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Joined 2026-09-21 Message
Incoming links: Temporary assumption
Assumption discharge 2026-10-25
Assumption discharge closes a designated temporary assumption together with its dependent subproof. Implication introduction rule turns a logical derivation of under into outside that temporary scope. Negation introduction rule turns a logical derivation of logical falsity under into . Disjunction elimination rule discharges its separate case assumptions while retaining any surrounding premises.
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 2026-10-25
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.
Logical derivation 2026-10-24
A logical derivation is a finite application of explicitly specified inference rules to premises or temporary assumptions. In natural deduction, which assumptions remain open is part of the logical derivation: a conclusion reached inside a temporary subproof cannot simply be used outside it.
Use only the displayed natural deduction rules. Let abbreviate . Indentation below marks temporary assumptions; the closing rule names the assumption discharged.
1 | ¬A assume
2 | | φ assume
3 | | A ∨-intro-L, 2
4 | | ⊥ ¬-elim, 3 and 1
5 | ¬φ ¬-intro, discharge 2–4
6 | A ∨-intro-R, 5
7 | ⊥ ¬-elim, 6 and 1
8 ¬¬A ¬-intro, discharge 1–7
9 A ¬¬-elim, 8There are no undischarged assumptions at line 9, so this proves the law of excluded middle. The use of double-negation elimination is explicit; an unproved appeal to the derived RAA rule is unnecessary.
For implication creation rule, keep available and discharge an otherwise unused antecedent:
1 ψ given
2 | φ assume
3 | ψ available from 1
4 φ → ψ →-intro, discharge 2–3For implication distribution rule, write . The logical derivation below keeps available and discharges exactly the two temporary assumptions:
1 φ → (ψ → ρ) given
2 | φ → ψ assume
3 | | φ assume
4 | | ψ →-elim, 3 and 2
5 | | ψ → ρ →-elim, 3 and 1
6 | | ρ →-elim, 4 and 5
7 | φ → ρ →-intro, discharge 3–6
8 (φ → ψ) → (φ → ρ) →-intro, discharge 2–7These justify both derived inference rules using Figure 2 primitives. Their rule forms retain any other surrounding assumptions; they are not unconditionally asserted conclusions after the displayed numerator is forgotten.
Let abbreviate the given negated existential statement, and choose a fresh variable that is not free in the undischarged premise. Work from an assumed to :
1 ¬∃x (Q(x) → P(x)) premise N
2 | P(u) assume
3 | | Q(u) assume
4 | | P(u) available from 2
5 | Q(u) → P(u) →-intro, discharge 3–4
6 | ∃x (Q(x) → P(x)) ∃-intro, witness u in 5
7 | ⊥ ¬-elim, 6 and 1
8 | Q(u) ⊥-elim, 7
9 P(u) → Q(u) →-intro, discharge 2–8
10 ∀x (P(x) → Q(x)) ∀-intro, 9, u freshThe occurrence of in the temporary assumption at line 2 is harmless because that assumption is discharged before universal quantifier introduction rule. No unproved implication-creation rule or quantifier-negation equivalence is needed.
Sequent 2026-10-25
A sequent records that the formula is derivable with undischarged assumptions drawn from . In natural deduction, a temporary assumption contributes to only until an appropriate discharge rule closes its scope.
Temporary assumption 2026-10-25
A temporary assumption is introduced for a hypothetical subproof and retained only within that scope. A suitable inference rule can discharge it while forming a conclusion valid under the surrounding assumptions. Its scope must be recorded even when the assumption is unused inside the subproof.