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Joined 2026-09-21 Message
Incoming links: Logical derivation
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.
Derived inference rule 2026-10-25
A derived inference rule abbreviates a logical derivation from the primitive inference rules. Its use retains the assumptions and side conditions required by that logical derivation; proving the abbreviation once allows subsequent proofs to use the rule without expanding every intermediate line.
Implication creation rule 2026-10-25
From a logical derivation of , infer under the same surrounding assumptions. To derive the rule, temporarily assume , retain the available , and discharge by the implication introduction rule. Its being unused is allowed.
Inference rule 2026-10-24
An inference rule licenses a conclusion from specified premises, possibly with side conditions. A derived inference rule is justified by a logical derivation using the primitive inference rules; using it subsequently abbreviates that logical derivation.
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.
Apply the complete logical derivation of the law of excluded middle in (a) to , obtaining . Then use the disjunction elimination rule, constructing each logical implication by an explicit hypothetical subproof:
1 Q ∨ ¬Q result of (a), φ := Q
2 | Q first case
3 | | P assume
4 | | Q available from 2
5 | P → Q →-intro, discharge 3–4
6 | (P → Q) ∨ (P → ¬Q) ∨-intro-L, 5
7 | ¬Q second case
8 | | P assume
9 | | ¬Q available from 7
10 | P → ¬Q →-intro, discharge 8–9
11 | (P → Q) ∨ (P → ¬Q) ∨-intro-R, 10
12 (P → Q) ∨ (P → ¬Q) ∨-elim, 1, cases 2–6 and 7–11Both case assumptions and both hypothetical assumptions have been discharged. Thus the conclusion is a theorem, not a logical implication that depends on assuming .
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.
Use the two derived inference rules whose complete proofs were supplied in (d). The following logical derivation has no nested hypothetical subproofs:
1 P → Q premise
2 Q → R premise
3 P → (Q → R) → creation from 2
4 (P → Q) → (P → R) → distribution from 3
5 P → R →-elim, 1 and 4The creation instance takes and . The distribution instance takes . The metarules are justified in (d); this requested flat use does not secretly invoke implication introduction rule in one of its own lines.