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Journal of Logic and Computation
Article . 1994 . Peer-reviewed
Data sources: Crossref
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Sequent Calculi for Normal Modal Propositional Logics

Sequent calculi for normal modal propositional logics
Authors: Heinrich Wansing;

Sequent Calculi for Normal Modal Propositional Logics

Abstract

This is a streamlined exposition of \textit{N. Belnap}'s display logic DL [J. Philos. Logic 11, 375-417 (1982; Zbl 0509.03008)]. DL can be seen as an encoding of a formalization introduced by \textit{S. Kripke} [Z. Math. Logik Grundlagen Math. 9, 67-96 (1963; Zbl 0118.013)] where finite systems of sequents \(S; \sigma_ 1 S_ 1; \sigma_ 2 S_ 2;\dots;\sigma_ k S_ k\) indexed by finite sequences \(\sigma_ i\) of natural numbers are derived. Such a system is interpreted as ``\(S\) is true in the original world, \(S_ 1\) is true in the world \(\sigma_ 1,\dots\)'' of a Kripke model. Formulas of DL are built up from the propositional variables and constants \textbf{0,1} by the connectives \(\neg\), \(\&\), \(\vee\), \(\supset\), \(\square\), \(\diamondsuit\). Structures are built from formulas by \({\mathbf I}\), \(\circ\), *, \(\bullet\). Sequents of DL are of the form \(X\to Y\) where \(X\), \(Y\) are structures. Structures are translated into standard modal language extended by a new connective \(P\) (sometimes in the past). There are several dozens of inference rules for the basic modal system \(K\), and additional rules for many of the popular modal systems allowing to embed these systems into DL. Inverse embedding uses a conservative extension result for \(P\) which is established using Kripke semantics. It is possible to visualize the rules of DL by comparing them to (very few) inference rules of Kripke-type systems. For examples for S5 only singleton indices are needed, and a system of sequents \(\to A;1\to A_ 1;\dots;n\to A_ n\) can be translated as display structure \(A^*\to \bullet A_ 1\circ \dots\circ \bullet A_ n\). If say \(A_ n\) is to be analyzed by an inference rule, the latter structure can be transformed into \(A^*_ n\to \bullet A\circ \bullet A_ 1\circ\dots\circ \bullet A_{n-1}\) (essentially as in the reviewer's paper in: Tr. Mat. Inst. Steklov 98, 88-111 (1968; Zbl 0169.299). For other systems it is necessary to use iterated \(P\) or \(\bullet\). The cut-elimination theorem is stated, and the reader is referred to Belnap's paper for a sketch of a proof. Some remarks on substructural logics are made.

Related Organizations
Keywords

sequents, cut-elimination, Cut-elimination and normal-form theorems, display logic, Modal logic (including the logic of norms), Kripke model, Kripke semantics

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
80
Top 10%
Top 1%
Top 10%
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