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.