Exercise 3: Why "Everyone Likes Someone" Differs From "Someone Everyone Likes" — Possible Solution ==================================================================== WHAT EACH STATEMENT ACTUALLY CLAIMS ------------------------------ ∀x ∃y Likes(x, y) says: pick any person x you like, and there is SOME person y (possibly different for each x you pick) that x likes. Each person is allowed their own, personal answer to "who do I like." ∃y ∀x Likes(x, y) says: there is ONE SPECIFIC person y, fixed in advance, such that every single person x likes that exact same y. All of the "liking" has to converge on one shared target. WHY THE ORDER OF THE QUANTIFIERS IS WHAT CREATES THIS DIFFERENCE ------------------------------ In the first statement, the ∃y comes after ∀x, meaning y is chosen separately for each x - it's allowed to vary from person to person. In the second, ∃y comes first, meaning y is chosen once, before x is even considered, and that same fixed y then has to work for every x. This is exactly the "everyone has a mother" vs. "one person is everyone's mother" example from this chapter, with "likes" standing in for "is the mother of." A CONCRETE GROUP: A LIKES B, B LIKES C, C LIKES A ------------------------------ Consider three people, A, B, and C, where A likes B, B likes C, and C likes A (a cycle, with no other likes). Checking ∀x ∃y Likes(x, y): does A like someone? Yes, B. Does B like someone? Yes, C. Does C like someone? Yes, A. Every person has someone they like, so this statement is TRUE. Checking ∃y ∀x Likes(x, y): is there one person liked by everyone? Check B: A likes B (yes), but does C like B? No, C likes A. So B isn't liked by everyone. Check C: B likes C (yes), but does A like C? No, A likes B. So C isn't liked by everyone either. Check A: C likes A (yes), but does B like A? No, B likes C. So A isn't liked by everyone either. No single person is liked by all three, so this statement is FALSE. WHY THIS EXAMPLE CLEANLY SEPARATES THE TWO STATEMENTS ------------------------------ The cyclic structure guarantees every person has exactly one person they like (satisfying the first statement easily), while simultaneously guaranteeing no person is liked by more than one other person (making the second statement fail completely) - a clean, minimal example where the gap between the two statements is as sharp as possible. WHY THIS WORKS AS AN ANSWER ------------------------------ It explains precisely what each statement claims and why the quantifier order is the source of the difference, then walks through a concrete three-person example, checking each statement explicitly against it, showing the first holds while the second fails.