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BELNAP–DUNN MODAL LOGICS: TRUTH CONSTANTS VS. TRUTH VALUES

Published online by Cambridge University Press:  22 February 2019

SERGEI P. ODINTSOV  and
STANISLAV O. SPERANSKI
SERGEI P. ODINTSOV*
Affiliation:
Sobolev Institute of Mathematics
STANISLAV O. SPERANSKI*
Affiliation:
St. Petersburg State University
*
*SOBOLEV INSTITUTE OF MATHEMATICS 4 KOPTYUG AVENUE 630090 NOVOSIBIRSK, RUSSIA E-mail: odintsov@math.nsc.ru
†ST. PETERSBURG STATE UNIVERSITY 29B LINE 14TH, VASILYEVSKY ISLAND 199178 SAINT PETERSBURG, RUSSIA E-mail: katze.tail@gmail.com
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Abstract

We shall be concerned with the modal logic BK—which is based on the Belnap–Dunn four-valued matrix, and can be viewed as being obtained from the least normal modal logic K by adding ‘strong negation’. Though all four values ‘truth’, ‘falsity’, ‘neither’ and ‘both’ are employed in its Kripke semantics, only the first two are expressible as terms. We show that expanding the original language of BK to include constants for ‘neither’ or/and ‘both’ leads to quite unexpected results. To be more precise, adding one of these constants has the effect of eliminating the respective value at the level of BK-extensions. In particular, if one adds both of these, then the corresponding lattice of extensions turns out to be isomorphic to that of ordinary normal modal logics.

Keywords

03B4503B5003Gxx06-XXmany-valued modal logicfirst-degree entailmentstrong negationalgebraic logic
Type
Research Article
Information
The Review of Symbolic Logic , Volume 13 , Issue 2 , June 2020 , pp. 416 - 435
Copyright
Copyright © Association for Symbolic Logic 2019 

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