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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.17724 |
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Table of Contents:
- 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.