Abstract

In the article the multimodal logic Tn with central agent interaction axiom is analysed. The Hilbert type calculi is presented, then Gentzen type calculi with cut is derived and the proof of cutelimination theorem is outlined. The work shows that it is possible to construct a Gentzen type calculi without cut for this logic.

Highlights

  • IntroductionMultimodal logics (Kn, Tn, S4n) is often used to model the behaviour of agents

  • Multimodal logics (Kn, Tn, S4n) is often used to model the behaviour of agents. They do not include knowledge of interaction between them, so they are often enriched with interaction axioms

  • C) If the left and right upper sequents of a mix are not axioms and both are lower sequents of logical rules, we must distinguish cases according to the rules applied

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Summary

Introduction

Multimodal logics (Kn, Tn, S4n) is often used to model the behaviour of agents. They do not include knowledge of interaction between them, so they are often enriched with interaction axioms. For simplicity the base logics is Tn. The aim of the article is to find the sequent calculi without cut for logics Tn with central agent interaction axiom. We define propositional formula in standart way, including operators ¬, ∨, ∧, ⊃. To get modal formula we add modal logic operator K(l) in similar way, where l is either c, meaning central agent, or a number of agent starting from 1. In the article capital latin letters (A, B, . . . ) means any modal formula, capital greek letters ( , , 1, ∗, ) means a (possibly empty) multiset of modal formulas (the order of the formulas in a multiset does not matter)

Hilbert type calculus
Gentzen type calculus with cut

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