Quine’s Substitutional Definition of Logical Truth and the Philosophical Significance of the Löwenheim-Hilbert-Bernays Theorem

Abstract

The Löwenheim-Hilbert-Bernays theorem states that, for an arithmetical first-order language L, if S is a satisfiable schema, then substitution of open sentences of L for the predicate letters of S results in true sentences of L. For two reasons, this theorem is relevant to issues relative to Quine’s substitutional definition of logical truth. First, it makes it possible for Quine to reply to widespread objections raised against his account (the lexicon-dependence problem and the cardinality-dependence problem). These objections purport to show that Quine’s account overgenerates: it would count as logically true sentences which intuitively or model-theoretically are not so. Second, since this theorem is a crucial premise in Quine’s proof of the equivalence between his substitutional account and the model-theoretic one, it enables him to show that, from a metamathematical point of view, there is no need to favour the model-theoretic account over one in terms of substitutions. The purpose of that essay is thus to explore the philosophical bearings of the Löwenheim-Hilbert-Bernays theorem on Quine’s definition of logical truth. This neglected aspect of Quine’s argumentation in favour of a substitutional definition is shown to be part of a struggle against the model-theoretic prejudice in logic. Such an exploration leads to reassess Quine’s peculiar position in the history of logic.