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? — Terrapin Station

We're not ignoring the semantic content of the statements. That they're logically equivalent is that they have the same semantic content.

That they're logically equivalent is that they have the same semantic content. — Michael

Logic is about form, not semantic content. You're arguing otherwise? That logical form is identical to semantics?

Logic is about form, not semantic content. You're arguing otherwise? That logical form is identical to semantics? — Terrapin Station

Read up on contraposition. Here's a simple summary:

In logic, contraposition is an inference that says that a conditional statement is logically equivalent to its contrapositive. The contrapositive of the statement has its antecedent and consequent inverted and flipped: the contrapositive of P → Q is thus ¬Q → ¬P. For instance, the proposition "All bats are mammals" can be restated as the conditional "If something is a bat, then it is a mammal". Now, the law says that statement is identical to the contrapositive "If something is not a mammal, then it is not a bat."

What would you say that has to do with my last post and my questions to you?

You seem to be saying that contraposition doesn't have anything to do with the meaning of the statements. I'm showing you that you're wrong.

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?

The part where it states that the statements are logically equivalent. As explained here, "Two sentences can be equivalent in a sense much stronger than that of material equivalence; they may be equivalent in meaning as well as having the same truth value. If they do have the same meaning, any proposition that incorporates one of them can just as well incorporate the other; there will not be—there cannot be—any case in which one of these statements is true while the other is false. Statements that are equivalent in this very strong sense are called logically equivalent".

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?

First you didn't quote that part. I asked "What part of what you quoted . . ." — Terrapin Station

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.

Understanding is different than agreement. — Terrapin Station

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.

I don't agree that logic deals with semantics other than formally. I had said that above. — Terrapin Station

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?

This is why I try not to debate with you. Most of the time it seems like you're trolling. I'm going to end this before it gets any further.

You can't go off script too much. I understand.

It depends on who is assigning meaning to those statements and what meanings they're assigning, doesn't it? — Terrapin Station

This is the bollocks of a loser out of his depth.

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. — unenlightened

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.

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. — andrewk

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.

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. — andrewk

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.

There's no paradox because that isn't really how evidence works. Like I said, anyone can weave any tale to retroactively explain the facts, but a good hypothesis predicts something that we didn't know, and then we look for it and find it. That doesn't bolster the logical necessity of the theory being true or anything, it could as well just be a coincidence, but it sure as hell is damned impressive, and bolsters confidence. We then keep the hypothesis, use it in more general theories, keep trying to make new predictions, which keep bolstering confidence for as long as they keep working out, until we're all but certain of them.

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.

There's no paradox because that isn't really how evidence works. Like I said, anyone can weave any tale to retroactively explain the facts, but a good hypothesis predicts something that we didn't know, and then we look for it and find it. That doesn't bolster the logical necessity of the theory being true or anything, it could as well just be a coincidence, but it sure as hell is damned impressive, and bolsters confidence. We then keep the hypothesis, use it in more general theories, keep trying to make new predictions, which keep bolstering confidence for as long as they keep working out, until we're all but certain of them. — Wosret

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.

Yes, when it's a million ravens it becomes almost obvious. I wonder what underlies this. I'm trying to formulate it in terms of statistical hypothesis testing.

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.

...a single datum doesn't help us with the universal. — andrewk

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.

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

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 0.5¹² chance of every egg being white. We look at the first egg and see that it is white. There's now a 0.5¹¹ × 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.

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. — andrewk

Does the above help address this?

My humble opinion on the matter. My logic is a bit rusty so bear with me

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.

One green apple will NOT suffice to prove either of these statements.

Paradox solved.

''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.

One green apple will NOT suffice to prove either of these statements.

Paradox solved. — TheMadFool

We're not talking about proof. We're talking about evidence. Not all evidence is proof.

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 0.512 chance of every egg being white. We look at the first egg and see that it is white. There's now a 0.511 × 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. — Michael

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).