Rule of Conjunction

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

If we can conclude both $p$ and $q$, we may infer the compound statement $p \land q$:

Sequent Form
The rule of conjunction is symbolised by the sequent:
 * $p, q \vdash p \land q$

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

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


 * Abbreviation: $\land \mathcal I$
 * Deduced from: The pooled assumptions of each of $p$ and $q$.
 * Depends on: Both of the lines containing $p$ and $q$.

Explanation
This means: if we can show that two statements are true, then we may build a compound statement expressing this fact, and be certain that this is also true.

Thus a conjunction is added to a sequent.

Also known as
This is sometimes known as:


 * the rule of and-introduction;
 * the rule of adjunction.

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

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

Also see

 * Rule of Simplification