Tautological equivalence. entence equivalence. Tautological vs logical equivalence

Transcription

1 entence equivalence Recall two definitions from last class: 1. A sentence is an X possible sentence if it is true in some X possible world. Cube(a) is TW possible sentence. 2. A sentence is an X necessity if it is true in all X possible worlds. a = a is a logical necessity. Tautological equivalence Two sentences can be shown to be tautologically equivalent using truth tables: A B C A (B C) (A B) (A C) T T F T F T T T T F T T F T T T T F F T F T T T F T T T F T F F F T F F F F T T Class notes Oct 11, 2002 p.1 Class notes Oct 11, 2002 p.2 Tautological vs logical equivalence Truth tables can also show that two sentences are not tautologically equivalent: a = b b = a a = b a = b b = a T F T F F T F F The above shows that a = b and a = b b = a are not tautologically equivalent. If two sentences are tautologically equivalent, they are logically equivalent. Thus, A (B C) is logically equivalent to (A C) (A B), and not just taut equivalent.

2 onsequence Consequence, equivalence, necessity We can also define different types of consequence: Sentence Q is an X consequence of sentences P 1,..., P n iff there are no X possible worlds where P 1,..., P n are true and Q is false. For X = tautological, logical, etc. Special case: Sentence Q is a tautologically consequence of sentences P 1,..., P n iff there is no row of a truth table where P 1,..., P n are true and Q is false. An argument with premises P 1,..., P n and conclusion Q is valid just in case Q is a consequence of P 1,..., P n. For sentences A and B, if A implies B and B implies A, A and B are equivalent. For proof, assume opposite. Then one T and other F. If a sentence is necessary (a validity), it is a consequence of any other sentence. If a sentence is impossible, any other sentence is a consequence of it. Class notes Oct 11, 2002 p.5 Class notes Oct 11, 2002 p.6 autological consequence examples A B A A B A is a consequence of A B: T F T F F T F F No atomic sentence tautologically implies another one: Green(grass) T all(jim) Green(grass) T all(jim) T F T F F T F T Tautological consequence examples 2 A B is a tautological consequence of A and B A, but not either A or B separately: A B A B A A B T T F T F F F T F T F F F F T F

3 autological vs logical consequence Tautological consequence logical consequence. (Why?) So now we know that A and B A logically imply A B. However, logical consequence tautological consequence a = b is a logical consequence of b = a, but not a tautological consequence. Likewise, Tautological consequence implies TW consequence, but not vice versa. Taut Con Fitch the program allows you to use the rule Taut Con when the sentence follows from the previous ones. 1 A 2 B A 3 A B Taut Con: 1, 2 This can save you the work of making a truth table and checking all the contents. Class notes Oct 11, 2002 p.9 Class notes Oct 11, 2002 p.10 na Con, FO Con The computer also has the more powerful rules Ana Con and FO Con. 1 a = b 2 Cube(a) 3 Cube(b) FO Con: 1, 2 Replacing FO Con with Taut Con doesn t work: a = b Cube(a) Cube(b) T T F Taut Con vs FO Con vs Ana Con We ll learn more about these later. For now: A world is tautologically possible iff the sentential connectives (,, ) work the same way. A world is FO (First Order) possible if above and equality (=) work the same way. A world is Analytically possible if above and TW predicates (Larger, etc.) work the same way. For practical reasons, Between and Adjoins don t have to work the same way.

4 arious con examples 1. Taut Con: A B implies B A 2. FO Con: a = b implies b = a 3. Ana Con: Larger(b, a) implies Smaller(a, b) Taut Con FO Con Ana Con. Ana Con is the strongest, but takes the longest. Substitution principle If P and Q are equivalent, then so are S(P ) and S(Q). Examples: B B, so A ( B C) A (B C). (A B) A B, so (A B) C ( A B) C. a = a b = b, so Cube(a) a = a Cube(a) b = b. Class notes Oct 11, 2002 p.13 Class notes Oct 11, 2002 p.14 egation normal form A sentence is in negation normal form if any negations apply only to atomic sentences. Examples of sentences in NNF: Green(grass) ( a = a Larger(a, b)) A B Examples of sentences not in NNF: (A B) Cube(a) A (B C) Transforming sentences into NNF 1. Move negations as far in as possible using DeMorgan equivalences: (A B) A B (A B) A B 2. Cancel out double negations using equivalence of A and A. Example: 1. (A B) 2. A B DeM 2 above 3. A B elim

5 istribution laws for, A (B C) (A B) (A C) A (B C) (A B) (A C) Compare the distribution law for and +: A (B + C) (A B) + (A C) However, + doesn t distribute over. Like DeMorgan s laws, we can use this to simplify complex sentences. Disjunction Normal Form More definitions: A sentence is a literal iff it is atomic or the negation of an atomic sentence. Grass(green), a = a A sentence is in disjunctive normal form iff it s in negation normal form, and any conjunctions are inside the disjunctions. Grass(green) A (B C) a = a Class notes Oct 11, 2002 p.17 Class notes Oct 11, 2002 p.18 oving sentences into DNF 1. Put sentence into NNF. 2. Use distribution rule A (B C) (A B) (A C) Example: 1. ((A B) C) 2. (A B) C DeMorgan 3. ( A B) C DeMorgan again 4. ( A B) C elim, now in NNF 5. ( A C) (B C) distribution Conjunction Normal Form A sentence is in conjunction normal form if it s in negation normal form and the disjunctions are inside the conjunctions. Sentences in CNF: A, a = b, A B, (A B) ( C D) Sentences not in CNF: A, (B C) A. The CNF procedure is similar to the DNF procedure: 1. Put sentence into NNF. 2. Use distribution rule A (B C) (A B) (A C).