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9.10: Derivabilidad y los cuantificadores

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    Template:MathJaxZach

    Teorema\(\PageIndex{1}\)

    Si\(c\) es una constante que no ocurre en\(\Gamma\) o\(A(x)\) y\(\Gamma \Proves A(c)\), entonces\(\Gamma \Proves \lforall{x}{A(x)}\).

    Prueba. Dejar\(\delta\) ser una derivación\(A(c)\) de\(\Gamma\). Al agregar una\(\Intro{\lforall{}{}}\) inferencia, obtenemos una prueba de\(\lforall{x}{A(x)}\). Ya que\(c\) no ocurre en\(\Gamma\) o\(A(x)\), se cumple la condición de variable propia. ◻

    Proposición\(\PageIndex{1}\)

    1. \(A(t) \Proves \lexists{x}{A(x)}\).

    2. \(\lforall{x}{A(x)} \Proves A(t)\).

    Prueba.

    1. La siguiente es una derivación\(\lexists{x}{A(x)}\) de\(A(t)\):

      9.10.1.png

    2. La siguiente es una derivación\(A(t)\) de\(\lforall{x}{A(x)}\):

      9.10.2.png

    ◻

    This page titled 9.10: Derivabilidad y los cuantificadores was last modified on Sun, 30 Oct 2022 11:57:07 GMT and is shared under a CC BY license and was authored, remixed, and/or curated by Richard Zach et al. (Open Logic Project) .