc., to denote variables; a, b, a1, b1, etc., to denote parameters; P, R, P1, R1, etc., to denote predicates; and Q to denote quantifiers.

A first-order language over the alphabet of S1 is the following collection: 1) An atomic formula of S' is any (m + 1)-tuple Pr1r2.. rm where P E PRm and

rieIV u IP. 2) A formula in S1 is an element of the following sets:

SO {X: X is an atomic formula}, (I+ -Snu {F(X1,.. ., Xp): Xi e S1 and F is any p-ary connective}

u {(Qjx)A: A e S', xe IV and Qj e QT},

S= US. n

For simplification we shall again restrict the connectives to the binary ones, and will make use of auxiliary symbols such as parentheses, commas, etc. In particular we are using the equality symbol = as a symbol in the metalanguage, not as a formal predicate symbol in S'.

We define a pure formula as a formula with no parameters. A formula has level r if it belongs to S1 - S1 1), r2 1.

The notion of replacement of a variable by a parameter is defined inductively as follows: given a formula A, a variable x and a parameter a, then

1) if A is atomic, [A]' is the result of replacing each occurrence of x in A by a; 2) [F(X1, X2)]x = F([X1 ] x, [X2 ] a); 3) [(Qx)A]x = (Qx)A; 4) [(Qv)A]' = (Qv)[A]' if y # x. The formula [A]' is said to be an instance of the formula A; a formula A is closed

(or is a sentence) if, for every x and every a, [A] x = A. We denote by S 1 the set of all sentences of S'.

A simultaneous substitution is defined inductively as:

5) [A]X .:Xr:-:fxr = [[A]X, '] if x1, x2,r. . ., Xr are all distinct. The notion of subformula is extended from the propositional case, adding also the

following clause: If X = (Qx)A, then for every parameter a, [A]x is a subformula of X.

By a universe we mean a nonempty set U; a U-formula is a formula as constructed in S 1 but replacing IP by the set U. The notions of U-subformula, U-substitution, etc., are easily adapted.

For a given universe U we write S '(U) for the set of all U-formulas and denote by Xu the formula in S1(U) corresponding to the formula X in S1. It is clear that the pure formulas are the same in S1 and S'(U) for any set U.




?6. Semantic definitions. The basic fact used in our development of propositional logics is that the truth values of formulas are determined by the truth values of their subformulas. Quantified formulas, however, may have infinitely many subformulas, and the relationship between truth values of formulas and their subformulas can be presented in several ways.

In our semantics we shall give one of the possible generalizations of the notion of quantifiers, which we call the distribution quantifiers. We divide the truth values of all subformulas into a finite number of certain special classes, and our semantics will then give the relationship between these classes and the value of the formula itself: this relationship is called the interpretation of the quantifier. We now formalize these ideas:

First we define a distribution function as any function with inputs in the set of quantified formulas in S'(U), for any given U, and outputs in the set Vn of all the 2n _ 1 n-tuples with coordinates 0 and 1, with at least one coordinate 1. Intuitively, each n-tuple in Vn will be interpreted as saying whether the truth value i, 0 < i < n, is absent or present in the set of all truth values displayed by the subformulas of the formula, according to whether the ith coordinate is 0 or 1, respectively.

Second, the interpretation function for the quantifier is a function with inputs in the Vn and outputs in the set of truth-values. The composition of these functions gives the truth value of the quantified formula.

We assume henceforth that every quantifier Q of the language S has a corresponding interpretation function aQ. The triple

L' = <S ', a?, JaQ: Q c- VI >

is called a first-order n-valued logic (associated with the propositional logic L = <S,?>).

We define the set S' of formulas in S'(U) with no initial quantifiers as the set of all formulas in S'(U) which are not of the form (Qx)A, for any quantifier Q.

Now a first-order valorization v from the set of sentences of S'(U), S'(U), into N, is defined as a function v: S'(U) -+ N such that

(V1) v is a p-valorization for the set S', and (V2) for each formula A and variable x, v((Qx)A) = qQ(DX v(A)), where qQ: Vn -+ N

is the interpretation function for the quantifier Q and Dx,: S1(U) -+ Vn is the distribution function relative to x and v, defined as

Dx v(A) = <d(O),.. .,d(i),.. ., d(n - 1)>,

where d(i) = 1 iff there exists K E U such that v([A]x) = i and d(i) = 0 otherwise. As an example we consider the interpretation functions UV and o3 for the classical

predicate logic; in this case V2 is the set {(0, 1), (1, 0), (1, 1)} and the functions are

0 1 UV 0 1 U3

(0,1) 1 (0,1) 1 (1,0) 0 (1,0) 0 (1, 1) 0 (1, 1) 1




where 1 = true and 0 = false. The symbols 0 and 0, 1 and 1 are not to be confused; here 0 and I mean absence or presence of the truth values 0 and 1 among the possible truth values of subformulas of a formula. Thus, for example, VxA(x) is false (value 0) in the case when the truth values of the subformulas of A present a distribution (1, 0) or (1, 1).

We remark that the set of all possible functions aQ has n(2-- 1) elements, since the cardinality of each set Vn is 2n- 1; that is the reason we consider only a finite number of quantifiers in our language. In the case of classical predicate logic, the remaining YQ functions are:

UQI ' Q2 '7Q3 '7Q4 '7Q5 'Q6

(0,1) 0 0 0 0 1 1 (1,0) 0 0 1 1 1 1

(1, 1) 0 1 0 1 0 1

which refer to the following combinations of quantifiers:

(Q1 x)A = (Vx)A A (Vx)- A, (Q4x)A = - (Vx)A,

(Q2x)A = (3x)A A (3x)-iA, (Q5x)A = n (Q2x)A,

(Q3x)A = -i (3x)A, (Q6x)A = - (Q1 x)A.

These combinations can be regarded as quantifiers themselves, in this case generated either by V or 3. The problem of understanding the relationship among quantifiers in n-valued logics in terms of interdefinability is very interesting and seems to constitute a difficult combinatorial problem.

We say that a quantifier V in L' is a universal quantifier when its interpretation function has the following property:

7(0,iO , *... j.n-1)D iff ji=Oforeveryie-N-D.

Similarly 3 is an existential quantifier if its interpretation function satisfies

3(30,iJil - ** inJ- 1) e D iff ji = I for some i e D.

We say that a logic L' is regular when it has among its quantifiers at least one universal and one existential quantifier. From now on we shall consider a fixed n- valued regular logic L' associated with the propositional logic L.

By an atomic valorization we mean an assignment of truth values for the atomic sentences of S1(U); as for the propositional cases, we have:

LEMMA 12. An atomic valorization can be uniquely extended to a valorization for S'(U).

PROOF. By induction on the level of formulas, for the quantified formulas using the fact that every quantifier has an interpretation function. OI

By a semantic interpretation of L' in a universe U #A 0 we mean a function I which associates

1) to each individual parameter of Li an element of U (not necessarily one-to- one), and




2) to each m-ary predicate symbol P a partition of the Cartesian product U x U x... x U = Um into n classes P*, P*, ... , P*_ 1 (corresponding to n truth values),

called a P-partition of U. The pair ' = <U, I> is called a structure for L'. Given the structure ' = <U, I>

we say that the atomic U-sentence Pu1 u2 ... Um has truth value i if the m-tuple (u1,u2, . .,um) belongs to the class P~' (where 1(P) = (P*,Pl*,.. ..I .n P*_)). If the truth value i belongs to D we say that Pu1 u2 um is true in V, or that ' is a model for Puiu2u. Um

LEMMA 13. A structure fY = <U, 1> defines a unique valorization for S1(U) and conversely.

PROOF. For one direction, Lemma 12 suffices; conversely, given S'(U), we define the function I for each m-ary predicate symbol P taking the partition (ul, u2,.. . , um) e P' ifif v(Pu1u2 Um) = i (the assignment of elements of U for the parameters is given by the definition of S'(U)). D

Let SP be the set of all pure sentences of S 1; as we have noted, SP is contained in S'(U) for every universe U. We say that a sentence X of SP is satisfiable if it is true in some structure V (or equivalently by Lemma 13, if there exists some valorization v such that v(x) E D). We say that X is valid if it is satisfiable in every structure 'V.

A set B of sentences of SP is satisfiable if all its elements are satisfiable in the same structure; a sentence X is said to be satisfiable in a universe U (respectively, valid in a universe U) if X is satisfiable (resp., valid) in every structure which has U as universe.

If A(cl,..., c,) is a sentence in S' -SP containing exactly the parameters C1,C2,. . . ,C, we say that A(c1,...,cr) is satisfiable (resp., valid) if there exists a structure V = <U,l> such that the sentence A' = A(I(c1),..., I(c,)) obtained from A by replacing each c' by I(cr) in U is true in V (resp., is true in every structure ' = (U, I)).

The next lemma, whose proof is omitted because it is completely analogous to the classical one (see [9, p. 51]), claims that it is enough to consider pure sentences when dealing with the notions of validity and satisfiability in regular logics.

LEMMA 14. Let L' be an n-valued regular logic; then, for each formula A(cc2, . . .,cr), if X1,X2,-. .,Xr are distinct variables which do not occur in A(c1, c2, . , cr) and A(x, x2,. . , x,) denotes the formula obtained on replacing c1 by X1, C2 by X2,..., cr by xr in A(c1,c2,. . .,cC), we have

1) A(c1, C2,.. , cr.) is satisfiable iff the pure sentence

(3x J)(3X2) ...

(3x,)A(xl x2) *Xr)

is satisfiable; and 2) A(c,,c2, . . ,cr) is valid iff the pure sentence

(Vx1)(Vx2) (Vxr)A(xvX2, XIXr)

is valid.

?7. Quantificational tableaux. Let S l * denote the set of all signed sentences of S 1 (i.e. S1* - {ai(X):X E S1 and 0 < i < n - 1}. If Q is a quantifier of L' with




interpretation function aQ we say that the (m + 1)-tuple (injl j2... jm), m < n, satisfies the quantificational condition for Q (in symbols, Dj(Q;j1,...,Ijm) holds) if the n-tuple <d(O), . .,d(j1), . . .,d(j2),. . .,d(jm),. . .,d(n - 1)>, defined as d(t) = 1 iff tE {Cil. . . ,jm} and d(t) = 0 otherwise, belongs to o41(i).

In other words, Dj(Q;jj,. . .,jm) holds if the n-tuple consisting of coordinates 1 in the places jl,... ,jm and 0 otherwise has the image i by the function oQ. As an example, the function a. shown in ?6 is such that a- 1 (0) = {(I 0), (1 1)} and hence the quantificational conditions DO(V; 0) and Do(V; 0, 1) both hold.

Let X be the formula (Qx) Y where Q is any quantifier; we define the rule ;i Q as a function which associates the following tree to the signed formula a'(X):

ai((Qx) Y)

+cjil(cl oj2([YIC2 o' * aj,([Y]xc)}

where l) jj,..., jm < n satisfy Di (Q; ji,...,j), and 2) if m > 2, then no parameter cr, 1 ? r ? m, has already appeared in the branches which contain aj([Y]X).

The symbols + and o should be understood in the same way as in the propositional case. Condition 2) is considered in order to avoid conflict between subformulas (see [9], for example, for an explanation of this proviso in the classical case).

If there exists no quantificational condition Di(Q;ij,...,Ijm) the rule ', Q is not defined. We denote by A' the set of all rules Xi Q for all quantifiers Q. The definitions of premise and X-consequence of a rule are the same as in the propositional case with the obvious modifications.

For example, our rules for the classical quantifiers will be (O = faise and 1 = true):

a, (VxY) (1) al[Y]x'

a0(]x Y) (2) a~xY

(3) a0[Y]c + {ao[Y]Cl o al [Y]a

(4) ao (]x Y)

a, [Y]x + {a, [Y]C1 o ao[Y]2

where in (3) and (4) cl and c2 occur with the proviso in clause 2) of the definition of the rule.

Rules (1) and (2) are the same as the usual tableau rules (see [9, Chapter V]), but (3) and (4) are usually given as:

(3s ao( VxY ) IA4'

al(]yx)

where c is a new parameter.




This proviso is necessary in rule (4'), for example, because if ]xY holds then [Y]C holds for some c. Thus we use a new parameter because we may have committed the parameters which have already appeared, unless we are sure that [Y]x holds for every c.

Our rule (4) takes into account the case where [Y]x holds for every c (a1([Y]x)) and the case where [Y]x holds for some but not every c (a1([Y] 1) and ao([Y] x)). In the usual presentation this information is condensed in the form of rule (4').

In this sense our quantifier rules are not the most simple in syntactic form, as are our propositional rules. Although it is possible to simplify the rules in some particular cases, we do not have a general procedure for these simplifications.

After defining all rules i'jQ. we define the quantificational systematic tableau (s- tableau, for short) of a formula ai(X) as the tree described as follows:

(Q1) We initiate the tree with the formula a'(X); this is node 1. (Q2) Given node s, if the formula in node s is of the form F(X1, X2) we extend the

tree as in the propositional case, changing propositional variable to atomic formula; that is, we use the rules niF with the obvious modifications. In this case formulas of the form ai(Qx)Z are considered as atomic from the point of view of the rules niF.

(Q3) If the node s is of the form ai(Qx) Y, we extend the tree according to the rule

i, Q and declare it to be used. In this case we add the node s to the end of each branch which passes through s.

(Q4) Suppose that the node s has been used; if all nodes where nonatomic formulas occur have already been used, the procedure stops. Otherwise we take as node s + 1 the leftmost node with least height and continue, using (Q2) or (Q3).

The notions of (nonsystematic) tableau, closed branch, closed tableau and proof are defined as immediate adaptations of the propositional case.

If the formula X of S1 is a theorem in L1 we write F1 X. Let B* = {ai,(XI), ai2(X2) ... Xair(Xr),... } be a denumerable (or finite) set of signed formulas; by an s-tableau for the set B* we mean a tree constructed as follows:

1) We place the formula ai,(X,) at the origin; this is node 1. 2) Suppose we are at node n; we then proceed exactly as in the case of an s-tableau

for a single formula, but after using (Q2) or (Q3) we adjoin ain+ 1(Xn+ 1) to the end of every open branch, and do not declare this node to be used.

3) We then apply (Q4).

We define a (nonsystematic) tableau for a set in a way similar to that of the propositional case.

The notions of syntactical and semantical consequence are defined in an analogous way to that of the propositional case. If X is a syntactical (semantical) consequence of B we write B 1 X (B k=1 X).

As an illustration, let us consider the three-valued predicate calculus J3 (cf. [3]), whose connectives and quantifiers are described by the following functions:

v 0 1 2 D 0 1 2 A O 1 2 V7

0 0 1 2 0 2 2 2 0 0 0 0 0 0 0 2 1 1 1 2 1 0 1 2 1 0 1 1 1 2 1 I 2 2 2 2 2 0 1 2 2 0 1 2 2 2 2 0




where N = {0, 1, 2} and D = {1, 2},

0 1 2 ]V

1 0 0 0 0 1 0 1 1 0 0 1 2 2 1 1 0 1 0 1 0 1 2 0 0 1 1 2 1 1 1 1 2 0

i.e., ] and V are interpreted by the following functions, respectively:

a(<d(0), d(l), d(2)>) = max{i: d(i) = O},

a(<d(O), d(l), d(2)>) = mini: d(i) # 0},

and the quantificational conditions are:

3o: Do(]; 0), 3 1: D, (3; 1), D, (3; , 1),

]2: D2(]; 2), D2(]; 0, 2), D2(]; 1, 2), D2(]; 0, 1, 2), V0: Do(V; 0), Do(V; 0, 1), DO(V; 0, 2), DO(V; 0, 1, 2), V1: D1 (V; 1), D1 (V; 1, 2),

V2: D2(V; 2).

The rules Ri Q (i e {0, 1, 2} and Q e {], V}) are the following:

(J- 1) ao((3x)A) ao(Ac)

(J-2) al((]x)A)

{al(Ac)} + {ao(Ac1) 0a(AC2)'

a2((]x)A) {a2Ac} + {aOAcj o a2Ac2} + {alAc3 oa2Ac4} + {a0Ac5 o aAc6 o a2Ac7}'

ao((Vx)A) {aoAc} + {aoAc1 o ajAc2} + {a0Ac3 o a2Ac4} + {a0Ac5 o ajAc6 o a2Ac7}'

(J-5) ~~~~~a 1((Vx) A)

({al(Ac)} + {al(Acl) a2(AC2)'

(J-6) a2((Vx)A (J-6) ~~~~~~~~a2(A c)

where, when applying the rules, all parameters are new in the tableau. The rules JtiF

are easily described from the tables.




The following tree is a proof, for A, B atomic, of

(Vx)V(Ax A Bx) D (]x)(VAx A VBx).

(1) ao((Vx)V(Ax A Bx) D (]x)(VAx v VBx)

(2) a1((Vx) V (Ax A Bx) (3) a2((Vx)V (Ax A Bx))

I 1 (4) aO((]x)(VAx v VBx)) (5) [4]

(6) a1(V(Aa A Ba) (7) a1(V(Ab A Bb))

* (8) a2(V(Ac A Bc))

*

(9) a2(V(Aa A Ba))

(10) al(Aa A Ba) (11) a2(Aa A Ba)

(12) a1(Aa) (13) a1(Aa) (14) a2(Aa) (18) ao(Aa)

(15) aj(Ba) (16) a2(Ba) (17) a1(Ba) (19) a2(Ba)

(20) [3] [3] [3] [3]

(24) ao(VAa v VBa) [24] [24] [24]

(28) ao(VAa) [28] [28] [28]

(32) ao(VBa) [32] [32] [32]

(36) ao(Aa) [36] [36] [36]

(40) ao(Ba) [40] [40] [40]

* * * *

?8. Model existence theorems for predicate logics. Let H be a set of signed U- sentences for some universe U; H is said to be a quantificational Hintikka set (related to the universe U) if the following conditions hold:

(C1) For each atomic sentence X in S1(U) and each k, j < n, k =# j, if ak(X) e H then aj(X) 0 H.

(C2) If Y = ai(F(Y1, Y2)) e H then some it-consequence of Y by the rule niF iS

contained in H.




(C3) If ai(Qx) Y E H then there exist jiJ2, . ."Ejm < n such that: a) the quantificational condition Di(Q; j1,... , jm) holds; b) for each formula [Y]', c E M, aj,([Y]') E H for some 1 < r < m; and c) for each jr, 1 < r < m, there exists some c E U such that aj,([Y]') c H. LEMMA 15. Let H be a quantificational Hintikka set; then there exists a valorization

v such that v(X) = i if ai(X) E H.

PROOF. We construct a valorization v by induction on the level of formulas as

follows:

1) If X is an atomic sentence of S'(U) and ai(X) E H, we take v(X) = i; if the

formula X does not occur (signed) in H, we take any arbitrary value for v(X).

2) We then have an atomic valorization, and by Lemma 12 can extend it to a

unique valorization v for S'(U).

We must show that, if ai(X) E H, then v(X) = i. For X atomic, this holds by

construction. For X nonatomic, by induction on level, we have:

3) If X is of the form F(Xl, X2) and ai(X) E H, using clause (C2) of the definition of

quantificational Hintikka sets, and using Theorem 4 with small modifications, we

obtain the result.

4) If X is of the form (Qx) Y and ai(X) E H, by clause (C3) of the definition of

quantificational Hintikka sets the condition Di(Q;jl,.. .,jm) holds for some

1 Jm < n, and for each jr there exists c e U such that aj,([Y]')

e H. Hence, by the induction assumption, v([Y]') = jr Moreover, all formulas

aj,([Y]') belong to H, for every c e U, and by the induction assumption they have

truth value js under v.

By the definition of Dj(Q;jj,. .jm) and by clause (V2) of the definition of

valorization, we have v((Qx) Y) = oQ(DXV(Y)) = i. The proof is complete. FO Let F denote a collection of sets of sentences of S'(U)*, and suppose F has finite

character. If B is a set of sentences in S'(U) (i.e., a set of unsigned U-sentences) we

say that B is F-consistent if there exists K e F such that K = {ai(X): x e B and i is

any truth-value in D}.

We say that F is an analytic consistency property, ACP for short, when every K e F

satisfies the following conditions:

(A1) K does not contain ai(X) and aj(X) if i =# j and X is an atomic sentence of

S 1(U).

(A2) If aiF(Yl, Y2) e K there exists some n-consequence C of aiF(Yl, Y2) by the

rule niF such that K u C e F.

(A) If ai(Qx)A e K then there exist j1j2 . . .jm < n such that Di(Q; jj, . .I jM) holds and there exists a set Z such that

a) for each formula [A]', c e U, there exists some r, 1 < r < m, such that aj,([A] ) e Z,

b) for each jr, 1 < r < m, there exists some parameter c such that aj,([A]x) c Z, and

c) K u Ze F.

In view of Theorem 14 we can restrict our next theorems to the pure sentences

with no loss of generality.

THEOREM 16 (Model existence). Let B be a set of pure sentences and F an ACP; if B

is F-consistent, then all the elements of B are simultaneously satisfiable in a

denumerable universe.




PROOF. Since B is F-consistent, there exists K E F which consists of the formulas of B signed with distinguished prefixes.

Let Z1, z2,... , Zr... be an enumeration of the formulas in K; we construct a quantificational Hintikka set defining the following sequence Mj:

1) M1 = Z1. 2) If the terms of the sequence up to Mn form a set which belongs to F, we extend

the sequence as follows: 2.1) If Mn is of the form aiF(Yl, Y2) we put M, ajK(YK), Z,"1 if ajK(YK) are the

elements of some it-consequence of ai(F(Yl, Y2) by lti,F which satisfies clause (A2) of the definition of ACP.

2.2) If Mn is of the form ai(Qx)A and jl,...,jIm are such that clause (A3) of the definition of ACP holds, we extend the sequence, putting M, aj,([A]x),... aj([A]x), Zn+1, where al,...,am are parameters which did not appear previously in the sequence.

Thus the set M of all terms of this sequence belongs to F and form a quantificational Hintikka set for the universe of parameters; then, by Lemma 15, B is satisfiable. LI

In Corollaries 17, 18 and 19, B can be any set of pure sentences. COROLLARY 17 (Completeness). B is satisfiable if there exists an open s-tableau for

some set aD(B) (i.e., if B = C u {X}, CF- X if C='1 X). PROOF. It is sufficient to show that the collection of all sets K of signedformulasfor

which there exist open s-tableauxforms an ACP. LI COROLLARY 18 (Compactness). B is satisfiable if all its finite subsets Bo are. PROOF. By completeness, using the fact that, if there exists a closed tableau for B,

then there exists a closed tableau for some finite subset Bo of B. LI COROLLARY 19 (Ldwenheim-Skolem). If B is satisfiable, then B is satisfiable in a

denumeable universe. PROOF. If B is satisfiable, by Corollary 17 there exists an open s-tableau for some

set aD(B); this open s-tableau has at least one open branch which contains all elements of B signed with distinguished prefixes. It is easy to see that this branch forms a quantificational Hintikka set; then Lemma 15 obtains. El

The last two theorems can be proved directly from the model existence theorem. It can also be used to derive interpolation lemmas for some families of many-valued logics, which ones depending basically on the connectives (this can be done, for example, adapting the proofs of Fitting in [4]).

Moreover, using our methods, several results of Smullyan [9], especially from Chapters VII, VIII and IX, can be extended to some first-order many-valued logics.

Acknowledgments. We are grateful to Newton C. A. da Costa, Justus Diller, Richard L. Epstein, and Edgard G. K. L6pez-Escobar for their comments and remarks and especially to an insightful referee for his excellent criticisms.

REFERENCES

[0] P. BOROWIK, On Gentzen's axiomatization of the reducts of many-valued logics, this JOURNAL, Vol. 48 (1983), pp. 1224-1225 (abstract).

[1] W. A. CARNIELLI, Sobre o metodo dos tableaux em l6gicas polivalentes finitdrias, Ph.D. thesis, University of Campinas, Brazil, 1982.




[2] , The problem of quantificational completeness and the characterization of all perfect quantifiers in 3-valued logics, Zeitschriftffur Mathematische Logik und Grundlagen der Mathematik (to appear).

[3] I. M. L. D'OTTAVIANO, Sobre uma teoria de modelos trivalente, Ph.D. thesis, University of Campinas, Brazil, 1982.

[4] M. FITTING, Model-existence theorems for modal and intuitionistic logics, this JOURNAL, vol. 38 (1973), pp. 613-627.

[5] H. RASIOWA and R. SIKORSKI, The mathematics of metamathematics, PWN, Warsaw, 1970. [6] I. ROSENBERG, The number of maximal closed classes in the set of functions over a finite domain,

Journal of Combinatorial Theory Series A, vol. 14 (1973), pp. 1-7. [7] A. M. SETTE, On the propositional calculus p', Mathematica Japonicae, vol. 16(1973), pp. 173-180. [8] R. M. SMULLYAN, A unifying principle in quantification theory, Proceedings of the National

Academy of Sciences of the United States of America, vol. 49 (1963), pp. 828-832. [9] , First order logic, Springer-Verlag, Berlin, 1968. [10] S. J. SURMA, An algorithm for axiomatizing very finite logic, Computer science and multiple-valued

logic: theory and applications (D. C. Rine, editor), North-Holland, Amsterdam, 1977, pp. 137-143.

DEPARTMENT OF MATHEMATICS

UNIVERSITY OF CAMPINAS

13081 CAMPINAS, BRAZIL



systematization of finite many-valued logics through the method of tableaux

Documents