Mathematical logic, algebra and number theory

On extensions of minimal logic with linearity axiom

D. M. Anishchenko^a, S. P. Odintsov<sup>b

a</sup> Novosibirsk State University, ul. Pirogova, 1, 630090, Novosibirsk, Russia
^b Sobolev Institute of Mathematics, pr. Koptyuga, 4, 630090, Novosibirsk, Russia

Abstract: The Dummett logic is a superintuitionistic logic obtained by adding the linearity axiom to intuitionistic logic. This is one of the first non-classical logics, whose lattice of axiomatic extensions was completely described. In this paper we investigate the logic $JC$ obtained via adding the linearity axiom to minimal logic of Johansson. So $JC$ is a natural paraconsistent analog of the Dummett logic. We describe the lattice of $JC$-extensions, prove that every element of this lattice is finitely axiomatizable, has the finite model property, and is decidable. Finally, we prove that $JC$ has exactly two pretabular extensions.

Keywords: Dummett's logic, minimal logic, linearity axiom, lattice of extensions, algebraic semantics, $j$-algebra, opremum, decidability, pretabularity.

UDC: 510.64

MSC: 03B20,03B70

Received June 23, 2024, published October 23, 2024

DOI: 10.33048/semi.2024.21.056