Abstract
The numerical syllogistic is the extension of the traditional syllogistic with numerical quantifiers of the forms at least C and at most C. It is known that, for the traditional syllogistic, a finite collection of rules, similar in spirit to the classical syllogisms, constitutes a sound and complete proof-system. The question arises as to whether such a proof system exists for the numerical syllogistic. This paper answers that question in the negative: no finite collection of syllogism-like rules, broadly conceived, is sound and complete for the numerical syllogistic.
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References
Corcoran, J.: Completeness of an ancient logic. Journal of Symbolic Logic 37(4), 696–702 (1972)
Martin, J.N.: Aristotle’s natural deduction revisited. History and Philosophy of Logic 18(1), 1–15 (1997)
Pratt-Hartmann, I., Moss, L.S.: Logics for the relational syllogistic (2008), ArXiv preprint server, http://arxiv.org/abs/0808.0521
Smiley, T.: What is a syllogism? Journal of Philosophical Logic 2, 135–154 (1973)
de Morgan, A.: Formal Logic: or, the calculus of inference, necessary and probable. Taylor and Walton, London (1847)
Boole, G.: Of propositions numerically definite. Transactions of the Cambridge Philosophical Society XI(II) (1868)
Boole, G.: Collected Logical Works: Studies in Logic and Probability, vol. 1. Open Court, La Salle, IL (1952)
Jevons, W.: On a general system of numerically definite reasoning. Memoirs of the Manchester Literary and Philosophical Society (3rd Series) 4, 330–352 (1871)
Jevons, W.: Pure Logic and Other Minor Works. Macmillan, London (1890)
Grattan-Guinness, I.: The Search for Mathematical Roots, 1870-1940: logics, set theories and the foundations of mathematics from Cantor through Russell to Gödel. Princeton University Press, Princeton (2000)
Murphree, W.: The numerical syllogism and existential presupposition. Notre Dame Journal of Formal Logic 38(1), 49–64 (1997)
Murphree, W.: Numerical term logic. Notre Dame Journal of Formal Logic 39(3), 346–362 (1998)
Hacker, E., Parry, W.: Pure numerical Boolean syllogisms. Notre Dame Journal of Formal Logic 8(4), 321–324 (1967)
Pratt-Hartmann, I.: On the computational complexity of the numerically definite syllogistic and related logics. Bulletin of Symbolic Logic 14(1), 1–28 (2008)
Fitch, F.B.: Natural deduction rules for English. Philosophical Studies 24, 89–104 (1973)
Suppes, P.: Logical inference in English: a preliminary analysis. Studia Logica 38(4), 375–391 (1979)
Purdy, W.C.: A logic for natural language. Notre Dame Journal of Formal Logic 32(3), 409–425 (1991)
Purdy, W.C.: Surface reasoning. Notre Dame Journal of Formal Logic 33(1), 13–36 (1992)
Purdy, W.C.: A variable-free logic for mass terms. Notre Dame Journal of Formal Logic 33(3), 348–358 (1992)
Fyodorov, Y., Winter, Y., Francez, N.: Order-based inference in natural logic. Logic Journal of the IGPL 11(4), 385–416 (2004)
Kuncak, V., Rinard, M.: Towards efficient satisfiability checking for Boolean algebra with Presburger arithmetic. In: Pfenning, F. (ed.) CADE 2007. LNCS, vol. 4603, pp. 215–230. Springer, Heidelberg (2007)
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Pratt-Hartmann, I. (2009). No Syllogisms for the Numerical Syllogistic. In: Grumberg, O., Kaminski, M., Katz, S., Wintner, S. (eds) Languages: From Formal to Natural. Lecture Notes in Computer Science, vol 5533. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01748-3_13
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DOI: https://doi.org/10.1007/978-3-642-01748-3_13
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