Elementary, My Dear Wason!

The Wason Selection Test Explained

(Maybe, Sort of, Perhaps)

(In Defense of the Kids)

There is a famous psychology experiment that was first run in the 1960's called the * Wason Selection Test*. Yes, that's

The discourteous say the test shows us that college kids just can't think (and most of the test subjects are university students). More charitable critics suggest the strange results are because of the way our brains are hardwired. Or perhaps its the way our reasoning is shaped by our society.

Well, before we go further, perhaps we should say just what the Wason Selection Test is.

Not Quite That Elementary

All right. Take four cards. Two of the cards are blue on one side and the other two are red. On the other side are numbers.

Next we lay the cards down with the number and color sides alternating like this:

7 | 4 |

Then we make this statement:

If the color on one side is red, the number of the other side is odd.

And we ask the following question:

Which cards must be turned over to see if the statement is TRUE?

We mean, of course, only the cards that are * necessary* to see if the statement is TRUE. You may need to turn over all the cards. But then maybe not. We just want the minimum.

The Wason Selection Test is, the psychologists tell us, a test for "abstract" reasoning. That is, there is no real-life application here. It's just a test on how well you think.

And yet 80% of college kids - quote - "get the answer wrong" - unquote. Hence the conclusion that college kids can't think.

But before we show you the - quote - "correct answer" - unquote - we'll look at the test again.

Interesting, Though Elementary

But not as an abstract problem. We pose it in a more practical - even lowbrow - manner.

So now look at * these* cards:

25 | Pepsi | 14 | Schlitz |

And this statement:

If you have a beer, then you must be at least 18.

Again we ask:

Which cards must be turned over to see if the statement is TRUE?

This problem, the professors tell us, is exactly the same as the first one.

But in this case **almost everyone gets the answer right!**

And their reasoning goes like this:

You don't need to turn the first card over. If you're an adult it doesn't matter what you drink. |

You don't need the next card either. If you drink a soda, you can be a kid or an adult. |

But you do need to turn over the third card. After all, if someone is fourteen, you have to make sure they're drinking a soda. |

You also need to turn over the last card. That is, if someone's drinking a beer, you must make sure they're an adult. |

So you only have to turn over the last two cards.

It Is Not Logical, Captain.

So why is the first problem - the "abstract" version - so hard to figure out and the second is so easy?

Well, one popular explanation is that the human mind evolved to detect cheating. And as the second problem has a possibility of cheating - an underage customer trying to sneak a swig - it's easy to figure out.

But the abstract version has no possibility of cheating. Ergo, the mind doesn't figure the problem out very well.

Now one author has said that if you read articles about the Wason Test, you * might* think the "it-might-be-cheating" answer has been universally accepted. That, though, is not really true. There is no real consensus as to what's going on.

But we've left out one other question.

Just what the heck * are* we testing?

It's How You Tell It

Note the way we phrase our statements.

IF the back of the card is red, THEN the number on the front is odd. |

IF you have a beer, THEN you must be at least 18. |

The Wason Selection Test, then, is testing if the subjects can correctly reason using a logical * IF-THEN* statement. In formal logic, an

Material implication is a statement with the form:

IF [Statement A], THEN [Statement B].

... which is usually represented by:

A → B

... and is sometimes read as "A implies B".

Now according to logic professors, an * IF-THEN* statement is only FALSE when the

We can summarize the * IF-THEN* statement - or rather

Truth Table: If-Then Statements (Material Implication) | ||

A | B | A → B |

TRUE | TRUE | TRUE |

TRUE | FALSE | FALSE |

FALSE | TRUE | TRUE |

FALSE | FALSE | TRUE |

And returning to the "abstract" Wason Test, we can write down the Truth Table as:

Truth Table: "Abstract" Wason Selection Test | ||

The back of the card is red. | The number is odd. | If the back of the card is red, then the number is odd. |

TRUE | TRUE | TRUE |

TRUE | FALSE | FALSE |

FALSE | TRUE | TRUE |

FALSE | FALSE | TRUE |

So let's look at the problem again:

7 | 4 |

If the back of the card is red, then the number on the front is odd.

So guided by the Truth Table, we can reason like this:

The first card has a 7. Since 7 is odd, the THEN part is TRUE. Therefore the overall IF-THEN statement is always TRUE. So you don't need to check the other side of this card. |

The second card has a blue side. Therefore the IF part of the statement is FALSE. So the overall IF-THEN statement is always TRUE and you don't need to check the other side of this card, either. |

The third card has an even number. So the THEN part of the statement is FALSE. So the whole IF-THEN statement may be TRUE or it may be FALSE. To determine which, we must check the color on the other side. We have to turn over this card. |

The fourth card has a red side. So the IF part of the statement is TRUE. So to determine if the IF-THEN statement is TRUE we must check the number on the other side. We have to turn this card over. |

So we only have to check Cards 3 and 4, just as in the practical test.

It's important to realize that although we first used informal reasoning for the practical version of the test, you can easily confirm that using the Truth Table gives you the same answer. The problems are indeed equivalent - or rather as philosophers say, they are * isomorphic*.

So when thinking out an * IF-THEN* problem, you only need when to know when the

Seems easy, doesn't it?

Give Me That Old Time Philosophy

Ain't Necessarily So

So if the Wason Selection Test is so easy and 80 % of college kids get it wrong, then it really does seem that college kids can't reason, doesn't it?

Well, to paraphrase Sportin' Life, this ain't necessarily so.

The sad truth is the * IF-THEN* Truth Table has caused considerable consternation for - to misquote Carl Sagan -

So what, we ask, is the beef?

The beef is that with our Material Implication * IF-THEN* Truth Table, you can come up with weird statements. Consider these sentences which the Truth Table tells us are all TRUE.

Truth Table: Weird IF-THEN Statements | ||

A → B | Truth Values | Overall Truth Value |

If George Washington was the first King of France, then Napoleon would have been the first US President. | FALSE → FALSE | TRUE |

If I break my leg today, then I will climb Mount Everest. | FALSE → FALSE | TRUE |

If I break my leg today, then I will NOT climb Mount Everest. | FALSE → TRUE | TRUE |

If 1 + 1 = 3, then C. S. Lewis wrote The Lion, the Witch, and the Wardrobe. | FALSE → TRUE | TRUE |

Well then, if philosophers don't like Material Implication, why don't they get rid of it?

Well, some have been working on it. But how to do it isn't so easy.

One obvious possibility is to allow three TRUTH values: TRUE, FALSE, and UNKNOWN. You can also try to mandate that the * IF* and

But so far no modifications of the * IF-THEN* conundrum has worked out very well. You end up with even bigger and more complex Truth Tables, and you still end up with oddball sentences.

So like it or not, the dang * IF-THEN* Truth Table is here to stay. That's because it really, really works (to see why it works, just click here.)

But what does any of this * brouhaha* have to do with the Wason Selection Test?

Changing the Rules

We mentioned that the Wason Selection Test is usually administered to college kids by professors. Naturally the professors use the professorial * IF-THEN* Truth Table we gave above. In other words, for the kids to get the "correct" answer, they have to use

There's just one wee-little problem, though.

IF-THEN statements in everyday conversation are not necessarily the IF-THEN statements of Material Implication. |

And this really goobers things up. Particularly since:

The professors don't tell the kids they are defining an IF-THEN statement as Material Implication. |

They don't teach the kids the Material Implication Truth Table. |

It Ain't (Bi)Conditionally So

Well, if * IF-THEN* statements in ordinary conversation aren't the same as Material Implication, we'd like to see what they are.

I thought you would as Captain Mephisto said to Sidney Brand. It's very simple really.

Let's say that a new restaurant is opening up. And the owners advertise the following:

If you are the 100th guest of the night, then you get your dinner free! |

What they mean, of course, is that * only* the 100th guest will get their dinner free. Any other guest has to pay. We understand what this "Conversational"

So now let's set up the * Truth Table* for what we will call our

Truth Table: Conversational IF-THEN | |||

Guest Number | Dinner Cost | 100th Guest → Free Dinner | If you are the 100th guest then you get your dinner free. |

100 | Free | TRUE → TRUE | ? |

100 | $25 | TRUE → FALSE | ? |

99 | Free | FALSE → TRUE | ? |

101 | $25 | FALSE → FALSE | ? |

Now we'll start reasoning from Line #1 and work down.

You were the 100th guest and you also got your dinner free. So the overall statement is TRUE. |

You were the 100th guest but you didn't get your dinner free. So the overall statement is FALSE. |

You were the 99th guest and you got a free dinner. Although you might be happy with this if you were the 99th guest, this isn't what the advertisement meant. So in the context of the ad, the overall statement is FALSE. |

You were the 101th guest and didn't get a free dinner. Bad luck, yes, but the overall statement is still TRUE. |

So the final Truth Table for this "Conversational" * IF-THEN* Statement is:

Truth Table: Conversational IF-THEN | |||

Guest Number | Dinner Cost | 100th Guest → Free Dinner | If you are the 100th guest then you get your dinner free. |

100 | Free | TRUE → TRUE | TRUE |

100 | $25 | TRUE → FALSE | FALSE |

99 | Free | FALSE → TRUE | FALSE |

101 | $25 | FALSE → FALSE | TRUE |

Note that with Material Implication the owner would have the option of giving any other guest a free dinner as well. But we understand that the advertisement really means only one prize will be given. That is, only one guest - the 100th - will be the one to eat for nothing. So from the * context* we know this particular

But what kind of * IF-THEN* Statement is it?

Well, turn to a logic textbook (if we can find one we can afford) and look up the Truth Table for the type of statement the author calls a * Biconditional*.

Biconditionals will be familiar if you've taken a math class. They are commonly used to express theorems that say such-and-such is TRUE * IF-AND-ONLY-IF* this-and-that is TRUE

A biconditional then is the same as an * IF-AND-ONLY-IF* statement. For this reason they are often abbreviated as

A IFF B

... or:

A ↔ B

And the Biconditional/IF-AND-ONLY-IF Truth Table is:

Truth Table: Biconditional (If-And-Only-If) Statements | ||

A | B | A ↔ B |

TRUE | TRUE | TRUE |

TRUE | FALSE | FALSE |

FALSE | TRUE | FALSE |

FALSE | FALSE | TRUE |

Yep, exactly what we were calling the "Conversational * IF-THEN* Statements".

The point, then, is that in ordinary conversation no one says "if-and-only-if". We just say "if". In fact, "if" in most conversations probably really means "if-and-only-if".

There is another way to understand the * IF-AND-ONLY-IF* statement as it relates to Material Implication. If only one prize is given and the 99th guest gets the dinner, but not the 100th, then the antecedent of the statement, "If you are the 100th guest then you get your dinner free" is TRUE and the consequent is FALSE. So

But in any case, whether we mean * IF-THEN* as Material Implication or

Back to the First

But remember the first problem? The * Abstract Wason Selection Test*?

Because the problem was an abstract test, there was, in fact, no context. So it's not clear to the kids what * IF* to use. Instead, they'll probably they'll use the Biconditional because that's what they most use in their everyday conversation.

OK, let's summarize what goes on in the Wason Selection Test:

The professor gives the kids a test asking to determine if an "IF-THEN" statement is TRUE or FALSE. |

He doesn't explain to them that the IF-THEN in the test is not the way they normally use IF-THEN. |

The kids get the wrong answer. |

That is, they get the answer wrong according to the professor.

Dr. Wason, What Does It All Mean?

Now lest we draw the ire of college professors everywhere, we should point out that the Wason Selection Test certainly shows us that people reason better when confronted with a problem in a familiar context. And sometimes "if" of ordinary speech does indeed mean the * IF-THEN* as in the logic textbooks.

But are contexts that involve deception easier to figure out? Well, that's still being debated, and the reader might want to experiment by creating some their own Wason Tests. (And to see another "practical" Wason Test - which may or may not be easy to solve - you can can click here).

But one thing the Wason Test * does* tell us is about

So a Humble CooperToons Suggestion is that a * New and Improved Wason Test* should be given as follows:

Give the test to the kids as usual, both the "abstract" and practical versions. |

Next give them a lecture on formal Material Implication pointing out it is only false when the IF part is TRUE and the THEN part is FALSE and that this is not the usual IF-THEN of ordinary conversation. |

Give them the abstract Wason Test again and see if more of them get it right. |

That might make the coming to conclusions a bit more elementary.

References

"Conditional Reasoning and the Wason Selection Task: Biconditional interpretation Instead of Reasoning Bias", Pascal Wagner-Egger
*Thinking and Reasoning*, Volume 13, Issue 4, pp. 484 - 505, 2007.

"Elementary, My Dear Wason: The Role of Problem Representation in the THOG Task", Cynthia S. Koenig and Richard A. Griggs, *Psychological Research*, Volume 65, Issue 4, pp. 289 - 293, 2001. This seems to be the first documented of the horrible pun but it seems to have occurred to others independently.