Notre Dame Journal of Formal Logic

Propositional Quantification in the Topological Semantics for S4

Philip Kremer

Abstract

Fine and Kripke extended S5, S4, S4.2 and such to produce propositionally quantified systems $\mbox{\bf S5\mbox{$\pi$}+}$, $\mbox{\bf S4\mbox{$\pi$}+}$, $\mbox{\bf S4.2\mbox{$\pi$}+}$: given a Kripke frame, the quantifiers range over all the sets of possible worlds. $\mbox{\bf S5\mbox{$\pi$}+}$ is decidable and, as Fine and Kripke showed, many of the other systems are recursively isomorphic to second-order logic. In the present paper I consider the propositionally quantified system that arises from the topological semantics for S4, rather than from the Kripke semantics. The topological system, which I dub $\mbox{\bf S4.2\mbox{$\pi$}t}$, is strictly weaker than its Kripkean counterpart. I prove here that second-order arithmetic can be recursively embedded in $\mbox{\bf S4.2\mbox{$\pi$}t}$. In the course of the investigation, I also sketch a proof of Fine's and Kripke's results that the Kripkean system $\mbox{\bf S4\mbox{$\pi$}+}$ is recursively isomorphic to second-order logic.

Article information

Source
Notre Dame J. Formal Logic Volume 38, Number 2 (1997), 295-313.

Dates
First available in Project Euclid: 12 December 2002

Permanent link to this document
http://projecteuclid.org/euclid.ndjfl/1039724892

Digital Object Identifier
doi:10.1305/ndjfl/1039724892

Mathematical Reviews number (MathSciNet)
MR1489415

Zentralblatt MATH identifier
0949.03020

Subjects
Primary: 03B45: Modal logic (including the logic of norms) {For knowledge and belief, see 03B42; for temporal logic, see 03B44; for provability logic, see also 03F45}

Citation

Kremer, Philip. Propositional Quantification in the Topological Semantics for S4 . Notre Dame J. Formal Logic 38 (1997), no. 2, 295--313. doi:10.1305/ndjfl/1039724892. http://projecteuclid.org/euclid.ndjfl/1039724892.


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References

  • [1] Bull, R. A., ``On modal logic with propositional quantifiers,'' The Journal of Symbolic Logic, vol. 34 (1969) pp. 257--63.
  • [2] Chellas, B., Modal Logic, Cambridge University Press, Cambridge, 1980.
  • [3] Dishkant, H., ``Set theory as modal logic,'' Studia Logica, vol. 39 (1980), pp. 335--45.
  • [4]Fine, K., ``Propositional quantifiers in modal logic,'' Theoria, vol. 36 (1970), pp. 336--46.
  • [5] Gabbay, D., ``Montague type semantics for modal logics with propositional quantifiers,'' Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 17 (1971), pp. 245--49.
  • [6] Gabbay, D., ``On 2nd order intuitionistic propositional calculus with full comprehension,'' Archiv für mathematische Logik und Grundlagenforsch, vol. 16 (1974), pp. 177--86.
  • [7] Gabbay, D., Semantical investigations in Heyting's intuitionistic logic, Reidel, Dordrecht, 1981.
  • [8] Ghilardi, S., and M. Zawadowski, ``Undefinability of propositional quantifiers in the modal system S4,'' Studia Logica, vol. 55 (1995), pp. 259--71.
  • [9] Gurevich, Y., and S. Shelah, ``Interpreting second-order logic in the monadic theory of order,'' The Journal of Symbolic Logic, vol. 48 (1983), pp. 816--28.
  • [10] Gurevich, Y., and S. Shelah, ``Monadic theory of order and topology in ZFC,'' Annals of Mathematical Logic, vol. 23 (1983), pp. 179--98.
  • [11] Henkin, L., ``Completeness in the theory of types,'' The Journal of Symbolic Logic, vol. 15 (1950), pp. 81--91.
  • [12] Jech, T., Set Theory, Academic Press, San Diego, 1978.
  • [13] Kaminski, M., and M. Tiomkin, ``The expressive power of second-order propositional modal logic,'' Notre Dame Journal of Formal Logic, vol. 37 (1996), pp. 35--43.
  • [14] Kaplan, D., ``S5 with quantifiable propositional variables,'' The Journal of Symbolic Logic, vol. 35 (1970), p. 355.
  • [15] Kreisel, G., ``Monadic operators defined by means of propositional quantification in intuitionistic logic,'' Reports on Mathematical Logic, vol. 12 (1981), pp. 9--15.
  • [16] Kremer, P., ``Quantifying over propositions in relevance logic: non-axiomatisability of $\forall p$ and $\exists p$,'' The Journal of Symbolic Logic, vol. 58 (1993), pp. 334--49.
  • [17] Kremer. P., ``On the complexity of propositional quantification in intuitionistic logic,'' The Journal of Symbolic Logic, vol. 62 (1997), pp. 529--44.
  • [18] Kripke, S., ``A completeness theorem in modal logic,'' The Journal of Symbolic Logic, vol. 24 (1959), pp. 1--14.
  • [19]Kripke, S., ``Semantical analysis of modal logic I, normal propositional calculi,'' Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 9 (1963), pp. 67--96.
  • [20] Kripke, S., [1963b], Semantical analysis of
  • [21] Lewis, C. I., and C. H. Langford, Symbolic Logic, 2d edition, Dover Publications, New York, 1932.
  • [22] Löb, M. H., ``Embedding first order predicate logic in fragments of intuitionistic logic,'' The Journal of Symbolic Logic, vol. 41 (1976), pp. 705--18.
  • [23] McKinsey, J. J. C., ``A solution of the decision problem for the Lewis systems S.2 and S.4 with an application to topology,'' The Journal of Symbolic Logic, vol. 6 (1941), pp. 117--34.
  • [24] McKinsey, J. J. C.,and A. Tarski, ``The algebra of topology,'' Annals of Mathematics, vol. 45 (1944), pp. 141--91.
  • [25] McKinsey, J. J. C., and A. Tarski, ``On closed elements in closure algebras,'' Annals of Mathematics, vol. 47 (1946), pp. 122--62.
  • [26] McKinsey, J. J. C., and A. Tarski, ``Some theorems about the sentential calculi of Lewis and Heyting,'' The Journal of Symbolic Logic, vol. 13 (1948), pp. 1--15.
  • [27] Montague, R., ``Pragmatics and intensional logic,'' Synthese, vol. 22 (1970), pp. 68--94.
  • [28] Murungi, R. W., ``Lewis' postulate of existence disarmed,'' Notre Dame Jornal of Formal Logic, vol. 21 (1980), pp. 181--91.
  • [29] Nerode, A., and R. A. Shore, ``Second order logic and first order theories of reducibility orderings,'' pp. 181--90 in The Kleene Symposium, edited by J. Barwise, H. J. Keisler, and K. Kunen, North-Holland, Amsterdam, 1980.
  • [30] Pitts, A. M., ``On an interpretation of second order quantification in first order intuitionistic propositional logic,'' The Journal of Symbolic Logic, vol. 57 (1992), pp. 33--52.
  • [31] Polacik, T., ``Operators defined by propositional quantification and their interpretation over Cantor space,'' Reports on Mathematical Logic, vol. 27 (1993), pp. 67--79.
  • [32] Polacik, T., ``Second order propositional operators over Cantor space,'' Studia Logica, vol. 53 (1994), pp. 93--105.
  • [33] Rabin, M. O., ``Decidability of second-order theories and automata on infinite trees,'' Transactions of the American Mathematical Society, vol. 131 (1969), pp. 1--35.
  • [34] Rabin, M. O., ``Decidable Theories,'' pp. 595--629 in The Handbook of Mathematical Logic, edited by J. Barwise, North-Holland, Amsterdam, 1977.
  • [35] Rasiowa, H., and R. Sikorski, The Mathematics of Metamathematics, Państwowe Wydawnictwo Naukowe, Warsaw, 1963.
  • [36] Scedrov, A., ``On some extensions of second-order intuitionistic propositional calculus,'' Annals of Pure and Applied Logic, vol. 27 (1984), pp. 155--64.
  • [37] Scott, D., ``Advice on modal logic,'' pp. 143--173 in Philosophical Problems in Logic: Some Recent Developments, edited by K. Lambert, Reidel, Dordrecht, 1970.
  • [38] Segerberg, K., An Essay on Classical Modal Logic, Filosofiska Institutionem vid Uppsala Universitet, Uppsala, 1971.
  • [39] Sobolev, S. K., ``On the intuitionistic propositional calculus with quantifiers'' (in Russian), Akademiya Nauk Soyuza S.S.R. Matematicheskie Zamietki, vol. 22 (1977), pp. 69--76.
  • [40] Shelah, S., ``The monadic theory of order,'' Annals of Mathematics, vol. 102 (1975), pp. 379--419.
  • [41] Tsao-Chen, T., ``Algebraic postulates and a geometric interpretation for the Lewis calculus of strict implication,'' Bulletin of the American Mathematical Society, vol. 44 (1938), pp. 737--44.
  • [42] Troelstra, A. S., ``On a second-order propositional operator in intuitionistic logic,'' Studia Logica, vol. 40 (1981), pp. 113--39.