Interpolation in Non-Classical Logics

TL;DR

This paper reviews key results on interpolation properties across various non-classical logics, emphasizing differences between Craig and deductive interpolation in diverse logical systems.

Contribution

It provides a comprehensive survey of interpolation phenomena in multiple non-classical logic families, highlighting distinctions and commonalities among them.

Findings

Interpolation properties vary significantly across non-classical logics.

Craig and deductive interpolation often differ in these systems.

The survey identifies open problems and patterns in interpolation behavior.

Abstract

This chapter surveys some of the main results on interpolation in several of the most prominent families of non-classical logics. Special attention is given to the distinction between the two most commonly studied variants of interpolation--namely, Craig interpolation and deductive interpolation. Our discussion focuses primarily on how these properties present in families of logical systems taken as a whole, particularly those comprising all axiomatic extensions of any of several notable non-classical logics. We consider a range of important examples: superintuitionistic and modal logics, fuzzy logics, paraconsistent logics, relevant logics, and substructural logics.