More on modal logics and deduction

TL;DR

This paper explores the persistence of deduction properties in modal logics beyond additive cases, introducing strongly non-additive varieties and showing their abundance and transformation capabilities.

Contribution

It introduces the concept of strongly non-additive varieties, proves their abundance among modal logics, and shows how any normal modal logic can be transformed into such a logic.

Findings

Existence of continuum many strongly non-additive minimal discriminator varieties.

Every normal modal logic can be injectively transformed into a strongly non-additive logic.

The class of modal logics with the local deduction theorem is non-elementary.

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.