Universal Introduction: Why One Named Case Is Not Enough

A line about one particular object cannot establish all objects

Universal introduction requires an arbitrary name, not just a true named instance. In the cited natural-deduction system, to infer ∀xFx from Fc, c must not occur in any undischarged assumption, and the generalization must replace all the relevant occurrences without variable capture.

A premise Fc concerns the particular object named c. You cannot use that premise and immediately conclude ∀xFx. “I did not specify which object c names” does not make a premise about c arbitrary in the proof-rule sense.

Definitions and rules: [1] [2].

This article uses the rule restrictions of forall x: Calgary. Other proof presentations may express an arbitrary-object condition with eigenvariables or different notation. Follow your system’s rule statement, rather than copying the visual symbol alone.

Refute an illicit particular-premise generalization

Take the complete original interpretation D={A,B}, with a naming A and F true only of A. Fa is true; ∀xFx is false because B is not F. The inference Fa therefore ∀xFx is invalid.

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One such interpretation is enough to disprove validity. Repeating Fa, giving a a new printed label, or saying that a was randomly selected does not change the premise’s logical strength. A claim about all objects requires premises or a derivation that applies to all of them.

Universal elimination runs in the other direction: from ∀xFx, you may derive an appropriate instance Fa. It is not reversible merely because the lines look alike.

Follow a complete six-line generalization proof

Now start with two universal premises: every P is Q, and every object is P. Choose a fresh constant c that occurs in neither premise. The following original proof derives that every object is Q.

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Download the complete proof and countermodel

Lines 3 and 4 follow from premises that apply to every domain object, so they do not privilege a particular c. The conclusion Qc depends only on the two universal premises, neither of which contains c. Generalization at line 6 is therefore licensed.

Definitions and rules: [1].

Semantically, any object satisfying the universal premises is P, and every P is Q; hence any object is Q. This explains the all-object reason behind the formal proof. We are not inferring a universal statement from a sampled successful object.

Audit open assumptions and every occurrence

Before applying ∀I, list the undischarged assumptions on which the line depends. If c occurs in any of them, stop: the line may rely on a special condition about c. A constant appearing only in an already discharged subproof is not by itself forbidden by this rule.

Generalize all relevant occurrences of the arbitrary name. From Rcc, the direct replacement of c by x gives ∀xRxx. It does not give ∀xRxc while leaving a c behind. Such a partial replacement could turn a statement about self-relation into one about relation to a fixed object.

Also avoid capture. Substituting a variable into a position already governed by a quantifier for that variable can change its binding. Use a fresh variable where needed, and check the result’s scope. The arbitrary-name check and substitution check are separate requirements.

Definitions and rules: [1].

Distinguish an open condition from a discharged one

1 | Pc             Assumption
2 | Pc             Reiteration, 1
3   Pc→Pc          →I, discharge 1–2
4   ∀x(Px→Px)      ∀I, 3

This second original proof has no premises. The assumption Pc is open at line 2, so generalizing Pc at that point would be impermissible. At line 3, conditional introduction has discharged that assumption. The formula Pc→Pc no longer depends on an open assumption about c, and replacing both c occurrences licenses line 4.

This does not license a conclusion ∀xPx. What survived the discharge is a conditional that any P object is itself P, not an unconditional P claim. When auditing a proof, read both the dependency status and the exact surviving formula.

Definitions and rules: [2].

Try three arbitrary-name checks

Use the cited ∀I rule, rather than an informal “one example represents everyone” shortcut.