Logical puzzle

[micromass]

The recent thread of @ProfuselyQuarky reminded me of this fun puzzle. It's called the Wason's Selection Test. I stole it pretty obviously from the following website, so do check it out: http://www.philosophyexperiments.com/wason/ About 75%-80% get this wrong.

You are a quality control technician working for a card games manufacturer. You have to ensure that cards have been produced in accordance with the following rule:

If a card has a circle on one side, then it has the colour yellow on the other side.

You already know for certain that every card has a shape on one side and a colour on the other side. Please indicate, taking this into account, which card or cards you definitely need to turn over, and only that or those cards, in order to determine whether the rule is broken in the case of each of the four cards below.

Card 1:

Card 2:

Card 3:

Card 4:

 

[ProfuselyQuarky]

I never saw this one before, micromass, but I really like it . . .

Well, I'll see if I feel the same way once we're officially told the correct answer

 

[mfb]

3 correct answers so far and 0 wrong answers. We can exclude the hypothesis "at least 75% get it wrong" with p<0.02, but physicsforums users are not representative, so it should not be surprising.

 

[ProfuselyQuarky]

3 correct answers so far and 0 wrong answers. We can exclude the hypothesis "at least 75% get it wrong" with p<0.02.

3 can hardly be considered a worthy sample size, though :)

 

[mfb]

So what? You can discover something new with a single event, if it is significant enough.
Now 4:1 (with some assumption about votes, because now it is not obvious any more).

 

[micromass]

3 can hardly be considered a worthy sample size, though :)

Never mind that PF has a special audience. On the site they linked, they do get numbers like 75%-80% get it wrong.

 

[ProfuselyQuarky]

Okay, fine. I didn't click the link (and I kind of/sort of just jumped to the puzzle, too, without reading the first sentences of your post).

 

[zoobyshoe]

Spoiler

I think you have to turn over cards 2, 3 and 4.

#1 has a square showing, but there's no rule about squares to check.

Cards 2 and 3 have to be checked for obvious reasons, but card 4 also has to be checked to make sure there's no circle on the reverse.

 

[micromass]

Spoiler

I think you have to turn over cards 2, 3 and 4.

#1 has a square showing, but there's no rule about squares to check.

Cards 2 and 3 have to be checked for obvious reasons, but card 4 also has to be checked to make sure there's no circle on the reverse.

Why turn over card 3?

 

[mfb]

Cards 2 and 3 have to be checked for obvious reasons, but card 4 also has to be checked to make sure there's no circle on the reverse.

How could card 3 violate the rule? Which shape would lead to a violation?

 

[zoobyshoe]

Why turn over card 3?

Yes, you're right.

 

[mfb]

I checked the linked website. They have a variant with alcoholic drinks (question 3 of 3), and there the fraction that gets it right is significantly larger: "I don't have to check what the 22-year-old drinks, but I have to check what the 17-year-old drinks".

 

[micromass]

I checked the linked website. They have a variant with alcoholic drinks (question 3 of 3), and there the fraction that gets it right is significantly larger: "I don't have to check what the 22-year-old drinks, but I have to check what the 17-year-old drinks".

Indeed. I have used this example to explain the truth table of $P\Rightarrow Q$ to new students. It worked perfectly.

 

[mfb]

That one is quite interesting as well.
One word of warning: If the conclusion is "therefore, some [foo] are [bar]", they only consider this as valid conclusion if you know for sure that some [foo] exist. The English language is not precise enough for that.

 

[SophiaSimon]

Spoiler: Spoiler

Ahh, I was initially wrong. I picked 2, 3, and 4. But you don't have to check 3. That bright yellow got my attention and I ran with it, tricky, tricky.

 

[Psinter]

I see when you guys post these puzzles and tell myself that I don't care, but then you post the phrases x% or y% of people get it wrong. Because I don't want to be with the wrong side I stay and solve it only to discover I got it wrong, get mad, and leave the forum.

but physicsforums users are not representative, so it should not be surprising.

That's until Psinter comes and gets it wrong.

Anyway, can someone explain the following to me and tell me whether I'm getting it wrong? The websites says that since $p\rightarrow q$ we can conclude that $q\rightarrow p$. But I remember in a course a professor asking a question like that and his answer was: We cannot conclude that $q\rightarrow p$ just because $p\rightarrow q$. Yet the answer to this puzzle says that we can do that. How come? Am I wrong? I think the puzzle answer is wrong.

Edit: I found this: https://en.wikipedia.org/wiki/Affirming_the_consequent

According to that, one answer of the puzzle, according to the website, is wrong. Because it is affirming the consequent. I think. Someone correct me if I'm wrong.

Spoiler: Affirming the consequent

It says: If a card has a circle on one side, then it has the colour yellow on the other side.

Then it says that because it has yellow, it must have a circle on the other side. That's affirming the consequent, you don't know whether your statement will be true.

 

[Enigman]

I see when you guys post these puzzles and tell myself that I don't care, but then you post the phrases x% or y% of people get it wrong. Because I don't want to be with the wrong side I stay and solve it only to discover I got it wrong, get mad, and leave the forum.

That's until Psinter comes and gets it wrong.

Anyway, can someone explain the following to me and tell me whether I'm getting it wrong? The websites says that since $p\rightarrow q$ we can conclude that $q\rightarrow p$. But I remember in a course a professor asking a question like that and his answer was: We cannot conclude that $q\rightarrow p$ just because $p\rightarrow q$. Yet the answer to this puzzle says that we can do that. How come? Am I wrong? I think the puzzle answer is wrong.

Edit: I found this: https://en.wikipedia.org/wiki/Affirming_the_consequent

According to that, one answer of the puzzle, according to the website, is wrong. Because it is affirming the consequent. I think. Someone correct me if I'm wrong.

Spoiler: Affirming the consequent

It says: If a card has a circle on one side, then it has the colour yellow on the other side.

Then it says that because it has yellow, it must have a circle on the other side. That's affirming the consequent, you don't know whether your statement will be true.

What the website actually says is :

Spoiler: Website quote

People almost always recognise that they have to turn over the card with the circle (P), but they fail to see that they also have to turn over the card with the colour red (not-Q). It is also common for people to think - mistakenly - that they have to turn over the card with the colour yellow (Q).

Spoiler: Also

You don't have to turn over the yellow card because it doesn't matter if it has a circle on the other side or a square. The rule is "P implies Q" and not "if and only if P then Q"

 

[reenmachine]

I see when you guys post these puzzles and tell myself that I don't care, but then you post the phrases x% or y% of people get it wrong. Because I don't want to be with the wrong side I stay and solve it only to discover I got it wrong, get mad, and leave the forum.

That's until Psinter comes and gets it wrong.

Anyway, can someone explain the following to me and tell me whether I'm getting it wrong? The websites says that since $p\rightarrow q$ we can conclude that $q\rightarrow p$. But I remember in a course a professor asking a question like that and his answer was: We cannot conclude that $q\rightarrow p$ just because $p\rightarrow q$. Yet the answer to this puzzle says that we can do that. How come? Am I wrong? I think the puzzle answer is wrong.

Edit: I found this: https://en.wikipedia.org/wiki/Affirming_the_consequent

According to that, one answer of the puzzle, according to the website, is wrong. Because it is affirming the consequent. I think. Someone correct me if I'm wrong.

Spoiler: Affirming the consequent

It says: If a card has a circle on one side, then it has the colour yellow on the other side.

Then it says that because it has yellow, it must have a circle on the other side. That's affirming the consequent, you don't know whether your statement will be true.

$P \rightarrow Q$ doesn't have the same truth table as $Q \rightarrow P$, but it does in the case of $\neg Q \rightarrow \neg P$.

Spoiler: Website quote

So what you have to do is flip the card when the premise of either $P \rightarrow Q$ or $\neg Q \rightarrow \neg P$ is true to see if the conclusion is false, because that's when the rule is broken.

 

[Borek]

@micromass - I think you skewed the result by referring to the ProfuselyQuarky post. Everyone who has seen the other thread thinks twice before answering here.

 

[PeroK]

And in the general population there would not necessarily be a clear understanding of precisely what the rule means.

Spoiler

It would be easy to interpret the rule (in an everyday use of the word "if") to mean that circle and yellow go together. In logical terms to interpret "if" as "if and only if". It's only doing maths or computer programming that leads to a concrete interpretation of something like this.

With the drinking example, the everyday context excludes the "if and only if" interpretation. People naturally interpret this example as "if".

This is perhaps as much about language interpretation than logic.

 

[zoobyshoe]

This is perhaps as much about language interpretation than logic.

Along the same lines, I felt it was about psychological priming. The instruction or rule is 'pseudo explicit'; it's written in explicit sounding language but actually fails to be explicit:

"Every X must be paired with a Y."

This pseudo-explicit language primes us to think this is the whole story, and strongly suggests the relationship is commutative, that "Every Y must be paired with an X."

An authentically explicit instruction would be, "Every X must be paired with a Y, but every Y does not need to be paired with an X." It is the lack of explicit full disclosure, so to speak, that leads people to assume the rule can be paraphrased, "Xs and Ys must always be paired with one another."

The sites explanation:

According to Leda Cosmides and John Tooby, the results of the Wason Selection Task demonstrate that the human mind has not evolved reasoning procedures that are specialised for detecting logical violations of conditional rules.

Is bogus, IMO. The error results from priming, from assumptions we're primed to have about how explicit instructions or rules are expected to be.

 

[Psinter]

What the website actually says is :

Spoiler: Website quote

People almost always recognise that they have to turn over the card with the circle (P), but they fail to see that they also have to turn over the card with the colour red (not-Q). It is also common for people to think - mistakenly - that they have to turn over the card with the colour yellow (Q).

Spoiler: Also

You don't have to turn over the yellow card because it doesn't matter if it has a circle on the other side or a square. The rule is "P implies Q" and not "if and only if P then Q"

$P \rightarrow Q$ doesn't have the same truth table as $Q \rightarrow P$, but it does in the case of $\neg Q \rightarrow \neg P$.

Spoiler: Website quote

So what you have to do is flip the card when the premise of either $P \rightarrow Q$ or $\neg Q \rightarrow \neg P$ is true to see if the conclusion is false, because that's when the rule is broken.

Nevermind, after much thinking I realized that the answer is right.

Spoiler

It was the wording of the explanation as to why the yellow card shouldn't be picked what threw me off. I thought that because it said:

The[/PLAIN] [Broken] card is showing yellow, therefore, we already know that the rule is upheld.

It was affirming the consequent with that statement. And also when it said:

We[/PLAIN] [Broken] already have the requisite yellow, so there's no problem even if there's a circle on the other side of the card.

I thought it should have said: ...there's no problem even if there's not...

Edit: But thanks for taking the time to reply.

 

[PeroK]

Nevermind, after much thinking I realized that the answer is right.

Spoiler

It was the wording of the explanation as to why the yellow card shouldn't be picked what threw me off. I thought that because it said:

It was affirming the consequent with that statement. And also when it said:

I thought it should have said: ...there's no problem even if there's not...

Or:

Spoiler

If you turn the yellow card over, what could you find that would break the rule?

 

[Psinter]

Or:

Spoiler

If you turn the yellow card over, what could you find that would break the rule?

Yup. I understand it a hundred times better when it is worded that way.

 

[Fervent Freyja]

Spoiler

I think you have to turn over cards 2, 3 and 4.

#1 has a square showing, but there's no rule about squares to check.

Cards 2 and 3 have to be checked for obvious reasons, but card 4 also has to be checked to make sure there's no circle on the reverse.

I thought the same...

 

[mfb]

The poll doesn't show individual vote sets (but it should store them). Assuming everyone selected at least card 2, we have 22 answers and between 12 to 14 (inclusive) of them are correct.

 

[twiz_]

Is it logical to assume the converse of that statement is true?

Whether it does or not makes or breaks my answer

Spoiler

However, I did end up solely picking 2, assuming that only the original statement is true.