Rule of Conjunction

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

The rule
If we can conclude both $$p$$ and $$q$$, we may infer the compound statement $$p \and q$$:
 * $$p, q \vdash p \and q$$

This is sometimes known as:


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

It can be written:
 * $${p \qquad q \over p \and q} \and_i$$


 * Abbreviation: $$\and \mathcal I$$
 * 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.

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

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