Questions: Open and Closed Formulas in First-Order Logic

Question 1

Which of the following is a closed formula (sentence) in first-order logic?

• A. P(x) ∧ Q(y)
• B. ∃y (y · y = x)
• C. ∀x ∃y (x + y = 0)
• D. ∀x P(x) → Q(z)

Answer: (C)

Option C is the only sentence: both x and y are bound by their respective quantifiers (∀x and ∃y), so there are no free variables. Option A has x and y free. Option B has x free (y is bound by ∃y but x has no quantifier). Option D has z free (even though x is bound by ∀x). A closed formula requires every variable occurrence to be within the scope of a quantifier that binds it.

Question 2

A logician writes the formula ∃y (y · y = x) and asks whether it is true or false. What information is needed to answer?

• A. Only the domain (the structure), since the existential quantifier handles y
• B. Both the domain (structure) and a specific assignment of a value to the free variable x
• C. Nothing — the formula is neither true nor false because it is open
• D. Only the value of y, since it appears in the predicate

Answer: (B)

This is an open formula: y is bound by ∃y, but x is free. The truth value depends on both the structure (which determines what values are in the domain) and the variable assignment (which assigns a specific value to x). For example, in the natural numbers, the formula is true when x is assigned a perfect square and false otherwise. Open formulas are not meaningless — they have truth values relative to a structure and assignment, not on their own.

Question 3

The truth value of a closed formula (sentence) in a given structure is determined solely by the structure itself, without reference to any variable assignment.

• T. True
• F. False

Answer: True

A sentence has no free variables — all variables are bound by quantifiers, which internally specify their range of values. The quantifiers handle the 'what does x refer to?' question internally, so no external assignment is needed. This is precisely why sentences are the natural objects of logical theorems, axioms, and model theory: '∀x (x > 0 → x² > 0)' is either true or false in the natural numbers, period, with no additional information needed.

Question 4

A free variable in a formula means the formula has no truth value and is logically meaningless until most variables are bound by quantifiers.

• T. True
• F. False

Answer: False

Free variables are not undefined — they are parameters awaiting an assignment. Given a structure and a variable assignment (a function mapping free variables to domain elements), any formula, open or closed, has a determinate truth value. The open formula P(x) is meaningful: it is true of exactly the elements x in the domain that satisfy P. In fact, open formulas are central to predicate logic — they define predicates, and substituting terms for free variables is how universal instantiation works.

Question 5

Explain the difference between a sentence (closed formula) and an open formula in first-order logic, and explain why sentences are the natural objects of logical axioms rather than open formulas.

Model answer: A sentence (closed formula) has no free variables — every variable is bound by a quantifier. Its truth value is determined entirely by the structure, with no variable assignment needed. An open formula has at least one free variable; its truth value depends on both the structure and an assignment of values to those free variables. Axioms are sentences because an axiom must make a definite claim that is true or false in a model — not a conditional claim that depends on what values free variables happen to have. If an axiom contained free variables, its truth would be assignment-dependent, which would undermine its role as a fixed foundational statement.

This distinction is fundamental to model theory. When we say 'the Peano axioms are true in the natural numbers,' each axiom is a sentence that can be evaluated as true or false without ambiguity. An open formula like 'x > 0' cannot serve this role — it is true for some elements of the domain and false for others. The shift from open to closed formulas is what allows logic to make structural claims rather than merely describing conditions on individual elements.