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"Forall"'s friend "if"

16 June, 2015 - 15:45
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We have already seen quite a few formulas of the general form ∀x :(P (x) ⇒ ... ). Indeed, this is a very useful idiom: If our domain is natural numbers but we want to say something about all primes, we simply write ∀n : (prime (n) ⇒ ... ). Don't be fooled; this formula is in no way suggesting that all numbers are prime!

Caution: This same construct using ∃ is usually a mistake. Consider ∃x :(P (x) ⇒ ... ). By choosing x to be any non-P element, this entire formula is true, without even glancing at what is inside the "..."l
Note: If you have to demonstrate that all ravens are black, ∀i : (isRaven (i) ⇒ isBlack (i)), there are two ways to do so: You can go out and find every raven and verify that it's black. Alternately, you can go and find every non-black item, and verify that it's a non-raven. Epistemologists, philoso phers dealing with how we humans come to learn and know things (about, say, raven colors), go on to ponder about real-world degrees-of-belief: If we have only looked at some ravens, and we find another raven and confirm it is black, does this increase our degree of belief about all ravens being black? If so, then whenever we find a non-black item which is a non-raven, this must also increase our degree of belief that all ravens are black. This leads to Hempel's (so-called) Paradox: if we are looking for evidence to choose between two competing hypotheses (say, "all non-black items are not ravens" versus "all non-orange items are not ravens"), then finding a purple cow increases our belief in both of these hypotheses, simultaneously!