On a multilattice analogue of a hypersequent S5 calculus

IF 0.6 Q2 LOGIC Logic and Logical Philosophy Pub Date : 2019-07-18 DOI:10.12775/LLP.2019.031
Oleg M. Grigoriev, Y. Petrukhin
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引用次数: 7

Abstract

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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超序S5微积分的多格模拟
本文提出了一种多格逻辑和模态逻辑S5的组合逻辑MML S5 n。MML s5n是Kamide和Shramko模态多格逻辑的扩展,是S4的多格模拟。我们在Restall的S5演算的精神上提出了MML S5 n的无切割超序列演算,并根据Kamide和Shramko的方法开发了MML S5 n的Kripke语义。此外,我们证明了将MML S5 n嵌入到S5中的定理,反之亦然。得到了MML s5n的完备性定理、切消定理、可判决性定理和插值定理。此外,我们还证明了MML s5n的对偶性原理。此外,我们引入了对Kamide和Shramko对S4的多格版本的序演算的一个修正(与Kamide和Shramko的原始版本相反),证明了必要性算子和可能性算子的可互定义性。最后,但并非最不重要的是,我们提出了hilbert式演算所有的逻辑问题,以及更大的一类模态多格逻辑。
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来源期刊
CiteScore
1.00
自引率
40.00%
发文量
29
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