Rule of Simplification

Axiom
The rule of simplification is one of the axioms of natural deduction.

This is two axioms 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 $q$.

Sequent Form
The rule of conjunction is symbolised by the sequents:
 * $(1): \quad p \land q \vdash p$
 * $(2): \quad p \land q \vdash q$

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

Tableau Form
In a tableau proof, the rule of conjunction can be invoked in the following manner:


 * Abbreviation: $\land \mathcal E_1$ or $\land \mathcal E_2$
 * Deduced from: The pooled assumptions of $p \land q$.
 * Depends on: The line containing $p \land q$.

Explanation
Note that there are two axioms here 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.

Demonstration by Truth Table
$\begin{array}{|ccc||c|c|} \hline p & \land & q & p & q \\ \hline F & F & F & F & F \\ F & F & T & F & T \\ T & F & F & T & F \\ T & T & T & T & T \\ \hline \end{array}$

As can be seen, when $p \land q$ is true so are both $p$ and $q$.