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There are infinitely many Diodorean modal functions1

Published online by Cambridge University Press:  12 March 2014

D. C Makinson
D. C Makinson*
Affiliation:
Worcester College, Oxford, England
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Extract

It is well known that the modal calculus S4 has infinitely many non-equivalent formulae in a single proposition letter (in standard terminology, infinitely many modal functions), whilst S5 has only finitely many. However, the situation regarding the intermediate modal calculi S4.2, S4.3, and Prior's Diodorean tense-logic D does not seem to have been settled. In this note we show that each of these systems, together with a certain proper supersystem D* of D, has infinitely many modal functions.

This is in contrast with the fact that in the intermediate propositional logics KC and LC, which correspond under the McKinsey-Tarski translations to S4.2 and S4.3, there are only finitely many non-equivalent formulae in a single proposition letter.2

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 31 , Issue 3 , 02 September 1966 , pp. 406 - 408
Copyright
Copyright © Association for Symbolic Logic 1996

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Footnotes

1

Based on a section of the author's D. Phil, thesis, Rules of truth for modal logic, submitted to the University of Oxford in June 1965.

References

[1]Dummett, M. and Lemmon, E., Modal logics between S4 and S5, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 3 (1959), pp. 250–264.CrossRefGoogle Scholar
[2]Nishimura, I., On formulas of one variable in intuitionistic propositional calculus, this Journal, Vol. 25 (1960), pp. 327–331.Google Scholar
[3]Prior, A. N., Time and modality (Oxford University Press), 1957.Google Scholar
[4]Prior, A. N., Diodorus and modal logic: a correction, Philosophical quarterly, vol. 8 (1958), pp. 226–230.CrossRefGoogle Scholar