# Translation of Enligh "every x, y" and "some x, y" in first order logic

I'm having trouble understanding why a conjunction is used in the first, and a conditional in the latter case. For example we have:

Every human is a mammal: $$(\forall x)(H(x)\to M(x))$$

Some animals are mammals: $$(\exists x)(A(x)\land M(x))$$

But why can't both be used exchangeably? The argument seems to be about the truth value of the statement when there are no humans/animals in the domain to begin with, but it's not very convincible to me.

• More or less every elementary logic book ever written explains this point. Which ones have you looked at? Commented May 18, 2020 at 12:12
• @PeterSmith Of course, if you read my last paragraph, that's exactly what those explanations generally are. Specifically in "Mathematical Methods in Linguistics" that I'm following there is more than half a page explaining just that. And again, as I have mentioned, that explanation doesn't sound convincible to me. Thus the question. Commented May 18, 2020 at 16:01

Regarding "Every human is a mammal", the traditional analysis of Categorical proposition assumes that "universal terms" (the predicates $$A, H, M$$) are always instantiated. Thus, "Every human is a mammal" is False only in case that there are some human that is not a mammal, i.e. when $$\exists x(Hx \land \lnot Mx)$$ is True.
Thus, if $$\exists x(Hx \land \lnot Mx)$$ is the negation of "Every human is a mammal", we have that this one is symbolized by $$\lnot \exists x(Hx \land \lnot Mx)$$, which is equivalento to: $$\forall x(Hx \to Mx)$$.
Regarding "Some animals are mammals", if we formalize it with: $$\exists x (Ax \to Mx)$$ we have that if there are no animals, then $$Ax$$ is False for every possible value of $$x$$ and thus $$Ax \to Mx$$ is always True.
This means that $$\exists x (Ax \to Mx)$$ is True also when there are no animals, contrary to our intuition about the truth value of "Some animals are mammals".
• Thank you. I'm still a bit confused though, especially when it comes to the existential quantification, since $(\exists x)(A(x)\to M(x))$ being true when there are no animals is just is "intuitive" to me as $(\forall x)(H(x)\to M(x))$ being true when there are no humans. So if we accept the vacuous truth with the universal quantification, what stops us from doing just the same with the existential quantification? Commented May 18, 2020 at 11:09