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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 3

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online