Material Implication

A precursor to argumentation.

Material implication is a fancy word for inference. The statement “P leads to Q” is true in all cases except when P and not Q. This might seem bizzare. How can both P and not P lead to Q? The key here is that they are not true simultaneously:

In one context, affirming P leads to Q.
In another context, denying P leads to Q.
In a third context, denying P means denying Q (or the reverse).

However: something true will not lead to something false (see the second row from the top in the following truth table):

$$ \begin{array}{|c|c|c|} \hline \textbf{P} & \textbf{Q} & \textbf{P} \implies \textbf{Q} \\ \hline 1 & 1 & 1 \\ \hline 1 & 0 & 0 \\ \hline 0 & 1 & 1 \\ \hline 0 & 0 & 1 \\ \hline \end{array} $$

One can also make logically equivalent expressions.