### Frequently misunderstood logic: modus tollens

Back when I was in upper school I studied Further Mathematics and one of the topics I loved was logic. We learnt about syllogism and modus ponens, and my favorite modus tollens.

Modus tollens is fun because it is often applied incorrectly in informal arguments to come to the wrong conclusion. Here's an example.

The other day I saw a tweet which read:

"A fear of weapons is a sign of retarded sexual and emotional maturity." - Sigmund Freud, General Introduction to Psychoanalysis (1952)

Simplifying that a little (and assuming we agree with Freud) it can be written "If a person is fearful of weapons, then that person is sexually and emotionally immature". If we write "fearful of weapons" as F and "sexually and emotionally immature" as I, this statement can be rewritten F => I (the => is read as implies and make the entire statement read as F implies I or If F, then I).

So if we come across someone who satisfies F (i.e. they fear weapons) then we know that I applies (i.e. they are sexually and emotionally immature).

Modus tollens tells us that if we come across someone who does not satisfy I (i.e. that person is sexually and emotionally mature) then we know that they do not satisfy F (i.e. they do not fear weapons). Symbolically that would be written ~I => ~F.

Modus tollens tells you that if the "then" side of an "if, then" is false then the "if" side must be also (this has to be the case because the "if, then" forces the "then" side to be true when the "if" side is true).

The common fallacy related to modus tollens is to think that ~F => ~I follows from F => I. That is, given Freud's statement some people will believe that the statement "People who are not fearful of weapons are sexually and emotionally mature". This is called the Denying the antecedent fallacy.

Next time you come across someone who doesn't fear weapons you'll know that Freud tells us nothing about their sexual or emotional maturity.

Unknown said…
"it is often applied incorrect in informal arguments"

See my post on "frequently misunderstood differences between adjectives and adverbs" ;)

### Your last name contains invalid characters

My last name is "Graham-Cumming". But here's a typical form response when I enter it: Does the web site have any idea how rude it is to claim that my last name contains invalid characters? Clearly not. What they actually meant is: our web site will not accept that hyphen in your last name. But do they say that? No, of course not. They decide to shove in my face the claim that there's something wrong with my name. There's nothing wrong with my name, just as there's nothing wrong with someone whose first name is Jean-Marie, or someone whose last name is O'Reilly. What is wrong is that way this is being handled. If the system can't cope with non-letters and spaces it needs to say that. How about the following error message: Our system is unable to process last names that contain non-letters, please replace them with spaces. Don't blame me for having a last name that your system doesn't like, whose fault is that? Saying "Your

### All the symmetrical watch faces (and code to generate them)

If you ever look at pictures of clocks and watches in advertising they are set to roughly 10:10 which is meant to be the most attractive (smiling!) position for the hands . They are actually set to 10:09.14 if the hands are truly symmetrical. CC BY 2.0 image by Shinji I wanted to know what all the possible symmetrical watch faces are and so I wrote some code using Processing. Here's the output (there's one watch face missing, 00:00 or 12:00, because it's very boring): The key to writing this is to figure out the relationship between the hour and minute hands when the watch face is symmetrical. In an hour the minute hand moves through 360° and the hour hand moves through 30° (12 hours are shown on the watch face and 360/12 = 30). The core loop inside the program is this:   for (int h = 0; h <= 12; h++) {     float m = (360-30*float(h))*2/13;     int s = round(60*(m-floor(m)));     int col = h%6;     int row = floor(h/6);     draw_clock((r+f)*(2*col+1), (r+f)*(row*2+1),

### The Elevator Button Problem

User interface design is hard. It's hard because people perceive apparently simple things very differently. For example, take a look at this interface to an elevator: From flickr Now imagine the following situation. You are on the third floor of this building and you wish to go to the tenth. The elevator is on the fifth floor and there's an indicator that tells you where it is. Which button do you press? Most people probably say: "press up" since they want to go up. Not long ago I watched someone do the opposite and questioned them about their behavior. They said: "well the elevator is on the fifth floor and I am on the third, so I want it to come down to me". Much can be learnt about the design of user interfaces by considering this, apparently, simple interface. If you think about the elevator button problem you'll find that something so simple has hidden depths. How do people learn about elevator calling? What's the right amount of