Whence the Laws of Probability?
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 wordsprobability laws of probability sum rule product rule Boolean algebra
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