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Math Help - predicate logic formalization

  1. #1
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    predicate logic formalization

    how would i formalize the following arguments using predicate logic where px means x is a potato head and a means adam,

    there is at most 2 potato heads
    adam is the only potato head
    there are no potato heads

    thanks for any help
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  2. #2
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    there is at most 2 potato heads
    First, note that existence of anything is not claimed. If some world is completely devoid of potato heads (as a result of war or pestilence...), then this statement is true in that world. The statement claims that the existence of three or more potato heads is impossible.

    Therefore, an equivalent, but, perhaps, less intuitive, was to say this is: It is not the case that there exist three p.h.'s and all of them are different. Note the last part because the formula \exists x_1\,\exists x_2\,\exists x_3.\,p(x_1)\land p(x_2)\land p(x_3) is true in a world with just one potato head.

    For the second statement, as a hint, look at the definition of the uniqueness quantifier.
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  3. #3
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    ok thanks but how do i formalize that they are all different, would i use ¬= to say that x does not equal y?
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  4. #4
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    Yes, e.g., to say that x_1, x_2 and x_3 are all different you can say x_1\ne x_2\land x_1\ne x_3\land x_2\ne x_3, where a\ne b is a contraction for \neg(a=b).

    Note that since \ne is not a transitive relation, x_1\ne x_2\land x_2\ne x_3 does not imply x_1\ne x_3.
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  5. #5
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    thanks for the help, i always have trouble formalizing
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