Does the sentence mean "for all x is true?

If so, and we are asked to prove that the sentence is not a tautology, does it mean that all we need to do is come up with some interpretation of what x and P(x) are that would make true and hence false?

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- Nov 15th 2013, 10:27 AMJohnDoe2013Predicate Logic: Clarification of Concepts
Does the sentence mean "for all x is true?

If so, and we are asked to prove that the sentence is not a tautology, does it mean that all we need to do is come up with some interpretation of what x and P(x) are that would make true and hence false? - Nov 15th 2013, 10:51 AMSlipEternalRe: Predicate Logic: Clarification of Concepts
The statement is false when its negation is true. What is its negation?

- Nov 15th 2013, 12:11 PMJohnDoe2013Re: Predicate Logic: Clarification of Concepts
Hi SlipEternal, the negation is that there exists some x for which P(x)->Q(x) is false.

I was trying to clarify the how the method of proof where the statement is considered a formula of a first-order language.

Since the statement has no free variables, the proof would only depend on the structure and not the valuation of x.

However, I'm not entirely clear on whether this meant that we could define any structure and interpret/define P(x) and Q(x) in any manner that would suit the proof. - Nov 15th 2013, 12:37 PMSlipEternalRe: Predicate Logic: Clarification of Concepts
A statement is logically valid if it is true in every interpretation. So, you need to show the existence of an interpretation where the statement is false.

- Nov 15th 2013, 12:54 PMemakarovRe: Predicate Logic: Clarification of Concepts
- Nov 15th 2013, 01:17 PMjohngRe: Predicate Logic: Clarification of Concepts
Hi,

"Every human is a man" is obviously false. That is "for any x, if x is a human, then x is a man" is false. There is an x such that x is a human__and__x is not a man. Clearly true for almost any x named Sue. - Nov 16th 2013, 06:46 AMHartlwRe: Predicate Logic: Clarification of Concepts
- Nov 16th 2013, 07:18 AMHartlwRe: Predicate Logic: Clarification of Concepts
If the sentence is a definition, it’s a tautology (alwaysTrue).

For ex:

for all x, (P(x)=Q(x))

is a tautology if it’s a definition, otherwise not.