First‐order Logical Validity and the Hilbert‐bernays Theorem

Abstract

What we call the Hilbert-Bernays (HB) Theorem establishes that for any satisfiable first-order quantificational schema S, there are expressions of elementary arithmetic that yield a true sentence of arithmetic when they are substituted for the predicate letters in S. Our goals here are, first, to explain and defend W. V. Quine's claim that the HB theorem licenses us to define the first-order logical validity of a schema in terms of predicate substitution; second, to clarify the theorem by sketching an accessible and illuminating new proof of it; and, third, to explain how Quine's substitutional definition of logical notions can be modified and extended in ways that make it more attractive to contemporary logicians.