Cut-free sequent and tableau systems for propositional Diodorean modal logics
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We present sound, (weakly) complete and cut-free tableau systems for the propositional normal modal logicsS4.3, S4.3.1 andS4.14. 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 formulaeX the “superformulae” involved are always bounded by a finite set of formulaeX*L depending only onX and the logicL. 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.
KeywordsMathematical Logic Modal Logic Decision Procedure Computational Linguistic Completeness Proof
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- R. A. Bull,An algebraic study of Diodorean modal systems Journal of Symbolic Logic 30(1), 1965, pp. 58–64.Google Scholar
- R. A. Bull,Review of ‘Melvin Fitting, Proof Methods for Modal and Intuitionistic Logics, Synthese Library, Vol. 169, Reidel, 1983’ Journal of Symbolic Logic 50, 1985, pp. 855–856.Google Scholar
- R. A. Bull andK. Segerberg,Basic modal logic, In D. Gabbay and F. Guenthner, editors,Handbook of Philosophical Logic, Volume II: Extensions of Classical Logic, pp. 1–88. D. Reidel 1984.Google Scholar
- M. Fitting,Proof Methods for Modal and Intuitionistic Logics, volume 169of Synthese Library, D. Reidel, Dordrecht, Holland 1983.Google Scholar
- M. Fitting,First order modal tableaux Journal of Automated Reasoning 4, 1988, pp. 191–213.Google Scholar
- R. I. Goldblatt,Logics of Time and Computation, CSLI Lecture Notes Number 7, CSLI Stanford 1987.Google Scholar
- G. E. Hughes andM. J. Cresswell,Introduction to Modal Logic Methuen, London 1968.Google Scholar
- G. E. Hughes andM. J. Cresswell,A Companion to Modal Logic Methuen, London 1984.Google Scholar
- E. J. Lemmon andD. Scott,An Introduction To Modal Logic American Philosophical Quarterly, Monograph Series, Basil Blackwell, Oxford 1977.Google Scholar
- W. Rautenberg,Modal tableau calculi and interpolation Journal of Philosophical Logic 12, 1983, pp. 403–423.Google Scholar
- W. Rautenberg,Corrections for modal tableau calculi and interpolation by W. Rautenberg, JPL 12 (1983) Journal of Philosophical Logic 14, 1985, p. 229.Google Scholar
- W. Rautenberg, Personal communication, December 5th, 1990.Google Scholar
- K. Segerberg,An essay in classical modal logic (3 vols.), Technical Report Filosofiska Studier, nr 13, Uppsala Universitet, Uppsala 1971.Google Scholar
- T. Shimura,Cut-free systems for the modal logic S4.3 and S4.3GRZ Reports on Mathematical Logic 25, 1991, pp. 57–73.Google Scholar
- S. Valentini,A syntactic proof of cut elimination for GL lin Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 32, 1986, pp. 137–144.Google Scholar
- J. F. A. K. van Benthem andW. Blok,Transitivity follows from Dummett's axiom Theoria 44 1978, pp. 117–118.Google Scholar
- P. Wolper,Temporal logic can be more expressive Information and Control 56, 1983, pp. 72–99.Google Scholar
- J. J. Zeman,Modal Logic: The Lewis-Modal Systems, Oxford University Press 1973.Google Scholar