The Truth Is Sometimes Simple

Note: The following version of this paper does not contain the proofs of the stated theorems. A longer version, complete with proofs, is forthcoming. §1. Introduction. In "The truth is never simple" (1986) and its addendum (1988), Burgess conducts a breathtakingly comprehensive survey of the complexity of the set of truths which arise when you add a truth predicate to arithmetic, and interpret that predicate according to the fixed point semantics or the revision-theoretic semantics for languages expressing their own truth concepts. Burgess considers various sets that can be said to represent truth in this context, and shows that their complexity ranges from Π 1 1 or Σ 1 1 to Π 1 2 or Σ 1 2. Thus, enriching arithmetic with a truth predicate increases its complexity, which is otherwise only ∆ 1 1. In his survey, Burgess assumes that we have fixed some Gödel numbering (or some other kind of coding of sentences as numbers) effectively identifying a sentence with its Gödel number. Gödel numbering is useful because it allows the object language to mimic talk about sentences, even when our model's domain contains only numbers. And if we want a semantics for an object language expressing its own truth concept, it seems a minimal requirement that the object language be able to refer to or at least quantify over sentences. Gödel numbering not only allows us to satisfy this minimal requirement; it also allows the object language to say when one sentence is the conjunction of two sentences, when one sentence is the negation of another, when one sentence is a substitution instance of another, and so on. Thus Burgess's results do not show that truth alone accounts for the increase in arithmetic's complexity from ∆ 1 1 to Π 1 2. Rather, it is the interplay between truth and these rich syntactic resources that is responsible for the increase in complexity. In the current paper, we investigate, in various contexts, the complexity of truth rather than the complexity of truth + rich syntactic resources. And we discover that the truth, even in arithmetic, is sometimes simple. Gupta (1982) makes a similar point, not about the complexity of truth, but about the ability of a languages containing its own truth predicate to express the liar's paradox and other