Finite axiomatizability of logics of distributive lattices with negation

Logic Journal of the IGPL (forthcoming)
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Abstract

This paper focuses on order-preserving logics defined from varieties of distributive lattices with negation, and in particular on the problem of whether these can be axiomatized by means Hilbert-style calculi that are finite. On the negative side, we provide a syntactic condition on the equational presentation of a variety that entails failure of finite axiomatizability for the corresponding logic. An application of this result is that the logic of all distributive lattices with negation is not finitely axiomatizable; we likewise establish that the order-preserving logic of the variety of all Ockham algebras is also not finitely axiomatizable. On the positive side, we show that an arbitrary subvariety of semi-De Morgan algebras is axiomatized by a finite number of equations if and only if the corresponding order-preserving logic is axiomatized by a finite Hilbert-style calculus. This equivalence also holds for every subvariety of a Berman variety of Ockham algebras. We obtain, as a corollary, a new proof that the implication-free fragment of intuitionistic logic is finitely axiomatizable, as well as a new corresponding Hilbert-style calculus. Our proofs are constructive in that they allow us to effectively convert an equational presentation of a variety of algebras into a Hilbert-style calculus for the corresponding order-preserving logic, and vice versa. We also consider the assertional logics associated to the above-mentioned varieties, showing in particular that the assertional logics of finitely axiomatizable subvarieties of semi-De Morgan algebras are finitely axiomatizable as well.

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Umberto Rivieccio
Universidad Nacional de Educación a Distancia

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References found in this work

An algebraic approach to non-classical logics.Helena Rasiowa - 1974 - Warszawa,: PWN - Polish Scientific Publishers.
Nothing but the Truth.Andreas Pietz & Umberto Rivieccio - 2013 - Journal of Philosophical Logic 42 (1):125-135.
An infinity of super-Belnap logics.Umberto Rivieccio - 2012 - Journal of Applied Non-Classical Logics 22 (4):319-335.

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