Whence the Laws of Probability?

  • Anthony J. M. Garrett
Part of the Fundamental Theories of Physics book series (FTPH, volume 98)


A new derivation is given of the sum and product rules of probability. Probability is treated as a number associated with one binary proposition conditioned on another, so that the Boolean calculus of the propositions induces a calculus for the probabilities. This is the strategy of R. T. Cox (1946), with a refinement: a formula is derived for the probability of the NAND of two propositions in terms of the probabilities of those propositions. Because NAND is a primitive logic operation from which any other can be synthesised, there are no further probabilities that the NAND can depend on. A functional equation is then set up for the relation between the probabilities and is solved. By synthesising the non-primitive operations NOT and AND from NAND the sum and product rules are derived from this one formula, the fundamental ‘law of probability’.

Key words

probability laws of probability sum rule product rule Boolean algebra 


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  1. Aczél, J. 1963. Remarks on probable inference. Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae, Sectio Mathematica 6, 3–11.Google Scholar
  2. Aczél, J. 1966. Lectures on Functional Equations and Their Applications. Academic Press, New York, USA.zbMATHGoogle Scholar
  3. Cox, R. T. 1946. Probability, frequency and reasonable expectation. American Journal of Physics 14, 1–13.MathSciNetzbMATHCrossRefGoogle Scholar
  4. Cox, R. T. 1961. The Algebra of Probable Inference. Johns Hopkins Press, Baltimore, Maryland, USA.zbMATHGoogle Scholar
  5. Franklin, J. 1991. The ancient legal sources of seventeenth-century probability. In: The Uses of Antiquity, editor S. Gaukroger, Kluwer, Dordrecht, Netherlands, pp.123–144.CrossRefGoogle Scholar
  6. Jaynes, E. T. 1983. E. T. Jaynes: Papers on Probability, Statistics and Statistical Physics. Synthese Library 158. Editor R. D. Rosenkrantz, Reidel, Dordrecht, Netherlands.Google Scholar
  7. Jaynes, E. T. In preparation. Probability Theory: The Logic of Science. Cambridge University Press, Cambridge, UK. Provisional versions available on the World Wide Web at http: //bayes. wustl. edu/.
  8. Keynes, J. M. 1921. A Treatise on Probability. Macmillan, London, UK.zbMATHGoogle Scholar
  9. Kuntzmann, J. 1967. Fundamental Boolean Algebra (English translation). Blackie, London, UK.Google Scholar
  10. Shannon, C. E. 1948. A mathematical theory of communication. Bell System Technical Journal 27, 379–423 & 623–659. Reprinted in: The Mathematical Theory of Communication, editors C. E. Shannon & W.W. Weaver, University of Illinois Press, Urbana, Illinois, USA, 1949.MathSciNetzbMATHGoogle Scholar
  11. Smith, C. R. & Erickson, G. J. 1990. Probability theory and the associativity equation. In: Maximum Entropy and Bayesian Methods, Dartmouth, USA, 1989, editor P. F. Fougère, Kluwer, Dordrecht, Netherlands, pp.17–30.CrossRefGoogle Scholar
  12. Tribus, M. 1969. Rational Descriptions, Decisions and Designs. Pergamon Press, New York, USA.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1998

Authors and Affiliations

  • Anthony J. M. Garrett
    • 1
  1. 1.CambridgeUK

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