lunes, 14 de mayo de 2007

Tema 2 - Logica Proposicional

UNIDAD 2: Leyes de la Lógica proposicional

INTRODUCCION A LA LOGICA

Origen de la Lógica: La Lógica deriva su origen de la naturaleza misma racional del hombre; pues el hombre está dotado de una facultad natural para alcanzar con sus actos la verdad y para evitar el error; de donde puede también procurar la rectitud con unas reglas determinadas. Y ésta se llama Lógica Natural o Vulgar. Pero la misma naturaleza humana, en cuanto esencialmente defectible, también es el origen de la Lógica Artificial, o Científica, la cual se preocupa del modo de pensar rectamente y de evitar los errores que, con mucha frecuencia se evitarían con enorme dificultad por medio de nuestra lógica ordinaria.

La lógica estudia la forma del razonamiento, es una disciplina que por medio de reglas y técnicas determina si un argumento es válido.El razonamiento lógico se emplea en matemáticas para demostrar teoremas; en ciencias de la computación para verificar si son o no correctos los programas; en las ciencias física y naturales, para sacar conclusiones de experimentos; y en las ciencias sociales y en la vida cotidiana, para resolver una multitud de problemas. Ciertamente se usa en forma constante el razonamiento lógico para realizar cualquier actividad.

Proposición: Una proposición es un juicio declarativo en el cual se afirma algo, independientemente de lo que se afirma sea verdadero o falso.
Una proposición o enunciado es una oración que puede ser falsa o verdadera pero no ambas a la vez. La proposición es un elemento fundamental de la lógica matemática.

A continuación se tienen algunos ejemplos de proposiciones válidas y no válidas, y se explica el porqué algunos enunciados no son proposiciones. Las proposiciones se indican por medio de una letra minúscula, dos puntos y la proposición propiamente dicha. Ejemplo.

p: la tierraes plana.
q: -17 + 38 = 21
r: x > y-9
s: La selección vinotinto será campeona en la presente temporada de Futbol.
t: Hola ¿como estas?
w: Lava el coche por favor.

Las proposiciones se clasifican en: Proposiciones Atómicas o Simples y Proposiciones Moleculares o Compuestas.

Las proposiciones atómicas son aquellas que expresan una sóla idea y las proposiciones moleculares son aquellas proposiciones simples entrelazadas por un conectivo u operador lógico.
Conectivos u Operadores Lógicos:

1. Conjunción
2. Disyunción Inclusiva
3. Disyunción Exclusiva
4. Condicional
5. Bicondicional
6.Negación


2.1- Leyes de la Lógica: son expresiones formales o fórmulas Proposicionales cuya función veritativa es una tautología que se utiliza para organizar un cálculo axiomático.
Principios Lógicos Básicos:

En el cálculo de inferencia es necesario tener en cuenta los siguientes principios lógicos.
1- Identidad: esta ley permite hacer equivalencia entre dos proposiciones de un mismo argumento
2- No contradicción: una proposición no puede ser simultáneamente verdadera y falsa p Λ –p.
3- Tercer excluido: una proposición es verdadera o es falsa.
p V –p.
4-Doble negación: una proposición afirmativa equivale a la misma proposición negada dos veces.

LEYES DE INFERENCIA: Las leyes de inferencia que corresponden a formas de razonamiento elementales cuya validez es fácil de demostrar.

1-MODUS PONENDO PONENS (MPP)

p entonces q “Si llueve, entonces las calles se mojan” (premisa)
p“Llueve” (premisa)
__________________________________________________

q “Luego, las calles se mojan” (conclusión)

El condicional o implicación es aquella operación que establece entre dos enunciados una relación de causa-efecto. La regla ‘ponendo ponens’ significa, “afirmando afirmo” y en un condicional establece, que si el antecedente (primer término, en este caso p) se afirma, necesariamente se afirma el consecuente (segundo término, en este caso q).

2- MODUS TOLLENDO TOLLENS (MTT)

‘Tollendo tollens’ significa “negando, niego”, y se refiere a una propiedad inversa de los condicionales, a los que nos referíamos en primer lugar.

p entoncesq “Si llueve, entonces las calles se mojan”
¬q “Las calles no se mojan”
__________________________________________________

¬p “Luego, no llueve”

Si de un condicional, aparece como premisa el consecuente negado (el efecto), eso nos conduce a negar el antecedente (la causa), puesto que si un efecto no se da, su causa no ha podido darse.

Esto nos permite formular una regla combinada de las ambas anteriores, consecuencia ambas de una misma propiedad de la implicación; la regla ponendo ponens sólo nos permite afirmar si está afirmado el antecedente (el primer término de la implicación), y la regla tollendo tollens sólo nos permite negar a partir del consecuente (segundo término de la implicación); ambas consecuencias se derivan de que la implicación es una flecha que apunta en un único sentido, lo que hace que sólo se pueda afirmar a partir del antecedente y negar sólo a partir del consecuente.

3- DOBLE NEGACIÓN (DN)

p sí sólo sí p

El esquema representa, “p doblemente negada equivale a p”. Siguiendo el esquema de una inferencia por pasos, la representaríamos así:

¬¬ p “No ocurre que Ana no es una estudiante”
_____________________________________________________

p “Ana es una estudiante”

La regla ‘doble negación’, simplemente establece que si un enunciado está doblemente negado, equivaldría al enunciado afirmado.


4-CONJUNCIÓN Y SIMPLIFICACIÓN

conjunción (C): Si disponemos de dos enunciados afirmados como dos premisas separadas, mediante la adjunción, podemos unirlos en una sola premisa utilizando el operador Λ (conjunción).

p “Juan es cocinero”

q “Pedro es policía”
___________________________________

p Λ q “Juan es cocinero y Pedro es policía”


5-Simplificación (S): obviamente, es la operación inversa. Si disponemos de un enunciado formado por dos miembros unidos por una conjunción, podemos hacer de los dos miembros dos enunciados afirmados por separado.


p Λ q “Tengo una manzana y tengo una pera”

____________________________________________

p “Tengo una manzana”

q “Tengo una pera”

6-MODUS TOLLENDO PONENS (TP)

La disyunción, que se simboliza con el operador V, representa una elección entre dos enunciados. Ahora bien, en esa elección, forma parte de las posibilidades escoger ambos enunciados, es decir, la verdad de ambos enunciados no es incompatible, si bien, ambos no pueden ser falsos.

A partir de lo anterior, se deduce la siguiente regla, denominada tollendo ponens (negando afirmo): si uno de los miembros de una disyunción es negado, el otro miembro queda automáticamente afirmado, ya que uno de los términos de la elección ha sido descartado.


p V q “He ido al cine o me he ido de compras”

¬q “No he ido de compras”
________________________________________________________

p “Por tanto, he ido al cine”

7-LEY DE LA ADICIÓN (LA)

Dado un enunciado cualquiera, es posible expresarlo como una elección (disyunción) acompañado por cualquier otro enunciado.


p “He comprado manzanas”
_______________________________________________________

p V q “He comprado manzanas o he comprado peras”

8- SILOGISMO HIPOTÉTICO (SH)

Dados dos implicaciones, de las cuales, el antecedente de la una sea el consecuente de la otra (el mismo enunciado), podemos construir una nueva implicación cuyo antecedente sea el de aquella implicación cuya consecuencia sea el antecedente de la otra implicación, y cuyo consecuente sea el de ésta última, cuyo antecedente era consecuencia del primero.

Expresado de otro modo, si una causa se sigue una consecuencia, y ésta consecuencia es a su vez causa de una segunda consecuencia, se puede decir que esa primera causa es causa de esa segunda consecuencia, del mismo modo que, si una bola de billar roja golpea a otra bola blanca que a su vez golpea a una bola negra, la bola roja es causa del movimiento de la bola negra. Expresado en forma de inferencia lógica:

p entonces q “Si la bola roja golpea a la bola blanca, la bola blanca se mueve”

q entonces r “Si la bola blanca golpea a la bola negra, la bola negra se mueve”
_______________________________________________________

p entonces r “Si la bola roja golpea a la bola blanca, la bola negra se mueve”

9-SILOGISMO DISYUNTIVO (DS)

Dadas tres premisas, dos de ellas implicaciones, y la tercera una disyunción cuyos miembros sean los antecedentes de los condicionales, podemos concluir en una nueva premisa en forma de disyunción, cuyos miembros serían los consecuentes de las dos implicaciones. Lógicamente, si planteamos una elección entre dos causas, podemos plantear una elección igualmente entre sus dos posibles efectos, que es el sentido de esta regla.


p entonces q “Si llueve, entonces las calles se mojan”

r entonces s “Si la tierra tiembla, los edificios se caen”

p V r “Llueve o la tierra tiembla”
____________________________________________________

q V s “Las calles se mojan o los edificios se caen”


10- SIMPLIFICACIÓN DISYUNTIVA (SD)

Si disponemos de dos premisas que corresponden a dos implicaciones con el mismo consecuente, y sus antecedentes se corresponden con los dos miembros de una disyunción, podemos concluir con el consecuente de ambas implicaciones.

p V q “Helado de fresa o helado de vainilla”

p entonces r “Si tomas helado de fresa entonces repites”

q entonces r “Si tomas helado de vainilla entonces repites”

____________________________________________________

r Luego, repites


11-LEY CONMUTATIVA

Esta ley, no es válida para la implicación, pero sí para conjunción y para la disyunción. Una conjunción es afirmar que se dan dos cosas a la vez, de modo que el orden de sus elementos no cambia este hecho. Igualmente, una disyunción es presentar una elección entre dos cosas, sin importar en qué orden se presente esta elección. Así pues,

p Λ q sí y sólo sí q Λ p “«p y q» equivale a «q y p»”

p V q sí y sólo sí q V p “«p ó q» equivale a «q ó p»


12- LEYES DE MORGAN (DM)

Esta ley permite transformar una disyunción en una conjunción, y viceversa, es decir, una conjunción en una disyunción. Cuando se pasa de una a otra, se cambian los valores de afirmación y negación de los términos de la disyunción/conjunción así como de la propia operación en conjunto, como podemos observar aquí:

p Λ q p V q
___________ ____________
¬(¬p V ¬q) ¬(¬p Λ ¬q)

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