Different uses of the keyword 'any' while writing predicate logic?

Question

I came across two statements which highlight the my problem with the keyword 'any' in mathematical english.

The statement "Anyone who eats any pumpkin is a nutrition fanatic." translates to

$\forall x ((\exists y (PUMPKIN(y) \land EAT(x,y))) \rightarrow FANATIC(x))$

another statement "Anyone who buys any pumpkin either carves it or eats it." translates to

$\forall x \forall y (PUMPKIN(y) \land BUY(x,y) \rightarrow CARVE(x,y) \vee EAT(x,y))$

The first one uses existential quantifier for 'any' while the latter uses a universal quantifier.

If someone could guide me how the use of existential and universal quantifiers work here.

Answer 1

Actually, the change of quantifiers is because in the first sentence the quantifier is only part of the antecedent of the conditional, while in the second statement that quantifier quantifies over the whole conditional.

And, as it turns out, any statement

$\forall x (P(x) \rightarrow Q)$

is equivalent to the statement

$\exists x P(x) \rightarrow Q$

as long as Q does not contain any free variable x.

So in this case, since $Fanatic(x)$ does not contain a free variable $y$, we can bring the existential quantifier 'outside' the conditional, where it becomes a universal quantifier, and rewrite the first sentence as:

$\forall x \forall y ((Pumpkin(y) \land Eat(x,y)) \rightarrow Fanatic(y))$

And now you see that the first statement is really not inherently different from the second in terms of its quantifiers. Indeed, the difference is not at all due to any ambiguity of the word 'any'!