Equivalence of Definitions of Logical Consistence

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Theorem

Let $\mathbf H$ be a countable set (either finite or infinite) of WFFs of propositional logic.

The following statements are logically equivalent:

$(1): \quad$ $\mathbf H$ has a model.

$(2): \quad$ $\mathbf H$ is consistent for the proof system of propositional tableaus.

$(3): \quad$ $\mathbf H$ has no tableau confutation.

Proof

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Sources

• 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): $\S 1.11$: Compactness: Corollary $1.11.7$