Some intuition behind the contrapositive | Max's blogSome intuition behind the contrapositive<br>27 Jun, 2026<br>ContentsIntroduction<br>While quite a fundamental law in logic, the contrapositive is certainly not immediately intuitive (or at least it was not to me).<br>After someone else expressed a similar feeling, I decided to try and derive some intuition to explain the rule, and I think I have succeeded.<br>This blog could also be treated as an introduction to the contrapositive.<br>Just so everyone is on the same page, here is the rule in all its formal glory:<br>$$<br>P \to Q \leftrightarrow (\lnot Q) \to (\lnot P)<br>$$where $P, Q$ are logical propositions and $\lnot$ represents the following proposition being false.<br>Stated colloquially, if having one thing ($P$) gives us that we have another thing ($Q$), then if we do not have this other thing ($\lnot Q$), then we cannot have our original thing ($\lnot P$).<br>The explanation<br>Consider a fish, an alive and healthy fish. To spark your imagination, I have used all of my artistic talent to provide a diagram:<br>Those who are expert marine biologists, may have noted, that my diagram (while beautiful) does not consider the full situation of a healthy fish.<br>If we have a healthy fish, then we must have a body of water (I will call it a lake for conciseness), for the fish, otherwise it will die :( 1.<br>I have worked hard to produce another diagram to show this relation:<br>Therefore, if we have a healthy fish, then we have a lake, or in notation:<br>$F \to L$<br>where $F$ represents having a fish, and $L$ represents having a lake.<br>Take a moment to really make sure this makes sense to you, both the notational and picture form:<br>Now, fish lovers may wish to stop reading here, and I implore them to do so!<br>Bathe in the enjoyment that we have a healthy fish, and worry no more my friend.<br>However, in the unrelenting interest of science, we must now do something quite cruel&mldr; we must&mldr; take away the water :((. Expressed in picture form,<br>So, after quite the monumentous task of taking away all the lakes (how did you do it reader??) we must now consider the consequences of our actions.<br>It seems, dear reader, that you were so preoccupied with whether or not you could, you didn&rsquo;t stop to think if you should 2.<br>The situation for our healthy fish has now changed significantly.<br>Earlier we noted that in order to have a healthy fish, we needed a body of water for the fish.<br>However, now we have no body of water, therefore, our fish is no longer healthy (it is likely dead).<br>I provide a graphic (in two ways!!!) representation of this fact below:<br>While cruel, our actions have now illuminated the right hand side of the contrapositive rule.<br>If we have no lakes, then our fish is not healthy, or<br>$$<br>\lnot L \to \lnot F<br>$$Drawn in all of its horrifying detail:<br>Though our methods were unconventional, we have now managed to demonstrate an example of the contrapositive!<br>If we have a healthy fish, then we have lakes.<br>If we have no lakes, then the fish are not healthy.<br>$$<br>F \to L \leftrightarrow \lnot L \to \lnot F<br>$$Conclusion<br>I hope this example was somewhat illuminating and entertaining. As always, any questions/comments are welcome below or by email: max_a (at) e.email!<br>Every animal needs water to survive (see this comprehensive source: https://www.bbc.co.uk/bitesize/articles/z343f82), but it&rsquo;s easier to draw how fish depend on water. ↩︎

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