## Definition

A contradiction is a statement which is always false, independently of any relevant circumstances that could theoretically influence its truth value.

This has the form:

$p \land \neg p$

or, equivalently:

$\neg p \land p$

that is:

$p$ is true and, at the same time, $p$ is not true.

An example of a "relevant circumstance" here is the truth value of $p$.

The archetypal contradiction can be symbolised by $\bot$, and referred to as bottom.

### Inconsistent

Let $\mathcal L$ be a logical language.

Let $\mathscr P$ be a proof system for $\mathcal L$.

A collection $\mathcal F$ of logical formulas is inconsistent for $\mathscr P$ if and only if:

For every logical formula $\phi$, $\mathcal F \vdash_{\mathscr P} \phi$.

That is, every logical formula $\phi$ is a provable consequence of $\mathcal F$.

### Unsatisfiable

Let $\mathcal L$ be a logical language.

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

#### Unsatisfiable Formula

A logical formula $\phi$ of $\mathcal L$ is unsatisfiable for $\mathscr M$ iff:

$\phi$ is valid in none of the structures of $\mathscr M$

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

$\mathcal M \not\models_{\mathscr M} \phi$

#### Unsatisfiable Set of Formulas

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 $\mathscr M$:

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

### Unsatisfiable for Boolean Interpretations

Let $\mathbf A$ be a WFF of propositional logic.

$\mathbf A$ is called unsatisfiable (for boolean interpretations) if and only if:

$v \left({\mathbf A}\right) = F$

for every boolean interpretation $v$ for $\mathbf A$.

In terms of validity, this can be rendered:

$v \not\models_{\mathrm{BI}} \mathbf A$

that is, $\mathbf A$ is invalid in every boolean interpretation of $\mathbf A$.

## Also known as

This is also known as a logical falsehood or logical falsity.

The term contravalid proposition can also be seen.

Some sources use the term absurdity or absurdism, as the idea of a statement being both false and true at once is absurd.

Some sources use the terms inconsistency or unsatisfiable formula.

On $\mathsf{Pr} \infty \mathsf{fWiki}$, these terms are reserved for the analogous concepts for proof systems and formal semantics respectively.