Bulletin of the Section of Logic Volume 38:3/4 (2009), pp. 99–110 Jan Wole´nski THE PRINCIPLE OF BIVALENCE AND SUSZKO THESIS Abstract According to Suszko, every logic with structural consequence operation is biva- lent, that is, evaluates sentences as either true or false. Truth and falsehoods are logical valuations. On the other hand, the Suszko thesis (ST) admits alge- braic valuations as related to semantic correlates of sentences. Due to Frege’s axiom (materially equivalent sentences are identical), standard propositional cal- culus identifies logical and algebraic valuations. SCI (sentential calculus with identity) considers identity and equivalence as different connectives. Hence, we can ascribe separate semantic correlates to different true sentences. Thus, SCI is logically bivalent and algebraically multivalent. This paper argues that although ST can be defended, its justification is pragmatic, but not purely logical. Key words: extensionality, logical square, metalogic. Suszko introduced ST in the following manner ([12], p. 378): [...] in case of any logic considered as an inference relation ` [...], one can find sets V of zero-one valued functions defined for all formulas and, called here logical valuations, with the following adequacy property: where a1, . , an ` b are arbitrary formulas with n = 0, 1, 2,..., then a1, . , an ` b if and only if for all t in V , t(b) = 1 whenever t(a1) = ... = t(a1) = 1. In short, every logic is (logically) two-valued. This general statement can be easily exemplified in case ofLukasiewicz three-valued sentential logic [...]. Although structurality of ` (or the consequence operation related to `) is not explicitly mentioned, Suszko tacitly assumes this property (see 100 Jan Wole´nski [11]). Logical valuations should be contrasted with algebraic valuations. The idea is this ([12], p. 377/378): Lindenbaum and Tarski observed that the formalized language [I slightly change original symbolism - JW] L is an absolutely free [...] algebraic structure and, hence the fountain of the whole class K(L) of all algebraic structures similar to L. The connections between L and any structure A in K(L) are given by maps of L to A satisfying so called morphisms conditions and labeled here as as algebraic valuations of L over A. They are admissible reference assignments. The domain of them consists of all expressions of definite syntactic category: formulas (sentences), terms (names) and diverse kinds of formators. The size of codomains of algebraic valuations is not a priori limited. In particular, the formulas may have many algebraic valuations [...]. Thus, the logical valuations and algebraic valuations are functions of quite different conceptual nature. The former relate to truth and falsity and, the latter represent the reference assignments. The formulas play a double semantical role, in general. It is the Fregean axiom which amalgamates it into the inseparable unity. The Fregean axiom mentioned at the end of the quoted passage is the formula (FA) ∀p∀q((p ⇒ q) → (Φ(p) ⇔ Φ(q))) This principle results in the following situation [12], p. 378]: In the case of the truth-functional logic, the logical valuations and algebraic val- uations coincide (in accord with the Fregean Axiom) and are represented by 1 and 0. Clearly, then, the material equivalence is the identity connective. Obviously, any mul- tiplication of logical values is a mad idea and, in fact,Lukasiewicz did not actualize it. Indeed, he defined his three valued logic [...] making essential use of algebraic valuations to a suitable algebra on three element set {1, 1/2, 0} with 1 as the sole “distinguished” element. Actually,Lukasiewicz defined but a logical system of L3-tautologies. However, one may reformulate L3 as an inference relation ` [...]. The, the following features of Lukasiewicz’s logic L3 can be revealed: (a) L3 is two-valued as stated previously. (b) L3 is a classical logic in the sense of [Brown and Bloom]. (c) L3 is a particular strengthening of SCI, i.e., the sentential calculus with identity. The point (b) in the above passage alludes to the concept of classical logic (see [1], p. 44) as generated by the consequence operation Cn which is finite, negative (Cn({A, ¬A}) = L) and disjunctive (Cn({A})∩Cn({B}) = The Principle of Bivalence and Suszko Thesis 101 Cn({A∨B}). Suszko considered the argument ending with observations (a) - (c) as justifying ST. For Suszko (see [10]), the division of sentences into true and false as the fundamental classification of linguistic items evaluated by truth-values and he strengthened this view by ideas captured by (a) - (c). Another way of introducing ST proceeds via valuations in matrices (I follow Malinowski’s presentation in [7], pp. 72–73 with some simplifications). A matrix M is a pair hA, Dsi, where A is an algebra similar to a given propositional language L and Ds is a non-empty set of designated values. Logical valuations are zero-one functions (homomorphisms) from languages to matrices. Now, v(A, h) = 1, if h(A) ∈ Ds and v(A) = 0, if h(A) 6∈ Ds. Thus, a formula A takes 1 as its logical value, if a given homomorphism h belongs to Ds, otherwise A is evaluated by 0. However, Malinowski points out that its practical realization sometimes requires additional data about the structure of matrices and thereby is generally non-constructive. My task consists in examining ST. In particular, I would like to ask whether this thesis is sufficiently justified by metalogic (let me note that I neglect Non-Fregean logic almost entirely). By sufficient (meta)logical jus- tification, I understand the derivation of a justified statement from some formerly prescribed formal logical models. Since I am not so much inter- ested in formal matters, I simplify ST to the following assertion (∗) every extensional logic generated by Cn satisfying the Tarski con- ditions is bivalent. I replace structurality of Cn (or rules of inference) by extensionality (mutual substitution salva veritate of materially equivalent formulas), be- cause the former as defined as preserving substitutions is a formal coun- terpart of the latter. The Tarski conditions include the constraint that X ⊆ CnX. If this axiom is dropped, one can find counterexamples to (∗) (see [8]). The converse of (∗) is, of course, false, because there are intensional bivalent logics, for example, modal (alethic) or deontic. Consider the following diagram (D) 102 Jan Wole´nski ν @ @ @ α @ β @@ @ @ @ @ @ γ @ δ @ @ @ @ µ The interpretation (T - it is true, F - it is false): αTA, β − FA, γ¬FA, δ¬TA. We have the following facts (the symbol ` indicates that the formula occurring after it is a logical theorem of logic generated by (D), that is, (D)-logic): (1) ` ¬(α ∧ β) (truth and falsehood are contraries); (2) ` (α ⇒ γ) (truth entails non-falsehood); (3) ` (β ⇒ δ) (falsehood entails non-truth); (4) ` (α ⇒ ¬δ) (truth and non-truth are contradictories); (5) ` (β ⇔ ¬γ) (falsehood and non-falsehood are c contradictories); (6) ` (γ ∨ δ) (non-falsehood and falsehood are sub-contraries). The diagram (D) (more precisely, the figure determined by the points αβγδ) displays the logical square of operators “it is true” and “it is false that”, fully analogical to diagrams related to categorical or modal sen- tences. The theorems (1)-(6) are not axioms, but rather indications of various interrelations holding between sentences denoted by α, β, γ and δ. Note that T and F are extensional operators. Thus, (D) can be used for analysis ST as captured by (∗). This is the main reason for replacing structurality by extensionality, because it fits Suszko’s extensionalism. What about the points ν and µ? They are naturally interpreted as (i) TA ∨ FA and (ii) ¬TA ∧ ¬FA, respectively. Of course, (i) and (ii) are mutually contradictory. The latter suggests that there are sentences which are neither true nor false; “can be” expresses here logical possibil- The Principle of Bivalence and Suszko Thesis 103 ity, that is, consistency. Consider the universal generalization of (i), that is, (iii) ∀A(TA ∨ FA) and its negation (v) ∃A(¬TA ∧ ¬FA). Thus, the assertion that every sentence is true or false excludes the statement that some sentences are neither true nor false. On the other hand, the assertion (vi) ∃A(TA ∨ FA) (some sentences are true or false) is consistent with the statement (v). In general, we have (7) ` (α ∨ β ∨ µ)(⇔ ν ∨ µ), as a theorem of (D)logic. Call the items represented by µ as neutral or in- different. (7) states that every sentence is either true or false or indifferent. This theorem is completely independent of any particular interpretation concerning neutralities (indifferences). One can understand them as truth- value gaps (sentences lacking of truth-values at all) or having other logical values. Clearly, it does not matter how many other truth-values are ad- mitted, when someone decides to adopt many-valuedness as the proper interpretation of neutralities. By the way, a mixed model, that is, recog- nizing that some sentences are gappy with respect to truth-values, or have other assignments than truth and falsehood, is also possible. Similarly, one can also introduce dialetheias (paraconsistencies) provided that they are understood as items which do not violate general principles of (D). The most important observation is that (D)-logic does not exclude neutralities. Speaking otherwise, the factual existence of sentences which are neither true nor false is independent of (D)-logic. The diagram (D) is very simple and captures fairly elementary logical ideas. Since it is universal and has many interpretations (see [13] and [14] for further information about applications of (D)), we can try to use it in metalogic. Clearly, (7) has straightforward and very important conse- quences for (∗). In particular, bivalence is not entailed by (D)-logic. The principle of bivalence is usually stated as the conjunction (PB) ∀A(TA ∨ FA) ∧ ∀A¬(TA ∧ FA), which means that every sentence is either true or false.