### The lattice of Belnapian modal logics: Special extensions and counterparts

Sergei P. Odintsov, Stanislav O. Speranski

DOI: http://dx.doi.org/10.12775/LLP.2016.002

#### Abstract

Let K be the least normal modal logic and BK its Belnapian version, which enriches K with ‘strong negation’. We carry out a systematic study of the lattice of logics containing BK based on:

• introducing the classes (or rather sublattices) of so-called explosive, complete and classical Belnapian modal logics;

• assigning to every normal modal logic three special conservative extensions in these classes;

• associating with every Belnapian modal logic its explosive, complete and classical counterparts.

We investigate the relationships between special extensions and counterparts, provide certain handy characterisations and suggest a useful decomposition of the lattice of logics containing BK.

#### Keywords

algebraic logic; paraconsistent logic; many-valued modal logic; strong negation

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ISSN: 1425-3305 (print version)

ISSN: 2300-9802 (electronic version)