milne - The simplest Lewis-style triviality proof yet PETER...

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ANALYSIS 63.4, October 2003, pp. 000–000. © Peter Milne The simplest Lewis-style triviality proof yet? P ETER M ILNE In his celebrated ‘Probabilities of conditionals and conditional probabili- ties’ David Lewis showed that the identification of the probability of con- ditionals a c with the conditional probability of consequent given ante- cedent when that antecedent has non-zero probability, i.e. for all a and c , P ( a c ) = P ( c | a ), when P ( a ) > 0, trivializes the probability distribution in question. Lewis presented three triviality results: LEWIS 1 If P ( a c ) > 0 and P ( a ~ c ) > 0 then P ( c | a ) = P ( c ); LEWIS 2 P assigns non-zero probabilities to at most two of any set of pairwise inconsistent propositions; LEWIS 3 P takes at most four values. The premisses for Lewis’s results are (i) the probability expansion rules, P ( a c ) = P (( a c ) b ) + P (( a c ) ~ b ) = P ( a c | b ) P ( b ) + P ( a c |~ b ) P (~ b ), the first holding generally, the second when 0 < P ( b ) < 1; (ii) the family of probability distributions satisfying the identification of conditional probability and probability of conditional is closed under conditionalization, so that P ( a c | b ) = P ( c | a b ) when P ( a b ) > 0. Lewis used the second expansion rule, expanding the probability of the
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milne - The simplest Lewis-style triviality proof yet PETER...

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