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RAZONAMIENTO E INFERENCIA LÓGICA

Tal vez resultaría importante ver como definimos razonamiento lógico, para lo cual no haremos más que repetir lo que tantos textos de lógica computacional o matemática discreta expresan:


Existe una regla de inferencia si todo par ordenado del tipo ({p1 , p2 , p3 , …..,pn}, q) formado por un conjunto de proposiciones denominadas premisas y una proposición llamada q que constituye una conclusión, con la condición que si las premisas del conjunto son Verdaderas entonces la conclusión q también lo es.


Pero quizá, para que el tema resulte “un juego de niños” , podamos entrar de una manera eficaz y entretenida.

En un bar del barrio de la Boca, en la ciudad de Buenos Aires, estaban sentados tres amigos Jorge, Gabriel y Fabio. Fabio sentado como de costado miraba el mundo y la gente pasar sin dar, aparentemente, demasiada importancia a lo que hablaban sus dos amigos. De pronto Jorge afirmó: “si los dirigentes pensaran en el bien común, estaríamos un poco mejor”, Gabriel, para no ser menos agregó: “pero los dirigentes solo piensan en el bienestar de sí mismo”. Ante esta conversación irrumpió Fabio y dijo, “deduzco entonces que estamos mal”.

Casualidades las de la vida de los hombres, pero en otro lugar se da una situación muy parecida y con un diálogo semejante pero ahora se trata de Pablo, Javier y Cristian. Es ahora Pablo el que afirma; “si los dirigentes actuaran en beneficio de todos, estaríamos mucho mejor”, Javier no se queda en silencio y confirma; “realmente no estamos bien”. Y saliendo de su estado de autismo Cristian reafirma: “según ustedes dos, deduzco que la dirigencia actúa en beneficio propio”

.

Dos situaciones muy parecidas nos permiten analizar en cual de ellas la deducción es resultado de las opiniones previas de los interlocutores, por supuesto que no ponemos en duda el grado de credibilidad que tienen las opiniones de los amigos.

  • En la primera situación se dan dos premisas de las cuales Fabio obtiene una aparente conclusión. Llamemos A a la premisa de Jorge: “si los dirigentes pensaran en el bien común, estaríamos un poco mejor”, considerando p: los dirigentes pensaran en el bien común y como q: estaríamos un poco mejor, por lo tanto [math]A=(p \Rightarrow q)[/math].

    Por lo tanto la opinión de Gabriel es “los dirigentes solo piensan en el bienestar de sí mismo” que en base a la representación anterior se trata de [math]\sim p[/math].

    Por último la deducción de Fabio estará dada por “estamos mal” que siguiendo con las representaciones anteriores será [math]\sim q[/math].

    Por lo tanto la expresión coloquial de la conversación es ésta:

    “si los dirigentes pensaran en el bien común, estaríamos un poco mejor”, “pero los dirigentes solo piensan en el bienestar de sí mismo”. “deduzco entonces que estamos mal”.

    Cuya representación simbólica es: [math][(p \Rightarrow q) \wedge \sim p] \Rightarrow \sim q[/math] la que resultará un razonamiento si es una tautología, es decir si la conclusión es verdadera resultando verdadera cada una de las premisas iniciales.

    Analicemos si [math]V( \sim q)=F \Rightarrow V(q)=V \Rightarrow V(p \Rightarrow q)=V [/math]cualquiera sea el valor de verdad de p por lo tanto [math]V((p \Rightarrow q) \wedge \sim p)[/math] puede ser verdadera o falsa por lo tanto no es tautología.

    Si [math]V( \sim q)=V[/math] entonces sí es una tautología aunque pudieren resultar falsas algunas de las premisas.

    Podemos construir una tabla de verdad:


    p q [math] \sim p[/math] [math] \sim q[/math] [math]p \Rightarrow q[/math] [math](p \Rightarrow q) \wedge \sim p[/math] [math][(p \Rightarrow q) \wedge \sim p] \Rightarrow \sim q [/math]
    V V F F V F V
    V F F V F F V
    F V V F V V F
    F F V V V V V

    Por otra parte podemos generar expresiones equivalente donde solo intervengan conjunciones, disyunciones y negaciones para así construir un circuito bivaluable y ver si su funcionamiento siempre es posible.

    [math][(p \Rightarrow q) \wedge \sim p] \Rightarrow \sim q \Leftrightarrow \sim [(p \Rightarrow q) \wedge \sim p] \vee \sim q \Leftrightarrow [ \sim( \sim p \vee q) \vee p] \vee \sim q \Leftrightarrow [(p \wedge \sim q) \vee p] \vee \sim q[/math]

    El circuito asociado es el siguiente:



    Con lo que se da una situación en que el circuito no funciona y es V(p)=F y el V(q)=V

    Por lo tanto la conclusión, si bien puede ser válida en la conversación y por el sentido común, no es válida como inferida de las dos premisas que corresponden a opiniones de Jorge y de Gabriel.



  • Vayamos ahora a la segunda situación en que se dan dos premisas de las cuales Cristian obtiene una aparente conclusión.

    Llamemos A a la premisa de Pablo: “si los dirigentes actuaran en beneficio de todos, estaríamos mucho mejor”, considerando p: los dirigentes actuaran en beneficio de todos y como q: estaríamos mucho mejor, por lo tanto [math]A = (p \Rightarrow q)[/math].

    Por lo tanto la opinión de Javier es “realmente no estamos bien”. que en base a la representación anterior se trata de [math]\sim q[/math].

    Por último la deducción de Cristian estará dada por “según ustedes dos deduzco que la dirigencia actúa en beneficio propio” que siguiendo con las representaciones anteriores resulta ser [math] \sim p [/math].

    Por lo tanto la expresión coloquial de la conversación es ésta:

    “si los dirigentes actuaran en beneficio de todos, estaríamos mucho mejor”, pero “realmente no estamos bien”, “según ustedes dos deduzco que la dirigencia actúa en beneficio propio”

    Cuya representación simbólica es: [math][(p \Rightarrow q)\wedge \sim q] \Rightarrow \sim p[/math] la que resultará un razonamiento si es una tautología, es decir si la conclusión es verdadera resultando verdadera cada una de las premisas iniciales.

    Analicemos si [math]V( \sim p)=F \Rightarrow V(p)=V \Rightarrow V(p \Rightarrow q)=V \quad si \quad V(q)=V[/math], en cuyo caso [math]V(\sim q)=F[/math] por lo tanto [math]V((p \Rightarrow q)\wedge \sim p)[/math] resultará falso; Si V(q)=F resultará [math]V(p \Rightarrow q)=F[/math] por lo tanto [math]V((p \Rightarrow q) \wedge \sim p)[/math] resultará falso lo que hace que la expresión en cuestión sea una tautología.

    Si [math]V(\sim p)=V[/math] siempre resultará verdadera la expresión dado que el consecuente así lo es

    Podemos construir una tabla de verdad:


    p q [math] \sim p[/math] [math] \sim q[/math] [math]p \Rightarrow q[/math] [math](p \Rightarrow q) \wedge \sim p[/math] [math][(p \Rightarrow q) \wedge \sim p] \Rightarrow \sim q [/math]
    V V F F V F V
    V F F V F F V
    F V V F V V V
    F F V V V V V

    Por otra parte podemos generar expresiones equivalente donde solo intervengan conjunciones, disyunciones y negaciones para así construir un circuito bivaluable y ver si su funcionamiento siempre es posible.

    [math][(p \Rightarrow q) \wedge \sim q] \Rightarrow \sim p \Leftrightarrow \sim [(p \Rightarrow q)\wedge \sim q] \vee \sim p \Leftrightarrow [ \sim(\sim p \vee q) \vee q] \vee \sim p \Leftrightarrow [(p \wedge \sim q) \vee q] \vee \sim p[/math]

    El circuito asociado es el siguiente:



    Con lo que se da una situación en que el circuito siempre funciona.

    Por lo tanto la conclusión es válida como inferida de las dos premisas que corresponden a opiniones de Pablo y de Javier.




Si bien no hemos mostrado más que el Modus Ponem creemos haber presentado dos situaciones muy parecidas pero una, a diferencia de la otra, resulta ser una inferencia lógica. Está en el lector construir otras situaciones de la vida real o como una fantasía de cotidianidad que le permita llegar a comprender algo más del razonamiento lógico inferencial.


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