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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 conjunction introduction rule combines derivations of and into a derivation of . Both conclusions must be available under the current assumptions.
The disjunction elimination rule is reasoning by cases: from , a subproof deriving the same conclusion under , and another subproof deriving under , conclude . It discharges the two case assumptions, not any surrounding premises.
The implication introduction rule derives by temporarily assuming and deriving ; it then discharges that occurrence of the assumption .
ThusA conclusion inside a box cannot simply be reused outside it without an appropriate inference rule. Any other undischarged assumptions continue to constrain the result. These are exactly the rules printed in Figure 1.[ref]
In proof by contradiction, assume the negation of a desired conclusion , derive logical falsity, and discharge the assumption to infer . A rule expressing this isHere the label identifies the assumption being discharged; any other open assumptions remain.
To derive this rule using the supplied rules, apply negation introduction to the subproof from to . This discharges and yields . Apply double-negation elimination to obtain . Thus the extra rule is admissible because Figure 1 explicitly includes double-negation elimination.[ref]
This is the classical form of proof by contradiction; simply deriving from an assumption leading to logical falsity uses negation introduction alone.
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).
A common-context sequent version of the disjunction elimination rule isThe second and third premises record exactly the two boxed subproofs of the natural deduction presentation. Their additional case assumptions do not occur in the conclusion: that is assumption discharge. The surrounding context remains throughout. A formulation with different surrounding contexts is equivalent after retaining their union by weakening.
In natural deduction, conjunction introduction combines derivations of and under the available assumptions into . Disjunction elimination takes and two subproofs of the same conclusion , one assuming and the other assuming , and concludes . Implication introduction takes a subproof deriving from a temporary assumption , and concludes .
The rectangular boxes mark the scopes of temporary assumptions. The disjunction elimination rule discharges its two case assumptions, while the implication introduction rule discharges the antecedent assumption. Other surrounding assumptions remain in force; intermediate results depending on the boxed assumptions cannot be exported arbitrarily. Schematically,Here the square brackets designate the assumptions discharged by the corresponding inference rule.
Use conjunction elimination to extract the three inputs, then apply the implication elimination rule three times:The implication introduction rule closes the only subproof, giving
Read chained logical implications as right-associative. Use the following natural deduction derivation; indentation bars mark nested subproofs and the last column records rules and discharged line ranges:Each application of the implication introduction rule discharges only its matching temporary assumption. Thus
Use negation introduction separately for each component, introducing the forbidden logical disjunction in each subproof:Both temporary assumptions have been discharged before the final conjunction introduction, so
Extract both negations, assume the logical disjunction, and use disjunction elimination to derive logical falsity in either case:The two case subproofs are siblings, not nested within one another. Therefore
The premise limits every possible existential witness to the objects named by ; it does not assert that these constants name distinct objects. Instantiate it at a fresh witness , then apply disjunction elimination. In the first case equality elimination rule transfers to ; in the second it transfers to , contradicting the assumed negation. The full natural deduction derivation isThe sibling case subproofs end before the witness subproof closes. At line 13, is absent from and from all surrounding assumptions, so no witness escape occurs. The final two applications of implication introduction giveThe logical falsity elimination rule is needed to complete the contradictory second case with the same conclusion as the first case.
The implication introduction rule opens a subproof under a temporary assumption , derives , then performs assumption discharge to conclude . The negation introduction rule instead derives logical falsity under a temporary assumption , discharges it, and concludes :The rectangular boxes delimit these subproofs and the scope of their temporary assumptions. A line depending on the boxed temporary assumption cannot simply be used outside the box. The relevant introduction rule exports the indicated logical implication or negation; other surrounding assumptions stay in force.
Write and . In the following natural deduction derivations, brackets name temporary assumptions, and cited ranges specify the subproofs discharged. The abbreviations denote the corresponding conjunction elimination rule, conjunction introduction rule, disjunction elimination rule and disjunction introduction rule.
First use the disjunction elimination rule backwards: split the premise into two cases and build the whole conclusion in each case.
Alternatively, use the conjunction introduction rule backwards on the conclusion. First obtain by cases, then obtain by a second case analysis, and finally combine the results:
The first proof goal transformation starts with case analysis; the second starts by splitting a conjunction goal. Counting only actual applications of the printed inference rules, both use nine steps: the first has four , two , two and one ; the second has four , two , one and two . The extra case analysis in the second exactly replaces one conjunction construction. If assumption lines are counted as well, the second display has two more lines because it opens four case subproofs rather than two. Thus the apparent difference in display length need not be a difference in inference count.
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.
The implication introduction rule opens a subproof with a temporary assumption , derives , and discharges that assumption to conclude . The negation introduction rule opens a subproof with a temporary assumption , derives logical falsity , and discharges it to conclude .
The boxes record the scopes and dependencies of the temporary assumptions. A result derived inside a box cannot be used outside it without the appropriate discharge rule; it may depend on the boxed assumption. Other, surrounding assumptions remain in force. This is how the natural deduction rules distinguish a conditional derivation from an unconditional assertion.
Proof by contradiction, or reductio ad absurdum, has the formdischarging the assumption . This is the classical rule that concludes directly, as opposed to the negation introduction rule, which concludes a negation.
To derive double-negation elimination using RAA without assuming double-negation elimination itself, useHere line 1 is the negation of line 2. Thus only negation elimination rule inside the subproof, followed by RAA, is needed.
1 ¬¬φ premise
2 | ¬φ assumption
3 | ⊥ ¬-elim 2,1
4 φ RAA 2–3Use 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 2–6Implication introduction assumes temporarily, derives , and discharges that assumption to conclude . Disjunction elimination starts from and derives the same conclusion in two separate branches, one assuming and the other assuming .
Now derive the needed or inside its respective subproof:Implication elimination supplies the branch-specific premise before conjunction introduction packages it for the final implication.
1 A → D premise
2 ¬A → E premise
3 (A ∧ D) → B premise
4 (¬A ∧ E) → B premise
5 A ∨ ¬A excluded middle
6 | A assumption
7 | D →-elim 1,6
8 | A ∧ D ∧-intro 6,7
9 | B →-elim 3,8
10 | ¬A assumption
11 | E →-elim 2,10
12 | ¬A ∧ E ∧-intro 10,11
13 | B →-elim 4,12
14 B ∨-elim 5,6–9,10–13Put . First derive , then , using the derived proof by contradiction rule. Indentation records the subproofs:Line 11 uses explosion; lines 5 and 12 discharge only their inner assumptions. Each RAA step can be expanded into negation introduction followed by double-negation elimination, as in part (ii).