More on modal logics and deduction

Abstract

We study the relation between additivity and deduction theorems in the algebraic semantics of congruential modal logic. Additivity of the modal operator is well-known to imply the local deduction-detachment theorem. Our main theme is that deduction properties of modal logic persist far beyond the additive setting. We introduce the notion of a strongly non-additive variety, and then we prove that there are continuum many strongly non-additive minimal discriminator varieties of Boolean frames; equivalently, continuum many strongly non-additive maximal congruential modal logics with deduction-detachment theorem. Moreover, every normal modal logic can be transformed, in an injective way, into a strongly non-additive one while preserving the (local) deduction theorem. Finally, we show that neither the class of congruential modal logics with the local deduction theorem nor its complement is elementary.

Theorems & Definitions (38)

• Definition 1.1
• Example 2.2
• Theorem 2.3
• proof
• Corollary 2.4
• proof
• Theorem 2.6
• proof
• Theorem 3.1
• proof