A Simple Logical Puzzle That Shows How Illogical People Are | Hacker News Reader

A Simple Logical Puzzle That Shows How Illogical People Are

nautil.us

46 pointsdnetesn60 comments

I think the vast majority of the confusion in this problem stems from the ambiguity of "if (but not only if)" versus "if and only if". In normal speaking people use the word "if" for both meanings, but logic puzzles always use it for just the first (having developed the "iff" shorthand for the second). I really doubt that the "socio-cognitive niche" of a cop in a bar is substantially more effective than simply adding "(but not only if)" to the original problem.

I really doubt this is the case. I don't think there would be any confusion if before a roadtrip I said something like "If we get hungry, we'll make a stop along the way". Nobody would assume that that precludes the possibility of making a stop to pee or something like that.

> I really doubt that the "socio-cognitive niche" of a cop in a bar is substantially more effective than simply adding "(but not only if)" to the original problem.

You might be correct about this, but I think that would be more of a function of specifically highlighting part of the problem and thus making it harder to skip over it in one's reasoning.

If you clean my house, I'll give you 50e. Do you think i'd give you 50e out of the blue if you refused to do the job ?

I feel this area of langage is one full of implicit meaning and shared context. Because in your example it is indeed a non exlusive if, while mine is.

Exactly, and this makes this purportedly logic problem a pyschologcial or sociological problem that is only tangentially related to the nominally pure logic of the question. Since we can with equal legitimacy infer either an exclusive or non-exclusive meaning to the "if", how we answer the question depends entirely on that choice.

Amongst undergraduates it is not implausible that when you tell them "this is a logic problem" they will tend to assume--based on the norms of undergraduate thought--that the researcher means "iff" when they say "if". I certainly would have as an undergrad.

To pretend that interpreting "if" as "iff" in an unusual situation is an error, is an error. The test subject is forced to guess what the researcher means, and for all kinds of reasons (not wanting to look stupid, etc) is unlikely to simply ask, "Do you mean if-and-only-if or not?"

Almost all supposed "logic problems" are in fact psychological problems of this kind. Any problem of the form, "What is the next number in this sequence..." for example, has an infinite number of valid answers, since no finite sequence deteremines its next element. Solving such problems requires making an informed guess about the state of mind of the person who designed the sequence, and infering which of the infinite number of possible patterns they thought was the obvious one.

> If you clean my house, I'll give you 50e. Do you think i'd give you 50e out of the blue if you refused to do the job ?

Ignoring the "out of the blue" part, which is entirely off-topic, actually yes: "How about if I wash your car?". As mentioned downthread, it would be a hugely bizarre interpretation to assume that there are no circumstances under which you would ever give me 50e other than cleaning your house. In this example, it's trivially possible that there are other jobs that I can perform that you might give me 50e for.

No. But also, I wouldn't assume that the only reason that you would conceivably give me 50e is because I cleaned your house.

The problem is that the question is poorly written, so it taps into the human brain's desire to extrapolate.

By saying "a card" in the general sense, it sounds like there might be more cards involved, and we are supposed to derive a rule for a larger set.

If you explicitly state that the only cards you are talking about are the cards shown here, things become a lot more clear.

To clarify the question: "When one of the cards here has an even number on one face, then its opposite face is blue. True or false?"

In my experience, it seems like lot of interviewers like to make themselves come across as "clever" by relying on poorly phrased questions which can be easily misinterpreted.

As I understand this problem, the challenge is determining the unclear phrasing of the task.

It could be non-commutative, which is how the professor set it up. (Even = blue on back side. Flip 8)

It could be commutative (I have to prove above + blue = even. Flip 8 and blue. How most people interpret)

It could even be strict (even = blue && odd != blue && green != even. Flip all)

As phrased, it's super vague, and since it's implied the problem is tricky, most people select the goal that at least includes the words (even + blue). The double brain mumbo-jumbo in the article seems like garbage. It's just a statistical goal assignment task with fuzzy bounds.

In my experience, that's not the case. People (or at least people not primed to take a tricky-logic-puzzle stance) think they understand the problem just fine, and quickly give the wrong answer. Asking about it afterwards (once you get past the defensive denials et al), it's usually revealed that they did in fact understand the problem. They just failed to grasp what a proof/disproof actually requires, because they've never had to think too deeply about the issue before.

Actually the way they mean it is simple:

IF even THEN blue.

To show that the rule holds then you need to check:

8 (it's even, so should be blue. Is it?)

and green (because if it's even, then it invalidates the rule that it should have been blue on the other side).

The other cards/numbers don't matter as to the validity of the rule.

The non-commutative interpretation requires that you also flip the green card. I think a lot of people failed the question even if correctly interpreting the words.

It's only unclear if you don't understand that if/then is not commutative. If I said that x/y = z, it wouldn't be "unclear" whether that implies that y/x = z, because the division operator simply isn't commutative.

The problem with this is that the 5 card also needs to be turned; it might have an even number on the back.

But if you take them at their word that each card has only one number, then yeah, it's 8 and green.

I guess I'm saying the article didn't make it clear that you could trust anything you didn't see in front of you.

>The problem with this is that the 5 card also needs to be turned; it might have an even number on the back.

"Each card has a number on one side and a color on the other side."

So 5 could not have an even number on the back given the rules. But your reasoning would make sense if you missed that statement.

The confusion doesn't arise from missing the statement you quoted, but whether this is to be treated as part of the problem statement that you are testing for truth.

If you can take it as absolute truth that each card has a number on one side and color on the other, then you only have to check which color. If on the other hand you must verify both that each card has one color/one number and that even cards have blue on the back, then you must turn over every card to ensure that each has the expected quality of opposite side (i.e. that if you see a number, the opposite side has a color and vice versa).

You are told each card has a number on one side and a color on the other. The statement is mentioned in other words in the article - but what I quote is from the very first sentence of the YouTube video used to demonstrate the problem.

The confusion comes from making an assumption that if a card is odd then it cannot also be blue.

  if even then blue //what is being asked
  if even then blue && if odd then !blue //what people assume

I understand the logic and confusion you are presenting. The confusion that I am referring to is what portion of the problem statement can be trusted and what must be verified.

The situation follows:

1) There are four cards with a number on one side and a color on the other.

2) If a card has an even number on one side, then it's blue on the other.

You are only allowing for confusion about the commutative nature of the conditional in (2); it is not commutative, but a common false assumption is that it is.

What the parent of this thread is referring to is that there can be confusion regarding whether (1) must also be verified. If (1)must be verified then every card must be flipped.

(1) only needs to be verified when it is not a part of the problem.

"If Jenny has 5 apples and Bob has 5 apples and Jenny gives Bob 2 apples - how many apples does Jenny now have?"

We don't question if Jenny went and bought n more apples to give to Bob and might have 3 Apples or 5 Apples or 5+n where n is the number of apples purchased minus 2 because it isn't part of the problem and isn't part of what is being asked. Maybe she won an apple lottery and that is why she gave Bob 2 apples. Maybe she gave Bob 2 apples and later lost her other 3 apples in a fire.

If you begin by questioning the validity of the question - Jenny can have 0...n apples. Which doesn't do us any good for solving the problem.

"There are four cards. Each card has a number on one side and a color on the other side." These details are taken as facts as a basis for solving the problem.

If you are questioning if there is a color and number on the sides. Are there really 4 cards? What if there is a 5th card I'm not told about? What if we took 4...n cards and I found a card that invalidates the problem? It's ignoring the problem and coming up with a new one to solve.

We are presented with an actual situation. 4 visible cards on a table in front of us. I am not going full Descartes and claiming we cannot trust anything we see.

We are told two things about the situation. You are assuming the first thing we are told can be trusted and the second can't.

I understand that the logic puzzle is only to verify the second thing we are told. I am not confused. The way that the puzzle is presented, though, allows for confusion, because we are presented with three things: 1) A physical setup which can be trusted (4 cards exist, we see 1 side of each card) 2) A statement that each card has one color and one number 3) Even -> blue on other side.

Given that one of the things we are expected to trust is a statement made to us and the thing we are expected to verify is a statement made to us, there is potential for confusion as to which statement(s) we are to verify. This potential is all that OP raised and all that I have been trying to explain.

Edit: To summarize in brief, as the problem is presented, it is not 100% clear that some statements told to you by the experimenter can/should be trusted.

Ah, I see the source of your confusion now.

The actual situation as given to us is not that there are 4 visible cars on a table. It is that there are 4 visible cars on a table that contain a number on one side and a color on the other side.

We are being told to verify the statement:

"Of the 4 cards placed on the table, which have a number on one side and a color on the other side, if the number is even then the other side must be blue."

If the color/number statement was left out of the original problem/video (it isn't) then your argument would be true, as it would be an assumption that is not listed and we would need to flip the 5 as well.

And I suppose the blue card since it could have green on the other side.

It seems, though that I was misinterpreting the original parent on this thread, though.

I don't retract anything I said, but apparently his confusion was different.

>If (1)must be verified then every card must be flipped.

Actually, only the 5,8 and green. The blue is good no matter what.

I saw this pedanticism in a Martin Gardner book originally, IIRC.

If (1) must be verified, how can you know that the blue card has a number on the other side?

To verify (1) you must prove that each card has a color on one side and a number on the other. The only way to do this is to see both sides of each card.

Oh, you mean actually verifying 1. I wasn't doing that. I was just doubting 1 being true, but interpreting the problem as asking you to verify 2, in which case you only need to turn over 5,8, and green.

Okay, now I'm really confused about what you meant with your original post in this thread.

If you doubt (1), how is it that you need to verify that the 5 has a color, but not that the blue has a number on the other side?

To verify (2) you only need to prove that every even card has a blue back. An odd may be blue or green, but an even must be blue. Hence the 5 is fine, it can have any color. The blue is fine, because it can have any number. But that 8 must have a blue back, and that green must have an odd number.

>The problem with this is that the 5 card also needs to be turned; it might have an even number on the back.

If you can't be sure that the cards have a color and a numbers, there's nothing outlawing a 5-6 card, which breaks (2).

So if I understand correctly:

You trust that if a card has a color on one side, it has a number on the other side.

You do not trust that if a card has a number on one side, it has a color on the other side.

Is this correct?

No. I don't make any assumptions as to anything I can't see.

However, regardless of what's on the back of blue, (2) will still be correct; the consequent is true, therefore the conditional is true.

Okay, I see where you're coming from now.

(1) may or may not be true. Understanding that we cannot trust (1), but that we don't care to verify it, then the 5 is necessary to prove (2).

That is much more subtle than I originally interpreted from you original post. Thanks for your patient explanation.

>The confusion doesn't arise from missing the statement you quoted, but whether this is to be treated as part of the problem statement that you are testing for truth.

You are not supposed to test for truth the problem premises. Just the if "even then blue" statement.

If you also had to test the premises then there would be no question and no asnwer -- because those require premises.

See sibling threads.

Short version:

You are presented with 4 cards on a table. You can see one side of each card.

You are told two things:

1) Each card has a number on one side and a color on the other side.

2) If a card has an even number, then its color is blue.

It is not unreasonable assume that you are expected to verify all things you have been told (i.e. all things you cannot verify by what you observe).

Since you know you can't trust the second thing you are told about the cards, why should you assume that the first thing you are told about the cards is true?

OP is saying that you cannot trust anything you've been told, and is attempting to verify (2) without trusting (1). I misinterpreted and assumed that OP was saying the goal to verify all statements that you've been told.

No, you're told these things:

1) Each card has a number on one side and a color on the other side.

2) Pick the minimum numbers of cards needed to verify the statement "If a card has an even number, then its color is blue".

>Since you know you can't trust the second thing you are told about the cards, why should you assume that the first thing you are told about the cards is true?

This reading doesn't make any sense. If you were indeed given (1) and (2) in that form, and indeed couldn't trust them both, then there would be no quiz at all. Just 2 random sentenses that might or might not be true.

In the setup, you are presented with four actual cards in front of you, two where you can see a number, and two where you can see a color. This is verifiable with your senses. You are then told two statements about these cards and asked to verify the second statement.

As OP of this thread pointed out, the card with 5 must be checked, because it's possible that it has an even number on its reverse side.

I originally misinterpreted the OP as saying that both (1) and (2) must be verified. You can see the two sibling branches that explore both of these in much more depth and back and forth.

>>"Each card has a number on one side and a color on the other side."

>So 5 could not have an even number on the back given the rules. //

Having "a number" on one side doesn't mean it only has one. Nor does having a colour on the other side mean it doesn't have more than one colour, nor indeed that it doesn't have colours on both sides.

It's like me saying "I have a coin in my pocket" it only tells you the number is >= 1 assuming I'm being truthful.

If you expect a precise answer you need to ask the question in a precise manner.

Won't argue semantics, but "a" and "one" are typically used interchangeably. A shirt with "a color" it's assumed to be "one color".

"Will you hand me a pencil" doesn't mean I want you to hand me at least 1 pencil but possibly 5.

I see where you are trying to come from with that angle though. You can feel free to apply different semantics to "a ___ "but I feel the general consensus is that it is equivalent to "one ____".

Note that "a colored shirt" is not the same as "a shirt with a color". As "A colored shirt" could be a shirt with multiple colors due to the ambiguity of "colored".

From the article:

>In one version of the task, one subject (always one—he spurned testing subjects in groups) is presented with four cards lying flat on a table, each with a single-digit number on one face and one of two colors on the other.

It specifies "one" for the colors.

It doesn't specify _only_ one colour ("each card is plain white except for" would appear to work well). If your question is going to be sloppy then you're going to need to be prepared to accept a range of solutions.

There's a difference between your examples and the question:

"Give me a shirt." is a request for one shirt.

"You've got a shirt in the wardrobe." is a statement that doesn't tell you how many shirts are in the wardrobe, just that there is one.

Similarly:

"Will you hand me a pencil." is asking for one pencil.

"I have a pencil." is telling you that I have at least one pencil.

Based on en-gb native use.

Valid point. There was no rule that said each card has a number one side and a color on the other.

> In one version of the task, one subject (always one—he spurned testing subjects in groups) is presented with four cards lying flat on a table, each with a single-digit number on one face and one of two colors on the other.

From the article. All the needed information is present.

I somehow split the difference on figuring out the logic and following the wrong intuition.

I took as truth each card has a color on one side and a number on the other (see others' comments for where this could be construed as part of the thing you must prove).

I knew I must flip the eight to confirm the color on its back.

I intuited that I should flip the blue.

I thought for a moment before deciding on my final answer, and concluded that I must flip the green to ensure that it does not have an even on the back.

I pondered a moment and somehow decided that order is important, and so that I should flip the 8 first, then the green, and finally the blue in that order (not sure on how I arrived at requiring an order).

I selected 8, blue, green, and was informed I was incorrect.

I thought and realized that the truth of the proposition did not exclude the possibility of odds having blue backs.

I arrived at the correct conclusion that I must flip the 8 and the green.

I was able to determine "flip the 8" due to my study of logic and AI: Modus ponens.

Unfortunately, I forgot about Modus Tollens, and forgot about the "Green" card.

The fact of the matter remains: the typical human does not study basic logic. And basic logic is hard to do without proper study. In fact, verifying that you must flip the 8 and green basically requires the human to "make the leap" and fully understand Modus Ponens and Modus Tollens.

There's an explanation of the solution here for those who don't want to play the video in the article: http://en.m.wikipedia.org/wiki/Wason_selection_task

Figuring out what to exactly click on in the 'interactive' video was far more challenging than the puzzle was.

Deleted comment

I agree. It is definitely not an issue of phrasing. I have had so many discussions where people fail to comprehend the concrete differences between converse, inverse, contrapositive, especially in relation to causality/mechanism. To me this seems to be especially bad in political discussions on facebook, but also, is even a critical issue among some PhD level synthetic organic chemistry students - where "mechanism" is a very important concept.

I am certain that these people would miss this problem on a first pass, second pass, and maybe even third or fourth pass.

> Then there's the issue of negation. "if P then Q" negated is "if not Q then not P".

No, "if P then Q" negated is "P and not-Q"

"If not-P then not-Q" is the inverse of "if P then Q".

"If not-Q then not-P" is the contrapositive of "if P then Q".

  S is true -> the negation of S is false
            -> the contrapositive of S is true
            -> (nothing at all about the inverse of S)

My formal logic was never great. Years since I did one of these. My first try was 8, then 8+green. Probably need to brush up on my formal logic. Especially since I've been trying (painfully) to read formal verification papers.

I did get the problem correct, but in figuring out the answer I got an intuitive sense of why the police officer statement of the problem might work better than the number/color statement of the problem. There are two things that I can see which seem way less vague than humans "evolving in a socio-cognitive niche".

1. The number/color statement of the problem doesn't have a cause and effect--the if/then statement of the problem doesn't indicate any actual causality between the number and the color. The only reason humans typically solve these kinds of problems "in the wild" is to understand the causality of a system so they can make intentional interactions with the system that produce effects they want and avoid effects they don't want. The police and underage drinking part has a clear cause/effect: if the person is drinking underage they should be arrested. Having a cause/effect to reason about is much easier to reason about than a correlation between causally disconnected systems.

2. People have prior experience with the underage drinking puzzle: at least in the US, people have probably had to think through some variation of part of this problem before to understand the who and where of drinking legality. Being pattern-matching machines, we solve logical puzzles by matching them to previously-solved logical puzzles.

In my experience just now I think the "old system" and "new system" makes sense. After reading the problem my initial reaction was "8 & Blue". I thought about it for a couple seconds and immediately realized it would be "8 and Green", which was the correct answer.

After taking a second to reason about the problem, the answer came and made sense. I guess that's why it's always good to take a second to choose your words, etc in day to day life?

I had the same experience (initially picking the wrong color card but then saying, wait, no, I'm a math major. Contrapositive, duh)... So, yeah it seems like the system1/system2 makes experiential sense. But: are we not using system 1 to evaluate the system1/system2 system?

> each with a single-digit number on one face and one of two colors on the other.

Is the subject ever told this? If not, you also need flip the five card over.

From the Wikipedia page on it, it seems so. The problem is described as:

  You are shown a set of four cards placed on a table,
  each of which has a number on one side and a colored
  patch on the other side. The visible faces of the cards
  show 3, 8, red and brown. Which card(s) must you turn
  over in order to test the truth of the proposition that
  if a card shows an even number on one face, then its
  opposite face is red?

Perhaps it's a consequence of being a programmer/mathematician and doing this for so long, but it seemed clear to me.

I would have turned the 5 card too. Why? Because it says to test the truth of the proposition. So it's saying all bets are off, whatever I told you may or may not be true. Only what you see (the table with the 4 cards) is real.

If you still assume that each of which has a number on one side and a colored patch on the other side holds, then no need to check 5. But if you focus on testing the final proposition, I'm not sure anymore of what I should assume...

Summary: poorly worded problem.

The proposition:

  Which card(s) must you turn over in order to test the
  truth of the proposition that if a card shows an even
  number on one face, then its opposite face is red?

The rest is given information, not the proposition.

The puzzles don't seem comparable, at least as presented by the article.

With the cards, you are given cards where you know if the if side of the problem is true or false and cards where you know the tend side of the problem is true or false.

With the people drinking beer, you are never given anyone cases to check where the then side of the problem is right or wrong (everyone is drinking or not drinking, there are no examples of people whose age you know but you don't know if they are drinking).

The people drinking problem is similar to if you gave people cards only with the numbers shown. I suspect the percentage of people getting the number part right would be much higher.

(Note, in the actual age/drinking situation, the experiment is designed to be the same; it just appears the author didn't explain that well in this piece.)

From skimming the study where they changed the wording[1], it seems like they did in fact use cards, just like the original study.

[1] http://cogpsy.info/wp-content/uploads/2013/08/Griggs_Cox_198...

I wonder whether they'd get different results if they used shapes rather than numbers, or by phrasing it slightly differently:

'All pyramids have red on the back, which cards do you need to turn over to verify this?'

IME people tend to turn off when numbers get shoved under their noses.

Restated: People are good at following an implication, but bad at deriving the contrapositive.

Maybe it is just me, but I don't see an issue at all, the puzzle is well stated, the answer is pretty easy to get when you understand the puzzle.

The trick, if there is one, is deciding what can be filtered out immediately (the 5), and what can be filtered out based on the direction of the implication (the blue).

Done.

(Maybe it helps that I studied logic - and math, lots of math - at uni, and that I read Fast&Slow recently, and therefore pay more attention than I used to? Dunno.)

"if A card (as in, at least one) shows an even number on one face, then its opposite face is blue".

This was my first interpretation, based on "deceptively easy" and "not turning any unnecessary cards". Which sentence did I miss that rules out this interpretation?

That "if an X" implies ANY X.

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