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# 8.10: Derivabilidad y los conectivos proposicionales

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Proposición$$\PageIndex{1}$$

1. Ambos$$A \land B \Proves A$$ y$$A \land B \Proves B$$.

Comprobante.

1. Ambos secuentes$$A \land B \Sequent A$$ y$$A \land B \Sequent B$$ son derivables:

2. Aquí hay una derivación del secuente$$A, B \Sequent A \land B$$:

Proposición$$\PageIndex{2}$$

1. $$A \lor B, \lnot A, \lnot B$$es inconsistente.

2. Ambos$$A \Proves A \lor B$$ y$$B \Proves A \lor B$$.

Comprobante.

1. Damos una derivación del secuente$$A \lor B, \lnot A, \lnot B \Sequent$$:

(Recordemos que las líneas de inferencia dobles indican varias inferencias de debilitamiento, contracción e intercambio).

2. Ambos secuentes$$A \Sequent A \lor B$$ y$$B \Sequent A \lor B$$ tienen derivaciones:

Proposición$$\PageIndex{3}$$

1. Ambos$$\lnot A \Proves A \lif B$$ y$$B \Proves A \lif B$$.

Comprobante.

1. El secuente$$A \lif B, A \Sequent B$$ es derivable:

2. Ambos secuentes$$\lnot A \Sequent A \lif B$$ y$$B \Sequent A \lif B$$ son derivables:

This page titled 8.10: Derivabilidad y los conectivos proposicionales is shared under a CC BY license and was authored, remixed, and/or curated by Richard Zach et al. (Open Logic Project) .