# Definition:Inconsistent (Logic)

(Redirected from Definition:Contradiction/Inconsistent)

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## Definition

Let $\LL$ be a logical language.

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

A collection $\FF$ 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 $\FF$.

This article is complete as far as it goes, but it could do with expansion.In particular: Include a variant comparable to Definition:Consistent (Logic)/Set of Formulas/Propositional LogicYou 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

**Inconsistent** collections 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 $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Also see

## Sources

- 2009: Kenneth Kunen:
*The Foundations of Mathematics*... (previous) ... (next): $\text{II}.11$ Some Strategies for Constructing Proofs: Definition $\text{II}.11.2$