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An algebraic study of Diodorean modal systems

  • R. A. Bull (a1)

Extract

Attention was directed to modal systems in which ‘necessarily α’ is interpreted as ‘α. is and always will be the case’ by Prior in his John Locke Lectures of 1956. The present paper shows that S4.3, the extension of S4 with

ALCLpLqLCLqLp,

is complete with respect to this interpretation when time is taken to be continuous, and that D, the extension of S4.3 with

ALNLpLCLCLCpLpLpLp,

is complete with respect to this interpretation when time is taken to be discrete. The method employed depends upon the application of an algebraic result of Garrett Birkhoff's to the models for these systems, in the sense of Tarski.

A considerable amount of work on S4.3 and D precedes this paper. The original model with discrete time is given in Prior's [7] (p. 23, but note the correction in [8]); that taking time to be continuous yields a weaker system is pointed out by him in [9]. S4.3 and D are studied in [3] of Dummett and Lemmon, where it is shown that D includes S4.3 and

CLCLCpLpLpCMLpLp.

While in Oxford in 1963, Kripke proved that these were in fact sufficient for D, using semantic tableaux. A decision procedure for S4.3, using Birkhoff's result, is given in my [2]. Dummett conjectured, in a conversation, that taking time to be continuous yielded S4.3. Thus the originality of this paper lies in giving a suitable completeness proof for S4.3, and in the unified algebraic treatment of the systems. It should be emphasised that the credit for first axiomatising D belongs to Kripke.

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References

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[1]Birkhoff, Garrett, Sub-direct unions in universal algebras. Bulletin of the American Mathematical Society, vol. 50 (1944), pp. 764768.
[2]Bull, R. A., A note on the modal calculi S4.2 and S4.3. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 10 (1964), pp. 5355.
[3]Dummett, M. A. E. and Lemmon, E. J., Modal logics between S4 and S5. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 5 (1959), pp. 250264.
[4]Harrop, R., On the existence of finite models and decision procedures for prepositional calculi. Proceedings of the Cambridge Philosophical Society, vol. (1958), pp. 113.
[5]McKinsey, J. C. C. and Tarski, Alfred, The algebra of topology, Annals of Mathematics, vol. 45 (1944), pp. 141191.
[6]McKinsey, J. C. C. and Tarski, Alfred, Some theorems about the sentential calculi of Lewis and Heyting. This Journal, vol. 13 (1948), pp. 115.
[7]Prior, A. N., Time and Modality. Oxford, 1957.
[8]Prior, A. N., Diodorus and modal logic: a correction. Philosophical Quarterly, vol. 8 (1958), pp. 226230.
[9]Prior, A. N., Tense-logic and the continuity of time. Studia Logica, vol. 13 (1962), pp. 133148.

An algebraic study of Diodorean modal systems

  • R. A. Bull (a1)

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