# Rule of Simplification

## Sequent

The **rule of simplification** is a valid argument in types of logic dealing with conjunctions $\land$.

This includes propositional logic and predicate logic, and in particular natural deduction.

### Proof Rule

- $(1): \quad$ If we can conclude $\phi \land \psi$, then we may infer $\phi$.
- $(2): \quad$ If we can conclude $\phi \land \psi$, then we may infer $\psi$.

### Sequent Form

The Rule of Simplification can be symbolised by the sequents:

\(\text {(1)}: \quad\) | \(\ds p \land q\) | \(\) | \(\ds \) | |||||||||||

\(\ds \vdash \ \ \) | \(\ds p\) | \(\) | \(\ds \) |

\(\text {(2)}: \quad\) | \(\ds p \land q\) | \(\) | \(\ds \) | |||||||||||

\(\ds \vdash \ \ \) | \(\ds q\) | \(\) | \(\ds \) |

## Explanation

The rule of simplification consists of two proof rules in one.

The first of the two can be expressed in natural language as:

- Given a conjunction, we may infer the first of the conjuncts.

The second of the two can be expressed in natural language as:

- Given a conjunction, we may infer the second of the conjuncts.

## Also known as

The Rule of Simplification can also be referred to as the **rule of and-elimination**.

Some sources give this as the **law of simplification for logical multiplication**.

Such treatments may also refer to the Rule of Addition as the **law of simplification for logical addition**.

This extra level of wordage has not been adopted by $\mathsf{Pr} \infty \mathsf{fWiki}$, as it is argued that it may cause clarity to suffer.