# Definition:Unsatisfiable/Set of Formulas

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## Contents

## Definition

Let $\mathcal L$ be a logical language.

Let $\mathscr M$ be a formal semantics for $\mathcal L$.

A collection $\mathcal F$ of logical formulas of $\mathcal L$ is **unsatisfiable for $\mathscr M$** iff:

- There is no $\mathscr M$-model $\mathcal M$ for $\mathcal F$

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

- $\mathcal M \not\models_{\mathscr M} \mathcal F$

## 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): $\mathrm{II}.7$ First-Order Logic Semantics: Definition $\mathrm{II.7.13}$ - 2012: M. Ben-Ari:
*Mathematical Logic for Computer Science*(3rd ed.) ... (previous) ... (next): $\S 2.5.2$: Definition $2.42$