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# Formal Logic Question watch

1. Hey guys,

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
2. Well, to begin with, do you think is possible or not possible?
3. 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?
4. (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?
5. 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?
6. Yes, although I would say that you cannot have a p such that and , 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 )
7. 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)?
8. (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. means that there does not exist an x such that p(x), hence it is equivalent to . Now we can see how it is different from
9. 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.
10. There are many statements where is true. For example, p(x) is the statement: x is even.

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