Rule of Implication

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

The rule
If, by making an assumption $p$, we can conclude $q$ as a consequence, we may infer $p \implies q$:
 * $\left({p \vdash q}\right) \vdash p \implies q$

This is sometimes known as:


 * The rule of implies-introduction;
 * Conditional proof (abbreviated CP).

It can be written:
 * $\displaystyle {\begin{array}{|c|} \hline p \\ \vdots \\ q \\ \hline \end{array} \over p \implies q} \to_i$


 * Abbreviation: $\implies \mathcal I$
 * Deduced from: The pooled assumptions of $q$.
 * Discharged assumption: The assumption of $p$.
 * Depends on: The series of lines from where the assumption of $p$ was made to where $q$ was deduced.

Explanation
This means: if we know that by making an assumption $p$ we can deduce $q$, then we can encapsulate this deduction into the compound statement $p \implies q$.

Thus it provides a means of introducing a conditional into a sequent.

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

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