Existential quantifier equivalent of a universal quantifier statement.

Question

My question is this, for the following sentence: Every pet has an owner , when translated into predicate logic gives $\forall x\enspace(P(x) \rightarrow O(x))$, but what is the equivalent with a $\exists x$ instead of $\forall x$? Is there a general rule when changing between the two?

Answer 1

The negation of a universal statement is an existential statement: $\lnot \forall x A \equiv \exists x \neg A$.

Thus, in your case this would be: $$ \forall x ( P(x)\to O(x))\equiv \lnot (\exists x\lnot (P(x)\to O(x))) \equiv \lnot (\exists x(P(x)\land \neg O(x)))\,. $$

Answer 2

Hint: Negate the negation of $\forall x . P(x) \Longrightarrow O(x)$, where:

$P(x)$ means x is a pet;

$O(x)$ means x has an owner.