# Material Implication

*A brief note explaining why the logical connective “implication” is defined the way it is.*

→ [preprint] ← **Keywords:** #implication #material-implication #logical-connective #induction #proof-by-contradiction

The logical connective “implication” has the following truth table:

\[ \begin{array}{c|c|c} \varphi & \psi & \varphi \rightarrow \psi \\ \hline 0 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 0 & 0 \\ 1 & 1 & 1 \\ \end{array} \]

This usually leaves students confused about what does this “implication” actually represent, especially because of the two first lines… Moreover (and somewhat surprisingly) the answer is not easy to find, and so in this brief note, I explain the way this connective is used in mathematics—which is somewhat different from the everyday meaning of the word “implication.”

**June 22, 2024.** *Got feedback? Great, email is your friend!*