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    jason_m's Avatar
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    Quote Originally Posted by Myst View Post
    Found that boring, no challenge. But I do like numbers so that part was ok




    Are you joking?! It's not just a trick question, and 7/0 != 0. It's not just a technicality. And it's not just convention.

    PS: I explained it above to Encrustacean.
    I've read your answer and researched the issue of vacuous truths now. E.g., here.

    This is what I believe:

    1) They're probably not completely arbitrary. For instance, P -> Q has to be equivalent to ~Q -> ~P. For this to be the case, when P is false and Q is false, ~Q -> ~P must be true. The only way they could be equivalent then is for P -> Q to be true when they're both false. So there is a reason for this one, and this might be the case for other vacuous truths, including the formula about the empty set above (Explaining your answer with a little more detail would be helpful to me as well...).

    2) They're part convention as well; there are other logical systems where vacuous truths are eliminated - e.g. here. In other words, if they weren't somewhat arbitrary, then we would be forced to accept them in almost any logical system that applies to the real world...

    EDIT: a better explanation than mine here. E.g.:

    Say we believe that all rubies are red, and we consider some some collection of rubies, called RR; say RR is all my rubies.

    We would like to conclude that all my rubies are red. This seems very reasonable, since all rubies are red. But with your idea, this conclusion might be false! At best we can say that all my rubies are red, if I have any rubies.

    This qualification doesn't add anything to the analysis. It doesn't illuminate any subtle point. It just complicates the discussion with an uninteresting special case.

    Since the purpose of formal logic is to model plausible reasoning as closely and as simply as possible, we agree to the convention that "all my rubies are red" is deemed to be true even when I have no rubies, so that we don't have to qualify a lot of claims with "… if there are any such rubies".
    Last edited by jason_m; 02-22-2017 at 04:58 AM.

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