Abstract
This paper is a contribution to natural logic, the study of logical systems for linguistic reasoning. We construct a system with the following properties: its syntax is closer to that of a natural language than is first-order logic; it can faithfully represent simple sentences with standard quantifiers, subject relative clauses (a recursive construct), and negation on nouns and verbs. We also give a proof system which is complete and has the finite model property. We go further by adding comparative adjective phrases, assuming interpretations by transitive relations. This last system has all the previously-mentioned properties as well.
The paper was written for theoretical computer scientists and logicians interested in areas such as decidability questions for fragments of first-order logic, modal logic, and natural deduction.
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References
van Benthem, J.: Essays in Logical Semantics. Reidel, Dordrecht (1986)
Böerger, E., Grädel, E., Gurevich, Y.: The Classical Decision Problem. In: Perspectives in Mathematical Logic, p. 1197. Springer, Heidelberg
van Dalen, D.: Logic and Structure, 4th edn. Springer, Berlin (2004)
Fitch, F.B.: Natural Deduction Rules for English. Philosophical Studies 24(2), 89–104 (1973)
Francez, N., Dyckhoff, R.: Proof-Theoretic Semantics for a Natural Language Fragment. In: Jaeger, G., Ebert, C., Michaelis, J. (eds.) Proceedings, MoL 10/11. LNCS (LNAI). Springer, Heidelberg (2010) (to appear)
Grädel, E., Kolaitis, P., Vardi, M.: On the Decision Problem for Two-Variable First-Order Logic. Bulletin of Symbolic Logic 3(1), 53–69 (1997)
Grädel, E., Otto, M., Rosen, E.: Undecidability Results on Two-Variable Logics. Archive for Mathematical Logic 38, 313–354 (1999)
Gurevich, Y.: On the Classical Decision Problem. Logic in Computer Science Column. The Bulletin of the European Association for Theoretical Computer Science (October 1990)
Lutz, C., Sattler, U.: The Complexity of Reasoning with Boolean Modal Logics. In: Wolter, F., et al. (eds.) Advances in Modal Logics, vol. 3. CSLI Publications, Stanford (2001)
McAllester, D., Givan, R.: Natural Language Syntax and First Order Inference. Artificial Intelligence 56 (1992)
Mortimer, M.: On Languages with Two Variables. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 21, 135–140 (1975)
Pelletier, F.J.: A Brief History of Natural Deduction. History and Philosophy of Logic 20, 1–31
Pratt-Hartmann, I.: A Two-Variable Fragment of English. Journal of Logic, Language and Information 12(1), 13–45 (2003)
Pratt-Hartmann, I.: Fragments of Language. Journal of Logic, Language and Information 13, 207–223 (2004)
Pratt-Hartmann, I., Moss, L.S.: Logics for the Relational Syllogistic. Review of Symbolic Logic 2(4), 647–683 (2009)
Vardi, M.Y., Wolper, P.: Automata-Theoretic Techniques for Modal Logics of Programs. Journal of Computer and System Sciences 32(2), 183–221 (1986)
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Moss, L.S. (2010). Logics for Two Fragments beyond the Syllogistic Boundary. In: Blass, A., Dershowitz, N., Reisig, W. (eds) Fields of Logic and Computation. Lecture Notes in Computer Science, vol 6300. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15025-8_27
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DOI: https://doi.org/10.1007/978-3-642-15025-8_27
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