Two proofs of the algebraic completeness theorem for multilattice logic

Grigoriev, Oleg; Petrukhin, Yaroslav

doi:10.1080/11663081.2019.1647654

Cited by 5 publications

(9 citation statements)

References 17 publications

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“…An algebraic completeness of MML n with respect to an S4-modal multilattices is left as an open problem in [28]. However, it was solved in [19]. But the formulation of the notion of S4-modal multilattice was changed in [19,Definition 41].…”

Section: What Is Modal Multillatice Logic?mentioning

confidence: 99%

“…However, it was solved in [19]. But the formulation of the notion of S4-modal multilattice was changed in [19,Definition 41]. The conditions (sub-multiplicative) and (sub-additive) were strengthened as follows:…”

Section: What Is Modal Multillatice Logic?mentioning

confidence: 99%

“…. , ≤ n (in a sense of [19,Definition 41]) is called S5-modal iff for each j n and x ∈ S it satisfies the following condition:…”

Section: What Is Modal Multillatice Logic?mentioning

confidence: 99%

“…It is interesting that, as follows from [19], the algebraic completeness theorem holds for Kamide and Shramko's original formulation of MML n , since it is generally not the case that the equalities…”

Section: What Is Modal Multillatice Logic?mentioning

confidence: 99%

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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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Section: What Is Modal Multillatice Logic?mentioning

confidence: 99%

Section: What Is Modal Multillatice Logic?mentioning

confidence: 99%

“…. , ≤ n (in a sense of [19,Definition 41]) is called S5-modal iff for each j n and x ∈ S it satisfies the following condition:…”

Section: What Is Modal Multillatice Logic?mentioning

confidence: 99%

Section: What Is Modal Multillatice Logic?mentioning

confidence: 99%

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On a multilattice analogue of a hypersequent S5 calculus

Grigoriev

Petrukhin

2019

LLP

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“…This logic was supposed to be a multilattice version of S4. However, as argued in [6,7], this point is a bit problematic. Namely, Kamide and Shramko exploited an incomplete S4 sequent calculus for the embedding purposes.…”

Section: Introductionmentioning

confidence: 99%

Modal multilattice logics with Tarski, Kuratowski, and Halmos operators

Grigoriev

Petrukhin

2021

LLP

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In this paper, we consider modal multilattices with Tarski, Kuratowski, and Halmos closure and interior operators as well as the corresponding logics which are multilattice versions of the modal logics MNT4, S4, and S5, respectively. The former modal multilattice logic is a new one. The latter two modal multilattice logics have been already mentioned in the literature, but algebraic completeness results have not been established for them before. We present a multilattice version of MNT4 in a form of a sequent calculus and prove the algebraic and neighbourhood completeness theorems for it. We extend the algebraic completeness result for the multilattice versions of S4 and S5 as well.

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Algebraic Completeness of Connexive and Bi-Intuitionistic Multilattice Logics

Petrukhin

2024

J of Log Lang and Inf

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In this paper, we introduce the notions of connexive and bi-intuitionistic multilattices and develop on their base the algebraic semantics for Kamide, Shramko, and Wansing’s connexive and bi-intuitionistic multilattice logics which were previously known in the form of sequent calculi and Kripke semantics. We prove that these logics are sound and complete with respect to the presented algebraic structures.

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Two proofs of the algebraic completeness theorem for multilattice logic

Cited by 5 publications

References 17 publications

On a multilattice analogue of a hypersequent S5 calculus

On a multilattice analogue of a hypersequent S5 calculus

Modal multilattice logics with Tarski, Kuratowski, and Halmos operators

Algebraic Completeness of Connexive and Bi-Intuitionistic Multilattice Logics

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