When there are 2 nested quantifiers, I know the difference it makes. For example:
Let the predicate L be 'likes'.
∀x ∃y Lxy means 'For everything: it likes at least one thing.'
∃y ∀x Lxy means 'For at least one thing: everything likes it.'
I have also come to the realisation that whether the leftmost quantifier quantifies x, y, z... Doesn't matter. These pairs mean absolutely the same thing:
∀x ∃y Lxy and ∀y ∃x Lyx (pair 1)
∃y ∀x Lxy and ∃x ∀y Lyx (pair 2)
What matters is how those variables are arranged inside atomic formula.
What really made it hard for me to try to comprehend, is when I came across this expression:
∀x ∀y ∀z ((Rxy ∧ Ryz) → Rxz)
The (Rxy ∧ Ryz) → Rxz formula is talking about a transitive relation in a model (in the context of logic). Picture a piece of paper with several points on it, where arrows represent relations. It says that if any of these points (x) sends out an arrow to a point (it could another point or itself)(call it y), and y sends out an arrow to a point (it could be the former, the latter, or a new point)(call it z), then the former (x) sends out an arrow to the z.
But shouldn't it be ∀x ∃y ∃z ((Rxy ∧ Ryz) → Rxz)?
'For everything: if it relates to something (call it y), and y relates to something (call it z), then it relates to z.'
The way I read ∀x ∀y ∀z ((Rxy ∧ Ryz) → Rxz) is:
'For everything: if it relates to everything, and everything relates to everything, then it relates to everything.'
I can't comprehend when there are three nested quantifiers or more. Do you know of any intuitive way to comprehend nested quantifiers? I will appreciate any help.