On Wright’s Inductive Definition of Coherence Truth for Arithmetic
In “Truth – A Traditional Debate Reviewed” (1999), Crispin Wright proposed an inductive definition of “coherence truth” for arithmetic relative to an arithmetic base theory B. Wright’s definition is in fact a notational variant of the usual Tarskian inductive definition, except for the basis clause for atomic sentences. This paper provides a model-theoretic characterization of the resulting sets of sentences "cohering" with a given base theory B. These sets are denoted WB. Roughly, if B satisfies a certain minimal condition (for each term t, B proves an equation of the form t = n, where n is a numeral), then WB is the Th(M), where M is the canonical model of the set At(B) of atomic sentences provable in B. The paper also shows that the disquotational T-scheme is provable (in a metatheory T) from Wright’s inductive definition just in case the base theory B is (provably in T) sound and complete for arithmetic atomic sentences.