Material implication (rule of inference)

An example: we are given the conditional fact that if it is a bear, then it can swim. Then, all 4 possibilities in the truth table are compared to that fact.

1. If it is a bear, then it can swim — T
2. If it is a bear, then it can not swim — F
3. If it is not a bear, then it can swim — T because it doesn’t contradict our initial fact.
4. If it is not a bear, then it can not swim — T (as above)

Thus, the conditional fact can be converted to ${\displaystyle \neg P\vee Q}$, which is "it is not a bear" or "it can swim", where ${\displaystyle P}$ is the statement "it is a bear" and ${\displaystyle Q}$ is the statement "it can swim".

Intuitionistic logic does not treat ${\displaystyle P\to Q}$ as equivalent to ${\displaystyle \neg P\vee Q}$ because

${\displaystyle P\to Q{\cancel {\Rightarrow }}\neg P\lor Q}$

Given ${\displaystyle P\to Q}$, one can constructively transform a proof of ${\displaystyle P}$ into a proof of ${\displaystyle Q}$. In particular, ${\displaystyle P\to P}$ holds in intuitionistic logic. If ${\displaystyle P\to Q\Rightarrow \neg P\lor Q}$ would hold, then ${\displaystyle \neg P\lor P}$ could be derived. However, the latter is the law of excluded middle, which is not accepted by intuitionistic logic (one cannot assume ${\displaystyle \neg P\lor P}$ without knowing which case applies).