The lattice of Belnapian modal logics: Special extensions and counterparts

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The lattice of Belnapian modal logics: Special extensions and counterparts

Logic and Logical Philosophy. 2016. Vol. 25. No. 1. P. 3–33.

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.

Language: English

DOI

On Belnapian modal algebras: Representations, homomorphisms, congruences, and so on

Speranski S. O., Siberian Electronic Mathematical Reports 2013 Vol. 10 P. 517–534

We obtain a bunch of principal results on Belnapian modal algebras (henceforth called BK-lattices) — these results may serve as a semantical basis for further investigation of the lattice of extensions of Belnapian modal logic (denoted by BK here). ...

Added: December 27, 2025

Belnap–Dunn modal logics: Truth constants vs. truth values

Odintsov S., Speranski S. O., Review of Symbolic Logic 2020 Vol. 13 No. 2 P. 416–435

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 ...

Added: December 26, 2025

Hintikka's independence-friendly logic meets Nelson's realizability

Odintsov S., Speranski S. O., Shevchenko I., Studia Logica 2018 Vol. 106 No. 3 P. 637–670

Inspired by Hintikka’s ideas on constructivism, we are going to ‘effectivize’ the game-theoretic semantics (abbreviated GTS) for independence-friendly first-order logic (IF-FOL), but in a somewhat different way than he did in the monograph ‘The Principles of Mathematics Revisited’. First we show that Nelson’s realizability interpretation — which extends the famous Kleene’s realizability interpretation by adding ...

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Некоторые предварительные условия для создания «многомировой теории всего» и развития интеллектуальной интуиции

Karpenko I., Логические исследования 2023 Т. 29 № 1 С. 9–29

The article questions some of the classical principles of thinking based on traditional intellectual intuition. In particular, the intuitive acceptability of the requirement of consistency, of the “anything follows from a contradiction” principle, of the law of the excluded middle, and some others are under challenge. The literature, especially logical one, devotes many studies to ...

Added: September 19, 2023

A Freedom beyond Conflict: The Logic of Internal Conflict and the Free Will in Maximus the Confessor

Lourié B., Scrinium: Journal of Patrology and Critical Hagiography 2018 Vol. 14 No. 1 P. 63–74

SummaryMaximus’ idea of appropriation of the divine will by deified humans, in any consistent interpretation, would mean their deprivation of their own freedom – exactly in the same manner as it could be in the case of servitude to sin. Maximus’ own logic, however, was paraconsistent when applied to the case of deification (whereas not ...

Added: October 8, 2018

The Logical Legacy of Nikolai Vasiliev and Modern Logic

Markin V., Springer, 2017.

This volume offers a wide range of both reconstructions of Nikolai Vasiliev’s original logical ideas and their implementations in the modern logic and philosophy. A collection of works put together through the international workshop "Nikolai Vasiliev’s Logical Legacy and the Modern Logic," this book also covers foundations of logic in the light of Vasiliev’s contradictory ...

Added: June 29, 2018

Potoses: Categorical Paraconsistent Universum for Paraconsistent Logic and Mathematic

Vasyukov V. L., Logical Investigations 2017 Vol. 23 No. 2 P. 76–95

It is well-known that the concept of da Costa algebra [3] reects most of the logical properties of paraconsistent propositional calculi Cn, 1< n <w introduced by N.C.A.da Costa. In [10] the construction of topos of functors from a small category to the category of sets was proposed which allows to yield the categorical semantics for da Costa's paraconsistent logic. ...

Added: June 29, 2018

L'opposition: analyse sémantique d'une notion flottante

Schang F., (France) 2012 Vol. 13 P. 65–85

A logical theory of oppositions deals with the relation between propositions and their truth values. On the basis of a formal semantics that proceeds by means of questions-answers, three theses are claimed in the following: (1) the concept of opposition usually refers to incompatibility, but our logical analysis focusses upon a broader relation of difference; ...

Added: October 31, 2014