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On Axiomatizations of General Many-Valued Propositional Calculi

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On Axiomatizations of General Many-Valued Propositional Calculi On axiomatizations of general many-valued propositional calculi Arto Salomaa Turku Centre for Computer Science Joukahaisenkatu 3{5 B, 20520 Turku, Finland asalomaa@utu.fi Abstract We present a general setup for many-valued propositional logics, and compare truth-table and axiomatic stipulations within this setup. Results are obtained concerning cases, where a finitary axiomatization is (resp. is not) possible. Related problems and examples are discussed. 1 Introduction Many-valued systems of logic are constructed by introducing one or more truth- values between truth and falsity. Truth-functions associated with the logical connectives then operate with more than two truth-values. For instance, the truth-function D associated with the 3-valued disjunction might be defined for the truth-values T;I;F (true, intermediate, false) by D(x; y) = max(x; y); where T > I > F: If the truth-function N associated with negation is defined by N(T ) = F; N(F ) = T;N(I) = I; the law of the excluded middle is not valid, provided \validity" means that the truth-value t results for all assignments of values for the variables. On the other hand, many-valuedness is not so obvious if the axiomatic method is used. As such, an ordinary axiomatization is not many-valued. Truth-tables are essential for the latter. However, in many cases it is possi- ble to axiomatize many-valued logics and, conversely, find many-valued models for axiom systems. In this paper, we will investigate (for propositional log- ics) interconnections between such \truth-value stipulations" and \axiomatic stipulations". A brief outline of the contents of this paper follows. In the next two sections we define formally the basics of the two stipulations and present some simple results of the related notion of a consequence. Section 4 applies the notions 1 in a classical framework, and also studies the case where the truth-functions constitute a functionally complete set. Section 5 (resp. Section 6) investigates cases where a finite model is (resp. is not) possible for an axiom system. We also pay attention to the minimal cardinality of the model. The last two sections contain some preliminary discussion and open problems. Remark. The topic of this paper was chosen because I had many discussions with Sandu Mateescu about many-valued logic, often in connection with mem- ories of Gr.C. Moisil. Moisil was one of the early investigators in the field, still often quoted in the literature, [6, 11, 2, 4]. 2 Many-valued propositional logics. Truth-table stipulation Our approach in defining many-valued propositional logics (calculi) is very gen- eral and suitable for all systems considered in the past. In many respects we follow the definitions of Alfred Tarski, [13]. We first define the notion of a propositional logic. Definition 1 Consider two sets of primitive signs: • A denumerably infinite set fp; q; r; : : :g called (propositional) variables, • f a1 ak g ≤ ≤ A finite set C1 ;:::;Ck called connectives. For 1 i k; we say that ai Ci is an ai-place connective. A finite sequence of primitive signs is a well-formed formula (wff)if it either ai consists of a propositional variable, or else equals Ci (α1; : : : ; αai ); for some i; 1 ≤ i ≤ k; where each αj; 1 ≤ j ≤ ai; is a wff. The set of all well-formed formulas is denoted by WFF . Let VA and NVA be two subsets of WFF , both closed under alphabetical variance of variables. The triple (W F F; V A; NV A) is called a propositional partial logic if VA and NVA are disjoint. If in addition WFF = VA [ NV A; then the triple is called a propositional logic (or a propositional calculus.) Some additional remarks are in order. Well-formed formulas contain also imprimitive signs (parentheses, comma) not made explicit in the definition. We could also add logical constants (such as T for truth) but this is not needed for our purposes. The notation VA (resp. NVA) is introduced with the word valid (resp, non-valid) in mind. For n ≥ 2; the numbers 1; : : : ; n are referred as truth-values. For i ≥ 1; any function mapping the Cartesian power f1; : : : ; ngi into the set f1; : : : ; ng is called an i-place truth-function in n values. Truth-functions are associated with connectives in a natural way. For instance, the function of two variables defined by f(1; 1) = f(1; 2) = f(2; 1) = 1; f(2; 2) = 2 2 is associated with the (2-valued) disjunction. (The semantic order of the truth- values will be defined below.) We are now in the position to define the notion of a many-valued propositional logic. Definition 2 A propositional logic (W F F; V A; NV A) is an n-valued proposi- tional logic, n ≥ 2; if the following four conditions are satisfied. 1. With each propositional variable p is associated a distinct numerical vari- able x. ai ≤ ≤ 2. With each connective Ci ; 1 i k; is associated an ai-place truth- ai function in n values, called the associate of Ci . 3. The following condition is satisfied for all wff's α; β; γ: Whenever β is the result of uniformly substituting a variable p in α with γ, then the associate of β is the result of uniformly substituting the associate of p with the associate of γ in the associate of α. 4. There is a natural number d; 1 ≤ d < n, such that VA equals the set of wff's whose associates assume a truth-value less than or equal to d, for all assignments of values for the associates of their variables. A propositional logic is a many-valued propositional logic if it is an n-valued propositional logic, for some n. The truth-values 1; : : : ; d (resp. d + 1; : : : ; n) are referred to as designated (resp. undesignated) truth-values. As an example, we consider some connectives in the Lukasiewiczthree-valued logic, [1]. (It was already used as an example in the Introduction.) The truth- values are 1; 2; 3, and 1 is the only designated truth-value. The truth-functions associated to the 1-place connective F1 and 2-place connective F2 are defined by, respectively, f1(x) = 3 − x + 1; f2(x; y) = min(x; y): The connectives F1 and F2 correspond to the 2-valued negation and disjunction, respectively. The two truth-functions yield, by Definition 2, a 3-valued proposi- tional logic (W F F; V A; NV A). Without going into the study of the set VA in this case, we mention that the wff F2(p; F1(p)), corresponding to the 2-valued law of the excluded middle is not in VA. This follows because f2(2; f1(2)) = f2(2; 2) = 2: Semantic studies concerning the truth-values lie outside the scope of this paper.Lukasiewicz interprets the intermediate truth-value 2 as "possible", "undecided" or "unknown". There are also various possibilities to generalize the ordinary 2-valued connectives to the many-valued case. The reference [9] contains a fairly exhaustive classification. 3 In Definition 2 we have introduced a truth-value stipulation for the set VA. This stipulation emphasizes the fact that we are dealing with a many-valued logic. Another common procedure for characterizing the set VA is the axiomatic method. Certain wff's, axioms, are chosen as assertable. Moreover, certain operations on wff's, rules of inference, are specified. In this axiomatic stipulation the set VA is defined to be the closure of the set of axioms under the rules of inference. In the axiomatic stipulation many-valuedness is not as clear as in the truth- vale stipulation. The following two general problems are basic in this set-up. 1. Is a given many-valued propositional logic axiomatizable? In other words, can we choose a finite subset AX of WFF , as well as a finite set OP of closure operations on WFF , such that the set VA equals the closure PR, \provable" wff's, of AX under the operations in OP ? 2. Does there exist a finite many-valued model for a given axiom system? More specifically, given an axiom system (W F F; AX; OP ), we are looking for integers n and d, and truth-tables for the connectives in WFF such that defining VA as the closure of AX under the operations in OP yields a many-valued propositional logic. The purpose of this paper is to present some results concerning these two problems. 3 Axiomatic stipulation We now return to the formal definition of the notions discussed above, as well as some other related notions. The set WFF of well-formed formulas is defined as in Definition 1. Definition 3 A triple AS = (W F F; AX; OP ) is referred to as an axiom sys- tem if AX is a finite subset of the set WFF , and OP is a finite set of closure operations on the set WFF such that alphabetic variance is one of the opera- tions in OP . The closure PR of AX under the operations in OP , in symbols CL(AX; OP ), is called the set of formulas provable under the axiom system AS. The axiom system AS is called inconsistent if PR = WFF . Otherwise, AS is consistent. If AS is consistent but CL(AX [fαg; AS) = WFF , for every wff α not in PR), then AS is complete. Consider two wff's α and β, as well as a subset U of WFF . The set of consequences of U in the axiom system AS is defined by CONS(U; AS) = CL(AX [ U; OP ): If CONS(U; AS) = WFF , we say that U is inconsistent with AS.
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