Language Proof and Logic The Logic of Boolean
Language, Proof and Logic The Logic of Boolean Connectives Chapter 4
4. 1. a Tautologies and logical truth 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)” “Same. Row(a, a)”? Tautology if every row of its truth table has a “T”. You try it, page 100
4. 1. b Tautologies and logical truth Tarski’s world necessities Logical necessities Tautologies
4. 2 Logical and tautological equivalence Two sentences S 1 and S 2 are: Logically equivalent if it is impossible that S 1 is true while S 2 is false, or S 1 is false while S 2 is true. Tautologically equivalent if S 1 and S 2 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)
4. 3 Logical and tautological consequence A sentence S is a logical consequence of sentences P 1, …, Pn if it is impossible that S is false while each of P 1, …, Pn is true. A sentence S is a tautological consequence of sentences P 1, …, Pn if in every row of the joint truth table, whenever each of P 1, …, Pn 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?
4. 4 Tautological consequence in Fitch Ana Con vs. FO Con vs. Taut Con You try it, p. 117
4. 5. a Pushing negation around 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. De Morgan’s laws allow us to bring any formula down to NNF: ((A B) C) (A B) C (A B) C ( A B) C
4. 5. b Commutativity, idempotence, associativity (A B) C ( ( B A) B) (A B) C (( B A) B) Commutativity: (A B) C ((B A) B) P Q Q P (A B) C (B A B) Idempotence: (A B) C (A B B) (A B) C (A B) P P P (A B) C Associativity: (A B) C P (Q R) (P Q) R P Q R Chain of equivalences
4. 6. a Conjunctive and disjunctive normal forms 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 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)) D (A B) C A B
4. 6. b Conjunctive and disjunctive normal forms 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 D) (B D)
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