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

Sergei P. Odintsov, Stanislav O. Speranski

Research output

1 Citation (Scopus)

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.
Original languageEnglish
Number of pages21
JournalReview of Symbolic Logic
DOIs
Publication statusE-pub ahead of print - 2019

Fingerprint

Modal Logic
Strong Negation
Kripke Semantics
Isomorphic
Term
Truth
Truth Value

Scopus subject areas

  • Logic
  • Philosophy

Cite this

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title = "Belnap–Dunn modal logics: truth constants vs. truth values",
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.",
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AB - 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.

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