Adding an Implication to Logics of Perfect Paradefinite Algebras

TL;DR

This paper extends the study of perfect paradefinite algebras by introducing an implication connective, analyzing its logical properties, and establishing axiomatizations and interpolation results for the resulting logics.

Contribution

It proposes a new implication for perfect paradefinite algebras, provides axiomatizations, and explores logical properties like interpolation and amalgamation.

Findings

The new implication connective admits the deduction-detachment theorem.

Axiomatizations for the expanded algebra and logics are provided.

The variety induced by the new algebra has the Maehara amalgamation property.

Abstract

Perfect paradefinite algebras are De Morgan algebras expanded with an operation that allows for the full behavior of classical negation to be restored. They form a variety that is term-equivalent to the variety of involutive Stone algebras. Their associated multiple-conclusion (Set-Set) and single-conclusion (Set-Fmla) order-preserving logics are non-algebraizable self-extensional logics of formal inconsistency and undeterminedness determined by a six-valued matrix, studied in depth by Gomes et al. (2022) from both the algebraic and the proof-theoretical perspectives. In the present paper, we continue that study by investigating directions for conservatively expanding these logics with an implication connective (essentially, one that admits the deduction-detachment theorem). We first consider logics given by very simple and manageable non-deterministic semantics whose implication (in isolation) is classical. These, nevertheless, fail to be self-extensional. We then consider the implication realized by the relative pseudo-complement over the six-valued perfect paradefinite algebra. Our strategy is to expand the language of the latter algebra with this connective and study the (self-extensional) Set-Set and Set-Fmla order-preserving and top-assertional logics of the variety induced by the resulting algebra. We provide axiomatizations for such new variety and for such logics, drawing parallels with the class of symmetric Heyting algebras and with Moisil's 'symmetric modal logic'. For the Set-Set logic, in particular, the axiomatization we obtain is analytic. We close by studying interpolation properties for these logics and concluding that the new variety has the Maehara amalgamation property.