
A Simple Logical Puzzle That Shows How Illogical People Are - dnetesn
http://nautil.us/blog/the-simple-logical-puzzle-that-shows-how-illogical-people-are
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femto113
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.

~~~
wutbrodo
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.

~~~
malka
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.

~~~
tjradcliffe
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.

------
t0mbstone
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.

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araes
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.

~~~
breadbox
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.

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ikeboy
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.

~~~
Nadya
>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.

~~~
greggyb
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).

~~~
Nadya
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

~~~
greggyb
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.

~~~
Nadya
(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.

~~~
greggyb
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.

~~~
Nadya
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.

E:

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

~~~
greggyb
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.

------
greggyb
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.

~~~
dragontamer
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.

------
akamaka
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](http://en.m.wikipedia.org/wiki/Wason_selection_task)

------
DIVx0
Figuring out what to exactly click on in the 'interactive' video was far more
challenging than the puzzle was.

------
nickpsecurity
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.

------
copsarebastards
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.

------
nemo44x
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?

~~~
dnautics
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?

------
stordoff
> 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.

~~~
Jtsummers
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.

~~~
alain94040
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.

~~~
Jtsummers
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.

------
Lawtonfogle
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.)

~~~
dcre
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...](http://cogpsy.info/wp-
content/uploads/2013/08/Griggs_Cox_1982_The-elusive-thematic-materials-effect-
in-Wasons-selection-task.pdf)

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6d0debc071
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.

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

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PeterWhittaker
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.)

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ups101
"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?

~~~
coldtea
That "if an X" implies ANY X.

