Material implication (rule of inference)

$${\displaystyle P\to Q\iff \mathrm{\neg }P\vee Q,}$$ where "${\displaystyle \iff }$" is a metalogical symbol representing "can be replaced in a proof with", P and Q are any given logical statements, and ${\displaystyle \mathrm{\neg }P\vee Q}$ can be read as "(not P) or Q". To illustrate this, consider the following statements:

• ${\displaystyle P}$: Sam ate an orange for lunch.
• ${\displaystyle Q}$: Sam ate a fruit for lunch.

Then, to say "Sam ate an orange for lunch" implies "Sam ate a fruit for lunch" (${\displaystyle P\to Q}$). Logically, if Sam did not eat a fruit for lunch, then Sam also cannot have eaten an orange for lunch (by contraposition). However, merely saying that Sam did not eat an orange for lunch provides no information on whether or not Sam ate a fruit (of any kind) for lunch.

Proof

Suppose we are given that ${\displaystyle P\to Q}$. Then we have ${\displaystyle \mathrm{\neg }P\vee P}$ by the law of excluded middle (i.e. either ${\displaystyle P}$ must be true, or ${\displaystyle P}$ must not be true).

Subsequently, since ${\displaystyle P\to Q}$, ${\displaystyle P}$ can be replaced by ${\displaystyle Q}$ in the statement, and thus it follows that ${\displaystyle \mathrm{\neg }P\vee Q}$ (i.e. either ${\displaystyle Q}$ must be true, or ${\displaystyle P}$ must not be true).

Suppose, conversely, we are given ${\displaystyle \mathrm{\neg }P\vee Q}$. Then if ${\displaystyle P}$ is true, that rules out the first disjunct, so we have ${\displaystyle Q}$. In short, ${\displaystyle P\to Q}$.

This can also be expressed with a truth table: