, ,

As I have stated repeatedly, probability does not apply to individual cases - only to populations (P) and samples randomly taken therefrom (S). Probability is simply the proportion of P or S that has a designated characteristic (C).

If we know that X% of P has C, then we can predict that approximately X% of S will have C; this is statistical deduction. If we know that Y% of S has C, then we can predict that approximately Y% of P will have C; this is quantitative induction. In both cases, the accuracy of the prediction increases as the size of S increases, until the error disappears completely when S is identical to P. By the way, this is what gives induction its distinctive (i.e., non-deductive) validity as a form of reasoning - it is self-correcting in the long run.

What we have primarily been discussing in this thread is a third type of inference. If we know that no S has C, then we can predict that no P has C, but it is meaningless to assign a probability value to such a prediction; this is crude induction. It is also self-correcting; in fact, it takes merely a single counterexample to falsify the prediction. Despite its obvious fallibility in this sense, it is the only way to "justify" a universal proposition inductively.

I'm definitely no mathematician, I learned probability from magic finding. I remember once mfing this dragon pet that had a 1 in 6000 chance of dropping. I was farming the area, when I saw someone else doing the same, and they told me that they didn't want to sign out, because they thought that it would reset their probability, and they've already killed like a thousand!

That's not how it works, each and every kill has a 1 in 6000 chance of dropping the pet. You could get it on your first kill, or your one millionth, but given a massive sample size, the rate is roughly 1 in 6000.

There is of course a sense in which the more you kill, the more likely you are to get it, but this isn't in a strictly probabilistic sense, as it includes variables that can't be known.

Similarly, surely if every raven is black, then the chances of any raven you encounter being black is 100% (minus albinos, or whatever). It can only be otherwise if there is a chance that the ravens could be any other color, but what kind of meaningful "probability" could this be other than just your own disposition of confidence in the face of not knowing an important variable?

Similarly, surely if every raven is black, then the chances of any raven you encounter being black is 100% (minus albinos, or whatever). It can only be otherwise if there is a chance that the ravens could be any other color, but what kind of meaningful "probability" could this be other than just your own disposition of confidence in the face of not knowing an important variable? — Wosret

I refer you to the Poker example I mentioned earlier. The cards have been dealt out and you have a pair of kings and your opponent either has a pair of aces or he doesn't, and so with this in mind you might say that the probability that he has a better hand than you is either 1 (if he has a pair of aces) or 0 (if he hasn't). But does that seem right to you? Not to me. It would be correct to say that the probability is 1/n, where n is the number of possible hands he can have.

Depends on what we're talking about. If talking about the probability of him having the cards that he actually does, then it is 100%, he already has them. If we're talking about the probability of him having this or that card, based on what we know, then we're really just saying that we know of no biases, or reasons to go one way or the other, so all things being equal, any guess is as good as any other.

Of course that's why you need a poker face, to keep all things equal.

If we're talking about the probability of him having this or that card, based on what we know, then we're really just saying that we know of no biases, or reasons to go one way or the other, so all things being equal, any guess is as good as any other. — Wosret

Yeah, and that's what we use probability for; to determine the likelihood that our guess is correct, based on the available epistemic criteria.

I have some questions about this. I don't see how H (hypothesis) logically implies E (evidence). — Arkady

Well spotted!

A couple of things:

The truth value of H and HE are the same, because H logically implies E

p(HE) = p(H)*p(E|H)

The probability of E given that H is in fact true is 1, because H logically implies E.

So by Bayes theorem

p(HE|B) = p(B|HE)P(HE)/P(B)

= p(B|H)p(H)/p(B) = p(H|B)

Also, in order for the posterior probabilities not to matter here (because E cancels out), H and K must somehow imply the same "E". But, how can a hypothesis and its negation imply the same observational consequences? — Arkady

Even better!

K is compatible with any evidence. p(E|K) is still 1, and (KE) and (K) have the same truth value. I certainly seems weaker, but I can see no reason that K does not logically imply E, just as it implies not(E).

You might regard this as a more formal statement:

K => E iff p(E|KB) = 1 for every B

Also, the construction P(H&E|B) to describe confirmation seems odd. Wouldn't P(H|E&B) be more appropriate? — Arkady

That's in all the equations!

Well spotted!

A couple of things:

The truth value of H and HE are the same, because H logically implies E

p(HE) = p(H)*p(E|H)

The probability of E given that H is in fact true is 1, because H logically implies E.

So by Bayes theorem

p(HE|B) = p(B|HE)P(HE)/P(B)

= p(B|H)p(H)/p(B) = p(H|B) — tom

My problem here is that I don't see how H logically implies E. Setting aside the propositional variables for a moment, I don't understand how this particular H ("all ravens are black") implies this particular E ("the sighting of another raven"). Again, I'm not sure what the E statement even means here.

K is compatible with any evidence. p(E|K) is still 1, and (KE) and (K) have the same truth value. I certainly seems weaker, but I can see no reason that K does not logically imply E, just as it implies not(E).

K is a particular hypothesis. To say that a particular hypothesis is "compatible with any evidence" means that there is nothing which can falsify K, even in principle. This is no bueno for a purportedly scientific/empirical hypothesis.

So, you are in essence saying that the same evidence would confirm "all ravens are black" as would confirm "not all ravens are black") (i.e. H and K, respectively, which constitute the hypothesis and its negation). I don't see how that could possibly be the case.

That's in all the equations!

Yes, I saw that. I wasn't reading your post correctly; the ordering threw me, which is why I deleted that paragraph from my post (seemingly long before you replied to it; not sure if the forum software is getting glitchy here). But, I agreed with your ultimate presentation of the logic of confirmation, i.e. P(H|E&B), so we're good on this point.

Much has been made in this discussion re: the the objective/subjective distinction, which you seem to think has been muddled in this thread.

Do you believe that an agent can have better or worse reasons for increasing or decreasing his confidence in a given hypothesis in the face of new evidence? That is, some types of evidence are "better" or "worse" than others?

Do you believe that an agent can have better or worse reasons for increasing or decreasing his confidence in a given hypothesis in the face of new evidence? That is, some types of evidence are "better" or "worse" than others? — Arkady

Objectively or subjectively? :D Yes, but I would prefer not to say that probability has anything to do with it, since we are clearly talking about one person's confidence. And I do think that someone's evaluation of evidence as "better" or "worse" typically has both objective and subjective aspects.

@aletheist, from a brief look at the various literature on probability, am I right in suggesting that you take the frequentist view and I take the (objective) Bayesian view (and Cox's theorem?)? From here:

While for the frequentist a hypothesis is a proposition (which must be either true or false), so that the frequentist probability of a hypothesis is either 0 or 1, in Bayesian statistics the probability that can be assigned to a hypothesis can also be in a range from 0 to 1 if the truth value is uncertain.

...

Broadly speaking, there are two interpretations on Bayesian probability. For objectivists, probability objectively measures the plausibility of propositions, i.e., probability corresponds to a reasonable belief everyone (even a "robot") sharing the same knowledge should share in accordance with the rules of Bayesian statistics, which can be justified by requirements of rationality and consistency.

In my non-technical opinion, my example of the randomly-selected eggs seems to be as described here.

Yes, although there is more to the frequentist view than what you quoted. As its own Wikipedia article states:

In the frequentist interpretation, probabilities are discussed only when dealing with well-defined random experiments (or random samples) ... For any given event, only one of two possibilities may hold: it occurs or it does not. The relative frequency of occurrence of an event, observed in a number of repetitions of the experiment, is a measure of the probability of that event ... A claim of the frequentist approach is that in the "long run," as the number of trials approaches infinity, the relative frequency will converge exactly to the true probability ...

The frequentist interpretation is a philosophical approach to the definition and use of probabilities; it is one of several such approaches. It does not claim to capture all connotations of the concept 'probable' in colloquial speech of natural languages. As an interpretation, it is not in conflict with the mathematical axiomatization of probability theory; rather, it provides guidance for how to apply mathematical probability theory to real-world situations.

Personally, I think that even the "objective" Bayesian approach is fundamentally subjective, because it claims to measure confidence, degree of belief, or plausibility, all of which will vary from person to person.

Personally, I think that even the "objective" Bayesian approach is fundamentally subjective, because it claims to measure confidence, degree of belief, or plausibility, all of which will vary from person to person. — aletheist

Then what about my example of the eggs? It certainly seems to make use of objective values and so won't vary from person to person, even though it's about confidence/plausibility.

If so then I think that the last issue of contention is my claim that evidence is anything that increases the objective Bayesian probability that a hypothesis is true. But then what else does it mean to count as evidence, I wonder?

Then what about my example of the eggs? It certainly seems to make use of objective values and so won't vary from person to person, even though it's about confidence/plausibility. — Michael

I raised no objections to what you said in the post that you linked about the probability of all ten eggs being white before they are produced; or even the probability of all ten eggs being white after eight of them have already come out white, but before the other two eggs are produced. My objection was to your claim that the same reasoning still applies after all ten eggs have been produced. At that point, either all ten eggs are white (p=1) or at least one of them is non-white (p=0).

What we are calculating here is not confidence/plausibility, but the proportion of outcomes that would occur in infinitely many random trials. If you were to use the device to produce a batch of ten eggs a million times, then the number of batches with all white eggs would be approximately 0.5¹⁰ x 1,000,000 = 977. If you were to use the device to produce a batch of two eggs a million times, then the number of batches with both white eggs would be approximately 0.5² x 1,000,000 = 250,000. However, these values tell us nothing about the actual color of the eggs in a single batch of ten or two after the device produces it.

I raised no objections to what you said in the post that you linked about the probability of all ten eggs being white before they are produced; or even the probability of all ten eggs being white after eight of them have already come out white, but before the other two eggs are produced. My objection was to your claim that the same reasoning still applies after all ten eggs have been produced. At that point, either all ten eggs are white (p=1) or at least one of them is non-white (p=0). — aletheist

I'm using Bayesian probability, as just mentioned. What I'm saying is that, given it's using objective values, it isn't something that will vary from person to person as you suggest. Hence it being objective Bayesian probability (as I understand it).

If so then I think that the last issue of contention is my claim that evidence is anything that increases the objective Bayesian probability that a hypothesis is true. But then what else does it mean to count as evidence, I wonder? — Michael

Lots of different people have lots of different ways of counting something as evidence, and then weighing it with other evidence. Very few people have any idea what "objective Bayesian probability" is, let alone how to use it as a tool for mathematically gauging their confidence or degree of belief. Again, I personally would prefer that everyone use the latter terms or "plausibility," rather than calling it "probability" at all.

Lots of different people have lots of different ways of counting something as evidence, and then weighing it with other evidence. Very few people have any idea what "objective Bayesian probability" is, let alone how to use it as a tool for mathematically gauging their confidence or degree of belief. — aletheist

They might not use the term "objective Bayesian probability", but what they understand evidence to be might be exactly this (i.e. increases the rational plausibility that the statement is true).

Or, if we assume that there's some proper, non-colloquial account of evidence (and surely we do if the problem of induction is a philosophical problem at all?), then I'm suggesting that my account is accurate (or at the very least, not without merit).

I'm use Bayesian probability, as just mentioned. What I'm saying is that, given it's using objective values, it isn't something that will vary from person to person as you suggest. Hence it being objective Bayesian probability (as I understand it). — Michael

My point is that someone's actual confidence or degree of belief is not objectively measurable. Again, most people will never do this calculation, and only those who agree with its underlying assumptions will attribute any validity to it at all. In any case, my main contention is that we should avoid calling it "probability," because doing so encourages confusing what we think with how things really are.

My point is that someone's actual confidence or degree of belief is not objectively measurable. — aletheist

But their actual confidence isn't relevant, as we're considering objective Bayesian probability, not subjective Bayesian probability. I believe that my example of the eggs (after their production) is an example of objective Bayesian probability, rather than an example of subjective Bayesian probability. It's objective Bayesian probability won't differ from person to person.

Again, most people will never do this calculation, and only those who agree with its underlying assumptions will attribute any validity to it at all.

Whether or not they attribute any validity to it all isn't relevant. What matters is whether or not it is valid.

So, my argument is that after checking each egg and finding it to be white the objective Bayesian probability that every egg is white increases, and that as evidence is whatever increases the objective Bayesian probability that a hypothesis is true (or, to be more liberal, that whatever increases the objective Bayesian probability that a hypothesis is true is evidence), each observation of a white egg is evidence that every egg is white.

And feel free to replace "probability" with "plausibility" if it really matters that much to you.

But their actual confidence isn't relevant, as we're considering objective Bayesian probability ... each observation of a white egg is evidence that every egg is white. — Michael

Your premiss here is that anything that increases the Bayesian plausibility of a proposition should count as evidence for its truth. Only someone who agrees with this - i.e., attributes validity to it - will actually be more confident about the truth of a proposition by virtue of such calculations. Your original example, where the observation a green apple somehow counts as evidence that all ravens are black, demonstrates the implausibility of this whole approach.

And feel free to replace "probability" with "plausibility" if it really matters that much to you. — Michael

If you had just said this in the beginning, then we might not have had much of an argument at all. :D

Your premiss here is that anything that increases the Bayesian plausibility of a proposition should count as evidence for its truth. Only someone who agrees with this - i.e., attributes validity to it - will actually be more confident about the truth of a proposition by virtue of such calculations. Your original example, where the observation a green apple somehow counts as evidence that all ravens are black, demonstrates the implausibility of this whole approach. — aletheist

I don't know what you mean by this. A thing can be evidence even if it isn't taken to be. That's why "ignoring evidence" is a thing.

Again, there's the example of the eggs I gave earlier. It's an application of Bayesian probability (as I understand it), but it has nothing to do with how confident any particular person is.

A thing can be evidence even if it isn't taken to be. That's why "ignoring evidence" is a thing. — Michael

This is simply what person A calls it when person B ignores something that person A counts as evidence, but person B does not. Is there some objective standard that dictates which of them is right? Bayesian plausibility somehow counts the observation of a green apple as evidence that all ravens are black, but I do not. I have other criteria for something to count as evidence, including relevance.

Again, there's the example of the eggs I gave earlier. It's an application of Bayesian probability (as I understand it), but it has nothing to do with how confident any particular person is. — Michael

Huh? Once the entire batch of ten eggs is produced, your calculations are only about how confident you are that all of them are white, as you observe them one by one. Again, objectively, either all of them white (p=1) or at least one of them is non-white (p=0).

Is there some objective standard that dictates which of them is right? — aletheist

The math in the example I gave of the white eggs.

Bayesian plausibility somehow counts the observation of a green apple as evidence that all ravens are black, but I do not. I have other criteria for something to count as evidence, including relevance.

To start, we're talking about my example of white eggs.

And also, given contraposition, they are relevant. If green apples are relevant to the proposition "if something isn't black then it isn't a raven" then it's relevant to the proposition "if something is a raven then it is black".

Huh? Once the entire batch of ten eggs is produced, your calculations are only about how confident you are that all of them are white, as you observe them one by one. Again, objectively, either all of them white (p=1) or at least one of them is non-white (p=0).

We're using objective Bayesian probability, not frequentist probability. And the point of the maths is to show that it has nothing to do with how confident any particular person is. I think you're conflating "confidence" in the ordinary sense with confidence in the sense meant in Bayesian probability. Here it describes it as "us[ing] the laws of probability as coherence constraints on rational degrees of belief" which isn't the same as just the strength of any particular person's conviction.

Again, the maths I used in the example shows that the Bayesian probability of every egg being white just is 0.5¹⁰, and this is true even if I believe otherwise (hence the objectivity).

If green apples are relevant to the proposition "if something isn't black then it isn't a raven" then it's relevant to the proposition "if something is a raven then it is black". — Michael

I do not consider the observation of a green apple to be relevant to either of these propositions. Bayesian plausibility somehow counts it as evidence that all non-black things are non-ravens, but I do not - in this case, mainly because of the sheer number of non-black things. Even if we limit it to actual non-black things and assume that the quantity is finite, it will still be so large that the change in your calculated value is vanishingly small - certainly not sufficient to count as genuine evidence in my book.

Again, the maths I used in the example shows that the Bayesian probability of every egg being white just is 0.5¹⁰, and this is true even if I believe otherwise (hence the objectivity). — Michael

Again, this is not yet Bayesian plausibility, it is frequentist probability - the proportion of infinitely many ten-egg batches that would consist entirely of white eggs. It only becomes Bayesian plausibility when you claim that the increase to 0.5⁹ when you observe that the first egg is white increases your confidence that all of the eggs in this particular batch are white.

So, you are in essence saying that the same evidence would confirm "all ravens are black" as would confirm "not all ravens are black") (i.e. H and K, respectively, which constitute the hypothesis and its negation). I don't see how that could possibly be the case. — Arkady

We have a situation where the observation of a green apple purportedly supports an enormous number of unrelated universal statements, including the statement "all ravens are black". The solution to this problem is to recognise that there is no such thing as epistemologically valuable corroborating evidence. It simply cannot exist.

The corroborating evidence E points everywhere and thus nowhere. For some psychological reason we see this in the case of green apples, but not in black ravens.

I do not consider the observation of a green apple to be relevant to either of these propositions. Bayesian plausibility somehow counts it as evidence that all non-black things are non-ravens, but I do not - in this case, mainly because of the sheer number of non-black things. Even if we limit it to actual non-black things and assume that the quantity is finite, it will still be so large that the change in your calculated value is vanishingly small - certainly not sufficient to count as genuine evidence in my book. — aletheist

If the principle holds when there are just two things to consider then it holds when there are a trillion things to consider.

And if you need the change to be sufficiently large enough then you just need to check a sufficiently large proportion of non-black things. With the example of eggs, having checked 9 of the 10 eggs should count as evidence.

Personally, I think that the sufficiency of the change is the measure of the strength of the evidence, not a measure of the fact that it's evidence. It is making a change at all that determines it to be evidence (even if weak).

Again, this is not yet Bayesian plausibility, it is frequentist probability - the proportion of infinitely many ten-egg batches that would consist entirely of white eggs. It only becomes Bayesian plausibility when you claim that the increase to 0.59 when you observe that the first egg is white increases your confidence that all of the eggs in this particular batch are white.

You said it isn't frequentist probability being that the probability (or plausibility, if you prefer) is 0.5¹⁰, not either 0 or 1. And, again, this number has nothing to do with the strength of my conviction. It's derived from the given fact that a random number generator was used to produce the eggs. It is entirely objective.

If the principle holds when there are just two things to consider then it holds when there are a trillion things to consider. — Michael

Turn that around - if it does not hold when there are a trillion things to consider, then it does not hold when there are just two things to consider. See, whether we accept Bayesian values as an objective measure of confidence or degree of belief is itself a subjective matter.

And if you need the change to be sufficiently large enough then you just need to check a sufficiently large proportion of non-black things. — Michael

What would be "sufficiently large"? Can we identify some objective threshold, above which the increase in calculated plausibility would count as evidence, and below which it would not? Or is it a matter that each person has to determine subjectively?

With the example of eggs, having checked 9 of the 10 eggs should count as evidence. — Michael

Would most people guess that the 10th egg is also white? Sure, but the probability in that individual case is no different than it was for each of the first nine - 0.5. It is just as likely that the 10th egg is non-white as that it is white; neither guess is objectively better than the other. If you flipped a coin that you knew to be fair and got heads nine straight times, would you bet on the next flip also being heads? Notice how our intuition goes the opposite way in this case, even though the two scenarios are probabilistically identical - I suspect that most people would guess that the 10th flip will be tails, even though (again) neither guess is objectively better than the other.

You said it isn't frequentist probability being that the probability (or plausibility, if you prefer) is 0.5¹⁰, not either 0 or 1. — Michael

No, you are not making the distinction between the probability that any batch of ten eggs would be entirely white (0.5¹⁰) and the probability that this particular batch of ten eggs actually is entirely white (either 0 or 1).

We have a situation where the observation of a green apple purportedly supports an enormous number of unrelated universal statements, including the statement "all ravens are black". — tom

But, that's part of the paradox. Green apples are not unrelated to the universal statement "all ravens are black." It confirms the (logically equivalent) contrapositive, i.e. that all non-black things are non-ravens.

The solution to this problem is to recognise that there is no such thing as epistemologically valuable corroborating evidence. It simply cannot exist.

Perhaps such evidence cannot exist, but your purported proof to that effect seems flawed. Unless you can address my specific concerns, I can't accept it.

Now that I think of it, I may have found another problem: you claim that H entails E, and so that p(he|b) = p(h|b). But, I'm not sure that this follows. Even if H entails E, unless they are necessary truths, the probability of their conjunction must be equal to or less than either of the conjuncts. P(H|E) or P(E|H) are not equal to P(H&E), even when H entails E (that is, P(E|H) = 1).

The corroborating evidence E points everywhere and thus nowhere. For some psychological reason we see this in the case of green apples, but not in black ravens.

Right...hence the paradox! It's counterintuitive (to put it mildly) that the observation of green apples confirms the hypothesis that all ravens are black.

This seemed to be Popper's view (as someone else pointed out). Let us consider for a moment the proposition that singular instances provide no confirmation of a universally-quantified hypothesis or statement (e.g. occurrences of white swans do not even marginally raise the probability of the hypothesis "all swans are white") by means of a thought experiment.

At the very least, this claim seems unintuitive under certain conditions. For instance, imagine that the world consists entirely of a carton of eggs, with a dozen egg cups, each containing exactly one egg. A "God's eye view" observer of the world formulates the hypothesis that "all eggs are white," and sets about inspecting each cup.

After the observer inspects, say, three of the eggs and finds that they're white, can he reasonably be more confident in the truth of his hypothesis to any degree whatsoever? After all, each cup which is found to contain a white egg is one less cup which can possibly hold a non-white egg (and we've stipulated that the world consists solely of this egg carton, so there is nowhere else for a non-white egg to hide). Does each observation of a white egg therefore confirm the hypothesis (even if only incrementally)? My intuition seems to say "yes," but of course, my intuition does not constitute any sort of rigorous proof. — Arkady

Upon further reflection, it occurred to me that my thought experiment (whether or not it presents a valid point) has limited applicability to the raven paradox. The universal statement under consideration is "all ravens are black." The contrapositive is "all non-black things are non-ravens."

Here are 4 possible observations, and how they (might) affect the hypothesis:
(1) black raven - confirms
(2) non-black raven - falsifies
(3) non-black non-raven - confirms
(4) black non-raven - neither confirms nor disconfirms

However, even in the very limited world of my thought experiment, a (4)-type observation would in fact confirm the hypothesis "all ravens are black." Assuming that we've already observed at least one white egg, observation of a white non-egg would confirm the hypothesis that "all eggs are white," because it would further diminish the probability of the carton containing at least one non-white egg (because the white non-egg, whatever it might be, is occupying space that might otherwise be occupied by a non-white egg).

So, thought experiments of this type (even if they succeed in demonstrating that universally-quantified hypotheses can be confirmed, which hardly seems to be the consensus here...) may not have much to do with the raven paradox specifically.

Assuming that we've already observed at least one white egg, observation of a white non-egg would confirm the hypothesis that "all eggs are white," because it would further diminish the probability of the carton containing at least one non-white egg (because the white non-egg, whatever it might be, is occupying space that might otherwise be occupied by a non-white egg). — Arkady

It's also occupying space that might otherwise be occupied by a white egg. So I think your original suggestion that it neither confirms nor disconfirms was accurate.

Actually, no, I think you're right.

But, that's part of the paradox. Green apples are not unrelated to the universal statement "all ravens are black." It confirms the (logically equivalent) contrapositive, i.e. that all non-black things are non-ravens. — Arkady

Green apples also "confirm" the universal statement "all ravens are white".

Now that I think of it, I may have found another problem: you claim that H entails E, and so that p(he|b) = p(h|b). But, I'm not sure that this follows. Even if H entails E, unless they are necessary truths, the probability of their conjunction must be equal to or less than either of the conjuncts. P(H|E) or P(E|H) are not equal to P(H&E), even when H entails E (that is, P(E|H) = 1). — Arkady

I covered that earlier.

Right...hence the paradox! It's counterintuitive (to put it mildly) that the observation of green apples confirms the hypothesis that all ravens are black. — Arkady

It's not counterintuitive it is just wrong. If green apples "confirm" "black ravens", they also confirm "white ravens".

Let us not forget that the universal statement "all ravens are black" is in fact false!