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