# Definition:Contradiction

## 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:

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

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): Chapter $1$: The Propositional Calculus $1$: $3$ Conjunction and Disjunction - 1973: Irving M. Copi:
*Symbolic Logic*(4th ed.) ... (previous) ... (next): $2$ Arguments Containing Compound Statements: $2.4$: 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 $(2)$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**contradiction** - 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 $1.19$ - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**contradiction** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**contradiction**