Rule of Simplification

Proof Rule
The rule of simplification is a valid deduction sequent in propositional logic:

This is two proof rules in one:
 * $(1): \quad$ If we can conclude $p \land q$, then we may infer $p$.
 * $(2): \quad$ If we can conclude $p \land q$, then we may infer $q$.

It can be written:
 * $\displaystyle {p \land q \over p} \land_{e_1} \qquad \qquad {p \land q \over q} \land_{e_2}$

Explanation
Note that here there are two proof rules in one. The first of the two tells us that, given a conjunction, we may infer the first of the conjuncts, while the second says that, given a conjunction, we may infer the second of the conjuncts.

At this stage, such attention to detail is important.

Also known as
This is sometimes known 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, as it is argued that it may cause clarity to suffer.

Also see

 * Rule of Conjunction

Technical Note
When invoking the Rule of Simplification in a tableau proof, use the Simplification template:



or:

where:
 * is the number of the line on the tableau proof where the Rule of Simplification is to be invoked
 * is the pool of assumptions (comma-separated list)
 * is the statement of logic that is to be displayed in the Formula column, without the  delimiters
 * is the line of the tableau proof upon which this line directly depends
 * should hold 1 for Simplification_1, and 2 for Simplification_2
 * is the (optional) comment that is to be displayed in the Notes column.