# Definition:Contradiction

## Contents

## 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.

## Also see

- Results about
**contradiction**can be found here.

## Sources

- 1965: E.J. Lemmon:
*Beginning Logic*... (previous) ... (next): $\S 1.3$: Conjunction and Disjunction - 1973: Irving M. Copi:
*Symbolic Logic*(4th ed.) ... (previous) ... (next): $2.4$: Statement Forms - 1977: Gary Chartrand:
*Introductory Graph Theory*... (previous) ... (next): Appendix $\text{A}.5$: Theorems and Proofs - 1980: D.J. O'Connor and Betty Powell:
*Elementary Logic*... (previous) ... (next): $\S \text{I}: 3$: Logical Constants $(2)$ - 2000: Michael R.A. Huth and Mark D. Ryan:
*Logic in Computer Science: Modelling and reasoning about systems*... (previous) ... (next): $\S 1.2.1$: Rules for natural deduction: Definition $1.19$