Chapter 4 : The Logic of Boolean Connec6ves. Not all English connec4ves are truth- func4onal
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1 Chapter 4 : The Logic of Boolean Connec6ves Not all English connec4ves are truth- func4onal Max was at home because Claire went to the library. Home(max) because WentToLibrary(claire) T T T T F T Hence the English connec6ve because is not truth- func6onal! More examples: while; it s necessary that; in virtue of; for the reason that; explains, etc. 3 1
2 Three Logical Rela4ons: Logical consequence, equivalence, & truth Logical consequence: P is a logical consequence of Q IFF it s impossible for Q to be true while P is false RECALL: This was the nodon of validity of an argument. Logical equivalence: P is logically equivalent to Q IFF it s impossible for Q and P to have different truth- values OR: P is a logical consequence of Q and vice versa! Logical truth: P is a logical truth IFF it s impossible for P to be false (i.e., P is necessarily true, that is, true in every possible/conceivable situa6on). 4 Finessing the general no4on of logical rela4ons Logical consequence! Tautological consequence Logical equivalence! Tautological equivalence Logical truth! Tautological truth Later (star6ng in Chapter 9), we ll introduce even more powerful technical no6ons finessing the general no6on: Logical consequence! FO- consequence Logical equivalence! FO- equivalence Logical truth! FO- validity (or, necessity) 5 2
3 Why finesse? Logical Truth: (single sentence) English Examples Either Obama is taller than his wife or he is not. FOL version would be a tautology: Taller(obama, michelle) V Taller(obama, michelle) More generally in terms of Ps and Qs: P V P Determined solely on the basis of truth- func4onal connec4ves (Taut- Con) If Samuel Clemens is idendcal to Mark Twain, then Mark Twain is idendcal to Samuel Clemens. FOL version would be a First- Order truth or FO- necessity: clemens=twain! twain=clemens (or: if a=b then b=a) Compare in terms of Ps and Qs: If P then Q Determined solely on the basis of TF connec4ves, quan4fiers, and the meaning of the = predicate (FO- Con) All bachelors are unmarried FOL version would be an analy6cal truth or necessity: x (Bachelor(x)! Unmarried(x)) Compare in terms of Ps and Qs: P Determined on the basis of all the above apparatus AND the meanings of all predicates. (Ana- Con) Compare: Tarski- World Necessity (or, TW- necessity): Cube(a) V Tet(a) V Dodec(a) 6 Rela4onship among Necessi4es 7 3
4 Rela4onship among Possibili4es 8 Tautologies A tautology is a logical truth that is true in all possible situa6ons en6rely due to the truth- condi6onal connec6ves it contains and nothing else! Alterna6vely: A tautology is a logical truth whose truth- table contains True in all of its rows under the main connec6ve. (Because of this tautologies are some6mes called Truth- Table- necessity or TT- necessity) Example: If A, B, C are sentences of FOL, then the following is a tautology: (A ( A (B C))) B 9 4
5 Construc4ng TRUTH- TABLES Determine how many dis6nct atomic sentence types there are in order to calculate the number of rows you need for the table. number of rows = 2 n, where n is the number of dis6nct atomic sentences. Law of the Excluded Middle: P V P (Either P is true or its nega6on is true but not both) Set up the reference columns according to the number of dis6nct atomic sentences. (Open Boole) 10 From logical consequence to tautological consequence An FOL sentence P is a tautological consequence of sentences Q n Q m IFF there is no row, in their joint TT, in which Q n Q m are true and P is false: 11 5
6 From logical equivalence to tautological equivalence FOL sentences P and Q are tautologically equivalent IFF there is no row, in their joint TT, in which P and Q have different truth value (i.e., if their truth- tables are exactly the same) 12 Truth- Table vs. Fitch Proof method Tautological rela6ons can be demonstrated by proof methods. Either by actually construc6ng the official proofs in F, Or by using the sogware Fitch s Taut- Con mechanism. Revisit the uses of Taut- con, FO- con, Ana- con mechanisms 13 6
7 Pushing nega4on around Subs4tu4on of equivalents: If P and Q are logically equivalent: P Q then the results of subs6tu6ng one for the other in the context of a larger sentence are also logically equivalent: S(P) S(Q) A sentence is in nega6on normal form (NNF) IFF all occurrences of apply directly to atomic sentences. Any sentence built from atomic sentences using just,, and can be put into NNF by repeated applica6on of the De- Morgan Laws and Double Nega6on. Sentences can ogen be further simplified using the principles of associa6vity, commuta6vity, and idempotence. 14 Associa4vity 1. (Associa6vity of ): P (Q R) (P Q) R P Q R 2. (Associa6vity of ): P (Q R) (P Q) R P Q R 15 7
8 Commuta4vity 3. (Commuta6vity of ): P Q Q P As a result, any rearrangement of the conjuncts of an FOL sentence is logically equivalent to the original. For example, P Q R is equivalent to R Q P. 4. (Commuta6vity of ): P Q Q P As a result, any rearrangement of the disjuncts of an FOLsentence is logically equivalent to the original. For example, P Q R is equivalent to R Q P 16 Idempotence 5. (Idempotence of ): P P P More generally (given Commuta6vity), any conjunc6on with a repeated conjunct is equivalent to the result of removing all but one occurrence of that conjunct. For example, P Q P is equivalent to P Q. 6. (Idempotence of ) P P P More generally (given Commuta6vity), any disjunc6on with a repeated disjunct is equivalent to the result of removing all but one occurrence of that disjunct. For example, P Q P is equivalent to P Q. 17 8
9 Distribu4ve Laws For any sentences P, Q, and R: 1. Distribu6on of over : P (Q R) (P Q) (P R) 2. Distribu6on of over : P (Q R) (P Q) (P R) 18 DNF and CNF A sentence is in disjunc6ve normal form (DNF) if it is a disjunc6on of one or more conjunc6ons of one or more literals. A sentence is in conjunc6ve normal form (CNF) if it is a conjunc6on of one or more disjunc6ons of one or more literals. Distribu6on of over allows you to transform any sentence in nega6on normal form into disjunc6ve normal form. Distribu6on of over allows you to transform any sentence in nega6on normal form into conjunc6ve normal form. Some sentences are in both CNF and DNF: e.g., P Q (why?) 19 9
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