What if the elements were finite, and we were to examine every single one? Wouldn’t our conclusion then be “logical”? If not, why not?
**Libertarian: **
I have two questions for you: 1) are you drinking and posting, and 2) have you ever used the handle ratchet?
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**RM Mentock ** asks
As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.
– **Albert Einstein, ** (1879-1955)
Father:
For further reading on this topic, try doing a search on “Hempel’s Raven Paradox”. Here is one link: http://www.westminster.edu/staff/brennie/Raven.htm .
Ziggy, correction noted. I was of course going from memory (and just ask David B. if you think you can count on memory!; – of course, that’s another thread…).
On this board, however, it gives rise to some terrible side potentials, such as a username of I’mthepurplecowgod& I’mgonnamoolikeit’s1999.
As I understand the conundrum as originally posted, the purple cow proves (slightly) the existence of diametrically opposed propositions (All A are B and All A are C). You state that this is a ‘mathematical certainty.’
Quite the contrary, as we have pointed out to you in several posts. The assertion that finding the purple cow does anything to prove the truth of either assertion about crows is not mathematically correct. It is the weak link in the argument, the one that shows that the supposed paradox isn’t.
As the site, the link for which was given two posts ago, shows, the only way that seeing a purple cow tells us anything about ravens is if the number of non-blacks is a known quantity substantially smaller than the number of ravens.
Indeed, as I think more on this, and the site referenced, I will take back one statement I made previously. I stated:
This is, of course, incorrect. I was using the definition of ‘proof’ implicit in the stated conundrum; in actuality, they are different things. Indeed, they are the point to the whole matter.
Evidence is a fact which tends to prove or disprove a conclusion. To prove something is to establish the truth of it. What is accepted as proof of a truth differs depending on the type of truth being proven. For example, O. J. was proven to have killed Nicole Simpson, but not proven to have murdered her, since to prove the tort of wrongful death only a preponderance of the evidence was needed (more likely than not); for the murder charge, no reasonable doubt could remain.
In mathematics, almost doesn’t count. One cannot ‘prove’ that a mathematical proposition is true by obtaining some evidence of it and saying, “well, it seems more likely than not.” Would that one could use such ‘proof’ in HS Geometry!
The conundrum listed in the OP asserts that there is a paradox caused by the fact that seeing a purple cow is proof (slightly) of the simultaneous statements that “All crows are black” and “All crows are white”. This mistakes the meaning of ‘proof’. The only proof of the assertion about all crows would be knowledge of the color of all crows, or a logical explanation as to why the set of all crows can’t contain more than one color. Failing that, there is no ‘proof’.
If I understand both Little Nemo and the website page, the assertion actually is that the purple cow is ‘evidence’ of both assertions, that is, it tends to prove each assertion. According to the site, the value of the evidence about the purple cow increases as the number of non-black items is decreased. Thus, if we have only two such items, then knowing there is a purple cow makes it more likely than it was before seeing the purple cow that there are no non-black crows. If, on the other hand, there are very many non-black items, then the purple cow helps us very, very little in knowing whether there are non-black crows.
I, on the other hand, assert that this shows incorrect understanding of ‘evidence’. Knowing that a member of the set Not-B is also a member of the set Not-A tells me nothing about the composition of A. It may reduce the chance that there is a member of A which is Not-B, but it doesn’t tell me anything about the unobserved (as yet) members of Not-B. I can observe 50 Not-B members, none of which is a member of A, and still have no idea if the next member of Not-B will or will not be an A. By contrast, the more A I observe who are also B, the more confident I am at predicting that the next A will also be a B. This is because the information about A is advanced by seeing members of A; the information about A is not advanced by seeing a member of Not-B which is not a member of A. Thus, the thing which makes the black crow evidence isn’t the fact it affects the chance of a Not-B crow existing (assuming the number of Not-B members to be finite); it is the fact that it affects the understanding of A, which is the group being tested.
And to totally understand what it means to say that the purple cow is not evidence of anything, I propose to you the following. Test the correctness of two assertions. Assertion 1: All crows are black. Assertion 2: All crows are white. I tell you that you have viewed all but two members of the Universal set, U, and seen not one single crow. I further tell you that ONE of the last two items is a crow. I am going to show you one further object, and then ask you to prove that either Assertion 1 or Assertion 2 is true. You then see a purple cow. Prove to me what color the crow is, showing your evidence and explaining how it demonstrates the color of the crow.
Didn’t think you could. 
DS, you’ve successful shown that you can’t prove that all crows are any given color using a purple cow. It’s too bad that isn’t the point of my OP.
The cow doesn’t prove anything. It * seems * to give some support. This sin’t mathematics. I’m not saying it does support; that may be the flaw; but you haven’t demonstrated that flaw.
–John
That means that your position is that “All crows are black” is a statement outside of the realm of logic and mathematics. That’s a pretty sterile argument, though. Myself, I’d rather include it, and consider the consequences.
Otherwise, we have these faceoffs, where neither side thinks the other is being reasonable. Just because it is impossible to examine every single crow doesn’t mean it wouldn’t be theoretically possible.
An exhaustive and complete search of all crows would constitute a “proof”? Then, is each individual crow part of the proof or not? Why not?
Next: we abstract the argument.
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RM Mentock supposes
No, that is not what I said. What I said is that the presumption that cows and the color of cows is evidence of anything about crows is not supported in the argument in the sense of logic or mathematics. There has been no formal argument put forward to even imply a relationship, although it is subsumed when the statement about cows is asserted to be evidence of some sort.
One may make statements about crows, and the color of crows, and assert that purple cows cause the color of crows to be black, and then further state that the existence of one purple cow proves the truth of the first premise. That would be logically consistent, although absurd. However, the problem presented lacks the formal statement about any relationship between cows, and the color of crows. The number of assumptions of fact, and causality necessary for consideration of cows, or unicorn, or even ravens as evidence is rather large. None of these assumptions are stated, and when the argument proceeds without them **that ** falls outside of the realm of logic.
Your next question is
Exhaustive examination ( more formally, proof by inspection) does fulfill the requirements of proof in formal logic. Each element of the process by itself is not pertinent. The aggregate is pertinent only if it is uniform in compliance with the definition of the premise.
Consider the counter example. I take 864500 marbles, two screws, and 15 feathers, and put them in a bag. In fact 864499 marbles are black, one is red, the feathers are each a slightly different shade of pink, and the screws are black. I show you the bag, and make the statement:
<p align=“center”>All the marbles in the bag are black.</p>
According to your version of logic, each marble that was black would be “evidence” that all were black. You wish further to assure me that the two screws would be evidence, and each and every feather. This is absurd, and if you don’t understand that, I want to play some logic games with you for cash.
Now let’s look for the tiny grain of wheat in your argument of chaff. Suppose I have a theory about objects in category x. I am quite sure that all objects in category x have characteristic z. I have the ability to examine a large number of elements of category x, and some ability to examine elements of a certain other category which I know are ~x. I can gain some degree of assurance about the make up of the categories and their characteristics by examining elements of both sets. That assurance is evidence in support of my theory if, and only if objects that are category x are selected by some means other than compliance with my theory, and still uniformly meets my criteria exactly. It is evidence that my theory is consistent with reality, in the sense of scientific evidence. It is not logical proof.
Category ~x is so far not described, and not germane to the proof, or evidence in a logical or scientific sense. If I can show some reason to believe that the existence of factor ~z is a sufficient condition for membership in the category ~x, I can widen my search include objects which are ~z, and examine them to ascertain if my supposition that they are ~x holds true. I can strengthen my theory by that experiment. Ten thousand iterations of my experiment by a hundred independent researchers would strengthen it further. A single counter example found by a fifth grader would prove it false.
Evidence is not proof, nor even a part of proof, in the logical sense. In the scientific sense, evidence is what we must rely on, since proof most often eludes us.
“…it doesn’t matter how beautiful your theory is, it doesn’t matter how smart you are – if it doesn’t agree with experiment, it’s wrong.”
–** R.P. Feynman **
Cool. How much?
That’s not a counter-example. You stepped into a logic trap there, when you made the example completely finite. Let’s just use “my” logic on your example. So, we would take all non-black objects out of the bag. We would see 15 pink things and one red thing. Would you not be able to say something logical and conclusive about the marbles that were in the bag? I would. They weren’t all black.
What if the red thing weren’t there? Then, all the marbles that were in the bag were black. (Note: in logic, this allows the empty set–although that’s not pertinent to this “counter-example.”)
The line of reasoning is only in dispute when it comes to collections that cannot possibly be exhaustively examined.
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These two posters look at the issue completely differently, with the same underlying willingness to accept the ‘paradox’ presented.
The conundrum IS a problem in mathematics: it is an example of both logical reasoning and set theory.
The purple cow is NOT evidence of anything except that the set Not-B can contain cows, and that the set of cows may have purple members.
To re-iterate:
A black crow proves that the assertion “All crows are white” is false. It is proof because it makes the statement untrue without any additional evidence.
A black crow is evidence that the assertion “All crows are black” is true, but does not prove it unless one knows that the crow is the only member of the set of crows. It is evidence because it fits the assertion, and does not disprove the assertion.
The purple cow is neither proof nor evidence. It does not provide us with useful information about either assertion, telling us nothing about the composition of A. It reduces the chance that the assertions are incorrect (by reducing the number of members of set Not-B), but that does not equate to helping prove either assertion, as my extreme example showed.
WallyM7 wrote:
Ah, but you could take an ordinary black crow and cover it with white paint, could you not? Or is that considered cheating?
Please excuse me, I did not understand at first. I have nothing further to add.
Sir, I have found you an argument. I am not obliged to find you an understanding. – **Samuel Johnson, ** (1709-1784)