Are these formal formulas equivalent?

The claim that either there is some object that is $P$ or there is some object that is $Q$, $\exists x_1 P(x_1) \lor \exists x_1 Q(x_1)$, is logically equivalent to the claim that there is some object that is either $P$ or $Q$, $\exists x_1 (P(x_1) \lor Q(x_1))$:

$\exists x_1 P(x_1) \lor \exists x_1 Q(x_1) \iff \exists x_1 (P(x_1) \lor Q(x_1))$.

Notice the same is not true for existential quantification with conjunction. For example, there exists a cat and there exists a dog, $\exists x \text{Cat}(x) \land \exists x \text{Dog}(x)$, but this does not exactly mean there exists something which is both a cat and a dog, $\exists x (\text{Cat}(x) \land \text{Dog}(x))$.

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As a caveat to the preceding image, it would be true to say that all objects are both a cat and a dog, $\forall x (\text{Cat}(x) \land \text{Dog}(x))$, if and only if all objects are a cat and all objects are a dog, $\forall x \text{Cat}(x) \land \forall x \text{Dog}(x).$


The existential quantifier distributes over disjunction. So the two sentences are indeed logically equivalent.

Tags:

Logic

Predicate Logic

First Order Logic

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