Category:Logical Consistency

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This category contains results about Logical Consistency.
Definitions specific to this category can be found in Definitions/Logical Consistency.

Let $\LL$ be a logical language.

Let $\mathscr P$ be a proof system for $\LL$.


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 $\FF$ be a collection of logical formulas.


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

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

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