Questions: Universal Quantification: Meaning and Scope

Question 1

In the natural numbers, which of the following statements is true?

• A. ∀x ∃y (y = x + 1) only — for each x there is a successor, but no single y is the successor of all x
• B. ∃y ∀x (y = x + 1) only — one fixed y is the successor of every natural number
• C. Both statements are true in the natural numbers
• D. Neither statement is true in the natural numbers

Answer: (A)

∀x ∃y (y = x + 1) is true: for any x, we can always find its successor (x+1). The y depends on x and is different for each one. ∃y ∀x (y = x + 1) is false: it claims one fixed y simultaneously equals 1+1, 2+1, 3+1, and so on — a contradiction. Quantifier order is not interchangeable. When ∃ is inside ∀'s scope, the existential witness can depend on the universal variable. When ∃ is outside, it must be chosen first, independently of all x.

Question 2

A logician asserts '∀x (x > 0) is true.' Under which domain is this statement TRUE?

• A. All real numbers (ℝ)
• B. All integers (ℤ)
• C. The set {−1, 0, 1}
• D. The positive real numbers (ℝ⁺)

Answer: (D)

∀x (x > 0) is true if and only if every element of the domain is greater than 0. In ℝ, ℤ, or {−1, 0, 1}, there are elements ≤ 0 that falsify it. In the positive reals, every element is strictly greater than 0, so the claim holds. This illustrates the fundamental domain-dependence of universal statements: the same formula can be true in one structure and false in another. Evaluating a universal claim always requires specifying the domain first.

Question 3

The statement ∀x (Unicorn(x) → HasHorn(x)) is false because there are no unicorns to verify it against.

• T. True
• F. False

Answer: False

This is vacuously true, not false. ∀x (Unicorn(x) → HasHorn(x)) says: for every object x, if x is a unicorn, then x has a horn. If no unicorns exist in the domain, the antecedent Unicorn(x) is false for every x, making the conditional true for every x. The universal statement holds. An empty conjunction is true by convention, and the conjunction-interpretation of ∀ gives the same result. Vacuous truth is logically consistent and appears constantly in mathematics when quantifying over empty sets.

Question 4

The truth value of ∀x (x > 0) can differ depending on which domain of interpretation is chosen for x.

• T. True
• F. False

Answer: True

Universal statements are evaluated relative to a structure — specifically, relative to the domain that x ranges over. ∀x (x > 0) is true in the positive reals, false in all integers (since 0 and negatives are included), and false over all reals. Logical truth in first-order logic is always relative to an interpretation. This domain-sensitivity is what distinguishes model-theoretic semantics from syntactic proof: you cannot determine whether a first-order formula is true without knowing the structure it is evaluated in.

Question 5

Explain why ∀x ∃y (y > x) and ∃y ∀x (y > x) have different truth values in the natural numbers, and what this reveals about quantifier scope.

Model answer: ∀x ∃y (y > x) is true: for any natural number x, we can always find a larger one (e.g., y = x + 1). The y depends on x and can vary. ∃y ∀x (y > x) is false: it claims there exists one fixed y that is greater than every natural number simultaneously — but there is no largest natural number, so no such y exists. The order matters because when ∃ appears inside ∀'s scope, the existential witness is chosen after x is fixed, allowing dependence. When ∃ appears outside, y must be chosen first, before x is known.

Scope determines what each variable 'knows about' at the time it is bound. Inner variables can depend on outer ones; outer variables cannot depend on inner ones. This asymmetry means ∀x ∃y is generally a weaker claim than ∃y ∀x — the latter requires a single witness that works uniformly for all x, which is a much stronger demand.