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Math Help - Simple logic truth table question

  1. #1
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    Simple logic truth table question

    Please help with this question, my English is not that good:

    Here is a truth table for the new symbol *


    P * Q
    T F T
    T T F
    F T T
    F T F


    Given this truth-table, what ordinary English expression can this new truth-functional connective (*) be used to symbolize?




    One other broad question:

    Suppose there are two sentences a and b. On every interpretation that a is true, b is false. What can you conclude (if anything) about the following argument? Briefly explain.

    ~(a -> b)
    .:. ~b v a

    -> :material conditional
    v :disjuction/or

    Many thanks for help
    Last edited by apterosss; April 12th 2013 at 11:24 AM.
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  2. #2
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    Re: Simple logic truth table question

    Quote Originally Posted by apterosss View Post
    Please help with this question, my English is not that good:
    Here is a truth table for the new symbol *
    P Q P * Q
    T T F
    T F T
    F T T
    F F T

    Given this truth-table, what ordinary English expression can this new truth-functional connective (*) be used to symbolize?

    I have most often have seen this as \begin{array}{*{20}{c}} P&Q&\;&{P|Q} \\ \hline  T&T&\;&F \\  T&F&\;&T \\   F&T&\;&T \\   F&F&\;&T \end{array}.

    WVO Quine called this the alternative denial operator, first discovered by CS Peirce.
    These are equivalent:
    \begin{array}{*{20}{c}}  {\neg P}&{\mathop  = \limits^{df} }&{P|P} \\   {P \wedge Q}&{\mathop  = \limits^{df} }&{(P|Q)|(P|Q)} \\   {P \vee Q}&{\mathop  = \limits^{df} }&{(P|P)|(Q|Q)} \\   {P \to Q}&{\mathop  = \limits^{df} }&{P|(Q|Q)} \end{array}
    Last edited by Plato; April 12th 2013 at 01:24 PM.
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    Re: Simple logic truth table question

    Plato - apologies, something happened with my spacing.
    Yes your table is equivalent to what I tried to put; please see edit.

    What would the English phrase be for this? Is there a sentence this could be symbolised with?

    Thanks
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    Re: Simple logic truth table question

    Quote Originally Posted by apterosss View Post
    Given this truth-table, what ordinary English expression can this new truth-functional connective (*) be used to symbolize?
    A * B can be read, "A and B cannot both be true" or "A contradicts B" (note that this is the same as "B contradicts A", i.e., * is symmetric).

    Quote Originally Posted by apterosss View Post
    Suppose there are two sentences a and b. On every interpretation that a is true, b is false. What can you conclude (if anything) about the following argument? Briefly explain.

    ~(a -> b)
    .:. ~b v a
    This argument is valid without assuming anything about truth of a and b in interpretations.
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  5. #5
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    Re: Simple logic truth table question

    Quote Originally Posted by apterosss View Post
    What would the English phrase be for this? Is there a sentence this could be symbolised with?

    Actually there are probably many names for this.

    As I said, WVO Quine called this the alternative denial operatorr, first discovered by CS Peirce.
    I don't remember is Perice had a name for it.
    It also goes by the name stroke function,

    The chart I posted shows that the other functions can be defined using this operator.

    It in effect: "It is false if both statements are true and true otherwise".
    Last edited by Plato; April 12th 2013 at 01:22 PM.
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    Re: Simple logic truth table question

    Hello, apterosss!


    Here is a truth table for the new symbol \star

    . . \begin{array}{ccc} P & \star & Q \\ \hline T & F & T \\ T & T & F \\ F & T & T \\ F & T & F \end{array}

    Given this truth-table, what ordinary English expression
    can this new operation (\star) be used to symbolize?

    \text{We have: }\:\sim(P \wedge Q) \:\Longleftrightarrow\: \sim\!P\:\vee \sim\!Q

    . . "P\text{ or }Q\text{ is false."}
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    Re: Simple logic truth table question

    Thank you all for your help
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    Re: Simple logic truth table question

    Sorry I mistyped this one - still don't understand the answer.

    One other broad question:

    Suppose there are two sentences a and b. On every interpretation that a is true, b is false. What can you conclude (if anything) about the following argument? Briefly explain.

    ~(a -> c)
    .:. ~b v c

    -> :material conditional
    v :disjuction/or
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  9. #9
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    Re: Simple logic truth table question

    Quote Originally Posted by apterosss View Post
    Suppose there are two sentences a and b. On every interpretation that a is true, b is false. What can you conclude (if anything) about the following argument?

    ~(a -> c)
    .:. ~b v c
    This argument is valid in this sense. The formula ~(a -> c) is equivalent to a /\ ~c. Then use the assumption.
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