Definition:Unsatisfiable/Formula
Jump to navigation
Jump to search
Definition
Let $\LL$ be a logical language.
Let $\mathscr M$ be a formal semantics for $\LL$.
A logical formula $\phi$ of $\LL$ is unsatisfiable for $\mathscr M$ if and only if:
- $\phi$ is valid in none of the structures of $\mathscr M$
That is, for all structures $\MM$ of $\mathscr M$:
- $\MM \not\models_{\mathscr M} \phi$
 | This article is complete as far as it goes, but it could do with expansion. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. 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 {{Expand}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Also known as
Unsatisfiable formulas are also referred to as:
- contradictions
- logical falsehoods
- logical falsities
- inconsistent formulas.
Because the term contradiction also commonly refers to the concept of inconsistency in the context of a proof system, it is discouraged as a synonym of unsatisfiable formula on $\mathsf{Pr} \infty \mathsf{fWiki}$.
The next two of these terms can easily lead to confusion about the precise meaning of "logical", and are therefore also discouraged on $\mathsf{Pr} \infty \mathsf{fWiki}$.
Also see
- Definition:Bottom (Logic), a symbol often used to represent contradictions in logical languages.
- Definition:Tautology
- Definition:Satisfiable Formula
- Definition:Falsifiable Formula