Definition:Contradiction

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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

A set $\FF$ 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 $\FF$.


Unsatisfiable

Let $\LL$ be a logical language.

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


Unsatisfiable Formula

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 $\mathscr M$:

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


Unsatisfiable Set of Formulas

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 $\mathscr M$:

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


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:

$\map v {\mathbf A} = \F$

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

A contradiction is also known as:

a logical falsehood or logical falsity
a contravalid proposition
an absurdity or absurdism, as the idea of a statement being both false and true at once is absurd.

Also seen:

an inconsistency
an unsatisfiable formula.

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


Also see

  • Results about contradiction can be found here.


Sources

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