The Raven Paradox
Hempel's raven paradox is the following:
(1) All ravens are black.
Via contraposition, this is logically equivalent to:
(2) Everything that is not black is not a raven.
Given the logical equivalence, any evidence in support of (2) is also evidence in support of (1). For example, a green apple is evidence in support of (2); it is not black and not a raven. Therefore, it is also evidence in support of (1).
So, green apples support the claim that all ravens are black. Which is quite unintuitive, hence the paradox.
(1) All ravens are black.
Via contraposition, this is logically equivalent to:
(2) Everything that is not black is not a raven.
Given the logical equivalence, any evidence in support of (2) is also evidence in support of (1). For example, a green apple is evidence in support of (2); it is not black and not a raven. Therefore, it is also evidence in support of (1).
So, green apples support the claim that all ravens are black. Which is quite unintuitive, hence the paradox.
Comments (312)
A green apple is evidence in support of (2) in terms of matters of fact. It is not evidence in support of (1) in terms of matters of fact.
If they're logically equivalent then how can something support (2) in terms of matters of fact (whatever that means) but not (1)?
In natural language, a green apple is relevant to (2), but it isn't relevant to (1). Formally accurate language guarantees the equivalence of (1) and (2). Natural language merely shows that they are two different strings of symbols.
And it's still the case that (1) is true iff (2) is true. Therefore evidence that supports the truth of (2) also supports the truth of (1).
(1*) For every x, if x is a raven, then x is black.
Do you see that 'a green apple' is irrelevant to (1*)?
The logical form of (2) would be:
(2*) For every x, if x is not black, then x is not a raven.
Do you see that 'a green apple' is not related to (2*)?
You should be able to distinguish the talk about formalization from the talk about matters of fact.
No. Green apples are evidence that support this claim. They're not black and not ravens.
(3) There exists some x such that x is green, and x is an apple.
Do you think that (3) is relevant to (1*) and (2*) in terms of formalization?
https://youtu.be/EfYH7a4RBAo
What's even more annoying is that intuition seems to say that a black raven is not evidence that non-black things are non ravens.
Unless someone wants to reform logic fairly drastically, the only thing to do is to agree with Hume that 'evidence' is all just habit and has no logic, and then follow Popper and @Cavacava and look to falsification.
And green apples are also evidence that all ravens are white.
Thus 'all dragons breathe fire' cannot be supported by any number of fire breathing dragons, but is falsified by a single non-fire-breathing dragon. This fits with the Venn diagram approach, and also with Popper.
If that were the case then much of science would have to be dismissed. From a finite number of observations we infer general rules of nature that are applicable to everything of that type.
So you seem to address the problem of induction by denying induction as rational.
The evidence is falsification, but more than that, a theory needs to be able to make a novel falsifiable prediction. The theory needs to do more than merely ad hoc an explanation to an observation, or rely entirely on the information given in experience, but must be able to predict something is the case, or will be the case based on the implications of the theory.
I don't think so. You just have to preface your universals with 'As far as I know', or some such trope. Science is provisional, but you don't have to dismiss it. The general rule applies until you find an exception, and then progress is made. I mean nearly all ravens are black still, until someone discovers the vast antarctic flock of white ravens, or the red ravens of Mars.
Then green apples support the claim "as far as I know, all ravens are black".
That's not as clear as I'd like it to be. But consider, 'as far as I know all dragons breathe fire'. Given that my knowledge extends to no dragons at all, my evidence is nil, no matter how many non-fire-breathing non-dragons I have seen. 'As far as I know' needs to be a non zero distance wrt dragons or ravens, and not non-dragons and non ravens. The evidence makes the claim an existential one, or it is not evidence.
But the point of the paradox is that this isn't the end of it. Green apples are evidence that as far as I know all things that aren't black aren't ravens. Which entails that they're evidence that as far as I know all ravens are black.
Evidence can only apply to existential claims.
Isn't this an inductive premise, though? In the sense that it cannot be deductively proven that all ravens are black? What if there was an albino raven, or a raven spray-painted lime green?
Since it is an inductive premise, then, we shouldn't be surprised when green apples fail to give us any substantial deductive information regarding the qualities of ravens.
I think the analysis is complete if one points out that 'there are ravens and all ravens are black' is no longer equivalent to 'There are non-black things and all non-black things are non-ravens'. And this asymmetry is the scientific escape from the logical paradox.
Because for logic, there is no problem in saying that all dragons breathe fire and no dragons breathe fire... As long as there are no dragons.. But the science of dragons is in its infancy
What would the argument be for that?
They're logically equivalent because of the law of contraposition, and evidence for one is evidence for the other because they're logically equivalent.
(forall x) (not black x) --> (not raven x)
Prima facie, it seems to be no evidence at all, since the claim is universal, and a single datum doesn't help us with the universal.
But I wonder if Bayes' Law can help us. Using P(A) to indicate the probability of A, and | do denote the conditional, we aim to prove that P(Black | Raven) =1, ie Probability that something is Black, given that it is a raven, is 1.
This is the same as P(~Black | ~Raven) = 1.
Now Bayes Law tells us that
P(~Black | ~Raven) = P(~Raven | ~Black) P(~Black) / P(~Raven)
Say we start with guess probabilities that there half the things in the world are ravens, and half of the things in the world are Black, and the two properties are independent (eg so that half of Ravens are Black), then observing a green apple will .................... aaargh, this seems to be leading to a dead end, and I'm late for work so I'm giving up for now.
I have a feeling that Bayes Law can somehow be used to interpret the observation of the green apple as evidence for the hypothesis, but right now I'm not getting there.
Can anyone see a way to make that argument, either using Bayes Law or something else.
If we can't make the interpretation then there's no paradox.
That can't be the argument for why evidence for one is evidence for the other just in case they're logically equivalent. After all, it's just a restatement of what it's supposed to be an argument for.
The thinking is that if a black raven is supporting evidence that all ravens are black (not proof, note), then by the same token a green apple is supporting evidence that all non black things are non-ravens.
It's like love and marriage, you can't have one without the other. Unless you use unenlightened's patent interpretation of scientific generalisations.
I'm pretty sure it's axiomatic. If statements X and Y are logically equivalent then they have the same truth value in every model. Therefore if some A is evidence that X is true then it must also be evidence that Y is true.
Compare with "I am a bachelor" and "I am an unmarried man". Given that they're logically equivalent, evidence for one is evidence for the other.
Well that kind of sucks for a support of it.
It seems to me that there's a problem with it, and we're right back at what I've been harping on in other contexts: for one, we're ignoring the semantic content of the statements. But can we really do that when we're talking about whether evidence for one is evidence of the other?
We're not ignoring the semantic content of the statements. That they're logically equivalent is that they have the same semantic content.
Logic is about form, not semantic content. You're arguing otherwise? That logical form is identical to semantics?
Read up on contraposition. Here's a simple summary:
What would you say that has to do with my last post and my questions to you?
Via contraposition, "if something is a bat, then it is a mammal" is logically equivalent to "if something is not a mammal, then it is not a bat". If one is true then ipso facto the other true. Therefore evidence that one is true is evidence that the other is true. It's really straightfoward.
I don't see how you're showing me that I'm wrong. What part of what you quoted addresses semantics rather than form?
With logically equivalent statements, evidence for one is evidence for the other. Again, it's really straightforward.
First you didn't quote that part. I asked "What part of what you quoted . . ."
Anyway, okay, so material equivalence isn't logical equivalence?
I quoted the part that said that they were logically equivalent. I assumed (naively, apparently) that you understood what that meant.
Understanding is different than agreement.
So you understand but don't agree that logically equivalent statements mean the same thing?
I don't agree that logic deals with semantics other than formally. I had said that above.
So you don't agree that "if something is a raven then it is black" means the same thing as "if something isn't black then it isn't a raven"?
It depends on who is assigning meaning to those statements and what meanings they're assigning, doesn't it?
You can't go off script too much. I understand.
This is the bollocks of a loser out of his depth.
Yes, I can readily accept that statement. What I'm having trouble with is finding a reason to believe the antecedent - that observation of a black raven is evidence for the proposition that all ravens are black. It is conclusive evidence for the proposition that SOME ravens are black, but I can't see why it should be any evidence at all for the ALL proposition.
Michael, you mean? It's not that difficult of a question or idea. It's just not on his script.
Well add a few million black ravens and no non-black ones, and you might start to see that no evidence at all multiplied sufficiently starts to become convincing - which means that either it is some evidence, or one is convinced by mere habit.
Well, if ¬X is evidence of ¬Y then is X not evidence of Y?
And this is the problem of induction, isn't it? We use an incomplete sample to make universal claims.
That's how it really works out, we aren't super logical machines, we're impressed by the predictive power, the explanation that allows for it, and the control over phenomena that this inevitably leads to. There is still the possibility that it is partially, or entirely wrong, and things have just been working out, but that seems unlikely when it keeps working, and keeps saying new stuff that keeps working out.
So, my hypothesis is that all ravens are black. I predict that if I find a raven then it will be black. I look and find a black raven. Therefore, I have evidence that supports my hypothesis.
We already knew that ravens were black... But why would all ravens be black? That isn't based on having saw a lot of black ravens, and obviously can't be.
That has to be based on a theory that predicts that all ravens will be black that you encounter for a reason that causes their being black that we were not aware of. Because they have a certain gene configuration say, so that albino ravens would be predicted, and the non-black raven proves the theory that all ravens are black. You know... except for the disgusting mutants.
The usual approach is to have a 'null hypothesis' that some parameter is zero (eg the impact of a certain drug on the chances of curing a disease), and then assess the probability of having observed the data we did if that hypothesis were true. If the probability of that observation is small enough, we reject the null hypothesis and conclude, with a stated level of confidence, that the parameter is nonzero.
Here, say we let the parameter be the proportion p of ravens that are not black. the trouble is that the null hypothesis that we wish to challenge via evidence is that p>0. I don't recall if there is a way to do that, as the hypothesis testing I've been involved with always involves a null hypothesis that the parameter of interest is zero.
What we could do is pick a value of p, say 1%, and make our null hypothesis be that p>1%. Then, given enough observations of black ravens, and none of non-black ones, we can reject that null hypothesis at a high level of confidence. That is, we can be very confident that the proportion of ravens that are non-black is less than 1%.
But I can't see any way of gaining any level of confidence that the proportion is zero. Perhaps somebody that has done wider and more varied hypothesis testing can comment.
Michael, you're right that this is the problem of induction. It never occurred to me before to wonder whether any statistical basis could be found for using the principle of induction, by considering it in terms of hypothesis testing. If not, that seems to lend even greater weight to Hume's insight.
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.
I'm going to try some maths (dangerous!).
Let's assume that we have 12 eggs and that they can be either white or brown. All other things being equal there's a 0.5[sup]12[/sup] chance of every egg being white. We look at the first egg and see that it is white. There's now a 0.5[sup]11[/sup] × 1 chance of every egg being white. Given that the second chance is greater than the first chance it then follows that our hypothesis is made more likely by the first successful observation. And assuming that evidence is anything that makes our hypothesis more likely it then follows that a single white egg is evidence that all eggs are white.
And so due to contraposition, a single white egg is evidence that all ravens are black.
Quoting andrewk
Does the above help address this?
These two logically equivalent statements are UNIVERSAL statements meaning they are both an ALL statement, one positive and the other negative.
This understanding is key to solving the paradox.
''ALL ravens are black'' is TRUE iff every raven you see is black. Observing a few ravens, so far as it's not ALL ravens, cannot PROVE this sratement.
''Everything that is not black is not a raven'' is TRUE iff every non-black thing is not a raven. Mind the word ''everything''. We must observe ALL non-black things in the universe.
[I]One[/i] green apple will NOT suffice to prove either of these statements.
Paradox solved.
We're not talking about proof. We're talking about evidence. Not all evidence is proof.
First, we need to look at why we're saying that the eggs can be either white or brown. Is it because we know that we have a collection of 12 eggs where some are white and some are brown? Is it because there are millions of eggs in the world and we know that those millions of eggs are either white or brown? We can't ignore factors like this when we're dealing with a scenario like you're presenting--and those aren't the only factors that are relevant.
If we know that we have 12 eggs where some are white and some are brown, then the more white ones we find, the greater the probability is that we'll come across a brown one.
If we just know that there are millions of eggs in the world and some are white and some are brown, then finding one white egg in random batch of 12 that we have on hand (where we can't see the others yet, of course) tells us nothing about what color the other eggs in the batch are likely to be. The probability for each egg would be whatever the proportion of white to brown eggs in general in the world, as long as our 12 were chosen randomly--the probability does change if we know the total number of eggs in the world, but with only 12 against millions, the change is negligible. Of course, it matters for this, too, whether the eggs were really chosen randomly (or "randomly" as the case may be).
But we don't know that some are white and some are brown. If we knew that some were brown then we wouldn't claim that they were all white.
So why start with "Let's assume that we have 12 eggs and that they can be either white or brown"? Why would we even say that unless we have some reason to believe that they can be either white or brown?
Because it's just an example to make sense of the maths. The fundamental point is that each successful observation makes the hypothesis more likely, and so is evidence for its truth.
The math about being white or brown would make no sense if we have no reason to believe that the eggs can be white or brown. The math wouldn't mean anything in that case.
In other words, this is what I was getting at earlier: the formalisms have no significance non-contextually. They only have signficance with respect to semantic, epistemic etc. contexts, and those semantic and epistemic contexts will have an impact on the implications of the formalisms.
Of course the math makes sense. We don't need actual examples to work with probabilities. The numbers suffice.
There is a x[sup]n[/sup] chance of the hypothesis "all Ys are Z" being true, where n is the number of Ys and x is the probability. If we find one example of a Y that is a Z then we know that x isn't zero, and the chance of the hypothesis "all Ys are Z" being true is now x[sup]n - 1[/sup] × 1. This is either equal to (if x is 1) or greater than (if x is less than 1) the original chance.
So we can either assume that the hypothesis isn't certain, in which case the post-observation chance is greater than the pre-observation chance, entailing that the observation is evidence of the hypothesis' truth, or we can assume that the hypothesis is certain, in which case asking for evidence is pointless.
Or if you prefer, consider the original example but use "white" and "not-white" rather than "white" and "brown" and change the probability to whatever you like (e.g. assume that it's far more likely to be not-white than white).
I don't agree that there's any way to come up with a number for claims such as that.
Quoting Michael
The issue here, which creates the appearance of a paradox, is with the notion of "equivalent". Logic proceeds by doing a very neat little trick, (which is extremely evident in mathematics), of making two things which are not the same, "equivalent". So the key to understanding the paradox is to understand what is meant by "logical equivalence". We know that (1) and (2) do not say the same thing, they are said to say equivalent things.
When we make two different things equivalent, we assign to them the same value. We do this by neglecting some qualities as accidentals. We can say the chair is one object, and the table is one object, so that they each have the value of one. They are equivalent, but not the same. They are each one. Likewise, (1) and (2) are equivalent by having some sort of logical value assigned to them, but they are not the same. Since they each state something which is qualitatively different from the other (though what is said is in some way equivalent), we cannot say that everything which is evidence of the truth of one is also evidence of the truth of the other.
Quoting Michael
I do not think that the law of contraposition entails that everything which is evidence of (1) is also evidence of (2), or vise versa, because it does not take into account what differentiates (1) from (2). Though they are equivalent, (1) and (2) are different. Because they each state something different, evidence of the truth of (1) is not necessarily evidence of the truth of (2). When we make them equivalent, they are equivalent based in some principle of logical validity, not in a principle of empirical truth. So evidence that (1) is true is not equivalent to evidence that (2) is true. We would need a different principle of equivalence to make this conclusion, one which does not exist, because (1) and (2) each say something different.
We do know that they say the same thing. That's what it means for them to be logically equivalent, and their logical equivalence is entailed by the law of contraposition.
Compare with "I am both a man and British" being logically equivalent to "I am neither not a man nor not British". Being that they're logically equivalent they mean the same thing, and so evidence for one is evidence for the other.
Two statements being logically equivalent is not the same as a table and a chair being equivalent (whatever "equivalent" means in this context).
Logically equivalent statements do say the same thing. That's just what it means to be logically equivalent.
See this:
Logical equivalence is a clearly defined term in logic. Two statements that are logically equivalent mean the same thing, and so have the same truth value in every model. That's just what it means to be logically equivalent.
Find replace all instances of ''proof'' with ''evidence''. My post still makes sense.
A single or even many, excepting ALL, cannot provide evidence (your preferred term) for a UNIVERSAL statement.
So you say. But I showed, with maths, that it does. What you say here is only true if "evidence" means "proof", but it doesn't.
Again, I think you're mistaken here. Logical equivalence is not defined as two statements which are logically equivalent "mean the same thing".
I'm not saying that the math doesn't work as a formalism. But the formalism has no significance devoid of context, and devoid of an actual number in this case. It's fine as a game we can play with math, but that's all it is as you're stating it.
The maths shows that the probability of our hypothesis being true increases with each successful observation. Therefore if evidence is whatever increases the probability of our hypothesis being true then a single successful observation is evidence.
You might say that two statements which are logically equivalent have the same truth value. But that is my point, they are equivalent according to this system of "value", and this does not imply that they "mean the same thing".
Two statements have the same truth value in every model iff they mean the same thing.
If you know you have a finite number of items, yes, but depending on how many items there are, it's not good evidence of the hypothesis being likely, with the normal connotations that "likely" has. It's not even good evidence of the hypothesis being likely if there are only two items to check.
They only mean the same thing when individuals assign the same meaning to them (keeping in mind that it's not going to literally be the same), and the truth value hinges on how the individual in question assesses the relationship of the proposition to what they consider the apt "truthmaker."
My argument is premised on the claim that evidence is whatever makes the hypothesis more likely (than it was before), not as whatever makes the hypothesis likely (which perhaps means greater than 50%?).
Look, the term "logical equivalence" has a very clear meaning in logic, and its meaning is such that if A is logically equivalent to B then A and B mean the same thing. And the law of contraposition is equally clear that if P then Q is logically equivalent to if not Q then not P.
So it simply follows from the fundamental principles of logic that if something is evidence that everything that is not black is not a raven then it is also evidence that all ravens are black.
I really feel I sorted this out earlier, but obviously not. I'll try one more time.
If they are taken to be analytic, they say exactly the same thing, that there are no non-white ravens. But such a declaration is not disproved by evidence as pictured above, because one simply declares that these white birds are not ravens, because they lack the essential quality of blackness. Likewise, there can be no supporting evidence, because it is analytic, and declares how the words are to be used.
Thusly, 'all electrons have a negative charge' is not disproved by finding a very similar particle with a positive charge, instead we give it a new name - 'positron'. Likewise we can decide to call those white birds, 'positavens' if we wish.
However, If we choose to define an electron by its mass and not its charge, and then we find a particle with the same mass and a positive charge, then we have to say, very well, it turns out that not all electrons have a negative charge.
Now the convention in logic is that 'all ravens are black' does not entail that there are any ravens, but only that there are no non-black ones. In Venn diagram terms it declares the emptiness of a region. And the contrapositive does exactly the same.
But science is not interested in emptiness and empty claims. When a scientist says all electrons have negative charge, he is saying under either analytic or synthetic interpretation that there are electrons. In Venn diagram terms, he is not merely depopulating a region, but also populating a region. Nothing there, but something here.
Under this interpretation, we have:
"All ravens are black" = "there are ravens, & there are no non-black ravens."
The contrapositive, though becomes:
"There are non-black things, & none of them are ravens."
These are not logically equivalent because They populate different regions of the Venn diagram.
You can just use 1) "if something is a raven then it is black" and the logically equivalent 2) "if something is not black then it is not a raven". The paradox still holds. Evidence for 2) is evidence for 1).
The problem with this is that they don't always mean the same thing to individuals in practice. You can't just ignore that and say that they have to mean the same thing to those folks, haha. That might make it much more neat and tidy and easy for us, but that's not actually how things work, and our job is supposed to be to talk about the world accurately.
Wasn't he criticizing you in that post? If not, he doesn't know what he's talking about.
How about an actual discussion though? Why are you so afraid to go off script at all? Philosophers shouldn't have taboo or sacred cow subjects, or subjects where they're simply dogmatic, should they?
I don't think so. @unenlightened, care to weigh in on this?
Oh. Well, that would look bad for him then.
Fuckin ell Micheal. You can if you want, but if you don't say anything that populates the world, you ain't saying anything about the world, and no evidence from the world applies. And then the statements are equivalent and there is no evidence for any of it and thus no paradox, because it is just a declaration about language.
A raven is evidence that there are ravens. If there aren't ravens, I really don't care what you say about them or what evidence you produce.
I don't understand this. If I were to say "if you are from Wales then you are a woman" then you can provide evidence (or even proof) against this claim by showing me that you're from Wales but not a woman. So evidence from the world certainly does apply to if/then claims.
It certainly can, but it doesn't necessarily, especially not based on simply playing a logic game that we've constructed. It depends on just what the claim is, the meaning assigned, the context, etc.
By the way, this is also the problem with arguments like the Gettier objections to jtb. Gettier's argument hinges on how we play particular logic games conventionally, but it makes the mistake of assuming that that's all there is to person's beliefs.
Only if I exist. Are you claiming I exist? Then there can be evidence.
And that's the point. According to the paradox, the existence of green apples is evidence for the claim "if something isn't black then it isn't a raven", and because of contraposition is also evidence for the claim "if something is a raven then it is black".
What people count as evidence for something doesn't hinge on logic games in that way. Most people would say--and I'd agree--that green apples have nothing to do with ravens. So green apples aren't evidence of anything about ravens.
As the maths shows, each successful observation increases the probability of the assertion being true, and so seems to me to count as evidence (even if weak evidence).
The paradox still holds even if the existence of green apples is only weak evidence that if something is a raven then it is black.
Again, I think you're mistaken. You've applied an unjustified condition to "mean the same thing".
Here's a little example. Let's say that 2+2=4. There is a logical equivalence between (2+2) and (4). Now we can have a look at what each of those means. I assume that (2+2) means that one group of two is added to another group of two. I also assume that (4) means one group of four. I see a difference between the meaning of (2+2) and the meaning of (4). You have applied an unjustified principle, to say that because the two are equivalent they have the same meaning.
Here is the result of your application of that unjustified principle. If we observe, and therefore have evidence, that one group of two objects was added to another group of two objects, we can claim to have evidence of a group of four objects. But if we have evidence that there is a group of four objects, this does not constitute evidence that a group of two objects was added to another group of two object.
Anything can be evidence of anything if you use a definition that stipulates that it's evidence, no?
What does that prove though?
Neither "2 + 2" nor "4" are truth-apt propositions. Therefore it's wrong to say that "2 + 2" and "4" are logically equivalent.
Again with the previous example, "I am both British and a man" is logically equivalent to "I am neither not British nor not a man". This is contraposition. Evidence for one is ipso facto evidence for the other.
What probability would you ascribe to Newton's theory of gravitation in 1915?
I didn't just use the word "equivalent". I used the term "logically equivalent" which has a strict definition in logic.
What I would like to see, is if you can justify the following claim. If this can be justified, then you might have an argument:
Quoting Michael
Newton's law of gravitation accumulated a vast amount of evidence in support over many years. Applying the purported probability calculus to it, it must have achieved the status of a highly probable theory by 1915.
Then what happened? Is it still a highly probable?
That's the paradox. The existence of green apples is relevant to and so is evidence for the truth of "if something is not black then it is not a raven". But because of the law of contraposition it is also evidence for the truth of "if something is a raven then it is black" even though it doesn't seem to be relevant.
What does your comment have to do with what you quoted from me? (Maybe it was an example of something without relevance? Haha)
No it isn't. I explained what I meant right after that. The purported evidence needs to have something to do with what it's evidence for. That's an explication of the first sentence.
Yes, and the existence of green apples does have something to do with the claim "if something is not black then it is not a raven", given that it isn't black and isn't a raven.
Not with a claim per a logic game.
It needs to have something to do with ravens--the creatures.
"Two statements have the same truth value in the same model iff they mean the same thing."
I should be asking you the same thing.
No, that would be material equivalence. We're discussing logical equivalence.
Quoting Terrapin Station
You seem to parse things as if you're a computer and you can only do exactly what the code you've been programmed with specifies.
Why must it have something to do with ravens? The claim is "if something is not black then it is not a raven". Therefore, to be relevant, surely it must have something to do with not being a raven? Which it does; apples aren't ravens.
The point you are missing is that if I don't exist, there can be no evidence as to my gender or my place of habitation. None at all. Not green apples, and not my genitals. Evidence can only be brought for or against existential claims.
"All dragons are Welsh." Look around and you will not find a non-Welsh dragon, though there may be stories. And of course there are those green apples in England. But this is nonsense. There are no dragons, and so no evidence is forthcoming about their nationality. "all dragons are Welsh" says nothing about the world, and therefore there is no evidence for or against it.
Why most people have that requirement for evidence is a good question maybe. It's not so easy to answer that. But whether we can answer that question or not, it's a fact that for most people, purported evidence needs to have something to do with what it's evidence for--so if it's purported evidence for something about ravens and the color they are, it needs to have something to do with the creature in question and their color.
Again, we're considering the claim "if something is a raven then it is black". You admitted that your existence as a Welsh man is evidence against the claim "if someone is Welsh then they are a woman". Presumably there is also possible evidence for it. And so by the same token, there is possible evidence for the claim "if something is not black then it is not a raven". According to Hempel, the existence of green apples is evidence for the claim "if something is not black then it is not a raven". Therefore it is also evidence for the claim "if something is a raven then it is black".
Terrapin, address the entirety of my comment. Your response doesn't make sense in context.
See, this shows that you're not reading what I'm writing. The claim isn't about the colour of ravens but about the colour of things that aren't ravens.
The claim "All ravens are black"
Evidence for that, for the vast majority of people, has to have something to do with ravens, the creatures, and their color, supposedly black.
"Everything that is not black is not a raven"
This doesn't have anything to do with the creatures in question--ravens, and their supposed color. It rather has to do with things that aren't black and things that aren't ravens.
"green apples support the claim that all ravens are black."
No they dont, because green apples have nothing to do with the creatures in question or their supposed color.
The logic game you're referencing has nothing to do with how most people think about this or what they count as evidence. Which is good evidence that the logic game in this instance doesn't very well capture how people actually think. It's not at all a "law of thought," at least not without further qualification/modification.
The existence of green apples is relevant to the proposition that if something is not black then it is not a raven.
The proposition that if something is not black then it is not a raven is logically equivalent to the proposition that if something is a raven then it is black.
Therefore, the existence of green apples is relevant to the proposition that if something is a raven then it is black.
No in terms of evidence it isn't, for most people. You can just ignore that, but philosophizing while ignoring something so clear and simple is going to result in philosophy that sucks because all it's going to capture is a very limited way to play a certain sort of game, where that has nothing to do with the actual world.
You just clearly don't understand what logical equivalence means, despite my many attempts to explain it to you. This really is a futile discussion.
You can't go outside of your computer program (which is what makes every discussion with you futile). That's not logically equivalent to most people in a "what-counts-as-evidence-for" context.
I'd say it's a knock down disproof. Unless you wish to make it definitional in some way, such that being Welsh means being a woman, and men cannot be Welsh. At which point the claim is definitional and says nothing about the world. And at that point evidence does not exist.
"Those particles are not electrons, because all electrons have a negative charge."
"Those white birds are not ravens, because all ravens are black."
It is a choice whether the statement is about the world or about the way we are going to talk. If it is about the world, then there will be evidence. But if it is about the world, it is not the same as the contrapositive, for reasons I've already gone into ad nauseam. The paradox relies on the ambiguity, and the refusal to choose whether the statement is actually making a claim about the world or not, but still applying rules of evidence as though it were.
Just to confuse you further, 'Welsh' is a contested term. Since I was born and brought up in England of English and Scottish parents, I do not consider myself 'Welsh', though in quasi-legal terms I am Welsh, by virtue of living here. On the other hand, Mrs un was born in England of Welsh and Caribbean parents but brought up mainly in Wales, and does consider herself Welsh, though many people consider her 'foreign' by virtue of the colour of her skin. And then there is the language issue... My existence as a Welsh man is highly contestable. :-O
I still don't understand this. So perhaps to keep it simple you could clarify which of these you disagree with:
1. The proposition that if something is a raven then it is black is logically equivalent to the proposition that if something is not black then it is not a raven.
2. The existence of green apples is evidence (even if weak) that the proposition that if something is not black then it is not a raven is true.
As I see it, 1 is confirmed by the law of contraposition and 2 is confirmed by the maths provided here (coupled with the seemingly reasonable definition of "evidence" given in that same post[sup]1[/sup]). I certainly don't see anything that can be construed as ambiguous.
The conclusion that the existence of green apples is evidence (even if weak) that the proposition that if something is a raven then it is black is true then follows.
[sup]1[/sup] Although saying that, the paradox holds even if we don't use the term "evidence". If the observation of a single white egg increases the probability that the proposition that if something is not black then it is not a raven is true then it increases the probability that the proposition that if something is a raven then it is black is true.
2. Is false. There is no such thing as evidence for a universal statement. What's more, you can't apply probabilities to universal statements.
In the case where the universal statement is an hypothesis - i..e a tentative explanation for some aspect of reality, you can't use probabilities either. The negation of an explanation is not an explanation, so the two are in different classes of objects. In such circumstances attributing (p) to an explanation and (1-p) to its negation is meaningless.
This is false for most people in the context of considering what counts as evidence for each statement. Unenlightened also makes a good point in that (b) isn't something that anyone would consider there to be evidence for really unless we've already stipulated that black things are necessarily not ravens. But that's a definitional issue, and not an evidential issue.
Quoting Michael
That's false for most people, too. There are a number of problems with it, including what I just said above (that (b) isn't something we'd have evidence for, as it's stipulative).
Also, green apples certainly wouldn't be considered evidence for (a) by the vast majority of people.
Quoting Michael
The law of contraposition has little to do with what people count as evidence or not, and the definition of evidence you gave ignores the relevance element as I described it, which most people require.
OK, I wasn't clear on your definition of "logical equivalence". If this is your definition of logical equivalence:
Quoting Michael
Then (1) and (2) of the op are not logically equivalent. That is the problem, you are assuming that they are logically equivalent, without adhering to your definition of "logically equivalent".
Let's assume that's something we're not stipulating, but it's instead a hypothesis that we feel we need to gather evidence for.
Well, first off, if we're checking it, we very well could be wrong. Maybe it's going to turn out to be that ravens are red and purple and all sorts of other colors, too.
So we check a green thing and see that it's an apple. That doesn't tell us anything about whether there are any non-black things that are ravens, it just tells us that one non-black thing isn't a raven. It only increases the probability that no non-black things are ravens if we assume that we're dealing with a finite set of colored items, and we don't at all know this.
The problem is that this (2) also allows that "Everything that is black is not a raven" is true as well. So it is impossible that (1) and (2) are logically equivalent, under that definition of logically equivalent.
I think that there is an equivocation here on what we mean by "probability." You are really talking about our (subjective) confidence in the truth of a proposition, rather than its (objective) likelihood. "All ravens are black" is either true (p=1) or false (p=0), regardless of what we think.
Quoting Terrapin Station
I agree. In this case, the relevance requirement is tied primarily to the subject (ravens), rather than the predicate (black). The fact that green apples are not black things is not nearly as relevant as the fact that green apples are not ravens.
Said another way, there is a relation between the perceived strength of evidence and the number of items in a collection to which we are attributing a universal property. There are vastly fewer ravens than non-black things, so each black raven that we encounter increases our confidence that "all ravens are black" by a much larger degree than each green apple that we encounter increases our confidence that "all non-black things are non-ravens." This may be the asymmetry of the two logically equivalent formulations that others have been trying to articulate. For most people, something counts as (even weak) evidence for the truth of a proposition only if it significantly increases our confidence.
That. If 1. then not 2.
"All ravens are black" declares the blue area to be empty. This is refuted by evidence that there is something in the blue area, but not confirmed or made more likely by anything appearing anywhere else. The contrapositive says exactly the same thing, and the same evidence rule applies.
"There are ravens, and they are all black" on the other hand, declares that there is nothing in the blue area and something in the turquoise. In this case, the contrapositive is not the same, and every black raven found in the absence of any non-black ravens can be said to support the compound statement. But again stuff appearing elsewhere in the diagram is irrelevant.
Given a shuffled deck of cards, what's the probability that the first card we turn over is the Ace of Spades? 1 in 52. Even though either it's the Ace of Spades or it isn't. And if the first card we turn over isn't the Ace of Spades then what's the probability that the next card is? 1 in 51.
So I don't understand what you mean by "probability".
No it doesn't. How have you derived that? Certainly not with the law of contraposition.
Quoting Metaphysician Undercover
Yes they are. It's guaranteed by contraposition.
I would count as evidence anything that increases the probability that the statement is true. As shown here, each successful observation increases the probability that the statement "if something is an egg then it is white" is true.
Then where does my math fail here? Each successful observation does increase the probability that the statement is true.
But given that original probability, why should each white egg found make it more likely that the next one is white rather than less likely?I looks like a reverse gambler's fallacy to me.
Edit. No, that last bit's wrong. It's the limited number of eggs that raises the probability, If there were only 12 ravens and eleven had been found to be black, your probabilities would work. It's knowing how many there are before you start looking at them that is problematic, along with making the existential claim that I have been pointing out all along.
As I said here, we don't even need to think about it in terms of evidence. The paradox arises even if we just think about it in terms of the probability that the statement is true. Given that the existence of green apples increases the probability that "if something isn't black then it isn't a raven" is true then it increases the probability that "if something is a raven then it is black", despite the fact that the existence of green apples is prima facie irrelevant.
Strictly speaking, probability only applies to the long run of experience, not to an individual case. In general, the probability is 1/52 that the top card of any randomly shuffled deck is the ace of spades; i.e., that is the value to which the proportion of cases where that happens will converge as the number of trials increases to infinity. However, in each individual case, the probability is either 1 (if it is the ace of spades) or 0 (if it is any other card). Again, you are confusing this (objective) fact of the matter with the (subjective) confidence that someone has before looking at the card.
Quoting Michael
But nothing that you think, say, or do can increase the (objective) probability that the statement is true; it is either true (p=1) or false (p=0) all along.
Quoting Michael
No, it increases your (subjective) confidence that the statement is true; and when you only have 12 things in a collection, each "successful" observation significantly increases that confidence. However, notice that it has no effect whatsoever on whether the 12th egg (or even the billionth) actually turns out to be white.
Quoting Michael
Saying this over and over does not make it accurate. The existence of green apples has no effect whatsoever on the probability that "if something isn't black then it isn't a raven" is true. The observation of a green apple might have a very small effect on one's confidence that "if something isn't black then it isn't a raven" is true; but since such confidence is subjective, the (lack of) relevance of the observation will in most cases prevent it from being counted as genuine evidence.
But I am right out of breath now, someone else have a go.
Again, I don't know what you mean by probability. Probability isn't simply limited to either there being a probability of 1 or a probability of 0. We can talk about the probability that I won the lottery yesterday being 1/x million (whatever it is) and we can talk about the probability that nobody won yesterday being (1 - 1/x million)[sup]the number of players[/sup].
I'm not talking about confirmation, i.e. proof. I'm talking about evidence. Evidence is just whatever increases the probability that the statement is true. Given that observing a white egg proves that the number of eggs isn't zero and that the probability that an egg is white isn't zero, each successful observation increases the probability that every egg is white (as the maths shows).
And to continue with my example of the pack of cards, imagine that we tear one of the cards. What's the probability that none of the intact cards is the Ace of Spades? It's certainly not correct to just say "either 1 or 0". It'll be 1/52. And each time you turn over a card and find it not to be the Ace of Spades the likelihood that none of them is the Ace of Spades increases (with the probability being 1/2 when you get to the last intact card). Because there are fewer opportunities for the hypothesis to be refuted.
What argument? You simply said:
Where's the argument to support your claim "[this is] not confirmed or made more likely by anything appearing anywhere else"? I've provided evidence against this claim.
It is limited to 1 or 0 in any individual (i.e., actual) case, where there is a determinate fact of the matter.
Quoting Michael
Not if we want to be precise in our language; this is a philosophy forum, not a casual conversation. Our lack of knowledge about whether you or anyone else won the lottery yesterday has no effect whatsoever on the associated probabilities. Either you won, or you did not. Either nobody won, or somebody did.
Quoting Michael
Please stop repeating this falsehood. The statement is either true or false, regardless of the evidence; i.e., the evidence has no effect whatsoever on the (objective) probability of the statement's truth, only our (subjective) confidence about it.
Quoting Michael
If you tore the ace of spades, p=1; if you tore some other card, p=0. You are confusing probability with epistemic uncertainty.
You seem to be missing the fact that we're talking about evidence for a contrapositive claim, not a different claim, so your analogies are false ones. Again, it's quite simple:
1. Evidence of white eggs increases the probability that "if something is an egg then it is white" is true.
2. "if something is an egg then it is white" is logically equivalent to "if something is not white then it is not an egg".
3. Therefore, evidence of white eggs increases the probability that "if something is not white then it is not an egg" is true.
Now, I assume that you accept 2, being that it's a simple principle of logic. So if you have an issue then it must be with 1. But as I've shown, each successful observation does increase the probability that the claim is true. Prior to any observation there's a (1/x)[sup]n[/sup] chance of every egg being white, but after a successful observation the probability increases to (1/x)[sup]n - 1[/sup].
You seem to be missing the fact that the contrapositive is the same claim - That the intersection of the set of nonblack things and ravens is empty. Both the the original and the contrapositive make the identical claim. And both have the same need for evidence to be of the intersection of the sets and not some other region. But you insist that evidence for some other region having contents is relevant. It isn't. That's what the logic says.
I'm not missing any fact. I provided a valid argument with true premises. Contrapositive claims are logically equivalent and evidence of white eggs increases the probability that all eggs are white.
The whole thing is about epistemology, so I don't understand your objection. The paradox is that if we observe a green apple then we can be more confident that all ravens are black.
This is the problem. This what is wrong, and since I cannot convince you, I'll just refer you to Hume.
Then where does my math fail? Prior to any observation the probability of the claim being true is (1/x)[sup]n[/sup]. After a successful observation the probability of the claim being true is (1/x)[sup]n - 1[/sup].
So you can continue to assert that it's wrong all you like, but I've provided actual evidence to support my claim that I'm right.
My objection was that probability is not the same thing as epistemic uncertainty.
Quoting Michael
At last! If we observe a green apple, then I suppose we can be very, very, very, very, very slightly more confident that all ravens are black. However, no one would take this kind of reasoning seriously; that almost infinitesimal increase in confidence would not lead anyone (except you, apparently) to count the green apple as evidence that all ravens are black - certainly not anywhere close to the same extent that observing a black raven would, and even that should only make us slightly more confident.
And yet it is perfectly ordinary to talk about the probability of the first card we turn over being the Ace of Spades being 1/52. So I dispute your claim that probability is somehow distinct from epistemic concerns. It's exactly for these kinds of situations that the maths of probability was developed.
It doesn't matter how weak the evidence is. The paradox is that there's evidence at all.
Sigh, I thought that we had made a breakthrough. Your math fails right here:
Quoting Michael
Observation has no effect whatsoever on the probability of the claim being true, since the claim is either true (p=1) or false (p=0) regardless of any observation (or lack thereof).
Quoting Michael
Like I said, this is a philosophy forum, not a casual conversation. Turning over the card has no effect whatsoever on whether it is the ace of spades.
And the paradox is resolved by recognising, with Hume, that there is no such thing as evidence for a universal proposition. The "paradox" reveals the absurdity of claiming empirical support exists.
The mild irony is however, that we know the statement that "all ravens are black" is in fact false, and no number of green apples is going to change that.
But there is evidence, as I've shown. After each successful observation the probability that the claim is true increases.
Imagine yourself an Englishman. You have seen {insert number} swans, all of which have been white. You conclude from this evidence that all swans are white with a probability of {insert number} Then you are convicted of unwarranted induction contrary to the rules of logic, and transported to Australia. Where you learn the error of your ways, taught by flocks of black swans.
Pooh-pooh? Suit yourself.
Bite what bullet? I have shown you with maths that the probability of the statement being true increases after each successful observation. At no point have you explained the error in this reasoning. You just ignore it.
Before you start flipping, the probability of the first two flips landing heads is 0.25. After the first flip, the probability of the first two flips landing heads is 0.5.
As such, the probability of the claim "the first two flips will land heads" being true increases after the first successful flip.
The problem is green apples have zero chance of being a raven (and black). Noticing a green apple simply doesn't say anything about ravens. Ravens don't depend on the green apple, unlike the result of a coin flip on a coin. We can't use a set of green apples to say the probablity of some unrelated state is more or less.
If you flipped green apples and they turned into ravens (black) half the time, your analogy would work.
Remains the same... if you also calculate the probabilities as sets, and compare them to each distinct throw then you get different things... yeah.
You've missed the part about contraposition. Given that green apples are evidence of the claim "if something isn't black then it isn't a raven", and given that "if something isn't black then it isn't a raven" is logically equivalent to "if something is a raven then it is black", if then follows that green apples are evidence of the claim "if something is a raven then it is black".
I have (repeatedly) explained the error in this reasoning - successful observations have no effect whatsoever on the (objective) probability of the statement being true. You just ignore it, so I will stop wasting my time now.
Yes, and I've repeatedly explained that you have a strange understanding of probability. It is perfectly correct to say that the probability of the top card of a shuffled deck being the Ace of Spades is 1/52. We don't simply say that the probability is either 1 or 0.
Out of curiosity, how do you deal with ontic uncertainty? Do you treat vagueness and propensity as elements of reality? Would you go as far as extending the principle of indifference to nature itself?
The problem here, as I see it, is that logic and probability as used in this thread depend on strict counterfactuality - the validity of the law of the excluded middle. So either side of the argument still presumes that it deals with a world that is crisp and particular, not vague and therefore also capable of being truly general.
In our mathematical models of probability - like coin flips, roulette wheels, packs of cards and other "games of chance" - the world is ontically determinate. Or at least we attempt to create mechanical situations that are as constrained, and therefore as determinate, as we care to make them. And in constraining nature to that degree, we then grant ourselves the privilege of maximising our own epistemic uncertainty. We can make it completely a matter of our own indifference that we don't know what the outcome of the next flip, spin, or shuffle, is going to be.
So there is a sly transfer from a real world with actual uncertainty (perhaps) to our ideal world where the world is made "ontically determinate" by an act of care, by deliberate design, and therefore we make it safe to assign all uncertainty to epistemic cause - that is, our own personal indifference about outcomes, our own lack of control about whether the next flip is heads or tails, the next swan black or not.
So there is a real danger then to take this rather artificially manufactured state of epistemic uncertainty - one modeled after games of chance - and use it to prove something about ontic reality. Just as it is a similar error to apply standard predicate logic to the real world without regard to the artificiality of the counterfactual determinism that is the LEM-style pivot of its modelling.
I have a black cat. But when it sits in the bright sun, it looks more chocolate brown. If I am reasoning about black cats, or black swans, or the ace of spades, I ignore such quibbles as a matter of indifference - for the sake of modelling. And yet back in the "real world", it could always be a (Gettier style) issue of whether some black swan is really black (their feathers too look chocolate brown in bright sun), or really a swan (maybe it is a plastic toy, or some visiting alien, that is the next example that crosses our path).
The OP, as I understand it, is concerned about how to model the world. So it talks about inductive evidence and the bolstering of states of belief (or epistemic certainty). And that in itself is best modeled, I would say, by Bayesian reasoning. So it is not really paradoxical that green apples might count as evidence in some inductively strengthening sense - even if at a huge remove. Instead it seems quite sensible that if As can be consistently B (apples keep turning up green), then by generalisation, it is more plausible that other As have their own consistent Bs (swans are black, fires are hot, cats have claws).
But predicate logic can't prove inductive beliefs, it can only sharpen their test by deductively isolating the putatively counterfactual. Swans either have blackness as a universal property ... or they don't. The problem then is that reality itself isn't so black and white. Instead - we might have good reason to believe - it is ontically vague and therefore only rises to the state of having certain well-formed propensities. Black swans are highly likely to be always black (given a certain shared history of genetic constraints). But also - as propensities express goals or purposes - at some level there will also emerge a degree of indifference. Blackness might be a matter of degree (some swans might be more chocolately than others - and evolution "doesn't care").
So green apples don't relate to black swans in any direct deductive logical fashion. Only in an inductive one. But deduction itself is founded on the un-reality of black and white counterfactuality. It is "pure model" that by design cuts the umbilical cord to the world it models (that being not a bug but a feature: the formal disconnect of the LEM is why it is so semiotically powerful a move).
And our standard models of probability - games of chance - do the same trick. They are ontically unreal in that they are manufactured situations where it is the absolute determinism (of a sign!: the suit of a card, the heads or tails of a coin, the number of a roulette slot) that underwrites a completely epistemic state of indifference (as to which sign we might next read off a device as "an unpredicted state of the world").
So we have an elaborate machinery of thought - one that by design excludes the very possibilities of ontic vagueness and ontic propensity. Both predicate logic and probability theory depend on it for their epistemic robustness. We can know the world to "be that way" because that is how we have constructed our formal acts of measurement that become all we know of the world. We reduce messy existence to some internalised play of marks - the numbers, or colours, or other values we read off the world as "facts".
But then in realising that is the semiotic game being played, this re-opens the question for metaphysics about what is really the case for ontic existence itself. If we could see past the very instruments of perception we have constructed for ourselves - these rational counterfactual modelling tricks - what would be the reality we then see?
Which is where we have to start constructing a better model - like an understanding of probability that is expanded by notions of ontic vagueness and ontic propensity (which of course is where Peirce comes in as a pioneer).
Only if we embrace a sloppy usage of "probability." You refuse to acknowledge the objective/subjective distinction. There is nothing strange about it.
We know the relevant set to do so.
By seeing one green apple, you niether know the number of green apples, number of ravens, their relationship to each other nor to the rest of the world.
You can put no probability on what the presence of a green apple means for ravens.
Yes you can. Observing a green apple increases the probability that "if something isn't black then it isn't a raven" is true from (1/x)[sup]n[/sup] to (1/x)[sup]n - 1[/sup], where n is the number of non-black things.
And so due to contraposition, observing a green apple increases the probability that "if something is a raven then it is black" is true.
You don't know the number of black or non-black ravens. In seeing one green apple, you can't tell if the probability of a black raven is 99.999999999999999999999999% or 0.000000000000000001%, or any of the numbers in between, before or after.
For all you know, all ravens might be white.
This doesn't address the point.
It shows your point is meaningless. You say that seeing a green apple allows you knowledge of the probability a raven is black, yet you do not name any relevant probability.
How exactly do you know the probability if you can't even define it (e.g. as 1/2 or 1/52, like the examples you keep bringing up, which supposedly reflect how you are using a green apple to tell the probability of a black raven)?
If by "vagueness and propensity" you mean what Peirce called 1ns and 3ns, then yes, that is the working hypothesis that I have currently adopted and continue to explore. So by "ontic uncertainty," I assume you mean what he called "absolute chance."
Quoting apokrisis
Again, it depends on exactly what you mean by that. As should be clear by now, I am opposed to using the term "probability" when what we really mean is (subjective) "confidence" or "degree of belief."
I don't need to know the actual probability. It could be 1 in 10 or it could be 1 in a trillion. Whatever the probability is, (1/x)[sup]n[/sup] is less than (1/x)[sup]n - 1[/sup].
Not if n=0... which you have no way of discounting or naming a probability for. You do need to know the actual probably or we can't tell what applies in a given situation.
We know that n isn't 0 given that we have an example of something that isn't black. So n is at least 1.
You don't. n may be 0 for black ravens. Knowing n= at least 1 for green apples doesn't give you number of black ravens.
The claim we're considering is "if something isn't black then it isn't a raven". Where n is the number of things that aren't black and 1/x is the probability that such things aren't ravens, the probability of the claim being true is (1/x)[sup]n[/sup] before any observation and (1/x)[sup]n - 1[/sup] after the first successful observation (e.g. observing a green apple).
Therefore, observing a green apple increases the probability that "if something isn't black then it isn't a raven" is true.
x is the problem. If there are non-black ravens, x=0 and the probability is incohrent. Currently, you have no definition of x, so you can't say what's probable or not.
Now if you knew x, the probability of a non-black raven in a certain area, you could say how probable it was. But in that case, you already know and so have no need for green apples.
No, if there are non-black ravens then x wouldn't be 1. And given that there's a non-black non-raven we know that the probability isn't 0.
So, again, the probability that "if something isn't black then it isn't a raven" is true is increased by observing a green apple.
The problem is when x=0, not when it equals 1. In the instance of non-black raven, x=0, as the probability of the non-black thing not being a raven is 0. And this is true whether we know about it or not.
For your probability to function, x cannot equal zero. Non-black ravens must be known to be impossible-- which renders the probability useless in the way you are using it. We would already know there were no non-black ravens.
Where is this x = 0 coming from? x can never equal 0. So, no, in the instance of a non-black raven, x doesn't equal 0.
But that's the whole point. In an instance of a non-black raven, the probability of a non-black thing that is not a raven is 0.
So unless you can discount that possible (i.e. black ravens are impossible), your probability cannot function.
So, probabilities in this case must be either 1 or 0?
Well, given the existence of a green apple we know that the probability of a non-black thing not being a raven isn't 0. So, using your logic, the probability of a non-black thing not being a raven is 1. Therefore, the existence of a green apple is proof that if something is a raven then it is black.
If you don't like the conclusion then you must accept that the probability of a non-black thing not being a raven is greater than 0 but less than 1.
No it isn't. A green apple is a non-black thing that is not a raven. So the probability isn't 0.
Yes, but that doesn't help you. That only gives you n. You still don't know x. The green apple doesn't tell you non-black ravens are impossible, which is what you need to avoid incohrenence to the probability.
Yep and yep.
Quoting aletheist
This is the tricky bit. I think you may be arguing towards the subjective as being real, and indeed the ultimately real. And I can't deny that Peirceanism heads in the direction of embracing frank idealism or panpsychism of that kind.
But I instead go in the other direction which would attempt to deflate "subjectivity" and reduce it to a scientific notion of pansemiosis. So there is a divide between the objective and subjective, the ontic and epistemic, the observables and the observer. But it is not a dichotomy of matter and mind, but matter and sign. The human observer becomes thus simply a highly complex and particular example of an ontologically general semiotic relation. And there is nothing causally mystic about a sign relation.
So when I talk about nature exhibiting the principle of indifference, it is in that deflationary sense. Nature really is a kind of mind, but only in the sense that minds are a kind of sign relation.
And with quantum physics especially, science now supports that. Quantum theory is not probabilistic in the standard sense (ie: there are hidden variables, so any sense of surprise is simply due to our epistemic ignorance of the details). Instead it supports the more radical view that events like the decay of a particle actually are just propensities. The world is set up in a general way as a state of constraint. Then after that, tychic chance or pure spontaneity takes over - because the world just doesn't care when some atom actually does go pop.
So when it comes to human concepts of probability, as I say, there is this sly trick going on. We impose deterministic constraints on the world (notions about what constitutes a fair coin toss, a fair roulette wheel spin, a fair shuffle of the deck) so as to ensure maximum epistemic uncertainty about the outcomes.
And that kind of modelling of randomness is great for pragmatic purposes. It really helps in constructing the human realm to divide our reality so clearly and counterfactually into the determined and the random.
But now getting back to metaphysics, we have to see past the very instruments we have constructed to "see the world more clearly". And that leads to the radical holism of firstness, secondness, thirdness, as you say. However then there is still a difference between accepting the traditional metaphysical dualism of objective and subjective as talking about world and soul, or matter and mind, and instead making the further radical break of following through pan-semiotically and seeing the mind as a species of sign.
You're talking nonsense.
No, you are ignoring the knowledge required to define a probability.
You see a green apple and say it must mean non-black ravens are unlikely, as if its presence meant there couldn't be that other non-black things of a particular type were likely.
Yet there might be a thousand white ravens sitting in the trees behind you.
I haven't said anything of the sort. What I've said is that observing a green apple increases the probability that "if something isn't black then it isn't a raven" is true.
That's sorts of true, but it has nothing to do with judging the chance of ravens being non-black.
An instance of a green apple means the probability of something non-black and not a raven is 1, and for anyone looking for something that is not a raven and not black, they have a higher chance of finding such an instance-- they might see this green apple we are talking about.
Doesn't tell us anything about the chance of non-black raven though, just that someone is more likely to find something that is not a raven and not black when it exists (as opposed to if it didn't).
Yes it does, due to contraposition. "if something isn't black then it isn't a raven" is logically equivalent to "if something is a raven then it is black". So, given that observing a green apple increases the probability that "if something isn't black then it isn't a raven" is true, it also increases the probability that "if something is a raven then it is black" is true. That's the paradox.
Now you are losing track of the quantifiers. Given the observation of a green apple, the probability that some non-black thing is a non-raven is 1. It tells us absolutely nothing about the probability that all non-black things are non-ravens; that is still either 0 or 1.
Look Michael, take (2) as a proposition. "Everything that is not black is not a raven.". Now take the proposition "Everything that is black is not a raven". Is there any contradiction evident here which would exclude those two propositions from being consistent within a particular model? I don't see any. This is model #1.
Now take a model which has as a proposition "All ravens are black". This is model #2. Model #2 is not consistent with model #1 because "all ravens are black" contradicts "everything that is black is not a raven.
The definition of logically equivalent is "Two statements have the same truth value in every model." It is evident that "all ravens are black" is not true in model #1, but true in model #2, yet "everything that is not black is not a raven" is true in models #1 and #2. Therefore "everything that is not black is not a raven" is not logically equivalent to "all ravens are black", according to the definition.
Quoting Michael
If contraposition guarantees a type of equivalence, then clearly it is other than "logically equivalent" as per your definition.
Here is the issue. Logically equivalent, as per the definition ensures that logically equivalent statements have the very same meaning, as you've been arguing. But two statements which are equivalent by contraposition do not necessarily have the very same meaning, as I've been arguing. Therefore logical equivalence, as per the definition, and "equivalence" as guaranteed by contraposition are not both the same form of "equivalence".
No, I am just taking exception to using the term "probability" for subjective confidence or degree of belief - as did Peirce, if I remember right. As a theist, physicalism is a non-starter for me; but as I [URL=http://thephilosophyforum.com/discussion/comment/51943]said[/URL] in the dualism thread, I am intrigued by Peirce's alternative of objective idealism, where mind is primordial and matter is the same "stuff" but with "inveterate habits."
This is easier to understand if you use the phrases "if something is not black then it is not a raven" and "if something is a raven then it is black". The models are consistent.
It is a simple fact of logic that P ? Q is logically equivalent to ¬Q ? ¬ P.
No, it's logical equivalence. As explained here, "In logic, contraposition is an inference that says that a conditional statement is logically equivalent to its contrapositive", and as explained here, "In logic, statements p and q are logically equivalent if they have the same logical content. This is a semantic concept; two statements are equivalent if they have the same truth value in every model".
There really isn't anything to argue here.
The claim is "if something is not black then it is not a raven". The probability that it is true isn't 0 if we have a green apple.
Also one about coins.
There are n non-black things. The probability that "if something is not black then it is not a raven" is true is between 0 and 1. If we have proof of a green apple then we have proof that the probability that the statement is true is between 1/n and 1. And if we have proof of 10 green apples then we have proof that the probability that the statement is true is between 10/n and 1.
Each proof of a green apple increases the minimum possible probability that the statement is true. And so due to contraposition, each proof of a green apple increases the minimum possible probability that the statement "if something is a raven then it is black" is true.
A minor terminological point here: in the context of hypothesis testing, "confirmation" generally means "make more likely," and is not to be confused with "verification," which is to demonstrate that the hypothesis is true. (In other words, verification is the limiting case of confirmation.)
https://plato.stanford.edu/entries/confirmation/
If there are a limited number of ravens in the world (which there almost certainly are), does that change whether observations of black ravens (or non-black non-ravens) at least incrementally confirm the universally-quantified hypothesis "all ravens are black"?
(Although, of course, I'm not suggesting that Hempel's theory has any bearing on the veracity of my attempt at explaining why it's evidence).
Yes, it does mention the raven paradox (there may even be an entire SEP article devoted to said paradox, though I may be misremembering).
The central two statements being discussed here are statements in deductive logic. Deductive logic is characterized by certainty in the process of inference.
Probability is a feature of inductive logic.
You're conflating inductive and deductive logic in your analysis.
That's right, there is nothing to argue here. So long as you separate logical equivalence, equivalence according to contraposition, and semantic equivalence, equivalence of meaning, and do not equivocate between them, as you have been doing, then there is no paradox, and nothing to argue.
I really don't understand what you're trying to argue here. I have provided references that show that P ? Q is logically equivalent to ¬Q ? ¬ P and that two statements that are logically equivalent have the same truth value in every model.
Which of these two things do you disagree with (contrary to the references)?
What we have been discussing is the raven paradox. I've demonstrated how the two statements are not what you call semantically equivalent. They are logically equivalent according to contraposition, but they do not have the same truth value in every model, so they are not semantically equivalent.
Now you have presented me with symbols, "P" and "Q". Unless these symbols are meant to symbolize something, how are we to discuss semantic equivalence? To say that the two statements have the same truth value in every model is meaningless, because P and Q are just symbols which don't represent anything, so there is no truth or falsity of those statements to discuss. How can we discuss whether "if P then Q" is true or false if the symbols have no meaning?
This is contradiction. If two statements are logically equivalent according to contraposition then ipso facto they have the same truth value in every model.
To make this clearer, contraposition is an inference that says that a conditional statement and its contrapositive have the same truth value in every model.
P is "X is a raven". Q is "X is black". So, P ? Q is "if X is a raven then X is black". Which is logically equivalent to ¬Q ? ¬P, "if X is not black then X is not a raven".
No that's not the case, we've already gone through this with the raven example. Reread my posts if you do not understand.
Quoting Michael
Now consider the other proposition "if X is black then X is not a raven". This is consistent with "if X is not black X is not a raven", but it is not consistent with "if X is a raven then X is black", so the two statements do not have the same truth value in every model. In this model, the one which holds that "if X is black then X is not a raven", the one may be true while the other is false.
Truth is not a matter of probability. It is either true (p=1) or false (p=0) that if something is not black then it is not a raven. The observation of a green apple is irrelevant; the proposition is still either true (p=1) or false (p=0). Our subjective confidence in the truth of the proposition is also irrelevant; it is still either true (p=1) or false (p=0). This is precisely why I consider it misleading to conflate degree of belief with probability.
Let me put it this way: I will concede that people ordinarily talk about "probability" when they really mean subjective confidence, if you will concede that people ordinarily do not count the observation of a green apple as evidence that all ravens are black. I think that the first is a mistake and the second is fine, you think that the first is fine and the second is a mistake.
To clarify, the model (often called a "world") in which these two propositions are consistent is one in which ravens do not exist; hence X is not a raven, regardless of whether X is black or not black. This goes back to the point about universal propositions not asserting the existence of anything.
Well I'm trying to follow the maths. Firstly, if there are a limited number of ravens, then there are some ravens. So we are not saying merely that there are no non-black ravens, but also that there are some black ravens. Then each black raven found in the absence of any white ones decreases the population of potential non-black ravens, and so increases the probability that they are all black.
One cannot put a figure on it though, without counting the ravens and also having some sensible notion of the probability of a raven being non-black, which one can only have if there actually are some non-black ravens, which by hypothesis there are not. I am ignoring here the inconvenience that eggs sometimes hatch into new ravens, as well as those red Martian ravens.
What it comes down to in the end, is not the adding up of black ravens at all, but exploring the range of potential non-black ravens and finding it to be empty; and that is the incremental evidence that there are no non-black ravens.
Edit. Hypothetically, one could explore the range of non-black things and find it to be raven-less just as well as exploring the range of ravens and find it to be non-black-less, and in this sense, black ravens are on a par with green apples evidentially although the limited number of non-black things is intuitively fair a bit bigger than the limited number of ravens, so the evidential significance would be proportionately less and possibly negligible.
If we have an egg-making device and know that there's a probability of 0.5 that any egg it makes is white (say we have an actual random number generator that if odd produces a white egg and if even produces a brown egg) then we know that there's a probability that every egg it makes, assuming it makes 10, being white is 0.5[sup]10[/sup].
So if after 8 eggs we have 8 white eggs then the probability of every egg being white is the probability that the next two eggs will be white, which is 0.5[sup]2[/sup].
This is exactly what the maths of probability is for. And we can use this reasoning even if the machine has already made the eggs (and we're just picking eggs out to check them).
And we don't even need to know the actual probability or the actual number of eggs. We know that the probability of a white egg isn't 0 (and that the number of eggs isn't 0) if we have one white egg, and so we know that 1/x[sup]n - 1[/sup] is greater than 1/x[sup]n[/sup]. The probability increases after each white egg.
The point about whether or not ravens exist was very relevant, but the argument was not formulated in a very logical way, which is necessary to reduce the appearance of paradox.
Indeed - when working with random samples, not individual cases. Returning to the deck of cards, if we anticipate drawing one from a truly random location in the stack, then the probability is 1/52 that it will be the ace of spades. Once we have actually drawn it, then it either is the ace of spades (p=1) or it is not (p=0). Shuffling and then taking the top card is not the same situation, because which card is on top is no longer random once the shuffling is done; at that point, it either is (p=1) or is not (p=0) the ace of spades, even before we look at it. On the other hand, before shuffling, the probability is 1/52 that the top card will be the ace of spades, assuming that the outcome of the shuffling is truly random.
Quoting Michael
So far, so good.
Quoting Michael
This is where I disagree - the reasoning is not the same. Once the machine has actually made the eggs, how many of them are white is a fact. If they are all white, then the probability that they are all white is 1; if any of them are non-white, then the probability that they are all white is 0. Our knowledge (or lack thereof) about how many are white vs. non-white is irrelevant to the associated probabilities.
The underlying idea here is that everything actual is subject to the principle of excluded middle, such that any given proposition about it is either true (p=1) or false (p=0). By contrast, anything general is not subject to the principle of excluded middle, such that intermediate probability values are possible for random samples thereof. A card in general is neither the ace of spades nor not the ace of spades; the probability that a randomly selected card in a standard deck is the ace of spades is 1/52. Unless all eggs are white, an egg in general is neither white nor non-white; in your example, the probability that a randomly selected egg is white is 0.5.
Likewise, unless all ravens are black, a raven in general is neither black nor non-black. The problem is that we have no way to determine the probability that a randomly selected raven is black, because we do not know what proportion of ravens is black. If it turns out that all ravens are black, then the probability that a randomly selected raven is black is obviously 1. For the contrapositive formulation, unless all ravens are black, a non-black thing in general is neither a raven nor a non-raven. Again, we have no way to determine the probability that a randomly selected non-black thing is a raven, because we do not know what proportion of non-black things are ravens. If it turns out that all ravens are black, then the probability that a randomly selected non-black thing is a raven is obviously 0.
Notice that the observation of a green apple can have no effect whatsoever on any of these probabilities. It only tells us that the probability that non-black non-ravens exist is 1; i.e., some non-black things are non-ravens.
And this is where I disagree. Probability is an epistemic concern. It is perfectly appropriate to use the maths of probabilities to determine the probability that every egg produced was white, not only to determine the probability that every egg produced will be white.
It's the exact sort of thing that people do in games like Poker, for example. The cards are all dealt out and you use the probabilities (if you're smart enough) to determine the best course of action.
As explained here:
I think such uses are as rigorous an application of probability as future-event prediction, so it's not right to accuse this of being a "sloppy" and "casual" use of the term.
It seems strange to suggest that I can't use the cards in my hand to determine the probability that my opponent has a pair of aces, and so it seems strange to suggest that I can't use the eggs I've checked to determine the probability that the remaining two eggs are white.
You were quoting someone else here, but it expresses precisely why I take exception to using the term "probability" in this way, rather than "confidence" or "degree of belief." It gives a false connotation of objectivity to what is a fundamentally subjective assessment.
Quoting Michael
We do not know the value of n, the total number of non-black things, or 1/x, the probability that a randomly selected non-black thing is a non-raven. Your equations presuppose that n is finite and that 1/x<1; i.e., that some ravens are non-black. If we include not just all actual non-black things in n, but all potential non-black things, then n is infinite, and the probabilities are identical before and after the observation of a green apple, regardless of the value of 1/x. If all ravens are black, then 1/x=1, so both probabilities are 1, regardless of the value of n.
Quoting Michael
The only way I can see that these two propositions are not logically equivalent is if the first one is treated as singular, rather than universal; and in that case, it is no longer logically equivalent to the original proposition, "All ravens are black."
Original: For all x, if x is a raven, then x is black.
Contraposition: For all x, if x is not black, then x is not a raven.
Singular: If a is not black, then a is not a raven.
The existence of x is not asserted by the first two, but the existence of a is asserted by the third.
I beg to differ! If there is such a thing as probabilistic support for a universal statement, then green apples do indeed support "all ravens are black". I have given the solution to this paradox earlier in the thread, so now let me prove it:
A well known result from probability calculus is:
p(he|b) = p(h|eb)p(eb)
Let h = "all ravens are black" i.e. the hypothesis
Let b = background knowledge e.g. all the ravens previously encountered
Let e = new evidence - the sighting of another raven
h logically implies e, so "h and e" is equivalent to h, so
p(h|b) = p(h|eb)p(eb)
Thus
p(h|eb)=p(h|b)/p(eb)
Do this again with an alternative hypothesis:
k = "NOT all ravens are black"
And divide one expression by the other, you get:
p(h|eb)/p(k|eb) = p(h|b)/p(k|b)
Now notice that no matter how h and k generalize under new evidence e, the evidence is incapable of affecting the ratio of their probabilities! What you are left with is the ratio of the prior probabilities, which you can have done nothing except arbitrarily set.
Thus there is no such thing as probabilistic support for a universal statement!
It wasn't meant to be. It was meant to be equivalent to "if something is a raven then it is black" (which is why this is the phrase I've been using since page 3/4). The paradox would still hold, as the paradox is about a seemingly unrelated piece of information being evidence.
Quoting aletheist
I think you must have misread. I said that "we have proof that the probability that the statement is true is between 1/n and 1", and so I wasn't presupposing that 1/x < 1.
I'm not sure the relevance of potential non-black things. Can't this just be about actual non-black things?
I don't know what you mean by giving a false connotation of objectivity. How, exactly, can one misinterpret the claim "there's a 0.5[sup]12[/sup] chance that every egg in that (closed) cartoon is a white egg"?
And it's not simply a subjective assessment, if by "subjective assessment" you're referring to subjective probability where "probability [is] derived from an individual's personal judgment about whether a specific outcome is likely to occur. It contains no formal calculations and only reflects the subject's opinions and past experience.". There are actual formal calculations in place. We know the number of cards and we know what cards we have. We might also know that there was no bias in the shuffling. There's a fixed formula then that can be used. The same with the eggs.
Please read what you quoted from me again.
Quoting aletheist
I said that the observation of a green apple only supports - in fact, proves - the particular proposition that some non-black things are non-ravens.
I have just proved that observational support for for a universal statement is impossible. If you think such support exists, and in particular that the observation of green apples provide support for any such statement, you have just been proved wrong.
And the paradox is solved of course.
You used universal propositions, not singular propositions, in the OP. Now you are claiming that the two propositions of interest are both singular - "if a is a raven, then a is black," and its contrapositive, "if a is not black, then a is not a raven." In this example, a is a green apple, so it is trivial to say that a is not black and not a raven; both propositions are true (p=1). A second observation of a green apple, call it b, would go with a different pair of singular propositions - "if b is a raven, then b is black," and its contrapositive, "if b is not black, then b is not a raven"; again, both are true (p=1). By definition, you cannot say anything general in a singular proposition.
Quoting Michael
A universal proposition does not assert the actual existence of anything in the subject class, so it must apply to all potential things in the subject class.
Quoting Michael
By believing that the actual color of the eggs is somehow indeterminate until one opens the carton. It is not; it is a fact that either they are all white (p=1) or that at least one is non-white (p=0), unless we are going to treat this as a quantum physics scenario like Schroedinger's cat where each egg is neither white nor non-white until one observes it.
Why do you keep addressing this to me? My statement that you quoted has absolutely nothing to do with universal propositions. Observation of a green apple merely proves that the particular proposition, "some non-black things are non-ravens," is true (p=1).
My apologies!
Anyway, the paradox is solved.
Hypothesis: All unenlightened's pockets are empty.
*checks all pockets, finds each and every one to be empty.*
"Hey Tom, all my pockets are empty. I just looked."
Sure, all ravens in Vienna in 1938 were also black. Perhaps you should consult an elementary text on universal statements?
The trouble with ravens is that there are lots of them and it's hard to know if you've seen them all, but if you did see them all, and they were all black, you'd have all the evidence you need.
The difference is that you actually observed all of your pockets. The OP is claiming that a single observation provides evidential support for a universal proposition. @Tom's proof shows that this is not the case - but it no longer applies once you have observed all members of the class, at which point you know whether the universal proposition is true (p=1) or false (p=0).
One more point - you also have to stipulate that this was true when the observations occurred. Even then, it is only strictly true if those observations were simultaneous; otherwise, something could have appeared in the first pocket that you checked by the time that you got to the last one. Furthermore, the fact that your pockets were empty then does not warrant the claim that they are still empty now and will remain empty in the future. This gets at my earlier comment about a universal proposition having to include all potential members in the class, not just its actual members.
Not really. Why would you think that? The contents (or lack thereof) of the first 16 pockets have no bearing whatsoever on the contents (or lack thereof) of the 17th pocket.
I'm afraid this is just bluster to save the point. A universal does not have to be eternal in scope, and if one cannot rule out pockets that fill themselves or ravens that turn black when looked at, then nothing can be said about anything. curb your skepticism a little.
To the extent that one has explored the logical space and found it empty of non.black ravens or pocketed stuff or whatever, to that extent it is probable that the space is empty. Bring in all the caveats you like to invalidate the observations, the principle holds if statistics and language mean anything at all.
Quoting aletheist
Check out the marbles in a bag scene. If there is one white one and sixteen black ones, would you bet on the white one being the last one out of the bag, or some other place?
I am fine with this common-sense approach for everyday living, especially if we substitute "one is confident" for "it is probable." What bothers me is the claim that we can meaningfully calculate a mathematical value for this "probability," as well as the claim that the observation of a non-black non-raven or a non-empty non-unenlightened's-pocket somehow affects the assessment.
Quoting unenlightened
That is a different scenario. If I knew nothing about the contents of the bag, and had already drawn 16 black ones, I might very well be tempted to bet that the last one would also be black - and I would be dead wrong.
In the proof I gave, observation of your own pockets would constitute background knowledge. The hypothesis would be the universal statement "all pockets are empty". I guess you could then set about gathering evidence.
Since you won't consult an elementary text, let me help you:
A universal statement is one in which no individual names occur.
Got it?
I don't think any reasonable person will interpret my claim in this way. It certainly isn't implied, as you suggest.
This doesn't seem right. If I say that all humans are shorter than 9 feet I'm not saying that all potential humans are shorter than 9 feet.
I switched to "if something is not black then it is not a raven" and "if something is a raven then it is black" pretty early on, and (unless I've been sloppy) stuck with it since.
And the above interpretation is wrong. When I say that the probability that "if something is not black then it is not a raven" is true is 0.5 I mean that that for any randomly selected non-black thing, the probability is 0.5 that it will not be a raven.
I think you are, unless you qualify it somehow. You are saying that anything taller than 9 feet cannot (ever) be human.
Quoting Michael
Then this is a universal proposition after all, rather than a singular proposition; and it is, in fact, logically equivalent to "all non-black things are non-ravens." Your use of probability in this case is unobjectionable to me; you are simply saying that exactly 50% of all non-black things are non-ravens.
How many non-black things do you need to select to show that your assertion that the probability of selecting a non-raven is 0.5?
Twice as many as there are non-black non-ravens.
Quoting aletheist
I don't think I am. If I say that nobody in my house is American I'm not saying that nobody in my house can ever be American.
Quoting aletheist
Wait, so you're saying that it's unobjectionable to claim that a universal proposition has a greater than 0 but less than 1 probability of being true?
Let's walk through this elementary probability problem.
Scenario 1. There are 17 marbles in a bag, but they could be any colour in any combination. You take out sixteen in turn, and they are all black. You now know that there are either sixteen black ones, and one non-black, or seventeen black marbles.
Now, scenario 2. How did the marbles get into the bag?
(a). Suppose they were picked at random from a container with equal quantities of each of 5 colours.
Then the chances of the last marble being black would be 0.2 But the chances of getting sixteen black marbles under (a) are 0.2^16. So (a) is rather improbable.
(b). Suppose they were picked at random from a container containing equal quantities of just black and white. Then the chances of the last marble being white would be 0.5 And the chances of getting sixteen black marbles would be 0.5^16 (0.000015, approx.). Still rather improbable.
(c). Suppose they were picked at random from a container with 99 black marbles for every 1 white marble. Then the chances of the last marble being black are 0.99 And the chances of getting sixteen black marbles are 0.99^16 (0.851 approx.).
It would take some rather complicated calculation to arrive at the most probable distribution of the marbles in the container, and thus the exact probability of the last marble being black, which are beyond this probability 101 course. But it should already be apparent that the the figure will come out to greater than 0.5
Therefore, probably, the last marble is black.
And therefore, probably, all the marbles in the bag are black.
And note, if it makes a halfpence' difference to you, that neither the bag nor the marbles have been named.
Fair enough, but what you are really saying then is that nobody in your house right now is American.
Quoting Michael
No, but I can see why you misunderstood me. The universal proposition is "if something is not black then it is not a raven"; i.e., "all non-black things are non-ravens." The proposition that I find unobjectionable is "for any randomly selected non-black thing, the probability is 0.5 that it will not be a raven." This is not the same (universal) proposition; it is instead a particular proposition, "some non-black things are non-ravens," with the additional information that the proportion of non-black things that are non-ravens is 50%.
You did not stipulate any knowledge of how the marbles got into the bag. All we knew was that the first 16 marbles that we took out were black. This information alone is insufficient to calculate a meaningful probability that the 17th marble will also be black. Most people would indeed be likely to bet on it being black in that scenario, but again, they would be wrong if it turned out to be white.
Probability is a complete red herring until someone states the prior and tells us how to update it.
Then they need to explain why we should set our credence to be equal to the probability. (hint look up the Principal Principle)
Then someone needs to explain how we can test a probability statement, what deviations from the expectation value we are willing to accept, and why.
Then someone might deign to explain why a "probable" theory is "probably true"
Then this
And, strange as it may seem, such probabilistic statements, which make no prediction about what will happen, are normative.
I'm glad you noticed that. This models the situation with ravens.
Quoting aletheist
Then you need to show where my admittedly incomplete calculation has gone wrong, because I think I have shown that the probability is greater than 0.5, and somewhere close to 0.9
Quoting aletheist
That's the nature of probability, that one can be wrong. The calculation is of the best bet not the certain bet.
The calculation is fine as far as it goes; the interpretation is the problem. The last marble in the bag is either black (p=1) or non-black (p=0), we just do not yet know which. You have basically invented a clever mathematical way of measuring your level of confidence in your guess that the 17th marble is black, based solely on the fact that the first 16 were black.
Quoting unenlightened
If one can be wrong, then one is really talking about (subjective) confidence or degree of belief, rather than (objective) probability.
I'm a damn smart dude, but I can't take the credit for inventing elementary probability theory. This is high school stuff that I assumed those discussing probable evidence would be familiar with. I'm amazed at the level of bluff and bluster that passes for argument and understanding in such matters.
Quoting aletheist
That is a pile of crap of biblical proportions that I am not going to even try and clear up.
That is certainly right. But it illustrates the bigger issue of how logic relates to the world - which you, as a student of Popper, would understand.
Popper nicely brought out how the universal and the individual (or singular) are formally reciprocal bounds ... or a dichotomy. So really, when it comes down to it, each "exists" only in distinction to its "other". Thus both universality and individuality remain always relative concepts. They are never standalone absolutes. And from that, we can understand the need for a triadic epistemology where the universal and the individual are the bounds "to either side" of the actual thing in question - the entity we invoke by calling out its proper name.
So here is my raven called Raven. We have the three things of some actual "bird" I own (a substantial instance), the form of that being (the generality that constitutes "a raven"), and the matter of that being (the individual materiality or collection of properties that allow Raven to be classed as an actual instance of a universal idea).
The world is thus hylomorphic. The debate about universals and singulars, generals and particulars, only repeats the metaphysical causal debate over the nature of substance at the level of the logical modelling of real things.
Anyway, getting back to the point about universal statements not naming individuals, this is how generality would be achieved - by managing to put as much distance as possible between the universal and individual sense of a word like "raven". And yet by the same token, the distancing achieved is only ever relative to itself, never actually absolute. But people then treat the logic as if it has achieved this absolute (deductively valid) status. And the same people look back at the world and see that it is still (inductively) relative to a history of observations.
As Popper says, scientific laws take the "negative" form of proscriptions or constraints when they are expressed as universal statements. The law of energy conservation sounds like it asserts an absolute generality of nature, but in practice it has to be cashed out in terms of the actual observation or measurement of its "other" - the particular or existential claim that "there are no perpetual motion machines". So universality - in practice, in the real world - obtains only by a failure to find otherwise. The absence of not-A as a particular, is inductive confirmation of the presence of A as a generality.
This reciprocal deal - the reason why scientific certainty boils down to lack of falsification - is why universals do get pushed in the direction of a-temporal and a-spatial statements. It is not good enough to talk about my empty pockets, or whatever, "right here and now". To be as absolute as possible, a statement would have to show that it has pushed away to the extreme margins of observation absolutely everything that could be considered individual or particular in relation to that statement. So that shifts us decisively out of the realm of the actual and into the realm of the possible.
That again is why we have to end up with a triadic or hylomorphic logical system. We want to speak about, and reason about, substantial or actual being. And to do that, we find ourselves having to strike out in both directions "beyond" the actual. We have to head towards universally constraining necessity by simultaneously manufacturing its complementary "other" of completely individual or particularised material possibility.
Thus the (Peircean) triad of firstness, secondness and thirdness - as possibility, actuality and necessity. Or individuals, proper names and universals.
And as I say, the thread seems to revolve around the fact that people can see that the absolutism implied by the standard syntax of logical form does not match the relativity of the world being described.
But this is not paradoxical. It is simply evidence of what I keep saying - that the work of logicians, if they hope to talk about existence with true metaphysical generality, is not done. And you only have to go back to Aristotle and Peirce to see how it is triadicy, or hierarchical organisation, that must be the next step to breath relativity back into the dyadic syntactical forms that have been frozen into static absolutism by Frege and others.
I hate to say it Tom, but this is why the many worlds interpretation, computationalism, digital physics and a whole bunch of other bandwagons are metaphysically doomed. They are "illogical" in this sense. They are extrapolations of a logical absolutism where the Universe is going to have to be a case of logical relativism - the kind of triadically self-organising "universal reasonableness" that Peirce was on to, and which Popper was following up on.
If I keep moving things around it certainly is going to seem so, but it really isn't.
I would say that aletheist has it bang on so far. So it is a shame to see you capitulate this way given the thread has been pretty instructive.
Aletheist is showing how our claims about objective reality always wind up being founded on subjectively reasonable seeming beliefs. People get used to talking about knowing stuff - like what's not in their pockets, or what they know they put in a bag, or included in a pack of cards. But in the end, that confidence is rather manufactured - a play of signs that we ourselves create to replace the world (whatever it is as the thing-in-itself).
My own key point in support of aletheist was to draw out how we do in fact go about manufacturing the "objective" grounds of our own certainty using games of chance. We make a physical determination (in shaping a die or printing a pack of cards) that then underwrites our claims to a concept of "randomness", or "accident", or "probability" based on a principle of indifference.
So we have a model of probability that is derived from subjective actions. We construct a machinery that divides the world sharply, counterfactually, into the bit we care absolutely about (some device like a coin that can only land on one of two sides), and then the bit we claim absolute indifference about (the spin that lands unpredictably). And then we compare this construction against the behaviour of the world to talk about "how the world actually is".
What is going on should be transparent. But people seem to hate their metaphysical realism being undermined even slightly.
Or as Peirce succinctly put it, "A particular proposition asserts the existence of something of a given description. A universal proposition merely asserts the non-existence of anything of a given description." (CP 5.155; 1903)
Yes, this was pretty much exactly the point of my egg thought experiment. So, if each black raven observed in the absence of white ones decreases the potential population of non-black ravens, thereby increasing the probability that they are all black, can we then not say that successive observations of black ravens confirms the hypothesis "all ravens are black" (contra some claims on this thread that no such confirmation can be had for universally-quantified propositions)?
I have some questions about this. I don't see how H (hypothesis) logically implies E (evidence). I understand the hypothetico-deductive mode of reasoning (which, in very general terms, science adheres to), i.e. posit a hypothesis, deduce observational consequences of said hypothesis, and perform a test to look for said consequences. However, in this case, I don't see how "all ravens are black" implies "the sighting of another raven." I'm not sure what the latter statement even means, exactly (H seems to imply only that, if one were to observe a raven, then said raven would be black).
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?
If you've already flipped a coin (your first of the day) and it's landed heads then what's the probability that the first two coins you flip today will be heads? Given that one of them landing heads is certain (it's already happened) the probability is the probability that the next one will land heads, which is 0.5.
So it would have been correct to say that the probability of "the first two coins I flip today will land heads" being true is 0.25 before your first flip, and correct to say that it's probability is 0.5 after the first flip (but before the second).
Might be related to the Monty Hall problem (or maybe it isn't, I don't know).
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.
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?
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.
Yeah, and that's what we use probability for; to determine the likelihood that our guess is correct, based on the available epistemic criteria.
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)
Quoting 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
Quoting Arkady
That's in all the equations!
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 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.
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.
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?
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.
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:
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.
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?
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 0.5[sup]10[/sup] 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 0.5[sup]2[/sup] 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'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).
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.
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).
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.
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.
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.
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.
Quoting Michael
If you had just said this in the beginning, then we might not have had much of an argument at all. :D
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.
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.
Quoting 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).
The math in the example I gave of the white eggs.
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".
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 0.5[sup]10[/sup], and this is true even if I believe otherwise (hence the objectivity).
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.
Quoting 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 0.5[sup]9[/sup] when you observe that the first egg is white increases your confidence that all of the eggs in this particular batch are white.
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.
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).
You said it isn't frequentist probability being that the probability (or plausibility, if you prefer) is 0.5[sup]10[/sup], 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.
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.
Quoting 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?
Quoting 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.
Quoting Michael
No, you are not making the distinction between the probability that any batch of ten eggs would be entirely white (0.5[sup]10[/sup]) and the probability that this particular batch of ten eggs actually is entirely white (either 0 or 1).
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.
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).
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.
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.
[s]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.[/s]
Actually, no, I think you're right.
Green apples also "confirm" the universal statement "all ravens are white".
Quoting Arkady
I covered that earlier.
Quoting 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!
Yup. Michael pointed this out fairly early on, IIRC.
You said:
My original concerns stand. K is a hypothesis which is supposedly compatible with "any evidence," which is completely at odds with its being falsifiable (indeed, this seems to smuggle your conclusion into the proof itself, thereby begging the question).
Also, I still suspect you're making an illicit move in proposing that if P(E|K) = 1, that P(K&E) = P(K). Unless K & E are both necessary truths, then the probability of their conjunction must be less than either conjunct alone (though, they probably should not be considered as statistically independent, so this point is debatable. Either way, I'm skeptical that E drops out so smoothly from the equations).
Yes, but evidence can be consistent with multiple hypotheses, which is called underdetermination, and is well-known in the philosophy of science. (I think that most treatments of this problem, even when they allow that non-black non-ravens confirm the hypothesis, treat such observations as very weak evidence.)
My problem was in saying that a given hypothesis and its negation should entail the same evidence. If H is "all ravens are black," and K is "not all ravens are black," K is not only amenable to confirmation, it is verified by a single instance of a non-black raven. Clearly, whatever we may think about the possibility of universally-quantified hypotheses being confirmed, an observation of a non-black raven decidedly does not confirm "all ravens are black."
Yes, I think we've all seen the white ravens, thanks. :D
A hypothesis can be confirmed by evidence but still turn out to be false. "Confirmation" is not equivalent to "verification." The observation of a single non-black raven falsifies the hypothesis that all ravens are black.
So, what happens to the millions of "confirmations"?
And, why am I still seeing them? I can literally look at green apples whenever I want.
Not sure what you're asking here. The apples are no less green because there are white ravens. We can accumulate evidence for a hypothesis which later turns out to be false (were this not the case, there would in fact be no difference between "justification" and "truth," at least with regards to empirical hypotheses).
It gets worse: every egg-sized block of empty space we observe also confirms the hypothesis that "all eggs are white" (again, assuming that we've already observed at least one white egg), because that is one less egg-sized block of space which could potentially contain a non-white egg.
Well, you were asking about p(HE|B), which, as I said, I had already covered.
The observation of a green apple, or a black raven - the corroborating evidence - is logically implied by H, and by K.
The "more formal statement" above is actually a definition of logical implication.
If you don't like the "not all ravens are black" fro some reason, then change it. There are several others you could chose, "all ravens are black except the white ones", "all ravens are black or white".
I know, but I had issues with your treatment of both H and K, which i will discuss in more detail below. In the meantime, I offer this correction to one of my points.
Quoting Arkady
I added bolding to my above quote, because the non-statistically independent nature of K and E (assuming that K entails E) is in fact the key here, at least according to some quick and dirty refresher research I did. I can flesh out my point, if need be, but suffice to say, I now agree that, if K entails E, then P(K&E) = P(K).
This is part of the sticking point. I don't see how K (i.e. "not all ravens are black") implies the observation of a black raven. It is at most logically consistent with this observation. More generally, I don't see how a statement and its negation both imply the same thing (at least with regards to empirical hypotheses).
Yes, I take no issue with your definition.
I don't have a problem with "all ravens are black"...
First it should be noted that nothing about evidence and confirmation is necessitated by classical logic, simply because these concepts do not belong in classical logic. That's not to say that a theory cannot be built on deductive foundations (that's what Hempel, who came up with the paradox, as well as a number of others, attempted to do). However, even with classical logic as a background there are various ways of going about it, and different models and starting assumptions will yield different results. And then there are various non-deductive theories of confirmation: Bayesian and even more exotic theories, such as two-parameter models. (A Popperian will just dismiss the challenge, since according to her there is no such thing as confirmation. And that's why few pay attention to Popperians :P)
However the fact that they both support the claim, does not mean that they provide equal support.
Consider what it would take to be certain that all ravens are black, one of the following needs to occur.
i) ravens need to be defined as being black
ii) we need to see all the ravens and all of them need to be black
iii) we need to see all the non-black objects, none of them can be ravens and I need to know ravens exist (I need to state that ravens exist here otherwise you could prove that all unicorns are pink by observing that all non-pink objects are not unicorns).
Case i is trivial. Case ii requires the observation of far fewer objects than case iii. Therefore one observation of a black raven supports the claim more than the observation of a non-black non-raven. The degree that an observation of a non-black non-raven supports the claim depends on how many non-black objects there are. If there are a finite number of non-black objects, then an observation of a non-black non-raven would support the claim some non zero amount. If you claim there are an infinite amount of non-black objects then the support for the claim from observing a non-black non-raven would be infinitely small. In either case, it would be hard to imagine any number of observations of non-black non-ravens changing your stance on whether all ravens are black.
For example, suppose there are no ravens. Then (2) is true, but the status of (1) is uncertain.
If an item or idea shares an absolute trait with its fellows, and without exception, then we can reasonably assume that everything which is lacking in that trait is not of that designation. We may do this by the process of elimination, categorization, and exclusion.
The Raven Paradox is highly conditional, as thre are few objects which share such an absolute similarity. When items are compared, such as Johnny Truants and black ravens, it is natural to seek evidence for the conclusion. When items other than ravens are mentioned, they give evidence in the context of the statement. I am not certain that this oddity should be considered a veritable paradox. In considering the problem, we should remember that abstractions between divers objects are a common construct of our existence. Exclusions are a crucial portion of evaluation. While the problem is counterintuitive, it may seem an oddment only according to our wonted schematics of awareness.