Logics of upsets of De Morgan lattices

Adam Přenosil

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arXiv · 2112.03884

Logics of upsets of De Morgan lattices

Adam Přenosil

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Abstract

We study logics determined by matrices consisting of a De~Morgan lattice with an upward closed set of designated values, such as the logic of non-falsity preservation in a given finite Boolean algebra and Shramko's logic of non-falsity preservation in the four-element subdirectly irreducible De Morgan lattice. The key tool in the study of these logics is the lattice-theoretic notion of an $n$-filter. We study the logics of all (complete, consistent, and classical) $n$-filters on De Morgan lattices, which are non-adjunctive generalizations of the four-valued logic of Belnap and Dunn (of the three-valued logics of Priest and Kleene, and of classical logic). We then show how to find a finite Hilbert-style axiomatization of any logic determined by a finite family of prime upsets of finite De Morgan lattices and a finite Gentzen-style axiomatization of any logic determined by a finite family of filters on finite De Morgan lattices. As an application, we axiomatize Shramko's logic of anything but falsehood.

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Adam Přenosil. 2021-12-07. Logics of upsets of De Morgan lattices. https://arxiv.org/abs/2112.03884

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