The lattice of Belnapian modal logics: Special extensions and counterparts

Odintsov, Sergei P.; Speranski, Stanislav O.

doi:10.12775/llp.2016.002

Cited by 9 publications

(5 citation statements)

References 33 publications

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“…Notice-as was proved in [13], the lattice of B3K • -extensions, in turn, is isomorphic to that of K-extensions, i.e., consisting of ordinary normal modal logics. At this point it is worth giving some historical background for our work.…”

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confidence: 72%

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Belnap–dunn Modal Logics: Truth Constants vs. Truth Values

Odintsov

Speranski

2019

The Review of Symbolic Logic

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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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confidence: 72%

“…Moreover the L-algebras in V (B3K), V BK • and V B3K • can be characterised up to isomorphism as follows: PROPOSITION 2.10 (see [13]).…”

Section: Algebraic Semanticsmentioning

confidence: 99%

Belnap–dunn Modal Logics: Truth Constants vs. Truth Values

Odintsov

Speranski

2019

The Review of Symbolic Logic

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“…Among the first papers regarding modal many-valued logics are Fitting's ones [14,15]. Modal extensions of FDE and related logics were studied by Goble [18], Priest [44,45], Odintsov and Wansing [41,40] as well as Odintsov, Skurt, and Wansing [37], Odintsov and Latkin [36], Odintsov and Speranski [38,39], Sedlár [55], Rivieccio, Jung, and Jansana [49].…”

Section: Introductionmentioning

confidence: 99%

On a multilattice analogue of a hypersequent S5 calculus

Grigoriev

Petrukhin

2019

LLP

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In this paper, we present a logic MML S5 n which is a combination of multilattice logic and modal logic S5. MML S5 n is an extension of Kamide and Shramko's modal multilattice logic which is a multilattice analogue of S4. We present a cut-free hypersequent calculus for MML S5 n in the spirit of Restall's one for S5 and develop a Kripke semantics for MML S5 n , following Kamide and Shramko's approach. Moreover, we prove theorems for embedding MML S5 n into S5 and vice versa. As a result, we obtain completeness, cut elimination, decidability, and interpolation theorems for MML S5 n . Besides, we show the duality principle for MML S5 n . Additionally, we introduce a modification of Kamide and Shramko's sequent calculus for their multilattice version of S4 which (in contrast to Kamide and Shramko's original one) proves the interdefinability of necessity and possibility operators. Last, but not least, we present Hilbert-style calculi for all the logics in question as well as for a larger class of modal multilattice logics.

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“…This article fills the gap. We discuss BPDL, a logic that adds program modalities to Odintsov and Wansing's [16] basic modal logic with Belnapian truth values BK (see also [17,18]). Our main technical results concerning BPDL (introduced in Section 3 of the article) are a decidability proof using a variation of the standard argument based on filtration (Section 4) and a sound and weakly complete axiomatisation (Section 5).…”

Section: Introductionmentioning

confidence: 99%