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Joined 2026-09-21 Message
Incoming links: Logical falsity elimination rule
Assume with . For the forward implication, open a fresh existential witness, instantiate domain exhaustiveness at it, and split into the equality cases. The equality elimination rule transfers its predicate to the corresponding constant, which supplies the required disjunct. Closing the witness scope is legal because the result contains only the named constants. Conversely each named instance gives the existential by the existential quantifier introduction rule, and cases combine those derivations. The constants need not denote distinct objects. A negated named instance can eliminate one case through contradiction and logical falsity elimination rule.
The logical implication associates to the right. Assume its two antecedents, then use nested applications of the disjunction elimination rule.
1 | P ∨ Q assume
2 | | P ∨ ¬Q assume
3 | | | P first case of 1
4 | | | P available from 3
5 | | | Q second case of 1
6 | | | | P first case of 2
7 | | | | P available from 6
8 | | | | ¬Q second case of 2
9 | | | | ⊥ ¬-elim, 5 and 8
10 | | | | P ⊥-elim, 9
11 | | | P ∨-elim, 2, cases 6–7 and 8–10
12 | | P ∨-elim, 1, cases 3–4 and 5–11
13 | (P ∨ ¬Q) → P →-intro, discharge 2–12
14 (P ∨ Q) → ((P ∨ ¬Q) → P) →-intro, discharge 1–13This proof uses logical falsity elimination rule, but does not need law of excluded middle or double-negation elimination.
Write for . The following derives Peirce's law directly from the printed primitive rules, including double-negation elimination:
| Line | Formula | Justification |
|---|---|---|
| 1 | Assume for the final implication | |
| 2 | Assume locally | |
| 3 | Assume inside the scope of 2 | |
| 4 | Negation elimination rule, 2 and 3 | |
| 5 | Logical falsity elimination rule, 4 | |
| 6 | Implication introduction rule, discharging 3–5 | |
| 7 | Implication elimination rule, 1 and 6 | |
| 8 | Negation elimination rule, 2 and 7 | |
| 9 | Negation introduction rule, discharging 2–8 | |
| 10 | Double-negation elimination, 9 | |
| 11 | Implication introduction rule, discharging 1–10 |
Let abbreviate , and group the three equality alternatives on the left. The proof uses the binary disjunction elimination rule twice, so no unproved three-way derived rule is needed.
| Line | Formula | Justification |
|---|---|---|
| 1 | Premise | |
| 2 | Assume for the first implication | |
| 3 | Assume for the second implication | |
| 4 | Assume with fresh witness name | |
| 5 | Universal quantifier elimination rule, 1 | |
| 6 | Assume the first outer case | |
| 7 | Assume the first inner case | |
| 8 | Equality elimination rule, 7 and 4 | |
| 9 | Negation elimination rule, 2 and 8 | |
| 10 | Logical falsity elimination rule, 9 | |
| 11 | Assume the second inner case | |
| 12 | Equality elimination rule, 11 and 4 | |
| 13 | Disjunction introduction rule, left, 12 | |
| 14 | Disjunction elimination rule, 6, 7–10 and 11–13 | |
| 15 | Assume the second outer case | |
| 16 | Equality elimination rule, 15 and 4 | |
| 17 | Disjunction introduction rule, right, 16 | |
| 18 | Disjunction elimination rule, 5, 6–14 and 15–17 | |
| 19 | Existential quantifier elimination rule, 3 and 4–18 | |
| 20 | Implication introduction rule, discharging 3–19 | |
| 21 | Implication introduction rule, discharging 2–20 |
The witness name is absent from and from every assumption remaining after line 19, so the existential elimination is legal. Each equality is used in the forward direction supplied by the printed equality rule. The three named constants need not denote distinct objects: the case proof requires only that they exhaust the domain. This is existential quantification over a finitely named domain followed by exclusion of the first case.
Use the disjunction elimination rule on the outer disjunction. In its left branch, assume and apply the disjunction elimination rule again: from a case assumption , the disjunction introduction rule gives ; from a case assumption , the premise and the modus ponens rule give , then the disjunction introduction rule gives . Discharging those two inner case assumptions yields under the left outer branch.
In the right outer branch, assume . A case assumption gives by the disjunction introduction rule. A case assumption , together with the premise , gives by the negation elimination rule, and then by logical falsity elimination rule. Discharge these inner case assumptions with the disjunction elimination rule.
Both outer branches now have the same conclusion. Discharge the two outer case assumptions with the disjunction elimination rule to obtain under exactly the stated premises.
Keep the four given premises in context and assume under label . The modus ponens rule applied to the first two premises gives and . Apply the disjunction elimination rule to as follows.
Under a case assumption , the premise gives by the modus ponens rule. Together with , the negation elimination rule gives , and the supplied logical falsity elimination rule gives . Under the other case assumption , the required conclusion is already available. Discharge both case assumptions with the disjunction elimination rule to conclude .
Finally discharge with the implication introduction rule, giving . The premise is not needed: a derivation is permitted to leave a premise unused.
Let be fresh, not free in any of the three universal premises. Universal quantifier elimination rule gives , and . Apply disjunction elimination rule to the last formula.
In the branch, apply disjunction elimination rule to . Its case contradicts by negation elimination rule, and logical falsity elimination rule then gives . Its case already gives that conclusion. Discharge those two inner case assumptions to obtain under the outer assumption. In the outer branch, the modus ponens rule applied to gives directly.
Discharge the two outer case assumptions by disjunction elimination rule. The conclusion now depends only on the universal premises, in which is not free. Universal quantifier introduction rule therefore gives . All case assumptions have been discharged, and the argument requires no classical reductio ad absurdum.
Peirce's law 2026-10-25
Assume and temporarily assume . A further assumption yields a contradiction and hence by the logical falsity elimination rule; discharge it to obtain . Applying gives , contradicting . Discharge the latter assumption to obtain , apply double-negation elimination, and finally discharge . This is a derivation from the primitive classical natural deduction rules. The principle is named for Charles Sanders Peirce.