# Rule of Assumption

## Sequent

The rule of assumption is a valid argument in many proof systems, including natural deduction.

### Proof Rule

An assumption $\phi$ may be introduced at any stage of an argument.

### Sequent Form

For structure-technical reasons, the rule of assumption is symbolised by the sequent:

$p \vdash p$

In this form it is usually referred to as the Law of Identity.

### Boolean Interpretation

The truth value of a propositional formula $\mathbf A$ under a boolean interpretation $v$ is given by:

$\map v {\mathbf A} = \begin{cases} \T & : \map v {\mathbf A} = \T \\ \F & : \map v {\mathbf A} = \F \end{cases}$

## Explanation

It does not matter whether the assumption is true -- all we are concerned about is making sure that any conclusion based on the assumptions made is as the result of a valid argument.

The introduction of an assumption $\phi$ into an argument by means of the Rule of Assumption can be interpreted in natural language as:

"Suppose it were true that $\phi$"

or:

"What if $\phi$ were true?"

## Also known as

Some sources refer to the Rule of Assumption as the rule of assertion.