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LOGICS FOR THE RELATIONAL SYLLOGISTIC

Published online by Cambridge University Press:  01 December 2009

IAN PRATT-HARTMANN  and
LAWRENCE S. MOSS
IAN PRATT-HARTMANN*
Affiliation:
School of Computer Science, University of Manchester
LAWRENCE S. MOSS*
Affiliation:
Department of Mathematics, Indiana University
*
*SCHOOL OF COMPUTER SCIENCE, UNIVERSITY OF MANCHESTER, MANCHESTER M13 9PL, UK
†DEPARTMENT OF MATHEMATICS, INDIANA UNIVERSITY, 831 EAST THIRD STREET, BLOOMINGTON, IN 47405-7106, USA
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Abstract

The Aristotelian syllogistic cannot account for the validity of certain inferences involving relational facts. In this paper, we investigate the prospects for providing a relational syllogistic. We identify several fragments based on (a) whether negation is permitted on all nouns, including those in the subject of a sentence; and (b) whether the subject noun phrase may contain a relative clause. The logics we present are extensions of the classical syllogistic, and we pay special attention to the question of whether reductio ad absurdum is needed. Thus our main goal is to derive results on the existence (or nonexistence) of syllogistic proof systems for relational fragments. We also determine the computational complexity of all our fragments.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 2 , Issue 4 , December 2009 , pp. 647 - 683
Copyright
Copyright © Association for Symbolic Logic 2009

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References

BIBLIOGRAPHY

Arnauld, A. (1662). Logic, or, the Art of Thinking (“The Port-Royal Logic”). tr. Dickoff, J., and James, P.Indianapolis: Bobbs-Merrill, 1964.Google Scholar
Corcoran, J. (1972). Completeness of an ancient logic. The Journal of Symbolic Logic, 37(4), 696–702.Google Scholar
De Morgan, A. (1847). Formal Logic: or, the Calculus of Inference, Necessary and Probable. London: Taylor and Walton.Google Scholar
De Morgan, A. (1860). On the syllogism, Part IV. Transactions of the Cambridge Philosophical Society, 10, 331–357.Google Scholar
Englebretsen, G. (1981). Three Logicians. Assen, The Netherlands: Van Gorcum.Google Scholar
Grädel, E., Kolaitis, P., & Vardi, M. (1997). On the decision problem for two-variable first-order logic. The Bulletin of Symbolic Logic, 3(1), 53–69.Google Scholar
Harel, D., Kozen, D., & Tiuryn, J. (2000). Dynamic Logic. Cambridge, MA: MIT Press.Google Scholar
Łukasiewicz, J. (1957). Aristotle’s Syllogistic (second edition). Oxford, UK: Clarendon Press.Google Scholar
Martin, J. N. (1997). Aristotle’s natural deduction revisited. History and Philosophy of Logic, 18(1), 1–15.CrossRefGoogle Scholar
McAllester, D. A., & Givan, R. (1992). Natural language syntax and first-order inference. Artificial Intelligence, 56, 1–20.Google Scholar
Merrill, D. D. (1990). Augustus De Morgan and the Logic of Relations. Dordrecht, The Netherlands: Kluwer Academic Publishers.Google Scholar
Moss, L. S. (to appear). Syllogistic logics with verbs. Journal of Logic and Computation. doi: 10.1093/logcom/exn086.Google Scholar
Moss, L. S. (2007). Syllogistic logic with complements. Manuscript, Indiana University.Google Scholar
Moss, L. S. (2008). Completeness theorems for syllogistic fragments. In Hamm, F., and Kepser, S., editors. Logics for Linguistic Structures. Berlin: Mouton de Gruyter, pp. 143–173.Google Scholar
Nishihara, N., Morita, K., & Iwata, S. (1990). An extended syllogistic system with verbs and proper nouns, and its completeness proof. Systems and Computers in Japan, 21(1), 760–771.Google Scholar
Papadimitriou, C. H. (1994). Computational Complexity. Reading, MA: Addison-Wesley.Google Scholar
Pratt-Hartmann, I. (2004). Fragments of language. Journal of Logic, Language and Information, 13, 207–223.Google Scholar
Smiley, T. J. (1973). What is a syllogism? Journal of Philosophical Logic, 2, 135–154.Google Scholar
Sommers, F. (1982). The Logic of Natural Language. Oxford, UK: Clarendon Press.Google Scholar
Westerståhl, D. (1989). Aristotelian syllogisms and generalized quantifiers. Studia Logica, XLVIII(4), 577–585.Google Scholar