Cut-free Sequent and Tableau Systems for Propositional Diodorean Modal Logics
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Raje Ev Gor'e
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We present sound, (weakly) complete and cut-free tableau systems for the propositional normal modal logics S4:3, S4:3:1 and S4:14. When the modality 2 is given a temporal interpretation, these logics respectively model time as a linear dense sequence of points; as a linear discrete sequence of points; and as a branching tree where each branch is a linear discrete sequence of points. Although cut-free, the last two systems do not possess the subformula property. But for any given finite set of formulae X the "superformulae" involved are always bounded by a finite set of formulae X L depending only on X and the logic L. Thus each system gives a nondeterministic decision procedure for the logic in question. The completeness proofs yield deterministic decision procedures for each logic because each proof is constructive. Each tableau system has a cut-free sequent analogue proving that Gentzen's cut-elimination theorem holds for these latter systems. The techniques are due to Hi...