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Math Help - Motivation for definition of conditional?

  1. #1
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    Motivation for definition of conditional?

    I understand why we define p-->q to be true if p is T and q is T.
    I understand why we define p-->q to be false if p is T and q is F.
    Since we can't say whether the conditional is true or false if the antecedent is not true, why is it defined to be true when p is true?
    Is it just so that the definition of the biconditional comes out the way we want it?
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    Re: Motivation for definition of conditional?

    Quote Originally Posted by lamp23 View Post
    I understand why we define p-->q to be true if p is T and q is T. I understand why we define p-->q to be false if p is T and q is F. Since we can't say whether the conditional is true or false if the antecedent is not true, why is it defined to be true when p is true?
    Is it just so that the definition of the biconditional comes out the way we want it?
    Consider the statement,
    "If it rains hard then the grass will be wet".
    Many an early riser has found the grass is very wet but knows absolutely that there has been no rain over night. Does that negate that statement? Absolutely not, there was just a very heavy dew that night. It simply means that there are other ways to wet the grass. It could have been a sprinkler system.
    The only way that conditional can be false is for T to imply F. In all of the other three cases the implication is true.
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    Re: Motivation for definition of conditional?

    Quote Originally Posted by Plato View Post
    Consider the statement,
    "If it rains hard then the grass will be wet".
    Many an early riser has found the grass is very wet but knows absolutely that there has been no rain over night. Does that negate that statement? Absolutely not, there was just a very heavy dew that night. It simply means that there are other ways to wet the grass. It could have been a sprinkler system.
    The only way that conditional can be false is for T to imply F. In all of the other three cases the implication is true.
    I agree that we cannot say the statement is necessarily false but we can't necessarily say it's true either.
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    Re: Motivation for definition of conditional?

    So, you don't agree that if it rains hard, then the grass will be wet?
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    Re: Motivation for definition of conditional?

    Quote Originally Posted by lamp23 View Post
    I understand why we define p-->q to be true if p is T and q is T.
    I understand why we define p-->q to be false if p is T and q is F.
    Since we can't say whether the conditional is true or false if the antecedent is not true, why is it defined to be true when p is true?
    Since you understand why T -> T = T, this leaves the question why F -> T = T. See the discussions in these two threads; they are mostly concerned with why implication is true when the premise is false.
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    Re: Motivation for definition of conditional?

    I checked out those two threads but feel it doesn't answer my question.
    If all we are given is p is F and q is T, I see why we can't necessarily say the conditional is false. Here a lot of people make the jump "then it must be true" but we also can't necessarily say the conditional is true.
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