Sunday, November 28, 2010

Is this a typo?

I'm reading through Probability: A Philosophical Introduction.  I have read this page about 10 times and have concluded there is either a typo in the book or a hole in my head. 

If A=>B, then if P(B)=1, P(A)=1

So if A entails B, then if the probability of B is 1, the probability of A is 1....huh?  Isn't that affirming the consequent?

A - John finished the race first place
B - John finished the race

John finished the race first place entails that John finished the race.

The probability (epistemic) that John finished the race is 1...we watched.  But suppose several horses crossed the line at the exact same time (as far as we can tell from the nose bleed section), so we have no evidence about first place other than the above entailment and the proposition "Either horse 1, 2, or 3 won first place."  So how in the world do we know that the probability of John finishing first place is 1?  Shouldn't our credence be .33 that John won first place while we wait for the announcement?  Actually... A=>B doesn't seem to be relevant at all.

Confirmed as a typo by Dr. Mellor himself.  He informed me that the fallacy of affirming the consequent only applies to material implication and not entailment (logical implication).  The reason isn't quite clear to me, but there are more important things to tackle.  Suffice it to say that A->B is not exactly the same as A=>B.


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