Recapture Results and Classical Logic

Abstract

An old and well-known objection to non-classical logics is that they are too weak; in particular, they cannot prove a number of important mathematical results. A promising strategy to deal with this objection consists in proving so-called recapture results. Roughly, these results show that classical logic can be used in mathematics and other unproblematic contexts. However, the strategy faces some potential problems. First, typical recapture results are formulated in a purely logical language, and do not generalize nicely to languages containing the kind of vocabulary that usually motivates non-classical theories—for example, a language containing a naive truth predicate. Second, proofs of recapture results typically employ classical principles that are not valid in the targeted non-classical system; hence non-classical theorists do not seem entitled to those results. In this paper we analyse these problems and provide solutions on behalf of non-classical theorists. To address the first problem, we provide a novel kind of recapture result, which generalizes nicely to a truth-theoretic language. As for the second problem, we argue that it relies on an ambiguity, and that once the ambiguity is removed there are no reasons to think that non-classical logicians are not entitled to their recapture results.