# 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

## Sources

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