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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:

$$\forall x \forall y [(Tet(y) \implies LeftOf(x,y)) \implies \forall z (Dodec(z) \implies LeftOf(x,z))]$$

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:

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enter image description here

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    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...
    – Double Knot
    Commented 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?
    – xoux
    Commented 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...
    – Double Knot
    Commented 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).
    – xoux
    Commented 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.
    – xoux
    Commented Jan 19, 2022 at 3:08

1 Answer 1

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The original sentence is an implication with "x is to the left of every tetrahedron" in the premise. The first translation does not work because it weakens the premise to "x is to the left of tetrahedron y", but keeps the conclusion "x is to the left of every dodecahedron". The ∀y placed in front then adds, "and this is true for any y". But we were entitled to infer something about x only when it was to the left of all tetrahedra, not when it is to the left of just one of them, even if that one can be any one of them. That is a much stronger implication because it infers the same conclusion from a much weaker premise.

To whittle down the clutter, the original claim was of the form ∀y [L(x,y)] ⟶ D(x), and the first translation replaced it with ∀y [L(x,y) ⟶ D(x)]. An implication holding when something is true for all y does not get us the implication holding for every y individually. If the world shall come to an end when all the stars turn red does not mean that when a single star turns red, any star, we should expect the end of days. The first translation is equivalent to the form ∃y [L(x,y)] ⟶ D(x) that infers D(x) from something about just one y instead of that for all y.

Above I abbreviated Tet(y) ⟶ LeftOf(x,y) as L(x,y), and in Tet(y) ⟶ LeftOf(x,y) both occurrences of y refer to the same tetrahedron in both translations. That is not the problem. The problem is that it sits in the premise of another implication. And moving ∀y from immediately in front of the premise to outside the implication it is the premise of illicitly replaces an inference "from all" by an inference "from any".

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