We are all familiar with optical illusions. These are situations where your eyes misperceive the nature of some image or physical object.
For some time now psychologists and cognitive scientists have been discussing the reality of cognitive illusions. These are situations where people just don't reason properly about some readily described situation. The Monty Hall problem is sometimes described as an example of such an illusion, which, indeed, is why I have been thinking about this recently.
Below the fold is an interesting example drawn from elemenatry logic. I found it in the book Inevitable Illusions by Massimo Piatteli-Palmarini.
We begin with a warm-up. Suppose I tell you that:
(1) All Ruritanians are rich; and (2) John is a Ruritanian.
What follows from this? Plainly, that John is rich. Agreed?
Now suppose I tell you that
(1) No fruit-picker is a sailor; and (2) All Ruritanians are fruit-pickers.
It follows that no Ruritanian is a sailor.
Very good. Now suppose I tell you that
(1) All politicians are thieves; and (2) No composer is a politician.
What, if anything, follows?
A whole lot of not much. We cannot say a thing about composers, other than that they are not politicians. They may or may not be thieves.
Nothing follows. The composers may or may not be thieves.
A common logical fallacy follows.
It is possible but not necessary that composers are thieves.
Can't say much...the composers aren't politicians and they may/may not be thieves.
The Democrats will claim Republicans are the victims of cognitive illusions and Republicans will claim that Democrats are.
Composers won't give a damn since nobody listens to their new music anyway.
1) All politicians are thieves; and (2) No composer is a politician.
What, if anything, follows?
Did the composers put you up to this? LOL!
Dave Briggs :~)
"What, if anything, follows?"
X answers "no composer is a thief" -> X isn't familiar with Venn diagrams?
Problem is, people read that last one and parse it as
"Only politicians are thieves"
But the question is: What would be the answer given by an ordinary person who knows nothing of formal logic. Or "What does the syllogism feel like it implies, but actually doesn't?"
I guess the (pseudo)implication is that "No composers are thieves".
A similar line of reasoning might go: "Everyone knows Harvard graduates are frauds, but I went to Yale, so you can trust me.", or indeed, "The Republicans are pro-war, but we're the Democrats, so if you're anti-war, vote for us!"
The difference then between optical illusions and cognitive illusions is that the latter go away when you realise they are illusions, while the former do not.
I don't know, I'll have to consult scripture to see what it says about composers.
Composers aren't thieves. They just borrow a lot. Some more than others though. Therefore no composer is a thief.
No politician is a composer.
"Good composers borrow; great composers steal" - attributed to many, including Igor Stravinsky.
What about this one? Is it a "Cognitive Illusion"?
In support of individualism, Adams Smith said: "If you are good to society, you are allow to profit from it. The invisible hand will take care of everything else."
Now, capitalists say: "It is God's whishes that I make profits, so, I am good to society."
Isn't that the fallacy of afirming the consequent?
If the set of politicians is not null then we know that there exists at least one thief who is not a composer. We cannot make any universal conclusions.
"The difference then between optical illusions and cognitive illusions is that the latter go away when you realise they are illusions, while the former do not."
Discuss with reference to Monty Hall.
It follows that some fruit-pickers are rich, which leads to the question of why, being rich, they don't switch to an easier occupation.
Well, you can say the some thieves are not composers.
Clearly, ScienceBlogs is not the right place to ask this question :D
Congratulations to Joshua Zelinsky and Rick Pikul for getting the right answer. The two statements above imply that there is at least one thief who is not a composer (assuming the set of thieves is not empty, of course). The illusion is that most people, even after thinking about it for a while, conclude that nothing follows from the two statements I gave.
No...There is STILL nothing that follows, without making further assumptions that are not in the problem, and may or may not be true.
Being strict about the answer does not make it an illusion - no one who answers "nothing follows from that" is wrong or misguided. Only if there is extra information, such as "there is at least one politician" does saying "nothing follows" become wrong.
Or do you mean that the illusion is that people don't think about the conditional results (what I'm calling the 'loose' answer) of the problem (If assumption X, then Y - but X may or may not be true), and only think of the known/real implications? If so, this still isn't so much of an illusion (in the way of optical illusion) as a trick or hack based on the normal methods of operation, or what people _assume_ to be the boundaries of the problem (that it is only the known/real results that matter, not all the possible ones) - still an illusion, but under the broader and looser definition of a deception or misapprehension (thus, unlike optical illusions, it DOES go away once you know what it is). For some things, like this problem, there are very few conditionals - but in a lot of other situations, there are very MANY possible conditionals, so the standard assumption (besides mathematicians, logicians, and some philosophers) is probably that the conditionals aren't relevant to the intended answer. Still an assumption problem, but by norms rather than in-built mechanisms.
To some extent, reminds me of http://xkcd.com/169/
Not sure where Monty Hall falls though - even knowing the problem, I'm sure some people still fall for it because of in-built (or in-trained) probabilistic reasoning beating thoughtful probabilistic reasoning..
Further, you made a small mistake: like Josh said, the assumption has to be that the set of _politicians_ is not null, not that the set of _thieves_ is not null. I assume this was just a typographic error.
Yes, I meant that the set of politicans is not empty. Sorry about that.
The illusion is simply that most people don't notice, until it is pointed out to them, that the statements (All A's are B's) and (No C's are A's) actually implies that (Some B's are not C's).
It is true that this follows only if the set of A's is nonempty, but I really don't think that's the sticking point for most people. If it is, I would just add that (If the set of A's is nonepty then some B's are not C's) is a perfectly reasonable conclusion, and ought to count as something that follows from the statements I gave.
(OT: there's a chess discussion over at Pharyngula; I can't make heads or tails of it, but you might be interested.)
I'm with dreikin. If one looks at the Venn diagram, it's quite clear what the various options are. Eg, a composer may be a thief or a not-thief, a politician - necessarily a thief - can't be a composer but a non-politician thief can be. But one wouldn't normally consider those conclusions to be particularly interesting precisely because they are so clear.
So, while the conclusion "nothing follows" may be wrong, I submit that the conclusion "nothing interesting follows" is right on.
I agree with Charles.
Saying "nothing follows" is merely shorthand for saying, "of course at least one thief is not a composer, but that is uninteresting unless you give me a reason why it should be construed as significant."
It's like asking the question: "What happens in this lane when the traffic light is red?"
"Wrong. When the light is red, the cars do not go."
"Uh, right. Like I said: nothing."
I think "interesting" and "significant" are value judgements that have little meaning here. The question only asked what follows, not what exciting follows. The statement that there is at least one theif who is not a composer (assuming set of politicians > 0) may not be interesting, but it is new information that you can't get from any one statement.
On another unrelated note, Bobby Fischer is dead at age 64. Whether this is interesting or significant is for you to decide.
"I think "interesting" and "significant" are value judgements that have little meaning here."
They certainly are the former, but they have the same meaning here as elsewhere. When someone posts a math/logic problem, IMO a reasonable assumption is that there is some subtlety - say, an "illusion". In this case, the answer was from set theory 101. To some of us, the only illusion seemed to be that there was something interesting to be gleaned from the problem.
Your comment suggests a rigor that seems totally missing in this thread. Although JK's answer was correct since it included a conditional, contrary to Jason's "award" comment RP's wasn't since it didn't. Jason blew his reiteration of the "right answer". And if one is to consider conditionals not contained in the initial problem statement, then why stop at that one? What if the intersection of non-composers and thieves is not null? Or if it is? Or if all the sets are empty? Etc.
Some of us consider the post to have been uninteresting and so stated. I infer from the seeming snarkiness that this caused offense. If so, sorry - it certainly wasn't my intent. I frankly gave both the problem and my comments what I considered to be appropriate attention, ie, not much.
Abstract: Margolis (1998) argues that the mental rotation essential for an illusory collision in a Tychonic system cannot be "seen," because if conscious imagery represented the rotation the illusion would be immediately dispelled. However, a number of research studies have demonstrated circumstances in which individuals can fail to discover quite simple insights on the basis of mental imagery alone. Rather than arising from a form of "imagery blindsight," the persistence of the illusion can be accounted for by existing cognitive theories of mental discovery.
There is no proof that no composer is a thieving politician and that no politician is a composer.
It implies that I, through much profound things in my brain, have decided to give you my crown.... its ok, i got a bigger one.... its got a ghecko on it... LOOK AT HIM SHAKE! Go, stevie, go!!!