Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-03T00:16:31.575Z Has data issue: false hasContentIssue false
  • English
  • Français

Validity Rules for Proportionally Quantified Syllogisms

Published online by Cambridge University Press:  14 March 2022

Henry Albert Finch
Henry Albert Finch*
Affiliation:
Pennsylvania State University
Get access Rights & Permissions [Opens in a new window]

Extract

Since the time, about a century ago, when DeMorgan (8), Boole (1) and Jevons (12), inaugurated the study of the logic of numerically definite reasoning, no one has been concerned to establish the validity rules for a very general type of numerically definite inference which is a strong analogue of the classical syllogism. The reader will readily agree that the traditional rules of syllogistic inference cannot even begin to decide whether the following proportionally quantified syllogism is a valid argument: at most 4/7 p are at most 2/3 not-m and at least 3/5 s are precisely 7/8 m, hence, at most 1/3 s are p. Likely reasons, of broad methodological interest, why this readily definable type of formal inference has been neglected will presently be proposed. Among the masters of twentieth century logic, Hilbert and Bernays (11) have, to be sure, meticulously shown how such numerically definite propositions as “at least 7 individuals have the property p” can be expressed in the predicate calculus with the use of the identity relation. They do not, since their concern with such propositions is indirect, proceed to establish a systematic calculus of classes whose “size” or extent is numerically definite. As we shall always notice, traditional logic and even ordinary Boolean algebra prefer to generalize about the laws governing the interrelationship of classes in inference systems without making any hypotheses concerning class “size” or extent except by the classical quantifier “some”, which quantifier admits the measure or extent of a class only in terms of the proposition that a given class contains or does not contain at least one member. Such a restricted admission of the relevance for inference of class size is among the fundamental reasons for the neglect of proportional quantifiers, i.e., quantifiers where numerical value is any rational number r such that 0/100 ≤ r ≤ 100/100. It can, of course, also be pointed out that the traditional syllogism and ordinary Boolean algebra do not i.e. measure, numerically define, the degree of inclusion of one class in another. Inclusion is taken either as total or partial, with no discrimination of the degree of partial inclusion.

Type
Research Article
Information
Philosophy of Science , Volume 24 , Issue 1 , January 1957 , pp. 1 - 18
Copyright
Copyright © 1957, The Williams & Wilkins Company

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Boole, G. Studies in Logic and Probability. Watts and Co. 1952, IV, 167 ff and passim.Google Scholar
2. Boole, G. Laws of Thought. Open Court, 1940, 312.Google Scholar
3. Borel, E. Théorie des Fonctions, Gauthier-Villars, 1914, Note V.Google Scholar
4. Church, A. The Concept of a Random Sequence. Bulletin of the American Mathematical Society, 46, 1940, 130.CrossRefGoogle Scholar
5. Churchman, C. W. Theory of Experimental Inference. Macmillan, 1948, Chapter X.Google Scholar
6. Copeland, A. H. Predictions and Probabilities. Erkenntnis, 1936, 189.CrossRefGoogle Scholar
7. Copi, I. M. Introduction to Logic. Macmillan, 1953, 196.Google Scholar
8. De Morgan, A. Formal Logic. Open Court, 1926, Chapter VIII.Google Scholar
9. De Morgan, A. Newton, Open Court, 1914, 130.Google Scholar
10. Halmos, P. R. The Foundations of Probability. American Mathematical Monthly, LI, 1944, 500.CrossRefGoogle Scholar
11. Hilbert, D. and Bernays, P. Grundlagen der Mathematik, I, Springer, 1934, 172.Google Scholar
12. Jevons, W. S. Pure Logic and Other Minor Works, Macmillan 1890, Chapter IV.Google Scholar
13. Kamke, E. Theory of Sets. New York, 1950.Google Scholar
14. Kneale, W. Probability and Induction. Oxford, 1949, 162.Google Scholar
15. Kolmogoroff, A. Grundbegriffe der Wahrscheinlichkeitsrechnung. Chelsea, 1946, Chapter I.Google Scholar
16. Lebesgue, H. Sur La Mesure des Grandeurs, Gauthier-Villars, no date.Google Scholar
17. Von Mises, R. Wahrscheinlichkeit, Statistik und Wahrheit, Springer, 1951, 106.CrossRefGoogle Scholar
18. Myhill, J. Note on Random Sequence. Journal of Symbolic Logic, 16, 1951, 236.CrossRefGoogle Scholar
19. Quine, W. Methods of Logic. Holt 1950, 89.Google Scholar
20. Rudin, W. Principles of Mathematical Analysis. McGraw-Hill, 1953, 199.Google Scholar
21. Ryle, G. Dilemmas. Cambridge, 1954, Chapter VIII.10.1017/CBO9781316286586CrossRefGoogle Scholar
22. Smith, Henry Bradford, Logic as the Art of Symbols reprinted in, A First Book in Logic, Crofts, 1938.Google Scholar
23. Tarski, A. Introduction to Logic. Oxford, 1941, 64.Google Scholar
24. Ulam, S. M. What is Measure. American Mathematical Monthly, 50, 1943, 597.CrossRefGoogle Scholar
25. Wald, A. Sur la notion de collectif dans le calcul des probabilités. Comptes Rendus des Séances de 1' Academie des Sciences, 202, 1936, 180 and same author's Die Widerspruchsfreiheit des Kollektivbegriffes in Selected Papers in Statistics and Probability, McGraw-Hill, 1955, 25.Google Scholar
26. Yule, G. U. On the theory of consistence of logical class-frequencies and its geometrical representation. Philosophical Transactions of the Royal Society of London, 197, (1901), 91.10.1098/rsta.1901.0015CrossRefGoogle Scholar
27. Yule, G. U., Notes on the Theory of Association of Attributes in Statistics. Biometrika, Volume II, 1903, 121.CrossRefGoogle Scholar