Lógica de Predicados Dr. Rogelio Dávila Pérez Profesor-Investigador Depto. de Tecnologías de la Información ITESM, Campus Guadalajara

Lógica de Predicados I. Sintaxis 1. Vocabulario (a) Constantes lógicas: (b) Conectores lógicos:,,,, = (c) Cuantificadores:, (d) Símbolos de puntuación: (, ),, (e) Símbolos predicados: R n, P m, Q s, … (f) Símbolos de función: f n, g m, r s, … (g) Símbolos constantes: a, b, c, … (h) Variables individuales: x, y, z 2. Términos (i) Un símbolo constante es un término. (ii) Una variable individual es un término. (iii) Si f n es un símbolo de función n-aria y, t 1, …, t n son términos entonces f n (t 1, …, t n ) es un término.

Lógica de Predicados 3. Fórmulas bien-formadas (fbfs) (a) La constante, llamada contradicción, es una fbf. (b) Si R n es un símbolo predicado n-ario, y t 1, …, t n son términos entonces R n (t 1, …, t n ) es una fbf. (c) Si y son fbfs, entonces,, y también fbfs. (d) Si es una fbf y x es una variable, entonces x. (x) y x. (x) son ambas fbfs. (e) Nada fuera de lo indicado en (a)-(d) es una fbf.

II. Reglas de Inferencia (a) Todas las reglas de inferencia de la lógica proposicional son válidas en la lógica de predicados. (b) Reglas del cuantificador Universal -Intro -Elim (Instanciación Universal) Lógica de Predicados (a) … (a) x.( (x) (x)) x. (x) (a) Solo en el caso de que (a) y (a) no sean premisas, y a no aparezca en las premisas.

(a) Reglas del cuantificador Existencial -Intro -Elim Lógica de Predicados (a) y. (y) (a) … y. (y) Sólo en el caso de que a no aparezca en.

Identidades de la lógica de predicados (a) v (b) Ley de Contraposición: (c) Leyes Distributivas: (i) v ( ) ( v ) ( v ) (ii) ( v ) ( ) v ( ) (e) Leyes de DeMorgan: (i) ( v ) (ii) ( ) v (f) x. (x) x. (x) (g) x. (x) x. (x)

Traduzca las siguientes oraciones a lógica (a) Monica likes some of her students. (b) Monica likes all her students. (c) All men are created equal. (d) Roses are red; violets are blue. (e) Some freshmen are intelligent. (f) All freshmen are intelligent. (g) No freshmen are intelligent. (h) One of the coats in the closet belongs to Sarah. (i) Some Juniors date only Seniors. (j) Not all birds can fly. Lógica de Predicados

(k) Every elephant has a trunk. (l) Adams is not married to anyone. (m) No freshmen are not serious. (n) Someone profited from the great depression. (o) All fish except sharks are kind to children. (p) Anyone with two or more spouses is a bigamist. (q) John married Mary and she got pregnant. (r) If all sophomores like Greek, then some freshmen do. (s) Everyone loves somebody and no one loves everybody, or somebody loves everybody and someone loves nobody.

Definition. An interpretation of a term consists in a non empty domain E, and an assignment function F: (i) To each constant symbol, it is assigned an individual from the domain. (ii) To each n-ary function symbol, it is assigned a mapping from E n E. (iii) To each n-ary predicate symbol, it is assigned a subset from E n. Predicate Logic Semantics

Definition. An interpretation of a wff consists of an mapping I: wffs {0,1} as follows: (i) I(P i (t 1, t 2,…,t n )) = 1 iff F( P i ), 0 otherwise. (ii) I ( ) = min(I( ),I( )) (iii) I ( ) = max(I( ),I( )) (iv) I ( ) = 0 if I( )=1 and I( )=0, 1 otherwise (v) I ( ) = 1-I( ) (vi) I ( x. (x)) = 1 if for all a E, I( (a))=1 (vii) I( x. (x)) = 1 if there is an F(a) E such that I( (a))=1. Predicate Logic Semantics

Predicate Logic Definition. An interpretation of a set of well-formed formulas, is called a MODEL of, if and only if, every wff in is true under that interpretation. Definition. A wff is called a logical consequence of a set of wffs, if and only if, is true in al models of ( |= ). Definition. A wff is satisfiable if it has a model, otherwise it is unsatisfiable.

Predicate Logic Example Lets define a simple language: Basic expressions: (i) Constant symbols: m, j, d and n. (ii) Predicate symbols: M 1, B 1, K 2, L 2. Sentences: M(d), B(j), K(j,n) and L(n,m) Evaluate the truth of the previous sentences according to the interpretations shown bellow.

Predicate Logic M 1 = A 1 = {x | x is a country in America} F 1 (m) =peru, F 1 (j)=chile, F 1 (d)= honduras, F 1 (n)= argentina F 1 (M)={colombia, belice, argentina, canada, nicaragua} F 1 (B)={x A 1 | x borders the pacific ocean} F 1 (K)={ | A 1 xA 1, x borders y} F 1 (L)={ | A 1 xA 1, x is bigger than y}

Predicate Logic M 2 = A 2 = {x | x is an integer} F 2 (m) =0, F 1 (j)=2, F 1 (d)= 9, F 1 (n)= -1 F 2 (M)={x A 2 | x is odd} F 2 (B)={x A| x is a perfect square} F 2 (K)={ | A 2 xA 2, and x > y} F 2 (L)={ | A 2 x A 2, x = y 2 }

The mother will die unless the doctor kills the child. If the doctor kills the child, the doctor will be taking life. If the mother dies, the doctor will be taking life. Therefore, the doctor will be taking life. Problems

If the soil is suitable for carrots, then it is deep, sandy and free of stones. The soil is not suitable for linseed if it is sandy or a heavy clay. Therefore the soil is not suitable for both carrots and linseed. Problems

Bank-notes all carry a metal strip. Anything with a metal strip can be detected by X-rays. Therefore, bank-notes can be detected by X- rays. Problems

All the birds are either chiff-chaffs or willow warblers. The birds are singing near the ground. Chiff-chaffs don t sing near the ground. Therefore the birds are all willow-warblers. Problems