Technical reports

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

**Rajeev Gore**

February 1993, 19 pages

**DOI:** 10.48456/tr-288

## Abstract

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 □ 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 logics. The techniques are due to Hintikka and Rautenberg.

## Full text

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## BibTeX record

@TechReport{UCAM-CL-TR-288, author = {Gore, Rajeev}, title = {{Cut-free sequent and tableau systems for propositional Diodorean modal logics}}, year = 1993, month = feb, url = {https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-288.pdf}, institution = {University of Cambridge, Computer Laboratory}, doi = {10.48456/tr-288}, number = {UCAM-CL-TR-288} }