# Equivalence of Definitions of Logical Consistence

Jump to navigation
Jump to search

This page has been identified as a candidate for refactoring.see talk pageUntil this has been finished, please leave
`{{Refactor}}` in the code.
Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Refactor}}` from the code. |

It has been suggested that this page be renamed.To discuss this page in more detail, feel free to use the talk page. |

## 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

This theorem requires a proof.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Sources

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