# 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.