existential quantifiers equivalence

Question

I am wondering if these are equivalent?

$\forall x . P(x) \rightarrow Q(x)$ and $\exists x . P(x) \vee Q(x)$

Also, I see a lot of theorems and proofs in the form of $\forall x . P(x) \rightarrow Q(x)$ but I rarely see $\exists x . P(x) \rightarrow Q(x)$? Is there some reason why? Is the existential form too weak to be useful?

Answer 1

No, they are not the same.

The first one can be written as

$\forall(x), \neg P(x) \vee Q(x)$

Suppose P(x) and Q(x) are fallacies, then this is always true (as $\neg P(x)$ is always true), whereas $\exists(x), P(x) \vee Q(x)$ is always false.

Answer 2

The two are not equivalent ...

Consider as $P(x)$ the formula $x=0$ and as $Q(x)$ the formula $x > 0$.

Clearly :

$\exists x ((x=0) \lor (x > 0))$

is satisfied in the set $\mathbb N$ of natural numbers : it is enough to choose $0$ as value for $x$.

But :

$\forall x ((x=0) \rightarrow (x > 0))$

is not : for $0$ as value for $x$, we have that $0=0$ is true while $0 > 0$ is false, and thus the conditional : $(x=0) \rightarrow (x > 0)$ is false.

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Regarding theorems of the form : $∃x.P(x) → Q(x)$ i do not think that there is some specific reason why they must be "less interesting" ...

Answer 3

Suppose we wish to say "$\color{red}{\text{All peas are in a queue.}}$"  To say "all" requires a universal quantifier.  Such a statement is true if the predicate holds for all things (peas or otherwise).  So we're claiming that anything we look at is in a queue whenever that thing is a pea.  This is a material implication.

Alternatively we can say it as: "anything is either not a pea or else it is in a queue."

$$\forall x\, \Big(P(x)\to Q(x)\Big) \\ \Updownarrow \quad \text{(by implication equivalence)} \\ \forall x\, \Big(\neg P(x) \vee Q(x)\Big) $$

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Suppose instead we only wish to claim that "$\color{red}{\text{Some pea is in a queue.}}$"   This is an existential quantifier, which is true whenever the predicate is true for at least one thing.  So we just need at least one thing to be both a pea and in a queue.  This is a conjunction.

$$\exists x\, \Big(P(x) \wedge Q(x)\Big)$$

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That's why we use implication with the universal and conjunction with the existential.