Slides / Bullets
• TT-consequence and logical consequence, redux
• Why does TT-consequence suffice for logical consequence?
• Suppose that Q is not a logical consequence of {P1,...,Pn}. Then there is some possible situation in which P1...Pn are true and Q is false.
• But every possible situation corresponds to some line in the joint truth-table
• Informal proofs and the Boolean connectives
• Cardinal rule: in an informal proof, you’re allowed to make any inference whose validity you can legitimately assume to be obvious to your audience.
• Some very obviously valid inferences
• Conjunction introduction: from any two premises P, Q, you may infer the conclusion P Q.
• Conjunction elimination: from the single premise P Q, you may infer either P or Q.
• Disjunction introduction: from any premise P, you may infer P Q for any Q.
• Other useful valid inferences
• Important tautological equivalences, such as
• De Morgan’s Laws:
• ¬(P Q) ¬P ¬Q
• ¬(P Q) ¬P ¬Q
• Idempotence:
• P P P P P
• Distribution:
• P (Q R) (P Q) (P R)
• P (Q R) (P Q) (P R)
• Important tautologies, such as P ¬P
• These can be introduced into your informal proofs at any time, since a logical truth follows from everything.
• Indirect proof: proof by cases
• Suppose we want to prove ¬(a = b), given the premise (Cube(a) Tet(b)) (Tet(a) Dodec(b)). How do we do it?
• Proof: Suppose Cube(a) Tet(b). Then it follows that ¬(a = b), since nothing can be both a cube and a tetrahedron. Suppose on the other hand that Tet(a) Dodec(b). Then again, it follows that ¬(a = b), since nothing can be both a tetrahedron and a dodecahedron. So in either case, ¬(a = b).
• What we’re relying on here is the following fact: if a sentence follows from P together with certain other premises, and the same sentence follows from Q together with those premises, then it follows from P Q and those same premises.
• When we say ‘Suppose...’, we’re beginning a subproof.
• An indirect proof is a proof that uses subproofs.