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    Hey guys,

    Any help you could provide with this is greatly appreciated.

    F = (AX)(p(s(X)) -> p(X)) and G = (EX)p(X)

    Where A = for all and E = exists

    Use as domain the set N = {0,1,2,...} of natural numbers.

    Having trouble with the following:

    1. Describe an interpretation where F ^ ¬G is true or argue that such an interpretation does not exist.

    2. Describe an interpretation where F ^ G and F ^ ¬G is true or argue that such an interpretation does not exist.

    3. Describe an interpretation where F ^ G ^ (EX)¬p(X) is true or argue that such an interpretation does not exist.


    Cheers

    Jack
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    Well, to begin with, do you think F \land ¬G is possible or not possible?
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    I do think it's possible. p(X) would have to be false so that ¬G returns true. Consequently p(s(X)) would also have to be false to satisfy the implication if p(X) is always false?
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    (Original post by Jack0)
    I do think it's possible. p(X) would have to be false so that ¬G returns true. Consequently p(s(X)) would also have to be false to satisfy the implication if p(X) is always false?
    Yep, well reasoned! I guess that now, to finish the question, you would need to give examples of p(X) and s(X) such that both p(s(X)) and p(X) are false for all X?
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    Yeah thanx Kolya. In that case would it mean that both F ^ G and F ^ ¬G being true is impossible since you can't have a p(X) that is both true and false at the same time?
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    Yes, although I would say that you cannot have a p such that \exists x p(x) and \nexists x p(x) , rather than talking about p(x). To me, p(x) is just a statement about natural numbers, and it requires the quantifier to give it a truth value. (This isn't what I did earlier, but I've changed my mind :p: )
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    Yeah, thanx for putting that more succinctly for me. For the last question is there a difference between ¬((EX)p(X)) as in the second question and (EX)¬p(X)?
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    (Original post by Jack0)
    Yeah, thanx for putting that more succinctly for me. For the last question is there a difference between ¬((EX)p(X)) as in the second question and (EX)¬p(X)?
    Yes, there is a difference. ¬(\exists x p(x)) means that there does not exist an x such that p(x), hence it is equivalent to \forall x (¬p(x)). Now we can see how it is different from \exists x (¬p(x))
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    Ok thanx for clarifying. I'm edging towards saying that there is no interpretation for question 3 because to me (EX)p(X) ^ (EX)¬p(X) doesn't make sense (i.e. I can't think of a valuation for an interpretation where it might hold). Struggling to argue it properly though.
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    There are many statements where \exists x (p(x)) \land \exists x (¬p(x)) is true. For example, p(x) is the statement: x is even.
 
 
 
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