Some intuition behind the contrapositive

Introduction

While quite a fundamental law in logic, the contrapositive is certainly not immediately intuitive (or at least it was not to me). After someone else expressed a similar feeling, I decided to try and derive some intuition to explain the rule, the results of which can be found here! This blog could also be treated as an introduction to the contrapositive.

Before we begin, I would like to show how the formal definition of the contrapositive looks. I like to think that definitions codify intuition, but it only goes one way. That is, we can turn intuition into definitions, but trying to gain intuition from a definition is a faux pas. Towards the end of this post we will revisit it, and hopefully, you will see your newfound intuition reflected in the definition.

The contrapositive is defined as the equivalence

where $P, Q$ are logical propositions and $\lnot$ represents the following proposition being false.

The explanation

Consider a fish, an alive and healthy fish. To spark your imagination, I have used all of my artistic talent to provide a diagram:

Those who are expert marine biologists, may have noted, that my diagram (while beautiful) does not consider the full situation of a healthy fish. 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. I have worked hard to produce another diagram to show this relation:

Therefore, if we have a healthy fish, then we have a lake, or codified:

where $F$ represents having a fish, $L$ represents having a lake, and $\to$ represents the idea that having $F$ gives you $L$. Take a moment to really make sure this makes sense to you, both the notational and picture form:

Now, fish lovers may wish to stop reading here, and I implore them to do so! Bathe in the enjoyment that we have a healthy fish, and worry no more my friend.

However, in the unrelenting interest of science, we must now do something quite cruel… we must… take away the water :((. Expressed in picture form,

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. It seems, dear reader, that you were so preoccupied with whether or not you could, you didn’t stop to think if you should ².

The situation for our healthy fish has now changed significantly. Earlier we noted that in order to have a healthy fish, we needed a body of water for the fish. However, now we have no body of water, therefore, our fish is no longer healthy (it is likely dead). I provide a graphic (in two ways!!!) representation of this fact below:

While cruel, our actions have now illuminated the right hand side of the contrapositive rule. If we have no lakes, then our fish is not healthy, or codified

Here $\lnot$ is used to represent the idea of there not being something, so $\lnot L$ means we don’t have lakes, and brackets are there just to make it clear what the $\lnot$ applies to. Drawn in all of its horrifying detail:

Though our methods were unconventional, we have now managed to demonstrate an example of the contrapositive! If we have a healthy fish, then we have lakes. If we have no lakes, then the fish are not healthy.

Here $\leftrightarrow$ means that $F \to L$ being true gives us that $(\lnot L) \to (\lnot F)$ is true, and $(\lnot L) \to (\lnot F)$ being true gives us that $F \to L$ is also true.

Conclusion

This may remind you of the definition we explored at the beginning of the post

They look pretty similar no? We have merely codified what our intuition said :).

I hope this example was somewhat illuminating and entertaining. As always, any questions/comments are welcome below or by email: max (at) max-amb.com!