Episode 44: I’m not just positive, I’m contrapositive

P —> Q = -Q—> -P

Now that we’re all clear on that, I guess we can call it day and go home, right?

No idea what I’m talking about, you say?  You think I started drinking too early, maybe?  That second one might be true, actually.  Let’s say that you are like me and have to shed a few pounds.  All the cookies did exactly what everyone told us and we gained some weight.  We could try dieting, but blah, that’s no fun.  Why not do some weight lifting and working out?  I got myself a small weight set, bench, mat, stability ball and went to it.  There’s a satisfaction in weightlifting, feeling the exertion, knowing I’m getting stronger.  Pretty soon I’ll be able to lift even the heaviest bag of cookies.  Then I get on the mat, do some push-ups, planks and use the stability ball for those great core workouts.  Yeah, good stuff!

I do this

Contraposition No Logo Cropped.png

because I’ve heard that if you want to lose weight you need a good fitness routine.  It follows then that if I work out regularly, I’ll lose weight.  Seems totally logical, right?  The weeks roll by, I’m certainly getting stronger and more flexible.  But I also notice I’m not really losing much weight.  Heck, I might have even gained a bit.  What happened?  Well, it was probably the lack of dieting, but this isn’t a healthy living blog, it’s a critical thinking blog.  We had some logic issues too.

I started this thinking that working out would make me lose weight.  Another way to look at this and test the logic is to say “If I did not lose weight, then I did not work out”.  Clearly, we know this is false as I did work out.  This is called a contrapositive and it’s a great way to test the truth of a statement.  If the contrapositive of the statement is false, then the original statement is false too.

What about

all that gibberish up top?  That is called symbolic logic and it’s how we show the idea of contraposition.  Much like the previous article on Fallacy of the Undistributed Middle, I thought this would give us a further introduction to thinking about statements logically.  If you read that article, you learned it was quite easy and this idea of contraposition is no different.

P and Q are the two parts of our statement.  P is equal to “work out” and Q is equal to “lose weight” with the arrow —> meaning “then” or “therefore”.  P —> Q is simply a way of saying “workout then lose weight” with symbols.  Easy as pie, right (although maybe with the tone of this article we should stay away from pie).  The second half is the contraposition, which means the reversing and negation of the original statement.  The dash symbol means “not”.  Since we have already defined P and Q, it’s easy to read -Q —> -P as “not lost weight then not worked out”. This we know is false because I was working out but I didn’t lose weight. Hopefully now that gibberish is clear.

We saw that analyzing the contrapositive of our weight loss technique showed us the logic was wrong.  However, consider this: human beings need to breathe air to survive.  If we examine the contrapositive we get: if it does not need to breathe air to survive then it’s not a human being.  This is certainly true.  If something exists that does not need air to live, then it is definitely not a human.  We’ve used the contrapositive to test the truth of our statement, and found it was true.  This was a lot easier than attempting to test it by not breathing – I don’t recommend that.

I have found many statements in life which, by using their contrapositive, can be proven false.  How often have you heard “Work hard and you’ll succeed”?  Now you know how to check that statement: if you did not succeed then you did not work hard.  However, maybe you worked really hard at making a new line of pencil toppers, but the market just wasn’t there for it anymore.  These types of statements also fall into the Ambiguity Fallacy because what is meant by “work hard” and “success” were never exactly defined.  There are many instances of this, such as “If you practice every day, you’ll be a great chess player”.  When we use our logic, we see that if you aren’t a great chess player then you didn’t practice every day.  Maybe you did though and everyone else just practiced twice a day or maybe there’s more to playing chess at a high level than just daily practice. Another statement proven false.

Today you took a step

into seeing how to apply some formal logic to evaluate the statements given to you.  You can see that such analysis doesn’t require a PhD in Philosophy or writing out pages of inscrutable scribblings.  Understanding the logic behind a statement and ways to check if those statements are true give us another quick and easy tool in our critical thinking toolset.  As you learn to think like this it becomes easier and almost second nature.  When I started this blog nearly a year ago, I found I had to think hard about some statements or I wasn’t able to quickly analyze and refute a point made.  A year later, I find it much easier.  Take your time, think over the concepts and start applying them.  If you want to go deeper into symbolic logic, trust me, it’s a journey.  I tried to keep this light and easy.  I think that’s enough for today, I’m off to the grocery store to find the heaviest bag of broccoli there is.