Abstract
This paper specifies classes of framesmaximally omnitemporally characteristic for Thomas' normal modal logicT +2 and for each logic in the ascending chain of Segerberg logics investigated by Segerberg and Hughes and Cresswell. It is shown that distinct a,scending chains of “generalized” Segerberg logics can be constructed from eachT +n logic (n ⩾ 2). The set containing allT +n and Segerberg logics can be totally- (linearly-) ordered but not well-ordered by the inclusion relation. The order type of this ordered setΓ isω *(ω + 1). Throughout the paper my approach is fundamentally semantical.
Similar content being viewed by others
References
K. Fine, An ascending chain ofS4 logics,Theoria 40 (1974), pp. 110–16.
G. E. Hughes, B(S4, 3, S4) unveiledTheoria, 41 (1975), pp. 11–34.
—,Omnitemporal logic and modal time,Reports on Mathematical Logic 8 (1977), pp. 41–61.
G. E. Hughes andM. J. Cresswell,An Introduction to Modal Logic. London: Methuen and Co., Ltd., 1968.
—,Omnitemporal logic and converging time,Theoria, 41 (1975), pp. 11–34.
N. Rescher, andA. Urquahart,Temporal Logic. Vienna and New York: Springer-Verlag, 1971.
K. Segerberg,An Essay in Classical Modal Logic. 3 volumes. Filosofika Studier, Nr 13. Uppsala: Philosophical Society and Department of Philosophy, University of Uppsala, Sweden, 1971.
-Disproof of a conjecture of Rescher and Urquhart, inP and Q: Mini-essays in Honor of Risto Hilpinen, Åbo Publication of the Group in Logic and Methodology of Real Finland, 1973.
W. Sierpiński,Cardinal and Ordinal Numbers, 2nd ed., rev. Warsaw and New York: Państwowe Wydawnictwo Naukowe, 1965.
I. Thomas,Modal systems in the neighborhood of T,Notre Dame Journal of Formal Logic 5 (1964), pp. 59–61.
B. C. van Fraassen,Formal Semantics and Logic, New York and London: The Macmillan Company, 1971.
M. M. Zuckerman,Sets and Transfinite Numbers, New York and London: Macmillan Publishing Co., Inc. and Collier Macmillan Publishers. 1974.
Author information
Authors and Affiliations
Additional information
I should like to thank Professor G. E. Hughes for helpful comments on an earlier draft of this paper.
Rights and permissions
About this article
Cite this article
White, M.J. On some ascending chains of brouwerian modal logics. Stud Logica 40, 75–87 (1981). https://doi.org/10.1007/BF01837557
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01837557