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Joined 2026-09-21 Message
Incoming links: Peirce's law
Charles Sanders Peirce 2026-10-24
An American philosopher associated here with Peirce's law.
Portrait of Charles Sanders Peirce in 1859
. Unknown photographer. Tuttle Collection, Institute for Studies in Pragmaticism, Texas Tech University. The linked Commons record identifies the subject and marks the image as public domain in the United States; it supplies further usage information. Source.
This instance of Peirce's law uses the supplied classical double-negation elimination rule. First derive while assuming , using explosion inside a further temporary assumption.This is a formal natural deduction proof from exactly the given premise; no unproved excluded-middle lemma is needed.
1. (P → Q) → P premise
2. ¬P assumption [i]
3. P assumption [j]
4. ⊥ ¬E, 2, 3
5. Q ⊥E, 4
6. P → Q →I, 3–5, discharging [j]
7. P →E, 1, 6
8. ⊥ ¬E, 2, 7
9. ¬¬P ¬I, 2–8, discharging [i]
10. P ¬¬E, 9Write 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 |