Element contents
Series: Elements in Philosophy and Logic
Propositional Quantifiers
Published online by Cambridge University Press: 23 April 2024
Summary
Propositional quantifiers are quantifiers binding proposition letters, understood as variables. This Element introduces propositional quantifiers and explains why they are especially interesting in the context of propositional modal logics. It surveys the main results on propositionally quantified modal logics which have been obtained in the literature, presents a number of open questions, and provides examples of applications of such logics to philosophical problems.
Keywords
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- Series: Elements in Philosophy and LogicOnline ISBN: 9781009177740Publisher: Cambridge University PressPrint publication: 16 May 2024
References
Ågotnes, T., van Ditmarsch, H., and French, T.. The undecidability of quantified announcements. Studia Logica, 104:597–640, 2016.CrossRefGoogle Scholar
Anderson, Alan Ross. An intensional interpretation of truth-values. Mind, 81:348–371, 1972.CrossRefGoogle Scholar
Anderson, Alan Ross, and Nuel, D. Belnap Jr., Enthymemes. The Journal of Philosophy, 58:713–723, 1961.CrossRefGoogle Scholar
Anderson, Alan Ross, Nuel, D. Belnap, Jr, and Michael Dunn., J. Entailment: The Logic of Relevance and Necessity, volume II. Princeton: Princeton University Press, 1992.Google Scholar
Anderson, C. Anthony. Bealer’s Quality and Concept. Journal of Philosophical Logic, 16:115–164, 1987.CrossRefGoogle Scholar
Anderson, C. Anthony. The lesson of Kaplan’s paradox about possible world semantics. In Almog, Joseph and Leonardi, Paolo, editors, The Philosophy of David Kaplan, pages 85–92. Oxford: Oxford University Press, 2009.CrossRefGoogle Scholar
Andrews, Peter B. A reduction of the axioms for the theory of propositional types. Fundamenta Mathematicae, 52:345–350, 1963.CrossRefGoogle Scholar
Antonelli, G. Aldo. and Richmond, H. Thomason. Representability in second-order propositional poly-modal logic. The Journal of Symbolic Logic, 67:1039–1054, 2002.CrossRefGoogle Scholar
Costa, Arló, , Horacio. First order extensions of classical systems of modal logic: The role of the Barcan schemas. Studia Logica, 71:87–118, 2002.CrossRefGoogle Scholar
Baaz, Matthias, and Preining, Norbert. Quantifier elimination for quantified propositional logics on Kripke frames of type . Journal of Logic and Computation, 18:649–668, 2008.CrossRefGoogle Scholar
Baaz, Matthias, and Veith, Helmut. An axiomatization of quantified propositional Gödel logic using the Takeuti–Titani rule. In Buss, Samuel R., Hájek, Petr, and Pudlák, Pavel, editors, Logic Colloquium ’98, volume 13 of Lecture Notes in Logic, pages 91–104. Cambridge: Cambridge University Press, 2000.Google Scholar
Baaz, Matthias, Ciabattoni, Agata, and Zach, Richard. Quantified propositional Gödel logics. In Parigot, Michel and Voronkov, Andrei, editors, Logic for Programming and Automated Reasoning. 7th International Conference, LPAR 2000, volume 1955 of Lecture Notes in Artificial Intelligence, pages 240–256. Berlin: Springer, 2000.Google Scholar
Bacon, Andrew. A Philosophical Introduction to Higher-Order Logics. New York: Routledge, 2024.Google Scholar
Bacon, Andrew, and Fine, Kit. The logic of logical necessity. In Padró, Romina and Weiss, Yale, editors, Saul Kripke on Modal Logic. Springer, forthcoming.Google Scholar
Bacon, Andrew, and Uzquiano, Gabriel. Some results on the limits of thought. Journal of Philosophical Logic, 47:991–999, 2018.CrossRefGoogle Scholar
Bacon, Andrew, Hawthorne, John, and Uzquiano, Gabriel. Higher-order free logic and the Prior–Kaplan paradox. Canadian Journal of Philosophy, 46:493–541, 2016.CrossRefGoogle Scholar
Badia, Guillermo. Incompactness of the fragment of basic second-order propositional relevant logic. Australasian Journal of Logic, 16:1–8, 2019.CrossRefGoogle Scholar
Barcan, Ruth C. A functional calculus of first order based on strict implication. The Journal of Symbolic Logic, 11:1–16, 1946.CrossRefGoogle Scholar
Barcan, Ruth C. The identity of individuals in a strict functional calculus of second order. The Journal of Symbolic Logic, 12:12–15, 1947.CrossRefGoogle Scholar
Bayart, Arnould. La correction de la logique modale du premier et second ordre S5. Logique et Analyse, 1:28–44, 1958.Google Scholar
Bayart, Arnould. Quasi-adéquation de la logique modale de second ordre S5 et adéquation de la logique modale de premier ordre S5. Logique et Analyse, 2:99–121, 1959.Google Scholar
Bealer, George. Property theory: The type-free approach v. the Church approach. Journal of Philosophical Logic, 23:139–171, 1994.CrossRefGoogle Scholar
Becker, Oskar. Zur Logik der Modalitäten. Jahrbuch für Philosophie und phänomenologische Forschung, 11:497–548, 1930.Google Scholar
Belardinelli, Francesco. Hans van Ditmarsch, and van der Hoek, Wiebe. Second-order propositional announcement logic. In Thangarajah, John, Tuyls, Karl, Jonker, Catholijn, and Marsella, Stacy, editors, Proceedings of the 15th International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2016), pages 635–643. New York: Association for Computing Machinery, 2016.Google Scholar
Belardinelli, Francesco. van der Hoek, Wiebe, and Kuijer, Louwe B.. Second-order propositional modal logic: Expressiveness and completeness results. Artificial Intelligence, 263:3–45, 2018.CrossRefGoogle Scholar
Besnard, Philippe. Guinnebault, Jean-Marc, and Mayer, Emmanuel. Propositional quantification for conditional logic. In Gabbay, Dov M., Kruse, Rudolf, Nonnengart, Andreas, and Ohlbach, Hans Jürgen, editors, Qualitative and Quantitative Practical Reasoning, Lecture Notes in Computer Science 1244, pages 183–197. Berlin: Springer, 1997.Google Scholar
Blackburn, Patrick. de Rijke, Maarten, and Venema, Yde. Modal Logic, volume 53 of Cambridge Tracts in Theoretical Computer Science. Cambridge: Cambridge University Press, 2001.CrossRefGoogle Scholar
Blackburn, Patrick. Braüner, Torben, and Lundbak Kofod, Julie. Remarks on hybrid modal logic with propositional quantifiers. In Hasle, Peter, Jakobsen, David, and Øhrstrøm, Peter, editors, The Metaphysics of Time: Themes from Prior, pages 401–426. Aalborg: Aalborg Universitetsforlag, 2020.Google Scholar
Boolos, George. To be is to be a value of a variable (or to be some values of some variables). The Journal of Philosophy, 81:430–449, 1984.CrossRefGoogle Scholar
Boolos, George. The Logic of Provability. Cambridge: Cambridge University Press, 1985.Google Scholar
Bradfield, Julian. and Stirling, Colin. Modal -calculi. In Blackburn, Patrick, van Benthem, Johan, and Wolter, Frank, editors, Handbook of Modal Logic, pages 721–756. Amsterdam: Elsevier, 2007.CrossRefGoogle Scholar
Büchi, J. Richard. On a decision method in restricted second order arithmetic. In Nagel, Ernest, Suppes, Patrick, and Tarski, Alfred, editors, Logic, Methodology and Philosophy of Science: Proceedings of the 1960 International Congress, pages 1–11. Redwood City, CA: Stanford University Press, 1962.Google Scholar
Bull, R. A. On possible worlds in propositional calculi. Theoria, 3:171–182, 1968.CrossRefGoogle Scholar
Bull, R. A. On modal logics with propositional quantifiers. The Journal of Symbolic Logic, 34:257–263, 1969.CrossRefGoogle Scholar
Carnap, Rudolf. Introduction to Semantics. Cambridge, MA: Harvard University Press, 1942.Google Scholar
Carnap, Rudolf. Meaning and Necessity: A Study in Semantics and Modal Logic. Chicago, IL: The University of Chicago Press, 1947.Google Scholar
Chagrov, Alexander. and Zakharyaschev, Michael. Modal Logic, volume 35 of Oxford Logic Guides. Oxford: Clarendon Press, 1997.CrossRefGoogle Scholar
Chellas, Brian F. Modal Logic: An Introduction. Cambridge: Cambridge University Press, 1980.CrossRefGoogle Scholar
Christensen, Ryan. Propositional quantification. Russell: The Journal of Bertrand Russell Studies, 31:109–120, 2011.CrossRefGoogle Scholar
Church, Alonzo. Review of The Liar by Alexandre Koyré. The Journal of Symbolic Logic, 11:131, 1946.Google Scholar
Church, Alonzo. Introduction to Mathematical Logic. Princeton, NJ: Princeton University Press, 1956.Google Scholar
Church, Alonzo. Mathematics and logic. In Nagel, Ernest, Suppes, Patrick, and Tarski, Alfred, editors, Logic, Methodology and Philosophy of Science. Proceedings of the 1960 International Congress, pages 181–186. Redwood City, CA: Stanford University Press, 1962.Google Scholar
Church, Alonzo. Russell’s theory of identity of propositions. Philosophia Naturalis, 21:513–522, 1984.Google Scholar
Church, Alonzo. Referee reports on Fitch’s “A Definition of Value”. In Salerno, Joe, editor, New Essays on the Knowability Paradox, pages 13–20. Oxford: Oxford University Press, 2009 [1945].CrossRefGoogle Scholar
Cohen, L. Jonathan. Can the logic of indirect discourse be formalised? The Journal of Symbolic Logic, 22:225–232, 1957.CrossRefGoogle Scholar
Copeland, B. Jack. Meredith, Prior, and the history of possible world semantics. Synthese, 150:373–397, 2006.CrossRefGoogle Scholar
Cresswell, M. J. Arnould Bayart’s modal completeness theorems. Logique et Analyse, 229:89–142, 2015.Google Scholar
Crossley, John N., and Humberstone, Lloyd. The logic of “actually”. Reports on Mathematical Logic, 8:11–29, 1977.Google Scholar
Davey, B. A. and Priestley, H. A.. Introduction to Lattices and Order. Cambridge: Cambridge University Press, second edition, 2002.CrossRefGoogle Scholar
deVries, Willem. Wilfrid Sellars. In Edward, N. Zalta, editor, The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, 2020.Google Scholar
Ding, Yifeng. On the logics with propositional quantifiers extending . In Bezhanishvili, Guram, D’Agostino, Giovanna, Metcalfe, George, and Studer, Thomas, editors, Advances in Modal Logic, volume 12, pages 219–235. College Publications, 2018.Google Scholar
Ding, Yifeng. On the logic of belief and propositional quantification. Journal of Philosophical Logic, 50:1143–1198, 2021a.CrossRefGoogle Scholar
Ding, Yifeng. Propositional Quantification and Comparison in Modal Logic. PhD thesis, University of California, Berkeley, 2021b.Google Scholar
Yifeng, Ding, and Holliday, Wesley H.. Another problem in possible world semantics. In Olivetti, Nicola, Verbrugge, Rineke, and Negri, Sara, editors, Advances in Modal Logic, volume 13. College Publications, 2020.Google Scholar
Dorr, Cian, Hawthorne, John, and Yli-Vakkuri, Juhani. The Bounds of Possibility: Puzzles of Modal Variation. Oxford: Oxford University Press, 2021.CrossRefGoogle Scholar
Došen, Kosta. Duality between modal algebras and neighborhood frames. Studia Logica, 48:219–234, 1989.CrossRefGoogle Scholar
Dunn, J. Michael, and Gary, M. Hardegree. Algebraic Methods in Philosophical Logic. Oxford: Clarendon Press, 2001.CrossRefGoogle Scholar
Ferreira, Fernando. Comments on predicative logic. Journal of Philosophical Logic, 35:1–8, 2006.CrossRefGoogle Scholar
Fine, Kit. For some proposition and so many possible worlds. PhD thesis, University of Warwick, 1969.Google Scholar
Fine, Kit. Propositional quantifiers in modal logic. Theoria, 36:336–346, 1970.CrossRefGoogle Scholar
Fine, Kit. In so many possible worlds. Notre Dame Journal of Formal Logic, 13:516–520, 1972.CrossRefGoogle Scholar
Fine, Kit. Properties, propositions and sets. Journal of Philosophical Logic, 6:135–191, 1977.CrossRefGoogle Scholar
Fine, Kit. Failures of the interpolation lemma in quantified modal logic. The Journal of Symbolic Logic, 44:201–206, 1979.CrossRefGoogle Scholar
Fine, Kit. First-order modal theories II – Propositions. Studia Logica, 39:159–202, 1980.CrossRefGoogle Scholar
Fine, Kit. The permutation principle in quantificational logic. Journal of Philosophical Logic, 12:33–37, 1983.CrossRefGoogle Scholar
Fitch, Frederic B. A logical analysis of some value concepts. The Journal of Symbolic Logic, 28:135–142, 1963. Reprinted in Salerno (2009).CrossRefGoogle Scholar
Fitting, Melvin. Interpolation for first-order S5. The Journal of Symbolic Logic, 67:621–634, 2002.CrossRefGoogle Scholar
Frege, Gottlob. Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle a. S.: Louis Nebert, 1879.Google Scholar
Frege, Gottlob. Über Begriff und Gegenstand. Vierteljahresschrift für wissenschaftliche Philosophie, 16:192–205, 1892.Google Scholar
Frege, Gottlob. Posthumous Writings. Oxford: Basil Blackwell, 1979. Edited by Hermes, Hans, Friedrich Kambartel and Friedrich Kaulbart.Google Scholar
Frege, Gottlob. Philosophical and Mathematical Correspondence. Oxford: Basil Blackwell, 1980. Edited by Gabriel, Gottfried, Hans Hermes, Friedrich Kambartel, Christian Thiel and Albert Veraart.Google Scholar
French, Tim. Bisimulation quantified modal logics: Decidability. In Governatori, Guido, Hodkinson, Ian, and Venema, Yde, editors, Advances in Modal Logic, volume 6, pages 147–166. College Publications, 2006.Google Scholar
French, Tim, and Reynolds, Mark. A sound and complete proof system for QPTL. In Balbiani, Philippe, Suzuki, Nobu-Yuki, Wolter, Frank, and Zakharyaschev, Michael, editors, Advances in Modal Logic, volume 4, pages 127–147. College Publications, 2003.Google Scholar
Friedman, Harvey. One hundred and two problems in mathematical logic. The Journal of Symbolic Logic, 40:113–129, 1975.CrossRefGoogle Scholar
Fritz, Peter. Propositional contingentism. The Review of Symbolic Logic, 9:123–142, 2016.CrossRefGoogle Scholar
Fritz, Peter. Logics for propositional contingentism. The Review of Symbolic Logic, 10:203–236, 2017.CrossRefGoogle Scholar
Fritz, Peter. Propositional quantification in bimodal S5. Erkenntnis, 85:455–465, 2020.CrossRefGoogle Scholar
Fritz, Peter. Propositional potentialism. In Federico, L. G. Faroldi and Putte, Frederik Van De, editors, Kit Fine on Truthmakers, Relevance and Non-Classical Logic, volume 26 of Outstanding Contributions to Logic, pages 469–502. Cham: Springer, 2023a.CrossRefGoogle Scholar
Fritz, Peter. Operands and instances. The Review of Symbolic Logic, 16:188–209, 2023b.CrossRefGoogle Scholar
Fritz, Peter. The Foundations of Modality: From Propositions to Possible Worlds. Oxford: Oxford University Press, 2023c.CrossRefGoogle Scholar
Fritz, Peter. Axiomatizability of propositionally quantified modal logics on relational frames. The Journal of Symbolic Logic, forthcoming.Google Scholar
Fritz, Peter. Nonconservative extensions by propositional quantifiers and modal incompleteness. Unpublished.Google Scholar
Peter, Fritz, and Jones, Nicholas K., editors. Higher-Order Metaphysics. Oxford: Oxford University Press, 2024.CrossRefGoogle Scholar
Fritz, Peter, and Lederman, Harvey. Standard state space models of unawareness. Electronic Proceedings in Theoretical Computer Science, 215:141–158, 2016. Proceedings of the 15th Conference on Theoretical Aspects of Rationality and Knowledge (TARK 2015).Google Scholar
Gabbay, Dov M. Montague type semantics for modal logics with propositional quantifiers. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 17:245–249, 1971.CrossRefGoogle Scholar
Gabbay, Dov M. On 2nd order intuitionistic propositional calculus with full comprehension. Archiv für mathematische Logik und Grundlagenforschung, 16:177–186, 1974.CrossRefGoogle Scholar
Gallin, Daniel. Intensional and Higher-Order Modal Logic. Amsterdam: North-Holland, 1975.Google Scholar
Garson, James W. Quantification in modal logic. In Dov, M. Gabbay and Guenthner, Franz, editors, Handbook of Philosophical Logic, volume II, pages 249–307. Dordrecht: D. Reidel, 1st edition, 1984.CrossRefGoogle Scholar
Gerson, Martin. An extension of S4 complete for the neighbourhood semantics but incomplete for the relational semantics. Studia Logica, 34:333–342, 1975a.CrossRefGoogle Scholar
Gerson, Martin. The inadequacy of the neighbourhood semantics for modal logic. The Journal of Symbolic Logic, 40:141–148, 1975b.CrossRefGoogle Scholar
Gerson, Martin. A neighbourhood frame for T with no equivalent relational frame. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 22:29–34, 1976.CrossRefGoogle Scholar
Ghilardi, Silvio, and Zawadowski, Marek. Undefinability of propositional quantifiers in the modal system S4. Studia Logica, 55:259–271, 1995.CrossRefGoogle Scholar
Givant, Steven, and Halmos, Paul. Introduction to Boolean Algebras. New York: Springer, 2009.Google Scholar
Goldblatt, Robert. The McKinsey axiom is not canonical. The Journal of Symbolic Logic, 56:554–562, 1991.CrossRefGoogle Scholar
Goldblatt, Robert. Quantifiers, Propositions and Identity. Admissible Semantics for Quantified Modal and Substructural Logics. Cambridge: Cambridge University Press, 2011.CrossRefGoogle Scholar
Goldblatt, Robert, and Kane, Michael. An admissible semantics for propositionally quantified relevant logic. Journal of Philosophical Logic, 39:73–100, 2010.CrossRefGoogle Scholar
Grover, Dorothy L. Topics in Propositional Quantification. PhD thesis, University of Pittsburgh, 1970.Google Scholar
Grover, Dorothy L. Propositional quantifiers. Journal of Philosophical Logic, 1:111–136, 1972. Reprinted in Grover (1992).CrossRefGoogle Scholar
Grover, Dorothy L. Propositional quantification and quotation contexts. In Leblanc, Hugues, editor, Truth, Syntax and Modality, pages 101–110. Amsterdam: North-Holland, 1973.CrossRefGoogle Scholar
Grover, Dorothy L. A Prosentential Theory of Truth. Princeton: Princeton University Press, 1992.CrossRefGoogle Scholar
Grover, Dorothy L., Joseph, L. Camp, Jr., and Nuel, D. Belnap, Jr. A prosentential theory of truth. Philosophical Studies, 27:73–125, 1975. Reprinted in Grover (1992).CrossRefGoogle Scholar
Gurevich, Yuri, and Shelah, Saharon. Monadic theory of order and topology in ZFC. Annals of Mathematical Logic, 23:179–198, 1983.CrossRefGoogle Scholar
Gärdenfors, Peter. On the extensions of S5. Notre Dame Journal of Formal Logic, 414:277–280, 1973.Google Scholar
Gödel, Kurt. Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik, 38:173–198, 1931.CrossRefGoogle Scholar
Halbach, Volker, and Welch, Philip. Necessities and necessary truths: A prolegomenon to the use of modal logic in the analysis of intensional notions. Mind, 118:71–100, 2009.CrossRefGoogle Scholar
Halmos, Paul, and Givant, Steven. Logic as Algebra. The Mathematical Association of America, 1998.CrossRefGoogle Scholar
Halpern, Joseph Y., and Rêgo, Leandro C.. Reasoning about knowledge of unawareness. Games and Economic Behaviour, 67:503–525, 2009.CrossRefGoogle Scholar
Halpern, Joseph Y., and Rêgo, Leandro C.. Reasoning about knowledge of unawareness revisited. Mathematical Social Sciences, 65:73–84, 2013.CrossRefGoogle Scholar
Hansson, Bengt, and Gärdenfors, Peter. A guide to intensional semantics. In Modality, Morality and Other Problems of Sense and Nonsense: Essays dedicated to Sören Halldén, pages 151–167. Lund: CWK Gleerup Bokförlag, 1973.Google Scholar
Heidelberger, Herbert. The indispensability of truth. American Philosophical Quarterly, 5:212–217, 1968.Google Scholar
Henkin, Leon. Completeness in the theory of types. The Journal of Symbolic Logic, 15:81–91, 1950.CrossRefGoogle Scholar
Henkin, Leon. A theory of propositional types. Fundamenta Mathematicae, 52:323–344, 1963.CrossRefGoogle Scholar
Hilbert, D., and Ackermann, W.. Grundzüge der theoretischen Logik. Berlin: Springer, second edition, 1938.CrossRefGoogle Scholar
Holliday, Wesley H. A note on algebraic semantics for S5 with propositional quantifiers. Notre Dame Journal of Formal Logic, 60:311–332, 2019.CrossRefGoogle Scholar
Holliday, Wesley H. Possibility semantics. In Fitting, Melvin, editor, Selected Topics from Contemporary Logics, pages 363–476. London: College Publications, 2021.Google Scholar
Holliday, Wesley H. Possibility frames and forcing for modal logic. The Australasian Journal of Logic, forthcoming.Google Scholar
Holliday, Wesley H., and Litak, Tadeusz. One modal logic to rule them all? In Bezhanishvili, Guram, D’Agostino, Giovanna, Metcalfe, George, and Studer, Thomas, editors, Advances in Modal Logic, volume 12, pages 367–386. College Publications, 2018.Google Scholar
Hughes, G. E., and Cresswell, M. J.. A New Introduction to Modal Logic. London: Routledge, 1996.CrossRefGoogle Scholar
Humberstone, Lloyd. From worlds to possibilities. Journal of Philosophical Logic, 10:313–339, 1981.CrossRefGoogle Scholar
Jónnson, Bjarni, and Tarski, Alfred. Boolean algebras with operators, part I. American Journal of Mathematics, 73:891–939, 1951.CrossRefGoogle Scholar
Jónnson, Bjarni, and Tarski, Alfred. Boolean algebras with operators, part II. American Journal of Mathematics, 74:127–162, 1952.CrossRefGoogle Scholar
Kaminski, Michael, and Tiomkin, Michael. The expressive power of second-order propositional modal logic. Notre Dame Journal of Formal Logic, 37:35–43, 1996.CrossRefGoogle Scholar
Kaplan, David. S5 with multiple possibility. The Journal of Symbolic Logic, 35:355–356, 1970a.Google Scholar
Kaplan, David. S5 with quantifiable propositional variables. The Journal of Symbolic Logic, 35:355, 1970b.Google Scholar
Kaplan, David. Demonstratives. In Almog, Joseph, Perry, John, and Wettstein, Howard, editors, Themes from Kaplan, pages 481–563. Oxford: Oxford University Press, 1989 [1977]. Completed and circulated in mimeograph in the published form in 1977.Google Scholar
Kaplan, David. A problem in possible-world semantics. In Sinnott-Armstrong, Walter, Raffman, Diana, and Asher, Nicholas, editors, Modality, Morality, and Belief, pages 41–52. Cambridge: Cambridge University Press, 1995.Google Scholar
Kesten, Yonit, and Pnueli, Amir. Complete proof system for QPTL. Journal of Logic and Computation, 12:701–745, 2002.CrossRefGoogle Scholar
Koyré, Alexandre. The liar. Philosophy and Phenomenological Research, 6:344–362, 1946.CrossRefGoogle Scholar
Kreisel, Georg. Monadic operators defined by means of propositional quantification in intuitionistic logic. Reports on Mathematical Logic, 12:9–15, 1981.Google Scholar
Kremer, Philip. Quantifying over propositions in relevance logic: Nonaxiomatisability of primary interpretations of and . The Journal of Symbolic Logic, 58:334–349, 1993.CrossRefGoogle Scholar
Kremer, Philip. Defining relevant implication in a propositionally quantified S4. The Journal of Symbolic Logic, 62:1057–1069, 1997a.CrossRefGoogle Scholar
Kremer, Philip. On the complexity of propositional quantification in intuitionistic logic. The Journal of Symbolic Logic, 62:529–544, 1997b.CrossRefGoogle Scholar
Kremer, Philip. Propositional quantification in the topological semantics for S4. Notre Dame Journal of Formal Logic, 38:295–313, 1997c.CrossRefGoogle Scholar
Kremer, Philip. Completeness of second-order propositional S4 and H in topological semantics. The Review of Symbolic Logic, 11:507–518, 2018.CrossRefGoogle Scholar
Kripke, Saul A. A completeness theorem in modal logic. The Journal of Symbolic Logic, 24:1–14, 1959.CrossRefGoogle Scholar
Kripke, Saul A. Semantical analysis of modal logic I: Normal modal propositional calculi. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 9:67–96, 1963a.CrossRefGoogle Scholar
Kripke, Saul A. Semantical considerations on modal logic. Acta Philosophica Fennica, 16:83–94, 1963b.Google Scholar
Kripke, Saul A. Semantical analysis of modal logic II: Non-normal modal propositional calculi. In Addison, J. W., Henkin, Leon, and Tarski, Alfred, editors, The Theory of Models. Proceedings of the 1963 International Symposium at Berkeley, pages 206–220. Amsterdam: North-Holland, 1965.Google Scholar
Kripke, Saul A. Review of Algebraic semantics for modal logics II by Lemmon, E. J.. Mathematical Reviews, 0205835 (34 #5661), 1967.Google Scholar
Kripke, Saul A. Is there a problem about substitutional quantification? In Evans, Gareth and McDowell, John, editors, Truth and Meaning: Essays in Semantics, pages 325–419. Oxford: Clarendon Press, 1976.Google Scholar
Kripke, Saul A. Naming and Necessity. Cambridge, MA: Harvard University Press, 1980 [1972]. First published in Semantics of Natural Language, edited by Davidson, Donald and Harman, Gilbert, pages 253–355, 763–769, Dordrecht: D. Reidel, 1972.Google Scholar
Kripke, Saul A. Review of Failures of the interpolation lemma in quantified modal logic by Kit Fine. The Journal of Symbolic Logic, 48:486–488, 1983.CrossRefGoogle Scholar
Kuhn, Steven. A simple embedding of T into double S5. Notre Dame Journal of Formal Logic, 45:13–18, 2004.CrossRefGoogle Scholar
Lemmon, E. J. The “Lemmon Notes”. An Introduction to Modal Logic. Oxford: Basil Blackwell, 1977 [1966]. In collaboration with Dana Scott, edited by Segerberg, Krister, completed and circulated in 1966.Google Scholar
Lewis, Clarence Irving, and Harold Langford, Cooper. Symbolic Logic. New York, NY: Dover, second edition, 1959 [1932]. Republication of the first edition published by The Century Company in 1932.Google Scholar
Lewitzka, Steffen. Denotational semantics for modal systems S3–S5 extended by axioms for propositional quantifiers and identity. Studia Logica, 103:507–544, 2015.CrossRefGoogle Scholar
Lesniewski, Stanisław. Grundzüge eines neuen Systems der Grundlagen der Mathematik. Fundamenta Mathematicae, 14:1–81, 1929.CrossRefGoogle Scholar
Lindström, Sten. Possible worlds semantics and the liar. In Almog, Joseph and Leonardi, Paolo, editors, The Philosophy of David Kaplan, pages 93–108. Oxford: Oxford University Press, 2009.CrossRefGoogle Scholar
Litak, Tadeusz. Modal incompleteness revisited. Studia Logica, 76:329–342, 2004.CrossRefGoogle Scholar
Lokhorst, Gert-Jan C. Propositional quantifiers in deontic logic. In Goble, Lou and Meyer, John-Jules, editors, Deontic Logic and Artificial Normative Systems, DEON 2006, volume 4048 of Lecture Notes in Artificial Intelligence, pages 201–209. Berlin: Springer, 2006.CrossRefGoogle Scholar
Łoś, Jerzy. Logiki wielowartościowe a formalizacja funkcji intensjonalnych. Kwartalnik Filozoficzny, 17:59–78, 1948.Google Scholar
Łukasiewicz, Jan. On variable functors of propositional arguments. Proceedings of the Royal Irish Academy, 54:25–35, 1951.Google Scholar
Łukasiewicz, Jan, and Tarski, Alfred. über den Aussagen-kalkül, Untersuchungen. Comptes Rendus des séances de la Societé des Sciences et des Lettres de Varsovie, 23:30–50, 1930. Reprinted in English translation by J. H. Woodger as “Investigations into the Sentential Calculus” in Alfred Tarski, Logic, Semantics, Metamathematics, pages 38–59, Oxford: Clarendon Press, 1956.Google Scholar
Löb, M. H. Embedding first order predicate logic in fragments of intuitionistic logic. The Journal of Symbolic Logic, 41:705–718, 1976.CrossRefGoogle Scholar
Löwenheim, Leopold. Über Möglichkeiten im Relativkalkül. Mathematische Annalen, 76:447–470, 1915.CrossRefGoogle Scholar
MacNeille, H. M. Partially ordered sets. Transactions of the American Mathematical Society, 42:416–460, 1937.CrossRefGoogle Scholar
Makinson, David. On some completeness theorems in modal logic. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 12:379–384, 1966.CrossRefGoogle Scholar
Makinson, David. Some embedding theorems for modal logic. Notre Dame Journal of Formal Logic, 12:252–254, 1971.CrossRefGoogle Scholar
Makinson, David. A warning about the choice of primitive operators in modal logic. Journal of Philosophical Logic, 2:193–196, 1973.CrossRefGoogle Scholar
McKinsey, J. C. C. A solution of the decision problem for the Lewis systems S2 and S4, with an application to topology. The Journal of Symbolic Logic, 6:117–134, 1941.CrossRefGoogle Scholar
McKinsey, J. C. C. On the syntactical construction of systems of modal logic. The Journal of Symbolic Logic, 10:83–94, 1945.CrossRefGoogle Scholar
McKinsey, J. C. C., and Tarski, Alfred. The algebra of topology. Annals of Mathematics, 45:141–191, 1944.CrossRefGoogle Scholar
McKinsey, J. C. C., and Tarski, Alfred. Some theorems about the sentential calculi of Lewis and Heyting. The Journal of Symbolic Logic, 13:1–15, 1948.CrossRefGoogle Scholar
Menzel, Christopher. Pure logic and “higher-order” metaphysics. In Fritz, Peter and Jones, Nicholas K., editors, Higher-Order Metaphysics. Oxford: Oxford University Press, 2024.Google Scholar
Christopher, Menzel, and Zalta, Edward N.. The fundamental theorem of world theory. Journal of Philosophical Logic, 43:333–363, 2014.Google Scholar
Meredith, C. A. On an extended system of the propositional calculus. Proceedings of the Royal Irish Academy, 54:37–47, 1951.Google Scholar
Muskens, Reinhard. Intensional models for the theory of types. The Journal of Symbolic Logic, 72:98–118, 2007.CrossRefGoogle Scholar
Myhill, John. On the interpretation of the sign ‘ ’. The Journal of Symbolic Logic, 18:60–62, 1953.CrossRefGoogle Scholar
Myhill, John. Review of Formal Logic by A. N. Prior, . The Philosophical Review, 66:117–120, 1957.CrossRefGoogle Scholar
Myhill, John. Problems arising in the formalization of intensional logic. Logique et Analyse, 1:78–83, 1958.Google Scholar
Nagle, Michael C., and Thomason, S. K.. The extensions of the modal logic K5. The Journal of Symbolic Logic, 50:102–109, 1985.CrossRefGoogle Scholar
Pacuit, Eric. Neighborhood Semantics for Modal Logic. Short Textbooks in Logic. Cham: Springer, 2017.CrossRefGoogle Scholar
Parry, W. T. Zum Lewisschen Aussagenkalkül. Ergebnisse eines Mathematischen Kolloquiums, 4:15–16, 1933.Google Scholar
Pascucci, Matteo. Propositional quantifiers in labelled natural deduction for normal modal logic. Logic Journal of the IGPL, 27:865–894, 2019.CrossRefGoogle Scholar
Pitts, Andrew M. On an interpretation of second order quantification in first order intuitionistic propositional logic. The Journal of Symbolic Logic, 57:33–52, 1992.CrossRefGoogle Scholar
Połacik, Tomasz. Operators defined by propositional quantification and their interpretation over Cantor space. Reports on Mathematical Logic, 27:67–79, 1993.Google Scholar
Połacik, Tomasz. Pitts’ quantifiers are not topological quantification. Notre Dame Journal of Formal Logic, 39:531–545, 1998a.CrossRefGoogle Scholar
Połacik, Tomasz. Propositional quantification in the monadic fragment of intuitionistic logic. The Journal of Symbolic Logic, 63:269–300, 1998b.CrossRefGoogle Scholar
Prawitz, Dag. Natural Deduction. A Proof-Theoretical Study, volume 3 of Stockhom Studies in Philosophy. Stockholm: Almqvist & Wiksell, 1965.Google Scholar
Prior, Arthur N. Escapism: the logical basis of ethics. In Melden, A. I., editor, Essays in Moral Philosophy. Seattle: University of Washington Press, 1958a.Google Scholar
Prior, Arthur N. Epimenides the Cretan. The Journal of Symbolic Logic, 23:261–266, 1958b.CrossRefGoogle Scholar
Prior, Arthur N. On a family of paradoxes. Notre Dame Journal of Formal Logic, 2:16–32, 1961.CrossRefGoogle Scholar
Prior, Arthur N. Objects of Thought. Oxford: Clarendon Press, 1971. Edited by Geach, P. T. and Kenny, A. J. P..CrossRefGoogle Scholar
Prucnal, Tadeusz. On two problems of Harvey Friedman. Studia Logica, 38:247–262, 1979.CrossRefGoogle Scholar
Quine, W. V. Philosophy of Logic. Cambridge, MA: Harvard University Press, second edition, 1986 [1970]. First edition published in 1970.CrossRefGoogle Scholar
Rabin, Michael O. Decidability of second-order theories and automata on infinite trees. Transactions of the American Mathematical Society, 141:1–35, 1969.Google Scholar
Ramsey, Frank P. Critical notice of Tractatus Logico-Philosophicus by Ludwig Wittgenstein. Mind, 32:465–478, 1923. Reprinted in Ramsey (1931).CrossRefGoogle Scholar
Ramsey, Frank P. Facts and propositions. Proceedings of the Aristotelian Society, Supplementary Volumes, 7:153–170, 1927. Reprinted in Ramsey (1931).Google Scholar
Ramsey, Frank P. The Foundations of Mathematics, and other Logical Essays. London: Routledge and Kegan Paul, 1931. Edited by Braithwaite, R. B. and Moore, G. E..Google Scholar
Ramsey, Frank P. On Truth. Original Manuscript Materials (1927–1929) from the Ramsey Collection at the University of Pittsburgh. Dordrecht: Kluwer, 1991. Edited by Rescher, Nicholas and Majer, Ulrich.Google Scholar
Rasiowa, Helena, and Sikorski, Roman. The Mathematics of Metamathematics. Warsaw: Państwowe Wydawnictwo Naukowe, 1963.Google Scholar
Rönnedal, Daniel. Semantic tableau versions of some normal modal systems with propositional quantifiers. Organon F, 26:505–536, 2019.CrossRefGoogle Scholar
Rönnedal, Daniel. The moral law and the good in temporal modal deontic logic with propositional quantifiers. Australasian Journal of Logic, 17:22–69, 2020.CrossRefGoogle Scholar
Richard, Routley, and Meyer, Robert K.. The semantics of entailment. In Leblanc, Hugues, editor, Truth, Syntax and Modality, pages 199–243. Amsterdam: North-Holland, 1973.Google Scholar
Russell, Bertrand. The theory of implication. American Journal of Mathematics, 28:159–202, 1906.CrossRefGoogle Scholar
Russell, Bertrand. Mathematical logic as based on the theory of types. American Journal of Mathematics, 30:222–262, 1908.CrossRefGoogle Scholar
Russell, Bertrand. Toward “Principia Mathematica”. 1905–08. The Collected Papers of Bertrand Russell. London: Routledge, 2014. Edited by Gregory, H. Moore.Google Scholar
Salerno, Joe, editor. New Essays on the Knowability Paradox. Oxford: Oxford University Press, 2009.CrossRefGoogle Scholar
Scott, Dana. Advice on modal logic. In Lambert, Karel, editor, Philosophical Problems in Logic. Some Recent Developments, pages 143–173. Dordrecht: D. Reidel, 1970.CrossRefGoogle Scholar
Scroggs, Schiller Joe. Extensions of the Lewis system S5. The Journal of Symbolic Logic, 16:112–120, 1951.CrossRefGoogle Scholar
Segerberg, Krister. An Essay in Classical Modal Logic, volume 13 of Filosofiska Studier. Uppsala: Uppsala Universitet, 1971.Google Scholar
Segerberg, Krister. Classical Propositional Operators. An Exercise in the Foundations of Logic. Oxford: Clarendon Press, 1982.Google Scholar
Sellars, Wilfrid. Grammar and existence: A preface to ontology. Mind, 69:499–533, 1960.CrossRefGoogle Scholar
Shapiro, Stewart. Foundations without Foundationalism: A Case for Second-order Logic. Oxford: Clarendon Press, 1991.Google Scholar
Shelah, Saharon. The monadic theory of order. Annals of Mathematics, 102:379–419, 1975.CrossRefGoogle Scholar
Shukla, Ankit, Biere, Armin, Seidl, Martina, and Pulina, Luca. A survey on applications of quantified boolean formulas. In IEEE 31st International Conference on Tools with Artificial Intelligence (ICTAI 2019), pages 78–84. IEEE Computer Society, Conference Publishing Services, 2019.Google Scholar
Skvortsov, D. Non-axiomatizable second order intuitionistic propositional logic. Annals of Pure and Applied Logic, 86:33–46, 1997.CrossRefGoogle Scholar
Sobolev, S. K. The intuitionistic propositional calculus with quantifiers (in Russian). Matematicheskie Zametki, 22:69–76, 1977.Google Scholar
Solovay, Robert M. Provability interpretations of modal logic. Israel Journal of Mathematics, 25:287–304, 1976.CrossRefGoogle Scholar
Stalnaker, Robert. A theory of conditionals. In Rescher, Nicholas, editor, Studies in Logical Theory, pages 98–112. Oxford: Blackwell, 1968.Google Scholar
Steinsvold, Christopher. Some formal semantics for epistemic modesty. Logic and Logical Philosophy, 29:381–413, 2020.Google Scholar
Stone, M. H. The theory of representation for Boolean algebras. Transactions of the American Mathematical Society, 40:37–111, 1936.Google Scholar
Suszko, R. Review of Many-Valued Logics and the Formalization of Intensional Functions by Jerzy Łoś. The Journal of Symbolic Logic, 14:64–65, 1949.CrossRefGoogle Scholar
Sørensen, Morten H. and Urzyczyn, Pawel. A syntactic embedding of predicate logic into second-order propositional logic. Notre Dame Journal of Formal Logic, 51:457–473, 2010.CrossRefGoogle Scholar
Tajtelbaum-Tarski, Alfred. O wyrazie pierwotnym logistyki. Przegląd Filozoficzny, 26:68–89, 1923. Reprinted in English translation by J. H. Woodger as “On the Primitive Term of Logistic” in Alfred Tarski, Logic, Semantics, Metamathematics, pages 1–23, Oxford: Clarendon Press, 1956.Google Scholar
Tang, Tsao-Chen. Algebraic postulates and a geometric interpretation for the Lewis calculus of strict implication. Bulletin of the American Mathematical Society, 44:737–744, 1938.Google Scholar
Tarski, Alfred. Zur Grundlegung der Bool’schen Algebra. Fundamenta Mathematicae, 24:177–198, 1935.CrossRefGoogle Scholar
Tarski, Alfred. Über additive und multiplikative Mengenkörper und Mengenfunktionen. Sprawozdania z Posiedzeń Towarzystwa Naukowego Warszawskiego, Wydział III Nauk Matematyczno-fizycznych (=Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, Classe III), 30:151–181, 1937.Google Scholar
Tarski, Alfred. The concept of truth in formalized languages. In Corcoran, John, editor, Logic, Semantics, Metamathematics, pages 152–278. Indianapolis: Hackett, 1983 [1933]. Translation of an article published in Polish in 1933 and German in 1935.Google Scholar
Tarski, Alfred. On the concept of following logically. History and Philosophy of Logic, 23:155–196, 2002 [1936]. Originally published in Polish and German in 1936.CrossRefGoogle Scholar
ten Cate, , , Balder. Expressivity of second order propositional modal logic. Journal of Philosophical Logic, 35:209–223, 2006.CrossRefGoogle Scholar
Thomason, S. K. Semantic analysis of tense logics. The Journal of Symbolic Logic, 37:150–158, 1972.CrossRefGoogle Scholar
Thomason, S. K. An incompleteness theorem in modal logic. Theoria, 40:30–34, 1974.CrossRefGoogle Scholar
Thomason, S. K. Categories of frames for modal logic. The Journal of Symbolic Logic, 40:439–442, 1975.CrossRefGoogle Scholar
Troelstra, A. S. On a second order propositional operator in intuitionistic logic. Studia Logica, 40:113–139, 1981.CrossRefGoogle Scholar
Uzquiano, Gabriel. Elusive propositions. Journal of Philosophical Logic, 50:705–725, 2021.CrossRefGoogle Scholar
van Benthem, Johan, and Bezhanishvili, Guram. Modal logics of space. In Aiello, Marco, Pratt-Hartmann, Ian, and van Benthem, Johan, editors, Handbook of Spatial Logics, pages 217–298. Dordrecht: Springer, 2007.CrossRefGoogle Scholar
van Ditmarsch, , , Hans. To be announced. Information and Computation, 292:1–42, 2023.CrossRefGoogle Scholar
van Inwagen, , , Peter. Generalizations of homophonic truth-sentences. In Schantz, Richard, editor, What Is Truth?, volume 1 of Current Issues in Theoretical Philosophy, pages 205–222. Berlin: De Gruyter, 2002.Google Scholar
Whitehead, Alfred North, and Russell, Bertrand. Principia Mathematica, Volumes 1–3. Cambridge: Cambridge University Press, 1910–1913.Google Scholar
Williamson, Timothy. Modal Logic as Metaphysics. Oxford: Oxford University Press, 2013.CrossRefGoogle Scholar
Williamson, Timothy. Menzel on pure logic and higher-order metaphysics. In Fritz, Peter and Jones, Nicholas K., editors, Higher-Order Metaphysics. Oxford: Oxford University Press, 2024.Google Scholar
Wittgenstein, Ludwig. Logisch-philosophische Abhandlung. In Oswald, Wilhelm, editor, Annalen der Naturphilosophie, volume 14, pages 185–262. Leipzig: Unesma, 1921.Google Scholar
Zach, Richard. Decidability of quantified propositional intuitionistic logic and S4 on trees of height and arity . Journal of Philosophical Logic, 33:155–164, 2004.CrossRefGoogle Scholar
Zdanowski, Konrad. On second order intuitionistic propositional logic without a universal quantifier. The Journal of Symbolic Logic, 74:157–167, 2009.CrossRefGoogle Scholar
Zhao, Zhiguang. Sahlqvist correspondence theory for second-order propositional modal logic. Journal of Logic and Computation, 33:577–598, 2023.CrossRefGoogle Scholar
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