Definition:Inconsistent (Logic)
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Definition
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
- $\FF \vdash_{\mathscr P} \paren {\phi \land \neg \phi}$
Also known as
Inconsistent sets of logical formulas are often called contradictory.
Likewise, a logical formula which is inconsistent by itself is often called a contradiction.
Since these terms are also often used to describe unsatisfiability in the context of a formal semantics, they are discouraged as synonyms of inconsistent on $\mathsf{Pr} \infty \mathsf{fWiki}$.