Temporal Logic pp 125-137 | Cite as

Temporal Modality

Part of the LEP Library of Exact Philosophy book series (LEP, volume 3)


In classical antiquity, Diodorus Cronus defined the necessary as that which is and always will be the case, and correspondingly, defined the possible as that which is or will be the case1. Using our standard notation for necessity and possibility, we can express these definitions in a tense calculus by:


Modal Logic Tense Structure Completeness Theorem Tense Logic Notre Dame Journal 
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  1. 2.
    For a comprehensive treatment of modal logic the reader should consult G. E. Hughes and M. J. Cresswell, Introduction to Modal Logic ( London, Methuen and Co., 1968 ).Google Scholar
  2. 5.
    For a proof, see Saul Kripke, Semantical Analysis of Modal Logic I, Normal Propositional Calculi, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 9 (1963), pp. 67–96.Google Scholar
  3. 6.
    In his paper: Modal Systems in the neighborhood of T, Notre Dame Journal of Formal Logic, vol. 5 (1965), pp. 59–61.Google Scholar
  4. 7.
    A. N. Prior, Past, Present and Future, Oxford University Press (1967), pp. 54–55.Google Scholar
  5. 11.
    A proof of this is to be found in R. A. Bull, An Algebraic Study of Diodorean Modal Systems, Journal of Symbolic Logic, vol. 33 (1968), pp. 27–38.Google Scholar
  6. 14.
    This formulation of deontic S5 differs from the system 0S5 of T. J. Smiley [Journal of Symbolic Logic, vol. 28 (1963), pp. 113–134], and from D5 of E. J. Lemmon Journal of Symbolic Logic, vol. 22 (1957),pp. 176–186].Google Scholar
  7. 15.
    See A. N. Prior, Time and Modality, Oxford University Press (1957) and R.A. Bull, An Axiomatization of Prior’s Modal Calculus Q, Notre Dame Journal of Formal Logic, vol. 5 (1964), pp. 211–214.Google Scholar

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© Springer-Verlag/Wien 1971

Authors and Affiliations

  1. 1.PittsburghUSA

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