2. The Logic of Boolean Connectives Chapter 4 Language, Proof and Logic

3. Tautologies and logical truth 4.1.a A sentence is: Logically possible if it could be true. “Bob is funny or Bob is not funny” “Bob is funny” “Bob arrived from the Earth to Pluto within 1 min” Logically necessary (logical truth) if it could not be false. “Bob is funny or Bob is not funny” “a=a” “if c=d and c is a grumber, then d is a grumber” How about: “Tet(a)  Cube(a)  Dodec(a)” “SameRow(a,a)”? Tautology if every row of its truth table has a “T”. You try it, page 100

4. Tautologies and logical truth 4.1.b Tautologies Logical necessities Tarski’s world necessities

5. Logical and tautological equivalence 4.2 Two sentences S1 and S2 are: Logically equivalent if it is impossible that S1 is true while S2 is false, or S1 is false while S2 is true. Tautologically equivalent if S1 and S2 have identical truth values in their joint truth table. Construct such tables for:  (A  B) vs.  A  B A vs. B  ((A  B)   C) vs. (  A  B)  C a=b  Cube(a) vs. a=b  Cube(b)

6. Logical and tautological consequence 4.3 A sentence S is a logical consequence of sentences P 1,…,P n if it is impossible that S is false while each of P 1,…,P n is true. A sentence S is a tautological consequence of sentences P 1,…,P n if in every row of the joint truth table, whenever each of P 1,…,P n is true, so is S. Is Cube(a) a logical consequence of Cube(b), a=b? Is it also a tautological consequence? Is A  B a logical consequence of B  A? How’bout vice versa? Is B a logical consequence of A  B,  A?

7. Tautological consequence in Fitch 4.4 Ana Con vs. FO Con vs. Taut Con You try it, p. 117

8. Pushing negation around 4.5.a The principle of substitution of logical equivalents: If P  Q, then S(P)  S(Q). Negation normal form (NNF): negation is applied only to atoms.  ((A  B)  C)   (A  B)  C   (A  B)  C  (  A  B)  C De Morgan’s laws allow us to bring any formula down to NNF:

9. Commutativity, idempotence, associativity 4.5.b Associativity: P  (Q  R)  (P  Q)  R  P  Q  R P  (Q  R)  (P  Q)  R  P  Q  R Commutativity: P  Q  Q  P P  Q  Q  P Idempotence: P  P  P P  P  P (A  B)  C  (  (  B  A)  B)  (A  B)  C  ((  B  A)  B)  (A  B)  C  ((B  A)  B)  (A  B)  C  (B  A  B)  (A  B)  C  (A  B  B)  (A  B)  C  (A  B)  (A  B)  (A  B)  C  (A  B)  C Chain of equivalences

10. Conjunctive and disjunctive normal forms 4.6.a Distribution of  over +: a  (b+c) = a  b + a  c Hence, e.g., (a+b)  (c+d) = (a+b)  c + (a+b)  d = a  c + b  c + (a+b)  d = a  c + b  c + a  d + b  d Disjunctive normal form (DNF): disjunction of one or more conjunctions of literals Conjunctive normal form (CNF): conjunction of one or more disjunctions of literals What are these? (A  B  C)  (A  D)  B (A  B)  C (A  (B  C))  D A  B

11. Conjunctive and disjunctive normal forms 4.6.b Distribution of  over  : P  (Q  R)  (P  Q)  (P  R) Allows us to bring any NNF to DNF Distribution of  over  : P  (Q  R)  (P  Q)  (P  R) Allows us to bring any NNF to CNF (A  B)  (C  D)  [(A  B)  C]  [(A  B)  D]  (A  C)  (B  C)  [(A  B)  D]  (A  C)  (B  C)  (A  D)  (B  D) (A  B)  (C  D)  [(A  B)  C]  [(A  B)  D]  (A  C)  (B  C)  [(A  B)  D]  (A  C)  (B  C)  (A  D)  (B  D)