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I was wondering if the following is a valid statement:

$\forall x \exists y\ (y\neq x)\ P(x,y)$, where P(x) is 'x is the parent of y' and the domain of x and y is the set of all animals.

I am specifically referring to the part '$(y\neq x)\ P(x,y)$'.

fast-and-last
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The expression isn't well-formed, though the intent is reasonably clear. The proper version is $\forall x\exists y(\lnot(x=y)\land P(x,y))$. And this sentence happens to be false.

André Nicolas
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  • Can you write $\forall x\exists y(\lnot(x=y)\rightarrow P(x,y))$? I mean, can the conditional operator be paired with an existential quantifier in the case of nested quantifiers? I ask this question because I already know that $\exists y\ Q(y)\rightarrow P(y))$ makes no sense - it is true even if Q(y) is false for every y in the domain. – fast-and-last Feb 22 '15 at 09:02
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    It does not mean the same thing. By picking $x=y$ we can make your proposed sentence true, irrespective of the existence of a parent relationship. – André Nicolas Feb 22 '15 at 09:05