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THE QUANTIFIED ARGUMENT CALCULUS

Published online by Cambridge University Press:  22 January 2014

HANOCH BEN-YAMI*
Affiliation:
Department of Philosophy, Central European University
*
*DEPARTMENT OF PHILOSOPHY CENTRAL EUROPEAN UNIVERSITY, 9 NÁDOR STREET 1051 BUDAPEST, HUNGARY

Abstract

I develop a formal logic in which quantified arguments occur in argument positions of predicates. This logic also incorporates negative predication, anaphora and converse relation terms, namely, additional syntactic features of natural language. In these and additional respects, it represents the logic of natural language more adequately than does any version of Frege’s Predicate Calculus. I first introduce the system’s main ideas and familiarize it by means of translations of natural language sentences. I then develop a formal system built on these principles, the Quantified Argument Calculus or Quarc. I provide a truth-value assignment semantics and a proof system for the Quarc. I next demonstrate the system’s power by a variety of proofs; I prove its soundness; and I comment on its completeness. I then extend the system to modal logic, again providing a proof system and a truth-value assignment semantics. I proceed to show how the Quarc versions of the Barcan formulas, of their converses and of necessary existence come out straightforwardly invalid, which I argue is an advantage of the modal Quarc over modal Predicate Logic as a system intended to capture the logic of natural language.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2013 

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References

BIBLIOGRAPHY

Barcan, R. C. (1946). A functional calculus of first order based on strict implication. The Journal of Symbolic Logic, 11, 116.Google Scholar
Ben-Yami, H. (2004). Logic & Natural Language: On Plural Reference and Its Semantic and Logical Significance. Aldershot, UK: Ashgate. Available from: http://philosophy.ceu.hu/publications/ben-yami/2004/15107.Google Scholar
Ben-Yami, H. (2006). Review of The Old New Logic: Essays on the Philosophy of Fred Sommers, Oderberg, D., editor. Cambridge, MA: MIT Press, 2005. Mind, 116, 197202.Google Scholar
Ben-Yami, H. (2009a). Generalized quantifiers, and beyond. Logique et Analyse, 208,306326.Google Scholar
Ben-Yami, H. (2009b). Plural quantification logic: A critical appraisal. Review of Symbolic Logic, 2, 208232.Google Scholar
Ben-Yami, H. (2012). Response to Westerståhl. Logique et Analyse, 217, 4755.Google Scholar
Ben-Yami, H. Truth and Proof without Models: A Development and Defence of the Truth-valuational Approach. Unpublished.Google Scholar
Boolos, G. (1984). To be is to be a value of a variable (or to be some values of some variables). Journal of Philosophy, 81, 430450.Google Scholar
Dunn, J. M., & Belnap, N. D. (1968). The substitution interpretation of the quantifiers. Noûs, 2, 177185.Google Scholar
Evans, G. (1977). Pronouns, Quantifiers, and Relative Clauses (I). Reprinted in his 1985. Collected Papers. Oxford, UK: Oxford University Press, pp. 76152.Google Scholar
Fitch, F. (1952). Symbolic Logic. New York, NY: Roland Press.Google Scholar
Francez, N. A Logic Inspired by Natural Language: Quantifiers as Subnectors. Forthcoming.Google Scholar
Frege, G. (1879). Begriffsschrift: Eine der Arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle A/S, Verlag von Louis Nebert.Google Scholar
Fuchs, N. E., Kaljurand, K., & Kuhn, T. (2008). Attempto controlled english for knowledge representation. In Baroglio, C., Bonatti, P. A., Maluszynski, J., Marchiori, M., Polleres, A., and Schaffert, S., editors. Reasoning Web: 4th International Summer School 2008. Lecture Notes in Computer Science, Vol. 5224. Berlin: Springer, pp. 104124.CrossRefGoogle Scholar
Geach, P. T. (1962). Reference and Generality: An Examination of Some Medieval and Modern Theories, (emended edition 1968), Ithaca, New York: Cornell University Press.Google Scholar
Jaśkowski, S. (1934). On the rules of suppositions in formal logic. Studia Logica, 1, 232.Google Scholar
Kripke, S. (1963). Semantical considerations on modal logic. Acta Philosophica Fennica, 16, 8394.Google Scholar
Lanzet, R. (2006). An Alternative Logical Calculus: Based on An Analysis of Quantification as Involving Plural Reference. Thesis submitted at Tel-Aviv University, Tel-Aviv.Google Scholar
Lanzet, R., & Ben-Yami, H. (2004). Logical inquiries into a new formal system with plural reference. In Hendricks, V. F., Neuhaus, F., Pedersen, S. A., Scheffler, U., and Wansing, H., editors. First-Order Logic Revisited. Berlin: Logos Verlag, pp. 173223.Google Scholar
LeBlanc, H. (1973). Semantic deviations. In LeBlanc, H., editor. Truth, Syntax and Modality. Amsterdam, London: North-Holland Publishing Company, pp. 116.Google Scholar
LeBlanc, H. (1983). Alternatives to standard first-order semantics. In Gabbay, D. M., & Guenthner, F., editors. Handbook of Philosophical Logic, Vol. I. Dordrecht: Reidel,pp. 189274.CrossRefGoogle Scholar
McKay, T. (2006). Plural Predication. Oxford, UK: Oxford University Press.Google Scholar
Menzel, C. (2012). Actualism. In Zalta, E. N., editor. The Stanford Encyclopedia of Philosophy, (Fall 2012 Edition). Available from: http://plato.stanford.edu/archives/fall2012/entries/actualism/.Google Scholar
Moss, L. (2010). Logics for two fragments beyond the syllogistic boundary. In Blass, A., Dershowitz, N., and Reisig, W., editors. Fields of Logic and Computation: Essays Dedicated to Yuri Gurevich on the Occasion of His 70thBirthday. Berlin: Springer,pp. 538564.Google Scholar
Neale, S. (2006). Pronouns and anaphora. In Devitt, M., and Hanley, R., editors. The Blackwell Guide to the Philosophy of Language. Oxford, UK: Blackwell, pp. 335373.CrossRefGoogle Scholar
Oliver, A., & Smiley, T. (2013). Plural Logic. Oxford, UK: Oxford University Press.Google Scholar
Pratt-Hartmann, I., & Moss, L. 2009. Logics for the relational syllogistic. Review of Symbolic Logic, 2, 647683.Google Scholar
Schwitter, R. (2010). Controlled natural languages for knowledge representation. In Chu-Ren, H. and Dan, J., editors. Proceedings of the 23rd International Conference on Computational Linguistics (COLING 2010). Beijing, China: Coling 2010 Organizing Committee, pp. 11131121.Google Scholar
Sommers, F. (1982). The Logic of Natural Language. Oxford, UK: Clarendon Press.Google Scholar
Strawson, P. F. (1950). On Referring. Reprinted in his 1971. Logico-Linguistic Papers. London: Methuen, pp. 127.Google Scholar
Strawson, P. F. (1952). Introduction to Logical Theory. London: Methuen.Google Scholar
Westerståhl, D. (2012). Explaining quantifier restriction: Reply to Ben-Yami. Logique et Analyse, 217, 109120.Google Scholar