The Principle of Bivalence and Suszko Thesis

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 bivalent, that is, evaluates sentences as either true or false. Truth and falsehoods are logical valuations. On the other hand, the Suszko thesis (ST) admits algebraic valuations as related to semantic correlates of sentences. Due to Frege’s axiom (materially equivalent sentences are identical), standard propositional calculus 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 valuations 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 multiplication 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 justification, 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 conditions is bivalent. I replace structurality of Cn (or rules of inference) by extensionality (mutual substitution salva veritate of materially equivalent formulas), because the former as defined as preserving substitutions is a formal counterpart 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 Woleński 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 sentences. 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 indifferent. (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. Since the second conjunct, that is, the formula ∀A¬(TA ∧ FA) follows from (1), (PB) is equivalent to (iii) in our model, but it must be stated as above, if (D)-logic is not explicitly assumed. One can say that (PB) results in restricting (D) to the triangle defined by the points αβµ. Accepting one of the equiva- lences TA ⇔ ¬FA, ¬T¬A ⇔ TA, ¬F¬A ⇔ FA is enough to achieve this 104 Jan Woleński triangle; note that the formula TA ⇔ F¬A (it follows from (D)-logic) does not suffice in this respect. Consequently, (∗) is not justified by (D), because (D)-logic does not entail (PB); thus the formula ∀A(TA ∨ FA) is absolutely crucial in our considerations. The focal point is that the principle of bivalence do not result from (D)-logic. Assume that we define the operation CnD related to (D)-logic. Since operators T and F as governed by (D) are extensional, this property plus CnD do not generate (PB). Moreover strengthening of (D)-logic by any means of first-order logic is ir- relevant here. The principle of bivalence, pace JanLukasiewicz, is a result of meta-logical decision. Bivalence may be introduced by using additional constraints, in particular, by pairing T and ¬T or F and ¬F. A more general strategy is suggested by Rosser and Turquette in [9], p. 22, where logical values are divided into designated and undesignated. The rule (n) for negation estab- lishes v(¬A) = 1, if v(A) 6∈ Ds and v(¬A) = 0, if v(A) ∈ Ds, where 0 is conventionally chosen as representing undesignated values; these equalities hold independently of the number of designated and undesignated values, provided that Ds and its complement Ds’ are non-empty and mutually disjoint. As a result we have (PB0) ∀A(1A ∧ 0A) ∧ ∀A¬(1A ∧ 0A), which means that every sentence is either designated or undesignated; 1 and 0 are abbreviations for “is designated” and “is undesignated”. If Ds = {T} and Ds’ = {F},(PB’) becomes (PB), but the antecedent of this assertion must be explicitly assumed. Although the rule (n) generalizes the two-valued negation, it does not reproduce all intuitions. For example, 1 inLukasiewicz’s three-valued logic we have v(A) = v(¬A), if v(A) = /2. Thus we cannot apply the valuation of negation introduced by (n) in every case, unless some additional qualifications are adopted, possibly related to neutralities. Obviously, the solution via (PB’) does not fit Suszko’s intentions, because he wanted to have truth as the designated value and falsehood as the undesignated one. On the other hand, since there is no automatic logical transition from his 1 and 0 to T and F, respectively, it is unclear whether (∗) assumes (PB) or (PB’). Note that Malinowski’s characterizations of ST (see above) does not appeal to truth and falsehood, but only do Ds and its complement. I will return to this question at the end of this paper. The Principle of Bivalence and Suszko Thesis 105

It is interesting to locate ST in a comprehensive map of logic, related to the problem of bivalence, proposed by Henryk Greniewski (see [4], pp. 136– 137). Let VER ⊂ L, VER 6= ∅ and FLS ⊂ L, FLS 6= ∅; verbally, the set of truths and the set of falsehoods are proper non-empty subsets of L (languages are set of sentences). The family {VER, FLS} is called bi- division of L; it is counterpart of the fundamental division in Suszko’s sense). Consider the following theses (some of them were already discussed in this paper): (A) ∀A ∈ L(A ∈ VER ∪ FLS) (every sentence is true or false); (B) ∀A ∈ L¬(A ∈ VER ∧ A ∈ FLS) (no true sentence is false); (C) The bi-division of L is enough for constructing logic; (D) ∃A ∈ L(A 6∈ VER ∪ FLS) (there are sentences which are neither true nor false); (E) ∃A ∈ L(A ∈ VER ∩ FLS) (there are sentences which are true and false); (F) The bi-division of L does not suffice for constructing logic. We have eight mutually disjoint combinations characterizing possible views about bivalence and many-valuedness: (I) (A), (B), (C); (II) (A), (B), (F); (III) (A), (C), (E); (IV) (A), (E), (F); (V) (B), (C), (D); (VI) (B),(D), (F); (VII) (C), (D), (E); (VIII) (D), (E), (F). Since some dif- ferences between particular combinations seem secondary, Greniewski proposes to distinguish three principal camps: (•) The Bivalentists ((I), (II)); (••) The Pseudobivalentists ((III), (V), (VII)); and (•••) The Antibivalen- tists ((IV), (VI), (VIII)). The Bivalentists accept PB (the conjunction of (A) and (B)), but they differ as far as the matter concerns whether the bi- division of L suffices for constructing logic. The Pseudobivalentists accept either the metalogical tertium non datur ((A)) or the metalogical principle of non-contradiction ((B)) and take the bi-division as sufficient or not. The Antibivalentists accept neutralities or dialetheias and deny that the bi-division adequately displays the basis of logic. Greniewski considers himself as a member of the camp (•), but in the version (II). Thus, according to Greniewski, standard logic (PC) is generally correct, but in some cases it requires to be supplemented by many-valued logics, because in some cases a more detailed classification of logical values should be introduced, for example, when we investigate probability, statistics or radioactivity. Since Greniewski accepts (PB), new 106 Jan Woleński logics are extensions of PC and should not be considered as its non-classical alternatives. Anyway, having a logic with 2n logical values, we can always construct its extension with 2n+1 logical values. Suszko’s idea is similar to some extent, because he assumes PB and proposes SCI as a more adequate propositional calculus. On the other hand, Suszko’s view is more radical than Greniewski’s proposal. Whereas the latter is local and oriented on precisely defined fragmentary tasks, Suszko’s distinction between logical valuations and algebraic valuations is global and intended as universal. Consequently, SCI is not conceived as a supplement of PC, but as the proper logic of propositions. Assuming Suszko’s explanations, Greniewski’s many-valued systems should not be called logic at all, but conceived as algebras of sentences. In fact, Greniewski explains his constructions by invoking Boolean algebra and nothing more is necessary for his exposition. Anyway, algebraic valuations in Suszko’s sense and Greniewski’s extensions of PC obviously exceed analytical tools provided by (D). This diagram and its logic show that one has to deny (PB) in order to incorporate the camps (••) and (•••) into the logical game. Leaving Greniewski’s logics aside, I concentrate, from the point of view related to (D), on algebraic valuations in the Suszko sense. Clearly, both kinds of valuations introduced by Suszko operate on different levels. Con- sequently, algebraic valuations defined on the bi-division (I deliberately use Greniewski’s apt terminology) of L require additional devices than avail- able in (D)-logic. In particular, neutralities cannot be considered as algebraically valued, because (D)-logic considers them as neither true not false, but the contrary statement is associated with (∗). The problem is that the distinction between logical valuations and algebraic valuations does not confirm by itself the view that the latter are not reducible to the former? This is the crucial point for any justification of (∗). In fact, this thesis characterizes the logic as extensional, having the Tarskian Cn (I will omit “Tarskian” below) and bivalent. I will consider extensionality and Cn very briefly. One can says that Tarski’s axiomatization of Cn (I refer to so- called general axioms) chooses the most elegant and useful mathematical consequence operation from infinitely many possibilities. Of course, this statement will be rejected by the proponents of non-monotonic logic; note that the earlier mentioned principle X ⊆ CnX belongs to controversial cases. Even if non-monotonic logicians agree that mathematics of Cn is simple and elegant, they probably observe that we need a more complex theory in order to cover scientific and commonsensical reasoning. Since The Principle of Bivalence and Suszko Thesis 107 the issue of Cn as the most proper consequence operation cannot be re- solved by purely objective criteria, let us stay with Suszko’s view that this consequence operation is basic. The principle of extensionality concerns syntax as well as semantics. The critics of extensionality as a mark of logic point out that this way of looking at logic ignores intensional contexts and their significant role in human communication in various fields. Hence, intensional logic is necessary. On the other hand, this criticism seems to overlook that extensionality makes syntax and semantics perfectly paral- lel. This fact makes possible to apply to syntax and semantics the same mathematical tools, in particular, proof-techniques based on the idea of recursivity. Speaking in the name of defenders of (∗), I would like to note that intensional languages can be subjected to formal theories, but we do not need to consider them as purely logical. Thus, I am inclined to think that extensionality and, so to speak, Cnity are very plausible features of logic, although chosen by pragmatic criteria, like simplicity, elegancy, etc. Lukasiewicz entirely deliberately defined functors acting on sentences interpreted by many-valued assignments as extensional. Hence, one can associate Cn with extensional many-valued languages and just this cir- cumstance decides that the issue of (PB) is so crucial for the discussed problem; incidentally, the general axioms for Cn do not entail (PB). Now we see that any defence of (∗) cannot be exclusively limited to justifying of (PB) as correct for some reasons. In particular, the proponent of ST must explain why many-valued assignments are not logical, but just algebraic, because combining extensionality (I drop Cnity here) together with (PB) leads to a very natural understanding of classical logic (see [3]). This strategy is reasonable, because it excludes many-valued logics and intensional logics as non-classical, but not necessary as logical systems. We see once again that Suszko went further in his understanding of classical logic. Per- haps the strongest argument against logical values outside the bi-division of L consists in observing that an intuitive interpretation of neutralities or dialetheias raises notorious difficulties (see [2], [5], [6], [7], [9]; almost every monograph or survey dealing with many-valued logic points out at this problem), but Suszko did not use this argument. He wanted to give a purely formal justification for ST. He did that for three-valued logic (see [8], p. 153, [11], p. 88) by two postulates (provided that atomic formulas and their negations are valued by 0), namely (x)v(A) 6= v(¬A) and (xx)v(A∨B) = 0 ⇔ v(A) = v(B) = 0 (the rest of connectives is defined by negation and disjunction). However, this formal machinery does not dis- 108 Jan Woleński tinguish (see above) between F and ¬T and, consequently, between (PB) and (PB’). Speaking otherwise, ¬0 (eventually 1) and 0 do not need be understood as truth and falsehood without further constraints. Returning to many-valued logic traditionally understood, we should note two things. Firstly, formal many-valued constructions are commonly regarded as logically correct Secondly, new developments, in particular, fuzzy logic motivate an optimism concerning the question of interpretation of other (than T and F) values. The above discussion, supplemented by metalogic based on (D), suggests that Suszko did not justify his thesis by purely logical devices. This does not mean that the idea of the bi-division as the fundamental classification of sentences is erroneous. On the contrary, cutting (D) to the triangle αβµ is a legitimate move, because resulting logic is the most universal, mathematically simple and indispensable in metalogic. Yet this conclusion is based on pragmatic constraints. Thus, we have good reasons to prefer (PB) (note that just (PB), not only (PB’)) as a reasonable metalogical principle, but we should not argue that this choice is dictated by pure logic. On the other hand, defending bivalence is fully independent of accepting ST, because the existence of neutralities is coherent with (D). In fact, Suszko’s advanced formal machinery only shows that sentences can be valued by various assignments, but it fails in proving that every extensional (or equipped with the structural Cn) logic is bivalent with respect to truth and falsehood. In fact, intensional logic can be bivalent, exactly in the same sense as extensional one. Similarly, non-monotonic or infinitary logic, that is, based on other conditions than imposed by Tarski, may satisfy (PB). Suszko certainly exaggerated when he said “Obviously, any multiplication of logical values is a mad idea”. The same exaggeration might be pointed out about dropping extensionality in favour of intensionality or changing some properties of Cn as signs of mad- ness. In general, emotional language is not a good device of communication in discussing logical matters. I am indebted to two anonymous referees for critical remarks which essentially contributed to the improvements in the paper. The Principle of Bivalence and Suszko Thesis 109

References

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