Definition:Consistent

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Definition

Let $\mathcal L$ be a logical language.

Let $\mathscr P$ be a proof system for $\mathcal L$.


Proof System

Then $\mathscr P$ is consistent if and only if:

There exists a logical formula $\phi$ such that $\not \vdash_{\mathscr P} \phi$

That is, some logical formula $\phi$ is not a theorem of $\mathscr P$.


Set of Formulas

Let $\mathcal F$ be a collection of logical formulas.


Then $\mathcal F$ is consistent for $\mathscr P$ if and only if:

There exists a logical formula $\phi$ such that $\mathcal F \nvdash_{\mathscr P} \phi$.

That is, some logical formula $\phi$ is not a provable consequence of $\mathcal F$.


Also defined as

Common confusion arises in the precise interpretation of consistency in logic.

On $\mathsf{Pr} \infty \mathsf{fWiki}$, consistency is placed in the realm of proof systems.

It is also common to place it in the realm of formal semantics.

That is, to define consistent as what is satisfiable on $\mathsf{Pr} \infty \mathsf{fWiki}$.

Because of the variation, any use of these terms in a source work should be treated with the utmost care and precision to determine the exact meaning.


Also see