### Cut-free sequent and tableau systems for propositional Diodorean modal logics

Rajeev Gor�
<span title="">1994</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/nzzezfqw65bhpbeksnoskjzuza" style="color: black;">Studia Logica: An International Journal for Symbolic Logic</a> </i> &nbsp;
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 nite set of
more &raquo; ... ulae X the \superformulae" involved are always bounded by a nite 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 Hintikka and Rautenberg.
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