Equality in Substructural Logics

Abstract

Abstract Standard axioms for equality, or standard definitions of equality in secondorder logic or set theory, permit to infer versions limited to equalities of structural rules otherwise banned in substructural logics. In the presence of settheoretic extensionality and a rather natural form of separation, these limited structural rules yield the full structural rules in question. It is suggested that this may be prevented by restricting substitution for individual variables in the principle of replacement of equal terms. These restrictions are parallel to the restrictions for structural rules in substructural logics.