Advertisement

Contributions to the Formal Theory of Probability

  • Karl Popper
  • David Miller
Part of the Synthese Library book series (SYLI, volume 234)

Abstract

Popper (1959), Appendices *iv and *v) has given several axiom systems for probability that ensure, without further assumptions, that the domain of interpretation can be reduced to a Boolean algebra. This paper presents axiom systems for subtheories of probability theory that characterize in the same way lower semilattices (Section 1) and distributive lattices (Section 2). Section 1 gives a new (metamathematical) derivation of the laws of semilattices; and Section 2 one or two surprising theorems, previously derived only with the help of an axiom for complementation. The problem of the creativity of the axioms is explored in Section 3, enlarging on (1963). In conclusion, Section 4 explains how these systems, and the full system of (Popper 1959), provide generalizations of the relation of deducibility, contrasting our approach with the enterprise known as probabilistic semantics.

Keywords

Boolean Algebra Distributive Lattice Formal Theory Axiom System Probabilistic Semantic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bendali, K.: 1982, ‘A “Definitive” Probabilistic Semantics for First-Order Logic’, Journal of Philosophical Logic, 11, 255–278.CrossRefGoogle Scholar
  2. Birkhoff, G.: 1973, Lattice Theory, 3rd edition, American Mathematical Society, Providence RI.Google Scholar
  3. Bridge, J.: 1977, Beginning Model Theory: The Completeness Theorem and Some of Its Consequences, Oxford Logic Guides, Clarendon Press, Oxford.Google Scholar
  4. Field, H. H.: 1977, ‘Logic, Meaning, and Conceptual Role’, Journal of Philosophy, 74, 379–409.CrossRefGoogle Scholar
  5. van Fraassen, B. C: 1981, ‘Probabilistic Semantics Objectified. I: Postulates and Logics’, Journal of Philosophical Logic, 10, 371–394.CrossRefGoogle Scholar
  6. Kalicki, J. and Scott, D. S.: 1955, ‘Equational Completeness of Abstract Algebras’, Koninklijke Nederlandse Akademie van Wetenschappen. Proceedings, Series A: Mathematical Sciences, 58 [= Indagationes Mathematicae, 17], 650–659.Google Scholar
  7. Leblanc, H.: 1979, ‘Probabilistic Semantics for First-Order Logic’, Zeitschrift für Logik und Grundlagen der Mathematik, 25, 497–509.CrossRefGoogle Scholar
  8. Leblanc, H.: 1981, ‘What Price Substitutivity? A Note on Probability Theory’, Philosophy of Science, 48, 317–322.CrossRefGoogle Scholar
  9. Leblanc, H.: 1983, ‘Alternatives to Standard First-Order Semantics’, in D. Gabbay and F. Gruender (Eds.), Handbook of Philosophical Logic, Vol. I, D. Reidel Publishing Company, Dordrecht, pp. 189–274.CrossRefGoogle Scholar
  10. Leblanc, H. and van Fraassen, B. C: 1979, ‘On Carnap and Popper Probability Functions’, The Journal of Symbolic Logic, 44, 369–373.CrossRefGoogle Scholar
  11. Paulos, J. A.: 1981, ‘Probabilistic, Truth-Value, and Standard Semantics and the Primacy of Predicate Logic’, Notre Dame Journal of Formal Logic, 22, 11–16.CrossRefGoogle Scholar
  12. Popper, K. R.: 1934, Logik der Forschung, Julius Springer, Wien.Google Scholar
  13. Popper, K. R.: 1959, The Logic of Scientific Discovery, Hutchinson, London. Greatly enlarged English translation of Popper (1934).Google Scholar
  14. Popper, K. R.: 1963, ‘Creative and Non-creative Definitions in the Calculus of Probability’, Synthese, 15, 167–186. (Some minor corrections are recorded in ibid. 21, 1970, p. 107.)CrossRefGoogle Scholar
  15. Popper, K. R.: 1966/1984, Logik der Forschung, J. C. B. Mohr [Paul Siebeck], Tübingen; 2nd and later editions of Popper (1934). (Note: Popper (1966/1984) contains material not in Popper (1959).)Google Scholar
  16. Popper, K. R.: 1968, ‘Birkhoff and von Neumann’s Interpretation of Quantum Mechanics’, Nature, 219, 5155 (17 August), 682–685.CrossRefGoogle Scholar
  17. Suppes, P.: 1957, Introduction to Logic, van Nostrand, Princeton, NJ.Google Scholar

Reference

  1. Suppes, P. and Zanotti, M.: 1982, ‘Necessary and Sufficient Qualitative Axioms for Conditional Probability’, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 60, 163–169.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1994

Authors and Affiliations

  • Karl Popper
    • 1
  • David Miller
    • 2
  1. 1.SurreyUK
  2. 2.Department of PhilosophyUniversity of WarwickUK

Personalised recommendations