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).