In one of Martin Gardner’s books, (Ah! Paradox!, I believe), he told a mind-boggling story about purple cows and white crows.
Proceeding logically:
You have the proposition
If you see a black crow, it confirms (very slightly) this proposition.
But the proposition
is the same thing reworded. So theoretically, anything you see that isn’t black and isn’t a crow slightly confirms the proposition.
Which is weird.
But even weirder is that fact that if you have a proposition
or
then something that is neither black or white, or a crow, like a purple cow, confirms both
and
simultaneously.
Which is just nonsense.
Can anyone explain where the logic fails here? Obviously, these two propositions can be easily disproved with reality; crows and albino crows,respectively, disprove both propositions. But what’s the logical flaw.
I put this here because I anticipate that not everyone will agree.
The idea that seeing a purple cow helps prove that crows are black seems paradoxical but is none the less true. The problem is that this single piece of evidence is so infintesimal that it has no real value. But imagine you were omniscient and could see everything in the universe. If you could see every non-black object in the universe simultaneously and none of them were crows then you would have proven that all crows are black.
The one logical flaw of the argument that “all crows are black” is the implicit assumption that an object such as a crow exists to have a color. If you were to postulate that all unicorns were black and examine every non-black object in the universe you would have equally proven your argument.
At first blush, I thought that was a type of non sequitur fallacy called “Denying the Antecedent”. Arguments of the form, “If A then B. Not A, therefore Not B” deny the antecedent, and are therefore invalid. For example, “If I am in Houston, then I am in Texas. I am not in Houston, therefore I am not in Texas.” (You might be in Dallas.)
But, stepping back a bit, it seems like nothing more than a classic false dilemma (or false dichotomy.) It is true that if all crows are black objects, then all nonblack objects are not crows. But they also might be not a lot of other things; in fact, exclusively nonblack things won’t be anything that is exclusively black.
That’s a simple induction fallacy drawn from a false premise.
That’s a type of non sequitur called an inconsistency fallacy, where two mutally contradictory propositions are made. Each disproves the other.
But to assert that seeing a purple cow sheds any light on the truth of the proposition that
is incorrect. Since the object observed is not a crow, we cannot test the correctness of the statement about the set of objects called crows. Only by seeing a crow can the truth of the statement about crows be tested. Therefore, the purple cow is irrelevant to the black/white crow controversy.
As I said before, DS, the existence of purple cows is relevant to determining what colors crows are.
Let’s scale down the problem a bit to illustrate. Let’s say there’s a car lot with a thousand cars in it and let’s accept the assumption that some of the cars are Toyotas. The lot owner divides up his cars by colors and keeps each different color car, regardless of its make, in a seperate showroom. Now suppose I made the proposition that “all the Toyotas in the lot are green.”
I then go to all the showrooms for every color except green. I examine nine hundred and fifty cars but none of them is a Toyota. I’m finally standing outside the door of the last showroom, the one with all the green cars. But I don’t need to go in. I’ve already proven that all the Toyotas on the lot are green without seeing a single Toyota or green car. Because I know there are Toyotas on the lot and if they weren’t green I would have seen them in another showroom. So looking at that Volkswagen back in the yellow showroom really did help prove my case. And by the same logic that demonstrates how a yellow Volswagen helps prove the existence of a green Toyota, on a much larger scale a purple cow helps prove the existence of a black crow.
Logic deals with the precise content of the statements only. The implication that a non black object which is not a crow implies anything whatsoever is an additional element never stated in the argument, but presumed by the OP to be relevant. Therefore the logical statement:
assumes an unstated premise that the existence of things that are not black, and are not crows is a resultant condition of the black characteristic of crows, or the non black characteristic of other objects.
This is assumed without support. Blackness is not assigned as sufficient for membership in the category of crows. It is a necessary condition only. Being a member of the set crow is not assigned as a necessary condition for being black, only a concomitant condition with respect the members of class crow.
Therefore, our argument now becomes:
All crows are black.
Some objects that are not black are not black because crows are black.
Therefore the existence of purple cows is evidence for the blackness of crows.
Even now our argument fails, since it does not indicate how to tell if crow blackness was causative in the particular case of purple cows, or in any other case. The assumption of evidentiary significance for purple cows contains another unstated assumption, namely that it is specifically cows which are made not black by crows. The assumption included the premise that crows and no other influence are the cause as well.
To become valid, our argument must have more statements, establishing that blackness is related in every case to the category of crows, and that non blackness is also related. Crows must be stated to be the causative agent in the coloration of each category of object that is considered evidence that crows are black.
<P ALIGN=“CENTER”>Tris</P>
You can only find truth with logic if you have already found truth without it.
– G. K. Chesterton, (1874 - 1936)
The discovery of a purple cow tells us nothing about crows. This simple bit of logic is not distorted by any bit of linguistic chicanery. The point to solving such ‘puzzles’ as postulated originally is to see where the chicanery has occurred.
The statement “all crows are black” is a statement about a set, the members of which are all the crows that exist. We now have the set A (crows) and the set Not-A (anything that isn’t a crow. We have asserted that “All A are B”. This means that A is a subset of the set B (All black things). By definition, there is no member of A that is not a member of B. The concomitant statement, “all non-black objects are not crows” is simply a recognition that there is no A that is Not-B.
Now we observe an object and see if it confirms our assertion regarding A’s membership. Remember, we are testing the hypothesis that All A are B. To know the truth of this statement, we would have to observe all members of the set A. Viewing members of the set Not-A tells us nothing about the membership of set A. Only if we could see all members of the set Not-B, and KNOW we had seen all members of the set Not-B, and determined that No Not-B is an A, could we use that information to assert that All A are B.
So we see a black crow. We have some confirmational evidence of the proposition asserted. Next we see a purple cow. All we know from this is that there are purple cows, because we know nothing about crows from it. This means that the purple cow neither confirms nor denies the truth of All A are B.
In the volkswagen example given, the only way that the person knows the answer before opening the door is if the person knows in advance that he has seen all Not-B before opening the door. This, of course, means that he knows the contents of the room, and if he knows those contents, then he already knows that all the cars are the appropriate color. In short, he already knows that he has seen a lot of A and that they are B, and all the rest did is confirm his suspicion that the other rooms didn’t have any A in them. It is the lack of an A anywhere else that gives him the answer, not the existence of the various Not-B’s he views along the way.
DS I don’t want to go over the same ground again. But basically you’re confusing evidence with proof. A single purple cow is not proof that all crows are black but it is one piece of evidence that could be used to construct that proof.
Nemo, I understand that, but I think you are missing the point of the supposed connundrum. The connundrum supposes that the purple cow is evidence of the proposition that All Crows are Black, AND evidence of the proposition that All Crows are White. In actuality, it is evidence of neither.
To argue that it is ‘evidence’ or ‘proof’ is tautological.
I think you are mistaking the counter argument. The logical step you take when you assume that cows, or the color of cows is evidence of anything with respect to crows, or their color is unsupported. The difference is not that it is evidence rather than proof, the contention is that it is neither proof, nor evidence. Making the argument itself is a logical fallacy, and assuming it without making the underlying supportive statements formally is another. Cows are not crows, being purple does not make other things black. Non black objects are not germane to crows if they are not crows, or in some way responsible for the color of crows.
It is also very likely that you assume that individual examples are elements of proof in a logical sense. That is not the case. Even the existence of a black crow, or a billion black crows is not logical evidence that all crows are black. They do constitute scientific evidence consistent with a theory that crows are black. Science is not a formal subset of logic, nor is it exclusively dependent of formal logic. Scientifically speaking, however the cows, whatever color, are irrelevant as well.
Observing large numbers of crows can provide supportive evidence for a theory about the color of crows. It does not qualify as proof, but the absence of any counter examples does give strong supportive evidence for the theory. Enough evidence to justify an in depth study of the mechanism which makes crows black, and the possibility that interruption of that mechanism would prevent the existence of a crow by itself. Over time, extensive investigation might bring the great majority of scientific theory to accept Nanobytes Theory of the identity of Crowism and Color. Thousands of years might pass without any counter example being shown. Still, without an exhaustive analysis of the ontology of blackness itself, the proof would be incomplete.
No doubt religious extremists, would find solace in the well know fact that Nanobytism was not proven. They would start www.WhiteCrow.com to hold debates on why athiest Nanobytists deny the possibility of White Crows. And the search for white crows would go on forever.
<P ALIGN=“CENTER”>Tris</P>
“This institution will be based upon the illimitable freedom of the human mind. For here we are not afraid to follow truth wherever it may lead, nor to tolerate any error so long as reason is left free to combat it.”
– Thomas Jefferson
Tris and DS, one of the beautiful things about mathematics is that not only does it provide absolutely correct answers, it also makes it possible to prove they are the correct answers. So what I wrote is not a matter for philosophical or religious debate, it’s an explanation of a mathematical certainty. Like the Monty Hall problem the logic of the situation may not seem readily apparent but regardless it is indisputably true.