In 1976, David Lewis proved that iIf we say that belief in a proposition X -> Y is defined as the amount of credence in Y given X, so that:

P(X -> Y) = P(Y|X)

...what follows is that the likelihood that X -> Y cannot mean what we think it means.

Lewis showed that since the chance that Y given X is defined as P(Y|X) = P( X & Y) / P(X), we have

P(X -> Y) = P(X & Y) / P(X).

Lewis showed, unfortunately, that this cannot fit our ideas of what tht likelihood X -> Y means, because he was able to derive

P( X -> Y ) = P(Y) from the above.

Interestingly, we can also use this theorem to derive the following:

If P( X -> Y) = P(Y) by Lewis’ theorem

then P(X->Y) P(X) = P(Y) P(X).

but P(X->Y) P(X) = P(Y|X) (P(X) = ( P(X & Y) / P(X) ) P(X) by substitution of the above, so

P(X->Y) P(X) = P(X)P(Y) = P(X & Y)

So Lewis’s theorem implies that

P( Y & X ) = P(X)P(Y).

This is interesting, since it implies that

(1) P(X -> Y) = P(Y|X), iff P( X & Y) = P(X)P(Y).

But if we look at the mathematics of probability, (1) is true only if X and y are fully independent variables. This suggests that the problem with defining P(X->Y) as P(Y|X) is that it fails to capture the causal dependence of Y on X that we often expect in the real world when we say that A implies B. If fact, such causal dependence is EXCLUDED by the above.

So the probability that X -> Y cannot in general possibly resemble “the probability of Y given X” unless X and Y are NOT causally associated.

In other words, P( X -> Y ) can be properly expressed by P(Y|X) iff X and Y are causally unrelated.

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Reference:

- Lewis, D. (1976) “Probability of Conditionals and Conditional Probabilities”,
*Philosophical Review*, 85: 297-315; reprinted in Harper et al. (eds.) (1981)*Ifs*, Dordrecht: D. Reidel.

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