PLEXUS Winter School 

Carlo Nicolai: “On Recapturing Classical Logic”

As a response to the semantic and logical paradoxes, theorists often reject some principles of classical logic. However, classical logic is entangled with mathematics, and giving up mathematics is too high a price to pay, even for nonclassical theorists. The so-called recapture theorems come to the rescue. When reasoning with concepts such as truth/class membership/property instantiation, if one is interested in consequences of the theory that only contain mathematical vocabulary, nothing is lost by reasoning in the nonclassical framework. In previous work I argued that this claim is misleading: recapture claims may hold for pure logics, but there’s no Liar or similar paradoxes for pure logics. As soon as some reasonable machinery to generate self-reference is in place, one can readily find counterexamples to them. The lectures will summarize the basic logical and philosophical issues underlying the debate and present some extensions.

The first lecture will be devoted to an overview of a reasonable setting to formulate the Liar and related paradoxes, recapture theorems, and their basic counterexamples. Among other things, I will introduce Kripke-Feferman theories of truth, their standard models based on fixed-point semantics, their internal nonclassical theories. In the second lecture, I will present several recapture claims and evaluate them both technically and conceptually. These include standard recapture strategies based on sharp syntactic distinctions such as the distinction between mathematical and non-mathematical vocabulary, recapture strategies based on restrictions by means of semantic notions such as groundedness or determinacy, and strategies relying conditions imposed on logical consequence such as JC Beall’s. The third lecture will be devoted to extensions of the results, open questions and invitation for further research.