Whence the Laws of Probability?

Abstract

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’.