| reductio ad absurdum |
Given both a sentence and its denial (at lines m and n), conclude the denial of any assumption appearing in the proof (at line k). |
| Annotation: m, n RAA (k) | |
| Assumption set: The union of the assumption sets at m and n, excluding k (the denied assumption). | |
| Comment: The sentence at line k is the assumption discharged (a.k.a. the REDUCTIO ASSUMPTION) and the conclusion is a denial of the discharged assumption. The sentences at lines m and n are denials of each other. | |
| Also known as: Indirect Proof (IP), ~Intro/~Elim. | |
| Examples. | |
| (a) | |
1 (1) P→Q A | |
2 (2) ~Q A | |
3 (3) P A | |
1,3 (4) Q 1,3 →E | |
1,2 (5) ~P 2,4 RAA (3) | |
| (b) | |
1 (1) P v Q A | |
2 (2) ~P A | |
3 (3) ~P→~Q A | |
2,3 (4) ~Q 2,3 →E | |
1,2,3(5) P 1,4 vE | |
1,3 (6) P 2,5 RAA (2) | |
| (c) | |
1 (1) P A | |
2 (2) Q A | |
3 (3) ~Q A | |
2,3 (4) ~P 2,3 RAA (1) |