Silogismo hipotético

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Reglas de transformación
Lógica proposicional
Reglas de inferencia
Modus tollendo tollens / ponens Modus ponendo ponens / tollens Introducción del bicondicional / eliminación Introducción de la conjunción / eliminación Introducción de la disyunción / eliminación Silogismo disyuntivo / hipotético Dilema constructivo / destructivo Absorción
Reglas de reemplazo
Asociatividad Conmutatividad Distributividad Doble negación Leyes de De Morgan Transposición Implicación material Exportación Tautología Introducción de la negación
Lógica predicativa
Generalización / instanciación universal Generalización / instanciación existencial
Lógica modal
v t e

En lógica, el silogismo hipotético es una forma de argumento válido que consiste en un silogismo con una sentencia condicional para una o ambas de sus premisas. En la lógica proposicional, el silogismo hipotético es una regla de inferencia válida (llamado también argumento cadena, regla de cadena, o el principio de transitividad de la implicación, y a veces abreviado SH).

El silogismo hipotético se puede escribir formalmente como:

{\begin{array}{cl}&P\to Q\\&Q\to R\\\hline \therefore &P\to R\\\end{array}}

donde la regla es que cada vez que las instancias de « $P\to Q$ » y « $Q\to R$ » aparecen en las líneas de demostración, « $P\to R$ » se puede colocar en una línea posterior.

Un ejemplo de silogismo hipotético es:

$P\to Q$ Si no me despierto, entonces no voy a ir a trabajar.
$Q\to R$ Si no voy a trabajar, entonces no me pagan mi sueldo.
${\therefore P\to R}$ Por lo tanto, si no me despierto, entonces no me van a pagar mi sueldo.

El silogismo hipotético está estrechamente relacionado al silogismo disyuntivo, ya que también es un tipo de silogismo, y también es el nombre de una regla de inferencia. Las relaciones transitivas son por su parte otro concepto cercano al de silogismo hipotético.

Es interesante resaltar que el silogismo hipotético es una de las reglas de lógica clásica, que no siempre es aceptado en ciertos sistemas de lógica no clásica.

Notación formal

La regla de silogismo hipotético puede escribirse en la notación subsiguiente:

(P\to Q),(Q\to R)\vdash (P\to R)

donde $\vdash$ es un símbolo metalógico que significa que $P\to R$ es una consecuencia sintáctica de $P\to Q$ , y $Q\to R$ en algún sistema lógico;

y expresado como una tautología verdad-funcional o teorema de la lógica proposicional:

((P\to Q)\land (Q\to R))\to (P\to R)

donde $P$ , $Q$ y $R$ son proposiciones expresadas en algún sistema formal.

Véase también

Referencias

Enlaces externos

Esta obra contiene una traducción total derivada de «Hypothetical syllogism» de Wikipedia en inglés, concretamente de esta versión, publicada por sus editores bajo la Licencia de documentación libre de GNU y la Licencia Creative Commons Atribución-CompartirIgual 4.0 Internacional.

Datos: Q1641477

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