On Wright's inductive definition of coherence truth for arithmetic
Jeffrey Ketland · Analysis · 2003
As the first illustration of a potential satisfier for the ‘platitudes for truth ’ in the appendix to his engaging recent discussion of the concept of truth (Wright 1999), Crispin Wright has proposed a notion of ‘truth conceived as coherence ’ for arithmetic. This paper attempts to clarify certain aspects of Wright’s proposal. Take the standard first-order language of arithmetic L. 1 Let B be some axiom system for arithmetic, which Wright calls the ‘coherence base’. With small notational modifications, Wright proposes the following inductive definition of the concept ‘coheres with B ’ (for L-sentences): (CAt) If ϕ is atomic, then ϕ coheres with B iff B � ϕ. (C¬) ¬ϕ coheres with B iff ϕ does not cohere with B. (C∧) ϕ ∧ ψ coheres with B iff ϕ and ψ cohere with B. (C∨) ϕ ∨ ψ coheres with B iff either ϕ or ψ coheres with B. (C→) ϕ → ψ coheres with B iff either ϕ does not or ψ does cohere with B. (C∀) ∀xϕ coheres with B iff, for each number n, ϕ(n) coheres with B. (C∃) ∃xϕ coheres with B iff, for some number n, ϕ(n) coheres with B. First, note that except for the basis clause (CAt) this is the same as the usual Tarskian inductive definition of truth for arithmetic: 2 (TAt) If ϕ has the form t = u, then ϕ is true iff val(t) = val(u). 3 1 The terms of L are defined recursively from a basis of variables, the constant 0 and the operation symbols s, + and ×. The numerals of L are written n, meaning 0 prefixed by n occurrences of the successor symbol s. The atomic formulas of L are equations of the form t = u (with t, u terms) and complex formulas of L are defined by recursion on complexity as usual. Below, Sent(L) is the set of Lsentences of L, AtSent(L) is the set of atomic L-sentences and Form(L) is the set of L-formulas. In the arithmetic formalization of semantics, we use SentL(x) and ClTmL(x) to mean arithmetic formulas expressing respectively that x is (the code of) a sentence of L or that x is (the code of) a closed term of