Category:Definitions/Inconsistent (Logic)

This category contains definitions related to inconsistent in the context of Logic.
Related results can be found in Category:Inconsistent (Logic).

Let $\LL$ be a logical language.

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

Definition 1

A set $\FF$ of logical formulas is inconsistent for $\mathscr P$ if and only if:

For every logical formula $\phi$, $\FF \vdash_{\mathscr P} \phi$.

That is, every logical formula $\phi$ is a provable consequence of $\FF$.

Definition 2

A set $\FF$ of logical formulas is inconsistent for $\mathscr P$ if and only if:

There exists a logical formula $\phi$ such that both

$\FF \vdash_{\mathscr P} \paren {\phi \land \neg \phi}$