**Slides / Bullets**
- Indirect proof 2: proof by contradiction
- An example: we want to prove ¬(SameRow(a, b) ∨ SameCol(a, b)) from the premises LeftOf(a, b) and FrontOf(a, b).
- Proof: To prove ¬(SameRow(a, b) ∨ SameCol(a, b)), we assume (SameRow(a, b) ∨ SameCol(a, b) and derive a contradiction. Suppose first that SameRow(a, b): this contradicts the premise FrontOf(a, b). Suppose on the other hand that SameCol(a, b): this contradicts the premise LeftOf(a, b). So in each case we have a contradiction, which means the original assumption that SameRow(a, b) ∨ SameCol(a, b) must be false.

- Proof by contradiction
- If we can derive a contradiction from a certain assumption, together with other premises, we can infer the negation of that assumption from those premises.
- A contradiction is something that is obviously logically impossible, i.e. logically can’t be true.
- For example: anything of the form P ∧ ¬P.
- The ‘contradiction’ symbol

- More examples
- An informal proof of one direction of one of De Morgan’s Laws: from (¬P ∨ ¬Q), infer ¬(P ∧ Q).
- Proof: Suppose for reductio that P ∧ Q. Given the premise that (¬P ∨ ¬Q), there are two cases. Case 1: ¬P; this contradicts the first conjunct of the assumption that P ∧ Q. Case 2: ¬Q; this contradicts the second conjunct. So in either case we have a contradiction. So we can conclude that ¬(P ∧ Q).

- For next week
- Read: Chapter 5; optionally, chapter 6.
- By hand, use the truth-table method to determine whether the conclusion of each of the following arguments is a tautological consequence of the premises:
- ¬(¬A ∧ B); A ∨ B; therefore ¬A (10%)
- A ∨ B; ¬B ∧ (C ∨ ¬A); therefore C (10%)

- Do: exercises 4.24 (20%); 5.8, 5.15, 5.17, 5.18 (15% each)

- Tautological consequence in Fitch