Abstract
Aristotle's syllogistic is extended to include denumerably many quantifiers such as ‘more than 2/3’ and ‘exactly 2/3.’ Syntactic and semantic decision procedures determine the validity, or invalidity, of syllogisms with any finite number of premises. One of the syntactic procedures uses a natural deduction account of deducibility, which is sound and complete. The semantics for the system is non-classical since sentences may be assigned a value other than true or false. Results about symmetric systems are given. And reasons are given for claiming that syllogistic validity is relevant validity.
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References
Anderson A. R. and N. D. Belnap, Jr., 1975,Entailment, vol. 1, Princeton, Princeton University Press.
Carnes, R. D. and P. L. Peterson, 1991, Intermediate quantifiers versus percentages,Notre Dame Journal of Formal Logic 32, 294–306.
De Morgan, A., 1847,Formal Logic, ed. by A. E. Taylor, London, Open Court, 1926.
Finch, H. A., 1957, Validity rules for proportionally quantified syllogisms,Philosophy of Science 24, 1–18.
Geach, P. T., 1976,Reason and Argument, Berkeley, University of California.
Johnson, F., 1991, Three-membered domains for Aristotle's syllogistic,Studia Logica 50, 181–187.
Łukasiewicz, J., 1957,Aristotle's Syllogistic, 2nd ed., Oxford, Clarendon Press.
Meredith, C. A., 1953, The figures and models of then-termed Aristotelian syllogisms,Dominican Studies 6, 42–47.
Peterson, P. L., 1985, Higher quantity syllogisms,Notre Dame Journal of Formal Logic 26, 348–360.
Peterson, P. L., 1991, Complexly fractionated syllogistic quantifiers,Journal of Philosophical Logic 20, 287–313.
Peterson, P. L. and R. D. Carnes, The compleat syllogistic: Including rules and Venn diagrams for five quantities, MS, Syracuse University.
Rescher, N. and N. A. Gallagher, 1965, Venn diagrams for plurative syllogisms,Philosophical Studies 26, 49–55.
Smiley, T. J., 1973, What is a syllogism?Journal of Philosophical Logic 2, 136–154.
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Johnson, F. Syllogisms with fractional quantifiers. J Philos Logic 23, 401–422 (1994). https://doi.org/10.1007/BF01048688
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DOI: https://doi.org/10.1007/BF01048688