Abstract
Boolean logic and set theory are so closely bound up that it is hard to say which is the foundation of which. Yet, we may say that the backbone of this theoretical complex is the Axiom of Comprehension. But, this Axiom, at the same time, is precisely the most vulnerable assumption in the whole of logical philosophy. While the binary-valued characteristic function of predicates is instrumental in expressing the Axiom of Comprehension, our analysis of scientific obervation, in particular in psychology, suggests that we have to introduce what we call a “propensity function.” This is, roughly speaking, a continuous-valued generalization of the characteristic function. Whereas the characteristic function leads to a distributive logic, we can derive a non-distributive logic from the propensity function. If we further introduce the assumption of compatibility (experimental results of two predicates do not depend on the order of the two observations), we can derive distributive logic, the usual laws of probability, set theory and the usual notion of extension of a predicate (Law of Comprehension). If we do not adopt the assumption of compatibility, our theory provides the framework for quantum mechanics, and hopefully, for future psychology. This way of deriving logic from a probability-like propensity function seems to be more desirable than the usual way of positing logical laws without unifying justification, because we can derive first a more general logic and then therefrom a more restrictive logic as a special case using a small number of clearly stated postulates. The necessity for modification of the Quinian definition of ontological commitment will be indicated.
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Reference
L.A Zadeh, ‘Fuzzy Sets’, Information and Control 8 (3) (1965), P. 338.
G. Frege, Grundgesetz der Arithmetik, Hildesheim: Georg Olms, 1962, p. 253.
W. V. Quine, ‘On Frege’s Way Out’, Mind, N. S. 64 (1955), 145–159.
S. Watanabe, ‘Logic of the Empirical World’, in C. Y. Cheng (ed.), Philosophical Aspects of the Mind-Body Problem, Honolulu: University of Hawaii Press, 1976, pp. 162–181. See in particular p. 169.
K. S. Fu, ‘Stochastic Automata as Models of Learning Systems’, in J. Tou (ed.), Computer and Information Science - II, New York: Academic Press, 1967, p. 177.
K. S. Narendra and M. A. L. Thathachar, ‘Learning Automata - A Survey’, IEEE Transactions on Systems, Man and Cybernetics, Vol SMC-4, No. 4, 1974, p. 323.
W. G. Wee and K. S. Fu, ‘A Formulation of Fuzzy Automata and Its Applications as a Model of Learning Systems’, IEEE Transactions on Systems Sciences, Cybernetics, Vol. SSC-5, 1969, pp. 215–223.
S. Watanabe, ‘Learning Process and Inverse H-Theorem’, IRE Transaction on Information Theory, Vol. IT-8, Sept. 1962, pp. 246–251.
W. Heisenberg, ‘Der Teil und das Ganze’, Munich: R. Piper, 1969, p. 91.
S. Watanabe, ‘Algebra of Observation’?, Progress of Theoretical Physics Supplement, No. 37–38, 1966, pp. 350–367.
S. Watanabe, ‘Modified Concepts of Logic, Probability and Information Based on Generalized Continuous Characteristic Function’, Information and Control 15 (1969), 7–21.
G. Birkhoff, Lattice Theory, New York: Am. Math. Soc., 1948.
C. S. Peirce, Collected Papers, Vol. III, Cambridge, Mass.: Harvard University Press, 1933, p. 440.
K. R. Popper, ‘The Propensity Interpretation of Probability’, British Journal for the Philosophy of Science 10 (1959), 25 - 42.
W. V. Quine, From a Logical Point of View, Cambridge, Mass.: Harvard University Press, 1953.
G. Birkhoff and J. von Neumann, ‘The Logic of Quantum Mechanics’, Ann. Math. 2nd Ser 37 (1936), p. 823.
K. Husimi, ‘Studies in the Foundations of Quantum Mechanics’, Proceedings of the Physical-Mathematical Society of Japan 19 (1937), 766–789.
S. Watanabe, ‘A Model of Mind-Body Relation in Terms of Modular Logic’, Synthese 13 (1961), 261–301.
S. Watanabe, ‘Causality and Time’, in Fraser and Lawrence (eds.), The Study of Time, New York, Springer-Verlag, 1975, p. 267.
S. Watanabe, ‘A Generalized Fuzzy Set Theory’, in IEEE-SMC Transactions, 1978, p. 756.
S. Watanabe, ‘Fuzzification and Invariance’, Proceedings of the International Conference on Cybernetics and Society, Tokyo, 1978, p. 947.
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© 1983 D. Reidel Publishing Company
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Watanabe, S. (1983). Theory of Propensity. In: Cohen, R.S., Wartofsky, M.W. (eds) Language, Logic and Method. Boston Studies in the Philosophy of Science, vol 31. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-7702-0_15
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DOI: https://doi.org/10.1007/978-94-009-7702-0_15
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