Time to address my argument, not to repeat yours.

What argument? You simply said:

"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.

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.

The argument is that if you make a claim about X where X = non-black ravens, and the claim is that there are none, that claim is not evidenced in any way by Y, where Y is any other thing whatsoever from black ravens to pink unicorns to green apples. It's like if I say my pockets are empty, and you show that yours are not. And I say so what, mine are empty. The only evidence is the nothing in my pockets, however many people with full pockets you cite.

Probability isn't simply limited to either there being a probability of 1 or a probability of 0. — Michael

It is limited to 1 or 0 in any individual (i.e., actual) case, where there is a determinate fact of the matter.

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)^{the number of players}. — 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.

Evidence is just whatever increases the probability that the statement is true. — 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.

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? — 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.

The argument is that if you make a claim about X where X = non-black ravens, and the claim is that there are none, that claim is not evidenced in any way by Y, where Y is any other thing whatsoever from black ravens to pink unicorns to green apples. It's like if I say my pockets are empty, and you show that yours are not. And I say so what, mine are empty. The only evidence is the nothing in my pockets, however many people with full pockets you cite. — unenlightened

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)ⁿ chance of every egg being white, but after a successful observation the probability increases to (1/x)^{n - 1}.

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: — Michael

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.

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. Bothe 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. — unenlightened

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.

and evidence of white eggs increases the probability that all eggs are white. — Michael

This is the problem. This what is wrong, and since I cannot convince you, I'll just refer you to Hume.

This is the problem. This what is wrong, and since I cannot convince you, I'll just refer you to Hume. — unenlightened

Then where does my math fail? Prior to any observation the probability of the claim being true is (1/x)ⁿ. After a successful observation the probability of the claim being true is (1/x)^{n - 1}.

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.

The whole thing is about epistemology, so I don't understand your objection. — Michael

My objection was that probability is not the same thing as epistemic uncertainty.

The paradox is that if we observe a green apple then we can be more confident that all ravens are black. — 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.

My objection was that probability is not the same thing as epistemic uncertainty. — aletheist

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.

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.

It doesn't matter how weak the evidence is. The paradox is that there's evidence at all.

Then where does my math fail? — Michael

Sigh, I thought that we had made a breakthrough. Your math fails right here:

Prior to any observation the probability of the claim being true is (1/x)ⁿ. After a successful observation the probability of claim being true is (1/x)^{n - 1}. — 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).

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. — 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 again you're back to this bizarre understanding of probability.

It doesn't matter how weak the evidence is. The paradox is that there's evidence at all. — Michael

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.

And the paradox is resolved by recognising, with Hume, that there is no such thing as evidence for a universal proposition. — tom

But there is evidence, as I've shown. After each successful observation the probability that the claim is true increases.

Then where does my math fail? — Michael

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 the bullet dude; induction is not logical. I have shown it to you every which way, with analysis and argument, and finally with an example from history, described with humour. I'm done.

Let say that instead of eggs, I'm standing there in each egg hole dealy flipping coins. The first one lands on heads, the second one lands on heads, the third one on heads. Does the probability of the next "egg" being heads increase each time? Decrease each time? Or remain the same?

Bite the bullet dude; induction is not logical. I have shown it to you every which way, with analysis and argument, and finally with an example from history, described with humour. I'm done. — unenlightened

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.

Let say that instead of eggs, I'm standing there in each egg hole dealy flipping coins. The first one lands on heads, the second one lands on heads, the third one on heads. Does the probability of the next "egg" being heads increase each time? Decrease each time? Or remain the same? — Wosret

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.

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.

If you flipped green apples and they turned into ravens (black) half the time, your analogy would work. — TheWillowOfDarkness

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".

The evidence thing I already destroyed. It doesn't count as evidence because not everything counts as evidence, particularly not to a scientific approach. Evidence has to be of a special, difficult to produce kind.

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. — Michael

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.

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. — aletheist

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.

My objection was that probability is not the same thing as epistemic uncertainty. — aletheist

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).

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. — Michael

Only if we embrace a sloppy usage of "probability." You refuse to acknowledge the objective/subjective distinction. There is nothing strange about it.