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E. J. Lowe, What is ‘conditional probability’?, Analysis, Volume 68, Issue 3, July 2008, Pages 218–223, https://doi.org/10.1093/analys/68.3.218
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1. The standard definition
Several years ago, I raised some questions concerning the standard ratio-based definition of conditional probability (Lowe 1996). According to that definition, the conditional probability of B given A, written ‘p(B|A)’, or alternatively ‘pA(B)’, is defined as follows: One question I raised was this: what entitles us to suppose that the expression ‘p(|)’, as defined by (1), signifies any kind of probability? Of course, it is easily shown that the value of p(B|A) must lie between 0 and 1 and thus within the numerical value range of a probability. But so too may the values of many other functions lie within this range. An absolute probability function, signified by an expression such as ‘p( )’, is a function of just one argument – that argument being a proposition – with a numerical value between 0 and 1. In (1), however, we are purportedly introduced to a different kind of probability function, which is a function of two arguments – both of them propositions – with a numerical value between 0 and 1. What is such a probability supposed to be a probability of? Not the probability of a proposition, clearly, since the function takes not a single proposition but a pair of propositions as its arguments. The answer will be offered that such a probability is simply the conditional probability of one proposition, B, given another proposition, A. Definition (1), however, throws no light at all on what is meant by saying this, beyond telling us that it is a way of talking about the ratio between the (absolute) probabilities of two propositions, (A & B) and A. Why call this ratio a ‘conditional’probability?
