Axiom:Rule of Implication
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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 $p \implies q$.
