Reading the universal quantifier as 'any' versus 'every'?

Question

Let $E(x,y)$ denote $x$ eats $y.$

We want to express:

Tom never eats only one thing.

So: $$∀x\;\bigg(E(\text{Tom},x) →∃z\;\Big(E(\text{Tom},z) \,∧\, z ≠ x\Big)\bigg).$$

If I translate it as "If Tom eats ANYTHING, then Tom eats something else too", then this sentence makes sense to me.

But if I translate it as "If Tom eats EVERYTHING, then Tom eats something else too", then this sentence differs in meaning.

I'm confused by how to make the distinction between reading the universal quantifier as "anything" versus as "everything".

Answer 1

I think of interpretation as a two-stage process. First, translate the symbols into "mathematical language" without referencing the quantified terms, and then coax it into natural language. For instance, $$\forall x\ E(T,x)$$ is "for everything, Tom eats it", which I revise to "Tom eats everything." But if it's something where $x$ is referenced in both sides of an implication, you want to treat it differently. For instance, your example $$\forall x(E(T,x)\to\exists z(E(T,z)\wedge(x\neq z))$$ would start off as "for everything, if Tom eats it, then there is something else that he eats that is different" which is more naturally translated as "For anything Tom eats, there is something different that he also eats." An example of where you might apply "everything" in an implication is "If Tom eats everything, he gets sick", which is $$(\forall x\ E(T,x))\to S(T)$$ This would be a good use of "everything", because $x$ is only quantified in the scope of the hypothesis.

Answer 2

Tom never eats only one thing.

So: $$∀x\;\bigg(E(\text{Tom},x) →∃z\;\Big(E(\text{Tom},z) \,∧\,z ≠ x\Big)\bigg).\tag1$$

This formalisation is correct, and can be read, "Everything is such that if Tom eats it then Tom eats something else too."

If I translate it as "If Tom eats ANYTHING, then Tom eats something else too", then this sentence makes sense to me.

This translation literally means $$\forall y\;\bigg(\exists x\;\Big(E(\text{Tom},x)\,∧\, y=x\Big) →∃x\;\Big(E(\text{Tom},x) \,∧\, x ≠ y\Big)\bigg),\tag2$$ which is logically equivalent to formalisation $(1)$, so it is also correct. Notice that

• formalisation $(1)$ does not contain a quantified antecedent, whereas formalisation $(2)$ does;
• here, "if anything" means "if something" instead of "if everything".

But if I translate it as "If Tom eats EVERYTHING, then Tom eats something else too", then this sentence differs in meaning.

This translation is wrong, and literally means $$\Big(\forall x\;E(\text{Tom},x) \Big)→∃x\;\Big(E(\text{Tom},x) \,∧\, \forall y\;x ≠ y\Big).$$

I would really like to know how the universal quantifier can be read out as both "Anything" and "Everything".

∀ always means everything, while ∃ always means something.

The word "anything" sometimes corresponds to ∃ instead of ∀: besides the above, here are more examples.