# Where should one place quantifiers when translating sentences into predicate logic?

I've been trying to learn formal logic but am somewhat confused. Given the following key, how would I symbolize the sentence "Everyone who trusts Ingmar trusts a vegetarian?"

Domain: people
Vx: x is a vegetarian.
Txy: x trusts y
i: Ingmar

In the text I'm using the correct answer is shown as ∀x[Txi→∃y(Txy&Vy)], but I was wondering what would be the effect of changing the scope of ∃y so that the sentence is instead symbolized as ∀x∃y[Txi→(Txy&Vy)]? Is there a difference between these two translations? From what I can tell, the first translation means that it's true for all x members of the domain that if x trusts Ingmar, then there is some y person such that x trusts y and y is a vegetarian. Whereas, if i'm interpreting this correctly, the second translation seems to say that it's true for all x people that there is some y person such that if x trusts Ingmar, then x also trusts y who is a vegetarian.

Your rephrasings of formulas in words are correct, but in this case moving the quantifier makes no logical difference (classically). You can verify this by converting formulas into equivalent form without implications using A → B = ¬A ∨ B, and then using the fact that quantifiers can be freely moved across conjunction and disjunction as long as the variables they bound are kept distinct, see prenex normal form. Therefore,

∀x[Txi → ∃y(Txy ∧ Vy)] = ∀x[¬Txi ∨ ∃y(Txy ∧ Vy)]

``````                        = ∀x∃y[¬Txi ∨ (Txy ∧ Vy)]  = ∀x∃y[Txi → (Txy ∧ Vy)].
``````

In other words, the two "symbolizations" are logically equivalent, although in words the second one sounds more awkward. In some non-classical logics the conversion formula A → B = ¬A ∨ B is invalid, but you do not have to worry about that, I suspect.

However, note that even classically things would be different if you had a quantifier on the premise of the implication rather than on its conclusion:

∀x[∃y(Txy ∧ Vy) → Txi] is not logically equivalent to ∀x∃y[(Txy ∧ Vy) → Txi].

The negation in ¬A ∨ B now prevents one from moving ∃y out.

• Interesting difference in terminology. I'd refer to the first part of the if-then as the antecedent of the conditional. The "premise" terminology I've usually seen restricted to refer to premises of an argument (which can, of course, be seen as a conditional of sorts). – Dennis Jun 13 '17 at 15:13
• @Dennis Please do not read too much into it, I was actually thinking of using the antecedent/consequent terminology but decided that it would sound too "fancy". – Conifold Jun 15 '17 at 18:29
• Oh, I definitely wasn't reading much into it. I was more curious if there were different conventions in other parts of the world since I have (perhaps mistakenly) somehow gotten the impression that you weren't US based. – Dennis Jun 15 '17 at 21:51