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).