# Definition:Unsatisfiable/Set of Formulas

Jump to navigation
Jump to search

## Definition

Let $\LL$ be a logical language.

Let $\mathscr M$ be a formal semantics for $\LL$.

A collection $\FF$ of logical formulas of $\LL$ is **unsatisfiable for $\mathscr M$** if and only if:

- There is no $\mathscr M$-model $\MM$ for $\FF$

That is, for all structures $\MM$ of $\mathscr M$:

- $\MM \not \models_{\mathscr M} \FF$

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

In sources where satisfiable is referred to as semantically consistent, **unsatisfiable** is correspondingly termed **semantically inconsistent**.

An **unsatisfiable** collection of logical formulas is also often referred to as **contradictory**.

However, this term is also commonly used to describe the notion of inconsistency in the context of a proof system.

It is therefore discouraged as a synonym of **unsatisfiable** on $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Also see

## Sources

- 2009: Kenneth Kunen:
*The Foundations of Mathematics*... (previous) ... (next): $\text {II}.7$ First-Order Logic Semantics: Definition $\text {II}.7.13$ - 2012: M. Ben-Ari:
*Mathematical Logic for Computer Science*(3rd ed.) ... (previous) ... (next): $\S 2.5.2$: Definition $2.42$