On Modalities and Quantifiers

• Autorzy: Fitting M.

Identyfikatory

DOI

10.3233/FI-2017-1610

• Języki publikacji: EN

Abstrakty

EN

In 1951 in his book An Essay in Modal Logic, Georg Henrik von Wright strongly called attention to the analogies between quantifiers and modal operators. In 1984 I published a paper in Synthese examining the analogy formally. Confession: the presentation in that paper was badly done, and there is a significant (though correctable) error. Its time to repair the damage, present the ideas in a better way, and continue the investigation further. There are natural sublogics of classical first-order logic that are direct analogs of standard, basic modal logics. The behavior of quantifiers can be given a possible world semantics, some analogous to normal models, some to regular models, and some to neighborhood models. The firstorder logics have axiom systems and generally also tableau systems, paralleling those of modal logics. Many have the interpolation property. This gives concrete substance to von Wright’s observations. But then, what is the crucial difference between modal operators and quantifiers? This turns out to be surprising in its simplicity, and leads to an interesting way of looking at the familiar Henkin style completeness proof for first-order logic.

Słowa kluczowe

EN

modality; quantifiers; first-order logic; axioms

• Wydawca: IOS Press
• Czasopismo: Fundamenta Informaticae
• Rocznik: 2017
• Tom: Vol. 156, nr 3/4
• Strony: 297--330
• Opis fizyczny: Bibliogr. 19 poz., rys., tab.

Twórcy

autor

Fitting M.

• Graduate Center of the City University of New York, Department of Computer Science, 365 Fifth Avenue, New York, NY 10016

Bibliografia

• [1] von Wright GH. An Essay in Modal Logic. North-Holland, Amsterdam, 1951. ASIN: B0018IMC1I.
• [2] Fitting MC. A Symmetric approach to axiomatizing quantifiers and modalities. Synthese, 1984;60:5–20. URL https://doi.org/10.1007/978-94-017-1592-8_1.
• [3] Mendelson E. Introduction to Mathematical Logic. Chapman and Hall, fourth edition 1997. ISBN: 10:0412808307, 13:978-0412808302.
• [4] Enderton HB. A Mathematical Introduction to Logic. Academic Press, New York, second edition 2000. ISBN: 10:0122384520, 13:978-0122384523.
• [5] Kripke S. Semantical analysis of modal logic II, non-normal modal propositional calculi. In: Addison JW, Henkin L, Tarski A (eds.), The Theory of Models. North-Holland, Amsterdam, 1965 pp. 206–220. doi:10.1017/S0022481200092434.
• [6] Montague R. Universal Grammar. Theoria, 1970;36:373–398. doi:10.1111/j.1755-2567.1970.tb00434.x.
• [7] Scott D. Advice on modal logic. In: Lambert K (ed.), Philosophical Problems in Logic. Reidel, 1970 pp. 143–173. URL https://doi.org/10.1007/978-94-010-3272-8_7.
• [8] Segerberg K. An Essay in Classical Modal Logic. Filosofiska Studier nr 13. Uppsala Universitet, Uppsala, 1971. (three vols.).
• [9] Chellas BF. Modal Logic, an Introduction. Cambridge University Press, 1980. ISBN: 0521295157, 9780521295154.
• [10] Arló-Costa H, Pacuit E. First Order classical modal logic. Studia Logica, 2006;84(2):171–210. URL https://doi.org/10.1007/s11225-006-9010-0.
• [11] Fitting MC. Proof Methods for Modal and Intuitionistic Logics. D. Reidel Publishing Co., Dordrecht, 1983.
• [12] Ohnishi M, Matsumoto K. Gentzen method in modal calculi, I. Osaka Mathematical Journal, 1957;9:113–130. URL https://projecteuclid.org/euclid.ojm/1200689157.
• [13] Ohnishi M, Matsumoto K. Gentzen method in modal calculi, II. Osaka Mathematical Journal, 1959;11:115–120. URL https://projecteuclid.org/euclid.ojm/1200689632.
• [14] Fitting MC. First-Order Logic and Automated Theorem Proving. Springer-Verlag, 1996. First edition 1990. Errata at http://melvinfitting.org/errata/errata.html.
• [15] Hansen HH. Monotonic Modal Logics. Illc prepublication series pp-2003-24, Institute for Logic, Language and Computation, University of Amsterdam, 2003. Master’s Thesis. URL https://eprints.illc.uva.nl/id/eprint/108.
• [16] Santocanale L, Venema Y. Uniform Interpolation for Monotone Modal Logic. In: Beklemishev L, Goranko V, Shehtman V (eds.), Advances in Modal Logic, volume 8. College Publications, 2010 pp. 350–370.
• [17] Koslow A. A Structuralist Theory of Logic. Cambridge University Press, 1992. URL https://doi.org/10.1017/CBO9780511609206.
• [18] Koslow A. The Modality and Non-Extensionality of the Quantifiers. Synthèse, forthcoming. 2014. doi:10.1007/s11229-014-0539-6.
• [19] Rasiowa H, Sikorski R. The Mathematics of Metamathematics, volume 41 of Monografie Matematyczne. Państwowe Wydawnictwo Naukowe–Polish Scientific Publishers, Warsaw, 1963. URL https://books.google.pl/books?id=67psAAAAMAAJ.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).