Slides / Bullets

• Informal proofs using conditionals
  • Basic valid steps:
    • Modus ponens: from P and P → Q, infer Q.
    • Biconditional elimination: from P and P ↔ Q or Q ↔ P, infer Q.
  • Other valid steps:
    • Modus tollens: from P → Q and ¬Q, infer ¬P
• • Useful equivalences:
    • Contraposition: P → Q and ¬Q → ¬P are logically equivalent.
    • P ↔ Q and ¬Q ↔ ¬P are also logically equivalent.
• Conditional proof
  • To derive a conclusion of the form P → Q from some premises, assume that P is true (in addition to those premises), and derive Q subject to that assumption.
• • An example:
    • Given the premises (Tet(a) ∧ Small(a)) → Small(b) and Tet(a), we want to prove Small(a) → Small(b).
    • Proof: Suppose that Small(a) is true. Then Tet(a) ∧ Small(a) by the second premise, and so by the first premise, Small(b). So by conditional proof we conclude that Small(a) → Small(b).
• • Another example: 8.4
    • Premises: (1) The unicorn, if horned, is elusive and dangerous. (2) If elusive or mythical, the unicorn is rare. (3) If a mammal, the unicorn is not rare. Conclusion: The unicorn, if horned, is not a mammal.
    • Argument. Suppose that the unicorn is horned, and assume for reductio that it is a mammal. By (1) it is elusive, so by (2) it is rare. But by (3) it is not rare: contradiction. Hence, if the unicorn is horned, it is not a mammal.
• • Proving biconditionals.
    • Biconditional introduction: If we can derive Q from the assumption that P (plus our premises), and we can derive P from the assumption that Q (plus our premises), then we can derive P ↔ Q from our premises.
    • Circles of proofs.
• For next week:
  • Read: chapters 7 and 8; optionally, chapter 9.
  • Do: exercises 7.6 - 7.8 (10% each); 7.11 (10%); 7.12 and 7.13 (20%); 8.3, 8.5, 8.6, 8.9 (10% each).