A paradox of evidence

[Tez]

Here's a well known paradox in the philosophy of science, that can be applied to general skeptical thinking. I've never been completely happy with the various resolutions that have been proposed (inlcuding my own!). See what you make of it.

Consider the hypothesis/claim:

All ravens are black

The hypothesis is well defined and falsifiable. Evidence for such a claim can be accumulated by going out and finding ravens, and then checking whether they are black.

Consider now the logically equivalent statement:

All non-black things are non-ravens

(More formally, the proposition "If raven then black" is logically equivalent to "If not black then non-raven".)

Consider the sort of evidence one may accumulate for this latter form of the claim. For instance, the coffee cup beside my computer is white, and it is not a raven. The key question is, does observing a non-black thing which is not a raven, provide evidence for the original assertion that all raven's are black, to which it is logically equivalent? Does my coffee cup being white and not a raven really reinforce my belief that all ravens are black?

It would certainly be strange if this were the case - it runs counter to our intuition, and, I suspect, our everyday practice. If such a process is flawed, however, then we need to be very careful when following what is a standard procedure of science - the testing of hypotheses by deducing their logical consequences, and then trying to determine whether such are compatible with our observations of the world around us.

 

[Stimpson J. Cat]

I think the problem here comes from the notion that any observation which is consistent with a falsifiable theory somehow constitutes supporting evidence for the theory. But of course, this is not the case.

Consider, for example, Special Relativity. Under most conditions its predictions are pretty much indistinguishable from those of Newtonian mechanics. Simple everyday observations which we have been making for centuries, which are consistent with both Newton's and Einstein's theory, did not constitute supporting evidence for SR, because they did not provide any new knowledge. What was needed was new evidence. Observations which had not been made before, which would not only contradict Newton's theory, but also confirm SR under conditions it had never been confirmed before.

With respect to the raven problem, we already know of all sorts of non-black objects which are not ravens. Simply observing more of these objects does not really add any new information. Likewise simply walking outside and seeing yet another black raven does not really add any new information. Adding meaningful supporting evidence to the theory would mean going to places where ravens have not been observed, and looking for non-black ravens there. In other words, we are already extremely confident that non-black ravens don't exist in any of the places we have already looked for them, so looking in those places more doesn't add much. We would need to go looking in places where we are not very confident that they don't exist.

So there's kind of a diminishing returns effect too. The more confident we become in a theory, the more difficult it becomes to find more supporting evidence for it. I'm sure you see the same effect in your field. After all, nobody would think that repeating basic experiments in QM, which have been repeated now for nearly a century, actually add any real additional supporting evidence to the theory. You have to work harder and harder all the time to come up with new experiments which actually have some chance of providing you with new information.


Dr. Stupid

 

[Patricio Elicer]

Consider now the logically equivalent statement: (More formally, the proposition "If raven then black" is logically equivalent to "If not black then non-raven

I've come to think in this apparent paradox for quite some time. I think the solution resides in a mis-statement or mis-understanding of the phrase.

When you say "If raven then black", there's the tacit fact that you are talking about ravens and not anything black.

So the claim should read: if you are a raven then you are a black raven, which is logically equivalent to if you are not a black raven, then you are not a raven

Another approach is to say if you are a raven then you are a raven "and" you are black. Equivalently, If you are not a raven "or" you are not black, then you are not a raven

 

[Tez]

What you say is certainly true about the practice of science Stimpy, but I'm not sure it addresses the issue head on. What if the statement under consideration was something for which I had little or no evidence for either its truth or validity? Is seeking evidence for some logically equivalent statement valid or not? In some circumstance it is, and in some it isn't.

For instance, the hypothesis All non-trivial zeros of the Riemann Zeta function have real part equal to 1/2 has been tested both by computing zeros ("if zero then real part equal 1/2"), but also by examining strips of "non-zeros" which lie close to the conjectured line of zeros ("if real part not equal to 1/2, then non-zero"). No one doubts the applicability of such a test, either back when the range of values over which it had been tested was very small, or today when its very large and little in the way of new information is gained. (Though its still being done!)

I think the issue with this latter example is that there is a level of uber-connectedness between the equivalent logical statements about the zeros of the Zeta function, that has little to do with formal logic. Its hard to make precise, but the difference (as I see it) is that there is a potential for "explanatory power" regarding the underlying "phenomenon" when one observes either type of "instance" of the Riemann zeros or non-zeros; something not true in the Raven case.

 

[Soapy Sam]

OK. What have we got here?

All ravens are black

All non-black things are non-ravens

"If raven then black" is logically equivalent to "If not black then non-raven"

We have three symbol strings and an assertion that they map onto one another in an equivalence described as "meaning", where "meaning" is derived by a symbol manipulation process named "logic", which satisfies certain operational criteria of human brains.

This claim (that the strings have equivalent meaning) is provable, if and only if we accept the rules of "logic" as both consistent and correct.

We can test the rules of logic, by applying either
1. Logic
or
2. Empirical observation.

Use of 1 is circular argument and therefore (by the rules of logic) void, unless the rule of logic condemning circular argument is itself wrong. In which case I give up.

Which leaves use of 2. It ain't perfect, but it's all we've got, except for belief with no test at all. Science is a best guess, with detailed notes, refined by observation.

It's not philosophy, thank Ed. Philosophy is concept manipulation in a linguistic context, playing with shadows on cave walls.

Some ravens are grey, by the way. In the right light.

 

[athon]

I think both Soapy and Patricio have answered this nicely here; we are looking at less a problem of logic, and more of a problem with language.

It becomes confusing when labelling the nature of the hypothesis; in that, they are not necessarily completely identical statements. The weakest part of science is not the process, but rather a way of detailing the principles of the method.

We know ourselves that we get caught up in semantics and definitions. It goes a step deeper here, as Patricio pointed out, in that the two statements only appear to be discussing the same thing. But the subject of each statement is subtly different.

Statement 1 seeks to predict the nature of impending discoveries of new ravens. You have a high probability that on finding a raven in a future event, it will be black. The subject is ravens, and the descriptor is their colour.

Statement 2 talks about non-black things. You have a high probability that on finding something that is not black, it will not be a raven. The subject is non-black things, and the descriptor which you use predict a future event is whether or not it is a raven.

This is a sticking point with science, namely because most semiotics carry baggage which is difficult to label. Which is why mathematics is so wonderful; it has the least connotation of all forms of language.

Athon

 

[FireGarden]

Theorem:
On the other side of every card that has a vowel, there is an even number.

Which of the following cards do you need to turn over to test this theory?

5..........A............T............6

Consider, for example, Special Relativity. Under most conditions its predictions are pretty much indistinguishable from those of Newtonian mechanics. Simple everyday observations which we have been making for centuries, which are consistent with both Newton's and Einstein's theory, did not constitute supporting evidence for SR,

Of course they do. They do not provide evidence that SR is a more accurate theory, but they do support SR

 

[athon]

Theorem:
On the other side of every card that has a vowel, there is an even number.

Which of the following cards do you need to turn over to test this theory?

5..........A............T............6

Ok, I'll buy.

Your subject is the vowel, and you are describing each vowel as having an even number on the reverse side. So you'd turn over the 'A'.

You have said nothing of whether an even number has a vowel on the other side, hence can't demonstrate anything by turning over the '6', and you have not described consonants at all, hence turning over the 't' would also demonstrate nothing, or odd numbers, so the '5' is pointless.

I'm not sure where this is going...

Athon

 

[Interesting Ian]

OK, dunno if you still have me on ignore (for some obscure reason), but I'll attempt an answer to your question anyway.

I don't know any official solutions to this problem, but here are my thoughts.

There are many many more non-ravens then there are ravens.

Let's suppose there are a trillion existents in the Universe; 999,999,000,000 are non-ravens, and 1,000,000 are ravens.

Thus the hypothesis is that there are 999,999,000,000 non-black things which are non-ravens, and 1,000,000 black things which are ravens.

You can prove the hypotheses that all ravens are black either by observing every single raven in existence and seeing they are all black -- 1,000,000 of them, or by observing all 999,999,000,000 non-black things in existence and observing that none of them are ravens..

Since there are 999,999 more non-black things than ravens you would need to observe 999,999 non-black things which are non-ravens to equal the degree of evidence of seeing one black raven.

 

[Interesting Ian]

I think both Soapy and Patricio have answered this nicely here; we are looking at less a problem of logic, and more of a problem with language.

Ummmm . .I don't think so.

 

[athon]

Ummmm . .I don't think so.

Did you read the rest of my post?

Your own example of counting each and every object does not address the problem.

Science is about making predictability out of an unpredictable situation. If I have ten blocks, and they are all red, my statement 'these ten blocks are all red' is an observation, not a prediction. Which is what you have done by counting all of the ravens and then (in another experiment) all of the black things.

If I have ten blocks, and I can see only nine that are red, I can say 'all ten blocks are red', as I have good evidence to make that prediction and be fairly sure I am right. I'm not making predictions about anything other than the colour (the descriptor) of the blocks (the subject of the experiment).

Tez was assuming that both statements were logically identical. They are not, as the subjects and descriptors differ in each. Hence, it is a language problem.

Let me take this a step further. I have ten objects; five balls and five blocks. Objectively, the balls are all red and the blocks are all blue.

My hypothesis is 'All balls are red'. The equivalent here is 'All non-red things are not balls'.

I can only see four balls, all red. Does the fact that the five blocks, all blue, increase my chances that the fifth ball is also red (which is the essence of the second hypothesis?) Not unless there is a link between the subjects, which again is the subject of another hypothesis which would make a statement of correlation.

Hence the hypothesis differ in subject.

Athon

(edited for clarity)

 

[FireGarden]

Ok, I'll buy.
[...]

You have said nothing of whether an even number has a vowel on the other side, hence can't demonstrate anything by turning over the '6', and you have not described consonants at all, hence turning over the 't' would also demonstrate nothing, or odd numbers, so the '5' is pointless.

I'm not sure where this is going...

Athon

Suppose I now turn over the '5' and there is an E on the back of it? Do you still think there is no need to turn over the 'T'?

 

[athon]

Suppose I now turn over the '5' and there is an E on the back of it? Do you still think there is no need to turn over the 'T'?

Not according to your hypothesis. You have made a prediction that vowels can be described as having an even number on the back. Your population is being observed by the face-up values, hence you are basing your observation on that (predictability being that turning over a vowel will display an even number). Turning over the 'A' and having an odd number has disproven your hypothesis. Turning over the 'A' and having an even number has added support to your hypothesis (but not disproven it).

In such a case, you are looking for falsification, remember.

Turning over the '5' and having an 'E' has not assisted you in your hypothesis, as you are out to make an accurate prediction of the back of an observed card more probable, as you had no idea the 'E' was there in the first place.

The end result is often the same, such as here, but you are extending the hypothesis into an area where you did not originally describe going, leaving room for assumptions. In simple examples as these, the results are identical. However as we know, when it gets more complicated you start to leave room for erroneous assumptions.

Athon
(I'm not sure if I described that well enough...)

 

[FireGarden]

You have made a prediction that vowels can be described as having an even number on the back

Actually, I used the term "on the other side". And I made no mention of "Face-up"




In fact, I did mis-word the question, but not in the above manner.

If you were merely trying to find a counterexample to the theorem, then there would be no point in turning over the '6'. That was what I intended.

But actually, to make this fit into Tez's thread,
Does turning over the 'T' and finding a 'Z' count as evidence in support of the theorem?

After all:
Vowel --> even on other side
is the same as
Not-even --> not-vowel on other side.

I think that this is more to do with the way the human mind works. It is easier for us to invisage, Venn diagram style, A-->B as "A is a subset of B" than it is for us to see the same picture as "Not-B is a subset of Not-A"

 

[Stimpson J. Cat]

Tez,

What you say is certainly true about the practice of science Stimpy, but I'm not sure it addresses the issue head on. What if the statement under consideration was something for which I had little or no evidence for either its truth or validity? Is seeking evidence for some logically equivalent statement valid or not? In some circumstance it is, and in some it isn't.

I'm not sure what you mean by "valid" here. The question is what conclusions we can draw from the evidence. Simply observing something which has been observed many times before, like that your blue jeans are not ravens, or that objects push back with equal and opposite force when you push on them, does not tell you anything you don't already know, so you can't really conclude anything new from them. Whether you are seeking evidence for the statement "all crows are black", or the equivalent statement "all non-black things are not crows", the fact remains that your observations have to convey some significant new information in order for them to constitute "supporting evidence". The fact that looking at your blue jeans and noticing that they are not crows, does not constitute supporting evidence for anything, has nothing to do with which of the logically equivalent statements you are trying to support. It supports neither of them, because it is simply something which you already knew.

I think the issue with this latter example is that there is a level of uber-connectedness between the equivalent logical statements about the zeros of the Zeta function, that has little to do with formal logic. Its hard to make precise, but the difference (as I see it) is that there is a potential for "explanatory power" regarding the underlying "phenomenon" when one observes either type of "instance" of the Riemann zeros or non-zeros; something not true in the Raven case.

If I am understanding you correctly, then this "explanatory power" is exactly what I was talking about. After all, it is not like a bunch of old wise men got together and decided what "supporting evidence" is. When we say that an observation supports a theory, what this really means is that the observation has provided us with information which we did not have before, but which was predicted by the theory. The important issue is not whether the observation is consistent with the theory, or even whether it is predicted by the theory, but whether or not it tells us something new.

Another way of looking at it is that we should not really be concerned with "supporting" or "refuting" theories at all. What we should be concerned with is learning new knowledge. The theories are just a good way of both expressing what we already have learned, and suggesting good places to look to learn new things.


Dr. Stupid

 

[athon]

Tez,

...the fact remains that your observations have to convey some significant new information in order for them to constitute "supporting evidence".

Dr. Stupid

Stimpy, I agree with everything you've said, but disagree with this one small point.

Every observation adds to an increasing pool of certainty. If you follow principles of inductive reasoning, you make an assumption of certainty based on the relative number of observations in relation to an objective quantity. So if you've seen one white swan, and know there are one hundred white swans out there, your prediction that all swans are white is weaker than if you have seen two white swans...and so on.

Practically, after a while each subsquent observation might not increase our knowledge pool (certainly not as much as an event where the hypothesis is falsified). But inductively, each observation does indeed add to our value of certainty, and makes our prediction that little more probable.

Athon

 

[Stimpson J. Cat]

FireGarden,

Consider, for example, Special Relativity. Under most conditions its predictions are pretty much indistinguishable from those of Newtonian mechanics. Simple everyday observations which we have been making for centuries, which are consistent with both Newton's and Einstein's theory, did not constitute supporting evidence for SR,

Of course they do. They do not provide evidence that SR is a more accurate theory, but they do support SR[/quote]

There is a difference between an observation being consistent with a theory, and an observation actually being supporting evidence for the theory. That is the whole point. Making observations when you already know what the outcome will be, conveys no new information. It tells you nothing. Such observations convey no more new information than turning over the 5 or T card in your example would.

If you want supporting evidence for a theory, you need to make observations which provide new information.

If I construct a new theory which is consistent with all known information, then simply repeating old experiments which do not provide any new information, does not provide supporting evidence for my theory. All it does is confirm what I already know, namely that my theory is consistent with all currently known information.


Dr. Stupid

 

[Gwyn ap Nudd]

All ravens are black.

All non-black objects are non-ravens.

These are considered logically equivalent because they are disproved by the same counter-example, a non-black raven.

But they are scientifically distinct because they have different domains. The logical equivalence is the assumption that the domains are mutally exclusive. Since the size of the universe is potentially infinite, it is easier to try to exhaust the smaller domain than the larger one.

On the other side of every card that has a vowel, there is an even number.

All vowels back even numbers.

has the same counter-example as

All (non even numbers) back (non vowels)

namely a card with at least one vowel, but no even numbers.

If the universe is 5/?, A/?, T/?, 6/? it small enough to exhaustively examine all the members. But it is not necessary to examine the 6/? card, since it cannot be a counterexample.

 

[Pragmatist]

To me, the problem is one of limits and/or boundary conditions. The bare statements about ravens don't specify any subsidiary conditions and therefore each is only true within assumed limits.

For example, is a raven "black" in infra-red light for example?

In reality we always need to specify the limits or boundary conditions under which an assessment is made. In pure logic we don't always specify such conditions because logic is mathematics and it is tacitly assumed that the laws of mathematics hold. But mathematics is artificial and has well defined boundary conditions. The real world does not have such (known) limits and so the problem is open to interpretation.

The confusion therefore arises from trying to apply a mathematically consistent statement with well defined limits to a situation which is not mathematically consistent and does not have well defined limits (i.e. the real world).

 

[FireGarden]

Stimpson

The important issue is not whether the observation is consistent with the theory, or even whether it is predicted by the theory, but whether or not it tells us something new.

[...]
There is a difference between an observation being consistent with a theory, and an observation actually being supporting evidence for the theory

This is not the way I see things.
Is there something inconsistent with my outlook, or is your outlook simply a preference?(I hesitate to say "eccentricity"!)

In your view, it seems that if GR had been conjectured before Newton, then the orbit of Mars would have been supporting evidence. But since Newton did in fact speak first, then the orbit of Mars is not supporting evidence. I don't see things that way.

The orbit of Mercury not only supports GR, but it refutes Newton. It's that last bit that makes it special as evidence. One counter-example is worth any number of confirmations.

BTW
How many proofs of Pythagorus' Theorem do you think there are? Only 1? or more than 300?

Such observations convey no more new information than turning over the 5 or T card in your example would.

But turning over the 5 or the T could potentially reveal something new. If there is a vowel on the other side, then they will be counter-examples. If we turn over a 100 such cards and fail to find a counter-example, then surely we must be more certain of the theorem than if we had only turned over 1 card. So each experiment must be adding something in the way of support to the theory.

I agree that there comes a point when people will say "so what, we have enough confirming observations." But that's people.

Gwyn ap Nudd
If, in a bigger sample, you had found 100 non-counter-examples (ie: confirmations) would you still consider the 101st confirmation as a piece of supporting evidence?

I would.
And, more to the point of the thread, I would consider turning over the A to find an even number just as much support for the theorem as turning over the 5 to find a non-vowel. That's because I consider both forms of the theorem to be equivalent.