Logicality and truth
Résumé
Our aim is to support the deflationist view on truth according to which the truth predicate is a mere expressive device that does not correspond to a 'substantial' property. Our strategy consists in showing that the truth predicate is logical, in a sense to be made precise. To make our point, we rely on the analysis of logicality in terms of invariance, which is reasonably well-accepted by philosophers of logic and logicians, following Tarski [1986], as well as by semanticists, following Barwise & Cooper's [1981] account of determiners in generalized quantifier theory. Deflationism about truth has it that truth attributions, unlike robust property ascriptions, do not have any explanatory force. The truth predicate provides us with a convenient way of increasing the expressive power of our language while keeping the syntax finite. In this respect, the truth predicate would be nothing but a logical alternative to infinitary logics. The idea that truth is a logical notion has been objected to on the basis of the so-called conservativity argument (see Shapiro [1998] and Ketland [1999]). For many theories T, adding to T its truth theory yields a non-conservative extension. Truth would have an explanatory power after all; it allows us to prove non-alethic facts about the base theory T. However, the argument might not be as conclusive as it looks: conservativity is a relative property, and failure of conservativity could be construed as revealing a lack of deductive power in the base theory rather than an intrinsic added value in the added theory (as we shall explained in the talk). We decide to follow a different route in order to ground the claim that truth is a logical notion. We shall consider the truth predicate as an interpreted predicate -- either as a Tarskian truth-predicate applied to an object-language which does not contain it or as a Kripkean predicate. And we shall argue that such a truth predicate does count as logical by standard semantic criteria of logicality, according to which an expression is logical iff its extension is invariant under a wide range of transformations. Here is a sketch for our main result. Consider an arbitrary abstract model-theoretic logic L. Extend L to L+Tr by adding an interpreted truth-predicate Tr together with arbitrary interpreted syntactic predicates (which make for definable sets of sentences to put the truth predicate to work). In parallel, expand the L-structures into so-called alethic expansions which contain a copy of the syntax of L (Tarskian case) or L+Tr (Kripkean case) together with suitable extensions for Tr and for the syntactic predicates. So each L-structure gets associated with exactly one L+Tr structure. We say that a logic L is logically complete iff there is an invariance criterion C such that all logical expressions in L satisfy C and all expressions satisfying C are definable in L by purely logical means. We say that a logic L is truth-complete iff for any class K of alethic expansions, if that class is elementary (i.e. definable by one sentence) in L+Tr, the associated reduct class K' is elementary in L, where K' is the class of structures M such that the alethic expansion of M belongs to K. We prove that a logic L is logically complete iff it is truth complete. Interpretation. Our theorem supports the following intuitive interpretation. Adding an interpreted truth predicate to a given logic might result in a substantial gain in expressive power: new classes of structures become definable. So it goes with first-order logic. But logics for which this happens are too weak. They do not satisfy the requirement of logical completeness which amounts to a form of functional completeness, if the invariance analysis of logicality is correct. To the contrary, if a logic is strong enough -- if it is logically complete -- the truth predicate turns out to be logical. It is already implicitly definable by purely logical means (where 'implicit definability receives a precise meaning from the previous definition of truth-completeness).