The Tarskian Turn: Deflationism and Axiomatic Truth

6. The Compositional Theory

6 The Compositional Theory In this chapter, we investigate a second axiomatic theory that is directly inspired by Tarski’s work on truth. Tarski’s definition of truth in a model inspires an axiomatic theory of truth that explicates the compositional nature of truth. We see that the compositional theory has several advantages when compared with the disquotational theory. 6.1 Clouds on the Horizon The following two propositions are typical illustrations of the proof-theoretic strength of the disquotational theory. Proposition 24 For all φ ∈ LPA: DT ⊢ T (φ) ∨ T (¬φ) Proof Already propositional logic alone proves φ ∨ ¬φ. Two restricted Tarskibiconditionals are T (φ) ↔ φ and T (¬φ) ↔ ¬φ. Combining these facts yields the desired result. n Proposition 25 (Tarski) DT  ∀φ ∈ LPA : T (φ) ∨ T (¬φ) Proof Take any proof P in DT. And suppose, for a reductio, that ∀φ ∈ LPA : T (φ) ∨ T (¬φ) belongs to P. P, being a finite object, can contain only finitely many restricted Tarski-biconditionals. Let us list these Tarski-biconditionals: 1. T (ϕ1) ↔ ϕ1 2. T (ϕ2) ↔ ϕ2 3. … . . . n. T (ϕn) ↔ ϕn 70 Chapter 6 Now we will construct a model M =: N, E which makes all sentences of P true. Let E consist precisely of those true arithmetical sentences ψ such that either ψ or ¬ψ is among {ϕ1, ϕ2, . . . , ϕn}. The model M makes all the arithmetical axioms true because it is based on N. And it makes all the restricted Tarski-biconditionals of P true by the construction of E. But for every true arithmetical sentence ϑ, such that neither ϑ nor ¬ϑ is in {ϕ1, ϕ2, . . . , ϕn}, the restricted Tarski-biconditional T (ϑ) ↔ ϑ is false in M. Thus, M |= ∀φ ∈ LPA : T (φ) ∨ T (¬φ). By one direction of the completeness theorem for first-order logic, this yields a contradiction with our supposition that ∀φ ∈ LPA : T (φ) ∨ T (¬φ) belongs to P. n So somehow DT proves all the instances of an intuitively plausible logical principle concerning the notion of truth, but it is unable to collect all these instances together into a general theorem. In a similar way, it can be shown that DT cannot prove: ∀φ, ψ ∈ LPA : (T (φ) ∧ T (ψ)) ↔ T (φ ∧ ψ). Something similar can be said for seemingly basic laws of truth concerning negation and the quantifiers. In summary, DTfails to fully validate our intuitions concerning the compositional nature of the notion of truth (i.e., the fact that the property of truth “distributes” over the logical connectives). In fact, our intuition that truth is compositional is (perhaps independent from but) just as basic as our intuition that truth is a disquotational device. Our truth theory has an obligation to either do justice to it or explain what is wrong with it. Earlier we insisted against inflationists about truth (such as Patterson) that proving many Tarski-biconditionals is a necessary condition for being a good truth theory.1 But now we see that deriving many Tarski-biconditionals is not a sufficient condition for being a good theory of truth. A good theory of truth must in addition do justice to the compositional nature of truth. The inability of DT to fully explicate the compositional nature of truth is a motivation for taking the principles that do explicate it as axioms of a theory of truth. We already know beforehand that if this is done, then there is a sense in which the resulting theory is stronger than DT. These principles of composition are of course contained in Tarski’s clauses for recursively explicating the notion of truth of a formula in a model. 1. Compare supra, section 2.2. The Compositional Theory 71 6.2 The Compositional Theory of Truth The axioms of our new axiomatic theory of truth can be directly read off from Tarski’s definition of truth in a model. Indeed, Tarski’s concept of a first-order model serves as the direct motivation for our new truth theory. Tarski’s axiomatic compositional theory of truth is denoted as TC.2 The axiomatic theory of truth TC is formulated in LT and consists of the following axioms: TC1 PAT ; TC2 ∀ atomic φ ∈ LPA : T (φ) ↔ val+ (φ); TC3 ∀φ ∈ LPA : T (¬φ) ↔ ¬T (φ); TC4 ∀φ, ψ ∈ LPA : T (φ ∧ ψ) ↔ (T (φ) ∧ T (ψ)); TC5 ∀φ(x) ∈ LPA : T ( ∀xφ(x)) ↔ ∀xT (φ(x)). So the idea behind this axiom system is straightforward.An explicit definition of the class of true atomic arithmetical sentences can be given by means of the arithmetical formula val+.3 Truth for complex arithmetical sentences can be reduced to truth of atomic arithmetical formulae through the compositional truth axioms...