Logical implication

An implication, also known as the material conditional, forms the conditional statement $p \rightarrow q$, read as “if $p$ then $q$”. The behavior of this logical operator is often initially unintuitive (it was for me and most of my peers in early discrete mathematics): the statement $p \rightarrow q$ is false only when $p$ is true and $q$ is false.

When $p$ is false, the first inclination one might have is to conclude the whole statement $p \rightarrow q$ is false. However $p$ is the conditional component of the implication, and we must get through it first before asking about the truth value of $q$. That is, if $p$ is true and $q$ ends up being false, then our whole implication breaks since it directly contradicts what the statement is claiming. However, if $p$ is false, then the implication is “not invoked”, or the conditional is not met; we don’t care about the truth value of $p$ for its own sake. This is the nature of vacuous truth, which seems to itself refer back to the pure definition of the material conditional; there seems to be a recursive trail when trying to find a way to the origin of the assigned truth under a false antecedent.