Evidence?

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. — Terrapin Station

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?

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? — Terrapin Station

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.

Because it's just an example to make sense of the maths. — Michael

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.

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. — Terrapin Station

Of course the math makes sense. We don't need actual examples to work with probabilities. The numbers suffice.

There is a xⁿ 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^{n - 1} × 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).

There is a xn chance of the hypothesis "all Ys are Z" being true, — Michael

I don't agree that there's any way to come up with a number for claims such as that.

We don't need to come up with a number. It just needs to not be zero (which we know it isn't if we find one example of a Y that is a Z). The math then follows. x^{n - 1} × 1 is greater than (or equal to) xⁿ.

Given the logical equivalence, any evidence in support of (2) is also evidence in support of (1). — 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.

They're logically equivalent because of the law of contraposition, and evidence for one is evidence for the other because they're logically equivalent. — 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 know that (1) and (2) do not say the same thing.. — Metaphysician Undercover

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.

They do not say the same thing, they say an equivalent thing. Therefore what they say is equivalent, not the same. Likewise the table and the chair are both "one" object. They are equivalent, as "one", but not the same

They do not say the same thing, they say an equivalent thing. Therefore what they say is equivalent, not the same. Likewise the table and the chair are both "one" object. They are equivalent, as "one", but not the same — Metaphysician Undercover

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:

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.

To be "equivalent" logically, does not mean the same thing as to be the "same". This is a mistaken assumption.

To be "equivalent" logically, does not mean the same thing as to be the "same". This is a mistaken assumption. — Metaphysician Undercover

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.

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

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.

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. — TheMadFool

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.

Two statements that are logically equivalent mean the same thing... — Michael

Again, I think you're mistaken here. Logical equivalence is not defined as two statements which are logically equivalent "mean the same thing".

We don't need to come up with a number. It just needs to not be zero (which we know it isn't if we find one example of a Y that is a Z). The math then follows. xn - 1 × 1 is greater than (or equal to) xn. — Michael

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.

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. — Terrapin Station

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.

Two statements that are logically equivalent mean the same thing, and so have the same truth value in every model. — Michael

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".

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". — Metaphysician Undercover

Two statements have the same truth value in every model iff they mean the same thing.

The maths shows that the probability of our hypothesis being true increases with each successful observation. — Michael

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.

Two statements have the same truth value in every model iff they mean the same thing. — Michael

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."

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. — Terrapin Station

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%?).

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." — Terrapin Station

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.

We know that (1) and (2) do not say the same thing..
— Metaphysician Undercover

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. — Michael

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.

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. — unenlightened

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).

and its meaning is such that if A is logically equivalent to B then A and B mean the same thing. — Michael

And we can certainly talk about situations where A and B mean the same thing to S.

And the law of contraposition is equally clear that if P then Q is logically equivalent to if not P then not Q.

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.

I would refer you back to unenlightened's response from earlier, but that would be childish of me.

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?