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Math Help - Sentential Logic Problem

  1. #1
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    Sentential Logic Problem

    Hi there,
    I would be extremely appreciative to anyone that could help/explain this reasoning a little to me.

    Q. Let P stand for the statement "I will buy the pants" and S for the statement "I will buy the shirt." What English sentencecs are represented by the following expression

    \neg(\mbox{P}\wedge\neg\mbox{S})

    correct answer from book [How to Prove It, 2006, Velleman] is...
    I won't buy the pants without the shirt.

    Now, to me, this doesn't make any sense.

    I would interpret the part inside the parenthesis to be:
    I will buy the pants but not the shirt.

    Now, outside, it's the negation of that entire statement, which requires a double negative on the shirt part, and from an english viewpoint doesn't make much sense to me.

    Only other way I can interpret this myself is:
    I will not buy both the pants and not the shirt.


    Thanks for any help.
    Last edited by lannett; October 7th 2009 at 01:28 AM.
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  2. #2
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    Hello lannett

    Welcome to Math Help Forum!
    Quote Originally Posted by lannett View Post
    Hi there,
    I would be extremely appreciative to anyone that could help/explain this reasoning a little to me.

    Q. Let P stand for the statement "I will buy the pants" and S for the statement "I will buy the shirt." What English sentencecs are represented by the following expression

    \neg(\mbox{P}\wedge\neg\mbox{S})

    correct answer from book [How to Prove It, 2006, Velleman] is...
    I won't buy the pants without the shirt.

    Now, to me, this doesn't make any sense.

    I would interpret the part inside the parenthesis to be:
    I will buy the pants but not the shirt.

    Now, outside, it's the negation of that entire statement, which requires a double negative on the shirt part, and from an english viewpoint doesn't make much sense to me.

    Only other way I can interpret this myself is:
    I will not buy both the pants and not the shirt.


    Thanks for any help.
    I'm afraid I agree with the answer in the book, but you need to juggle things about a bit first.

    The proposition 'I won't buy the pants without the shirt' can be re-written 'If I do not buy the shirt, then I won't buy the pants'. This is symbolised as:

    \neg S \Rightarrow \neg P

    which can be re-written without an implication sign as

     \neg(\neg S) \lor \neg P (draw up a truth table if you haven't seen this before)

    which, by De Morgan's Law, is logically equivalent to

    \neg(P \land \neg S)

    Grandad
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