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Incoming links: Eigenvariable condition
Eigenvariable 2026-10-31
A variable used as an arbitrary parameter in a natural deduction subproof, subject to an eigenvariable condition. For universal introduction, it must be free in no undischarged assumption. For existential elimination, a temporary witness must also be free in no conclusion exported from its subproof. These restrictions distinguish an arbitrary object from one constrained by a special hypothesis.
Eigenvariable condition 2026-10-25
An eigenvariable condition ensures that a variable chosen as arbitrary or as an existential witness cannot smuggle extra information into a quantifier inference. For the universal quantifier introduction rule, the generalized variable must not occur free in any undischarged assumption. For the existential quantifier elimination rule, the fresh witness must not occur free in the resulting conclusion or in the other undischarged assumptions. Temporary assumptions may contain it if they are discharged before the final inference.
Existential quantifier elimination rule Created 2026-10-25 Updated 2026-10-31
From , reason under a temporary assumption for a fresh witness . If this yields a conclusion not containing free, close the witness subproof and infer . The eigenvariable condition also requires that not be free in other undischarged assumptions. The rule does not authorize selecting one fixed named object as a witness or allowing the witness to escape its scope.
Existential witness 2026-10-31
An object, or a term denoting an object, that satisfies the predicate asserted by an existential quantification. In a natural deduction use of the existential quantifier elimination rule, a fresh eigenvariable represents an otherwise unspecified witness inside a subproof. The eigenvariable condition prevents it from occurring freely in the exported conclusion or the other undischarged assumptions; violating this restriction is witness escape.
The original scan repeats the premise and conclusion of part (i) exactly; it is not a missing or unreadable alternative exercise.[ref]
Digital systems, linear algebra, introduction to formal proof examination paper by the University of Oxford (2013). Printed p. 8, Question 8(a)(i) and (iii)
Here is a second formal ordering of the same argument. First introduce the fresh existential witness, derive the desired negation inside its scope, and then eliminate the witness:Thus, again,At the last existential elimination, occurs in neither the exported conclusion nor the remaining premise. The eigenvariable condition is therefore satisfied. This proof retains a different nesting of the subproofs from part (i).
Choose an arbitrary fresh eigenvariable . If held, existential quantifier introduction rule would contradict the premise. Derive by double-negation elimination, then generalize:
Table 1.
An arbitrary object satisfies the relation
. | Line | Open assumptions | Formula | Justification |
|---|---|---|---|
| 1 | 1 | Premise | |
| 2 | 1, 2 | Temporary assumption | |
| 3 | 1, 2 | existential quantifier introduction rule 2; term | |
| 4 | 1, 2 | negation elimination rule 3, 1 | |
| 5 | 1 | negation introduction 2, 4 | |
| 6 | 1 | double-negation elimination 5 | |
| 7 | 1 | universal introduction 6; fresh |
HenceAt the final universal introduction, is free in no open assumption: the only remaining premise is the closed first-order formula . Thus the eigenvariable condition is satisfied. The passage from double negation to the positive relation uses the paper's classical rule.
Assume that every object fails . A fresh existential witness for the premise then has both and its negation. Eliminate the witness before closing the assumption about all objects:
The conclusion isThe witness occurs neither in the conclusion of existential elimination nor in its other open assumptions, so the eigenvariable condition holds.
Introduce a fresh existential witness with . The universal premise gives . In the first case, equality elimination rule gives . In the second, it gives , contradicting ; explosion then supplies the common case conclusion . Close both cases and then the witness scope:
| Line | Open assumptions | Formula | Justification |
|---|---|---|---|
| 1 | 1 | Premise | |
| 2 | 1, 2 | Premise | |
| 3 | 1, 2, 3 | Premise | |
| 4 | 1, 2, 3, 4 | Temporary assumption | |
| 5 | 1, 2, 3, 4 | universal elimination 1; term | |
| 6 | 1, 2, 3, 4, 6 | Temporary assumption | |
| 7 | 1, 2, 3, 4, 6 | equality elimination rule 6, 4 | |
| 8 | 1, 2, 3, 4, 8 | Temporary assumption | |
| 9 | 1, 2, 3, 4, 8 | equality elimination rule 8, 4 | |
| 10 | 1, 2, 3, 4, 8 | negation elimination rule 9, 3 | |
| 11 | 1, 2, 3, 4, 8 | explosion 10 | |
| 12 | 1, 2, 3, 4 | disjunction elimination 5, 6, 7, 8, 11 | |
| 13 | 1, 2, 3 | existential elimination 2, 4, 12; fresh |
HenceFor both uses of equality elimination rule, the formula template is . At the final existential elimination, is fresh in all three original premises and absent from , so the eigenvariable condition holds. No additional assumption that the constants and denote distinct objects is needed.
A variable is fresh for the relevant context when it occurs free in none of its undischarged assumptions. For the existential quantifier elimination rule, it must also not occur free in the final conclusion or the existential premise: it is a new eigenvariable, allowed in the temporary witness assumption and formulas inside that witness subproof. Bound variable names can always be renamed to avoid collisions.
The premise guarantees some witness, without saying which object it is. Temporarily call one such witness and assume . If the subproof yields without depending on any special identity or extra property of , its conclusion holds whichever witness exists. The existential quantifier elimination rule therefore discharges and retains only the original existential premise and other open assumptions.This eigenvariable condition prevents inferring for a previously chosen object merely from existence. In particular, does not license outside the discharged witness subproof.
Choose an eigenvariable fresh for the premise. Use the existential quantifier introduction rule to contradict the premise under a temporary assumption :At line 6 the assumption has been discharged and the sole remaining premise has no free , satisfying the eigenvariable condition. Thus
A variable is fresh for the surrounding context when it does not occur free in any undischarged assumption in . Choose it also distinct from bound variables that would cause variable capture in the intended substitution. The universal quantifier introduction rule isIts justification is that the logical derivation of used no special information about , so it establishes the property of an arbitrary object. Semantically, changing the assignment of preserves the truth of ; therefore holds for every possible assigned object. Without the eigenvariable condition, the assumption could incorrectly be generalized to in an interpretation where only one of two objects satisfies .
The dot after gives it scope over the following implication: read the premise as . Choose a fresh arbitrary eigenvariable . A natural deduction proof isAt line 6, is free in no undischarged assumption, so the universal quantifier introduction rule is valid.
The proposed variant is not derivable in a sound first-order logic. For a countermodel, use the two-element domain , interpret both predicates as , and assign the free variable the value 0. Then is true, as is under the assignment, but is false at element 1. The premise's bound and the conclusion's free have different scopes.Attempting to generalize from in the variant would violate the eigenvariable condition, because remains free in the undischarged assumption .
Apply the negation introduction rule to an assumed existential statement and the existential quantifier elimination rule with a fresh witness :The eigenvariable condition holds: is absent from the surrounding premise, the existential assumption and the exported conclusion . Its local assumption is discharged before the final negation introduction rule.
Let and abbreviateAssume and eliminate the existential quantifier using a fresh eigenvariable . Inside that subproof, specialize at . Write . The premise supplies , so a disjunction elimination rule finishes the contradiction:Line 7 abbreviates two consecutive universal quantifier elimination rule applications; no new rule is used. The fresh witness is absent from , the assumed existential statement and the exported , so the existential quantifier elimination rule satisfies its eigenvariable condition. This is the self-application behind the barber paradox, written entirely with the supplied rules.
Choose fresh arbitrary eigenvariables . Under , use the equality elimination rule to transport to . For the reverse direction, first derive by the symmetry derivation above. In full natural deduction form:All temporary assumptions are discharged before either universal quantifier introduction rule application, so both eigenvariable conditions hold. No premises remain:
Choose a fresh parameter . The universal quantifier elimination rule supplies both the implication and its premise at that same parameter:The eigenvariable condition holds because is absent from both undischarged assumptions at line 6. Hence the universally quantified implication is proved.
1 ∀x. (P(x) → Q(x)) premise
2 | ∀x. P(x) assumption
3 | P(a) ∀-elim 2
4 | P(a) → Q(a) ∀-elim 1
5 | Q(a) →-elim 4,3
6 | ∀x. Q(x) ∀-intro 5, a fresh
7 (∀x. P(x)) → (∀x. Q(x)) →-intro 2–6The proposed version with merely in the antecedent is not derivable. For a counterexample, take a two-element domain with and both true precisely at zero. The universal premise holds. Under the assignment , is true but is false. By soundness, a formal proof cannot exist. Attempting the previous derivation would fail exactly at universal quantifier introduction rule: the parameter would occur in the undischarged assumption , so it would not be fresh.
Use negation introduction on the unwanted existential quantification, and existential quantifier elimination rule with a fresh witness :The witness name occurs neither in the surrounding assumptions nor in the conclusion of its subproof. This satisfies the eigenvariable condition; it is not an arbitrarily chosen fixed element assumed to witness the existential quantification. The result is .
1 ∀x. ¬P(x) premise
2 | ∃x. P(x) assumption
3 | | P(a) assumption, a fresh
4 | | ¬P(a) ∀-elim 1
5 | | ⊥ ¬-elim 3,4
6 | ⊥ ∃-elim 2,3–5
7 ¬(∃x. P(x)) ¬-intro 2–6Choose distinct fresh parameters . Within the assumption , equality elimination rule gives the forward implication directly. Derive the reverse equality explicitly before using it for the reverse implication:The equality assumption is discharged before either universal quantifier introduction rule. There are then no undischarged assumptions, so both eigenvariable condition requirements hold. The final universal theorem expresses that equal arguments are interchangeable in the predicate in both directions.
1 | a = b assumption
2 | | P(a) assumption
3 | | P(b) =-elim 1,2
4 | P(a) → P(b) →-intro 2–3
5 | a = a =-intro
6 | b = a =-elim 1,5; φ(u) := (u = a)
7 | | P(b) assumption
8 | | P(a) =-elim 6,7
9 | P(b) → P(a) →-intro 7–8
10 | (P(a) → P(b)) ∧ (P(b) → P(a)) ∧-intro 4,9
11 a = b → ((P(a) → P(b)) ∧ (P(b) → P(a))) →-intro 1–10
12 ∀y. a = y → ((P(a) → P(y)) ∧ (P(y) → P(a))) ∀-intro 11, b fresh
13 ∀x. ∀y. x = y → ((P(x) → P(y)) ∧ (P(y) → P(x))) ∀-intro 12, a freshOn a finite domain , says that at least one of holds. A derivation from a fresh generic witness shows how to obtain the same conclusion whichever witness works, just as disjunction elimination handles all cases. If empty domains were allowed, the existential premise would be false there; ordinary first-order semantics normally assumes a nonempty domain.
Writing the other open assumptions as , the existential elimination rule in sequent form isCombining this with a derivation of the existential premise gives the displayed natural-deduction version. The eigenvariable condition requires to occur free in neither nor , with capture-free substitution in . It is a temporary name for an unspecified witness, not information about a chosen object. Without freshness, the invalid inference would identify an arbitrary named object with that witness. The witness escape restriction prevents this error.
The two proofs differ in whether existential elimination is outside or inside negation introduction. For the first proof, choose fresh , derive the negation inside the witness subproof, and then discharge the witness:The conclusion at line 6 contains no free . For the second proof, open the universal assumption first, eliminate the existential to derive falsity, and then discharge the universal assumption:At line 6 of the second proof, neither the exported conclusion nor the other open assumptions contains free, satisfying the eigenvariable condition. Neither proof needs RAA or double-negation elimination.