# How to understand scope of action of a quantifier in a quantified sentence that is a translation of an English sentence?

Let's try to translate the English sentence:

"Everything to the left of every tetrahedron is also to the left of every dodecahedron"

Consider the predicates

Tet(x): x is a tetrahedron

LeftOf(x,y): x is to the left of y

Dodec(x): x is a dodecahedron

Consider the following attempt at a translation: Apparently this is not an acceptable translation.

A justification for it not being acceptable is that "we should pay close attention to the placement of the ∀y quantifier and note that ∀x (P ⟶ Q) is not equivalent to ∀x P ⟶ Q."

What is wrong with this translation? Is there an alternative explanation than the hint above?

Does the quantifer ∀y in the sentence ∀x ∀y (Tet(y) ⟶ LeftOf(x,y)) mean that "for every object y, if y is a tetrahedron, then for every object x, and for every object y, x is to the left of y"? Are the two occurrences of "for every object y" referring to different objects in this case?

By the way, the following two are apparently correct translations:  • Re your "Are the two occurrences of "for every object y" referring to different objects in this case?", no, the 2 y's are both in the same scope of ∀y(...), so they must refer to same object in your universe, otherwise there's no other rule can specify in your example... Jan 19, 2022 at 2:46
• Perhaps I need to specify my question as: why is it that the first translation is not an acceptable translation? Jan 19, 2022 at 2:47
• I think your hint ∀x(P ⟶ Q) is not equivalent to ∀xP ⟶ Q gets the main point here... This is a typical vacuous truth case for the latter while what we really mean is the former in most cases... For example, "if all shapes are Tretra then everything is to the left of a dodec" is vacuously true and meaningless in your world... Jan 19, 2022 at 3:00
• I get that those two general sentences are different. But how does that apply to the incorrectly translated sentence? To me it reads: for every object x and for every object y, if y is a tetrahedron then x is to the left of y (ie everything is to the left of every tetrahedron), then for every object z, if z is a dodecahedron then it is to the left of x (ie every object is to the left of every dodecahedron). Jan 19, 2022 at 3:05
• Something being vacuously true depends on the world (unless it is inherently vacuous). As far as I can see neither of those two sentences (∀x(P ⟶ Q), ∀xP ⟶ Q) is inherently vacuously true, so whether they are vacuously true or not depends on what world we're talking about. But it seems that the fact that there could be different worlds isn't the main point of issue here. Jan 19, 2022 at 3:08