Hi. Can you please explain how you used math on this paradox. I can't find it. Only if you have the time ofcourse.

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. — aletheist

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.

The first reference was here, but directed at a different claim about the colour of eggs. A more general form was here.

Also one about coins.

Perhaps a better way to look at it is the following.

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.

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. — Michael

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/

Ah, my mistake. Thanks.

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. — unenlightened

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"?

Oh, and that article is directly relevant to this. It mentions the raven paradox and even includes Hempel's theory that "(the observation report of) a white swan (directly) Hempel-confirms that all swans are white".

(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).

Thank you for taking the time to reply. My humble observation follows:

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.

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. — Michael

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.

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. — Metaphysician Undercover

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)?

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. — Michael

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?

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. — Metaphysician Undercover

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.

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?

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".

This is contradiction. If two statements are logically equivalent according to contraposition then ipso facto they have the same truth value in every model. — Michael

No that's not the case, we've already gone through this with the raven example. Reread my posts if you do not understand.

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". — 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.

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. — Michael

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.

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. — Metaphysician Undercover

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.

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 confirms the universally-quantified hypothesis "all ravens are black"? — Arkady

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.

It's not just "ordinary" talk. The actual mathematics of probabilities includes the fractions between 0 and 1. So I really don't know what you're talking about.

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 0.5¹⁰.

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 0.5².

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^{n - 1} is greater than 1/xⁿ. The probability increases after each white egg.

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. — aletheist

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.

The actual mathematics of probabilities includes the fractions between 0 and 1. — Michael

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.

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 0.5¹⁰. — Michael

So far, so good.

And we can use this reasoning even the machine has already made the eggs. — 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.

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. — aletheist

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:

Probabilities are usually defined in terms of the uncertainty in what’s known. Liar’s dice is a beautiful example (so are most card games for that matter); all of the dice are what they are, and yet in the picture above, if you’re the player on the left, then there’s a chance of 1 that all of your dice are 5’s, but there’s an even chance that your opponent’s dice could be any combination. From the left player’s perspective, there’s some chance that the dice on the right are, say, “1,2,3,4,5”, even though from the right player’s perspective, that chance is zero (the right player knows their dice are not “1,2,3,4,5”).

Long story short: probability is extremely subjective. Whether an event happened in the past or will happen in the future doesn’t make too much difference, it’s the knowledge you have about an event that defines its probability (for you).

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.

Long story short: probability is extremely subjective. — Michael

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.

But if you insist on your understanding, then I think that this post is still relevant. — 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.

Note that you shouldn't conflate "if something is not black then it is not a raven" with "everything that is not black is not a raven". — 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.

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. — aletheist

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!

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." — aletheist

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.

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. — 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?

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.

I don't know what you mean by giving a false connotation of objectivity. How, exactly, can one misinterpret the claim "there's a 0.5¹² 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.

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". — tom

Please read what you quoted from me again.

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. — 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 said that the observation of a green apple only supports - in fact, proves - the particular proposition that some non-black things are non-ravens. — aletheist

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.

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. — Michael

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.

I'm not sure the relevance of potential non-black things. Can't this just be about actual non-black things? — 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.

How, exactly, can one misinterpret the claim "there's a 0.5¹²" that every egg in that (closed) cartoon is a white egg? — 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.

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. — tom

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).

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). — aletheist

My apologies!

Anyway, the paradox is solved.