Hempel's Ravens

[Magical Realist]

"If two statements are logically equivalent, if they assert exactly the same thing, then any evidence for one is evidence for the other.

This principle appears to be truism. Consider the two statements “Terry and Judith are my parents” and “I am Terry and Judith’s child”. These statements are logically equivalent, they say the same thing. There is no evidence that would support one of them without supporting the other.

No matter how superficially plausible this principle, however, the Hempel’s Ravens paradox seems to show that it is false. The Hempel’s Ravens paradox uses the principle to prove the absurd conclusion that an observation of a green parrot is evidence that ravens are black. The only way of avoiding this clearly unacceptable conclusion is to reject the principle above.

The paradox goes like this:

Consider the two statements:

(1) “All ravens are black.”
(2) “Everything that isn’t black, isn’t a raven.”

These two statements say exactly the same thing. The first statement says that everything of a particular kind has a certain property. The second statement says that everything that lacks that property isn’t of that kind.

The two statements are therefore logically equivalent; they are true and false in exactly the same circumstances. If there is anything that is a raven but isn’t black then both (1) and (2) are false; oherwise, they are both true. As the two statements are logically equivalent, any observation that supports one will also support the other.

Suppose, then, that I observe a green parrot. This observation confirms (2), “Everything that isn’t black isn’t a raven”. A green parrot isn’t black and isn’t a raven. The observation is evidence that (2) is true.

Given what has been said so far, my observation of a green parrot must also confirm (1). (1) and (2) are logically equivalent, so any evidence for one is evidence for the other. My observation of a green parrot, then, is evidence for the statement, “All ravens are black”. In fact, any observation of something that isn’t black and isn’t a raven is evidence that ravens are black.

This, though, is absurd; there is no way that we can discover what colour a raven is without looking at a raven."---http://www.logicalparadoxes.info/hempels-ravens/

 

[wynn]

The paradox goes like this:

Consider the two statements:

(1) “All ravens are black.”
(2) “Everything that isn’t black, isn’t a raven.”

These two statements say exactly the same thing.

They do not.

Since there are black things that aren't ravens, the above two premises are not enough to establish that something is or isn't a raven; the above two premises are necessary, but not sufficient to establish that something is or isn't a raven.



This seems rather basic.

 

[C C]

[...quoted text...]This, though, is absurd; there is no way that we can discover what colour a raven is without looking at a raven."---[/color]http://www.logicalparadoxes.info/hempels-ravens/

One might just as much say that: "The fingers of a hand added to the 5 marbles that Sue is holding equals ten." And then complain that: "This is absurd; there is no way that we can know how many fingers a hand has unless we have looked at one. Therefore 5+5=10 is bonkers."

Such is confusing the abstract template with whatever empirical content is selected to fill its empty placeholders (or symbols), or losing track of a borderline distinguishing them. The former only has to be interconsistent in terms of its relational structure -- or the latter in regard to any larger system it fell out of which supports it. Whereas concrete entities or affairs, or universal concepts derived from them (like "all hands have five fingers" or "all ravens are black") are not so world or experience independent; are or can be contingent on what is yet to be found in experience or the world, when minus a provision like "normally", "excluding sports", etc.

For instance, the "template" of 5+5=10 underlying the statement above is not the culprit, but instead the potential ambiguity of the empirical item that was plugged into one of the quantitative placeholders of "5". So an AI unfamiliar with either the definition of a human hand or the practicality of everyday conventions then whines: "But I've never seen a hand. How many fingers does it have? And even if I had seen a hand in the past, how can I be certain that all hands always have that many fingers?" But this simply means either a poor empirical item was selected to fill one of the abstract five symbols; or an incomplete description of the hand was provided. Just as it was indicated that Sue had 5 marbles, it can be precisely indicated that the concerned "hand" has five fingers, or is just a "typical" hand associated with the majority.

 

[Magical Realist]

They do not.

Since there are black things that aren't ravens, the above two premises are not enough to establish that something is or isn't a raven; the above two premises are necessary, but not sufficient to establish that something is or isn't a raven.



This seems rather basic.

But the two statements aren't premises by which to conclude if something is or isn't a raven. They are simply two logical statements that are saying the same thing: that all ravens are black, or its inverse, that everything that isn't black is not a raven. The assumption is that you will be able to tell that something is not a raven by simply looking at it. Just as you will be able to tell if something is a raven. Thus, when you see a green parrot, you can SEE that it is not a raven, NOT because it is not black, but because it is a parrot.

 

[Magical Realist]

One might just as much say that: "The fingers of a hand added to the 5 marbles that Sue is holding equals ten." And then complain that: "This is absurd; there is no way that we can know how many fingers a hand has unless we have looked at one. Therefore 5+5=10 is bonkers."

Such is confusing the abstract template with whatever empirical content is selected to fill its empty placeholders (or symbols), or losing track of a borderline distinguishing them. The former only has to be interconsistent in terms of its relational structure -- or the latter in regard to any larger system it fell out of which supports it. Whereas concrete entities or affairs, or universal concepts derived from them (like "all hands have five fingers" or "all ravens are black") are not so world or experience independent; are or can be contingent on what is yet to be found in experience or the world, when minus a provision like "normally", "excluding sports", etc.

For instance, the "template" of 5+5=10 underlying the statement above is not the culprit, but instead the potential ambiguity of the empirical item that was plugged into one of the quantitative placeholders of "5". So an AI unfamiliar with either the definition of a human hand or the practicality of everyday conventions then whines: "But I've never seen a hand. How many fingers does it have? And even if I had seen a hand in the past, how can I be certain that all hands always have that many fingers?" But this simply means either a poor empirical item was selected to fill one of the abstract five symbols; or an incomplete description of the hand was provided. Just as it was indicated that Sue had 5 marbles, it can be precisely indicated that the concerned "hand" has five fingers, or is just a "typical" hand associated with the majority.

But what about the two statements about ravens being black was ambigious? In your example the statement "fingers of a hand" IS ambigious in that it leaves out the number of fingers a hand has. I can't see anything similarly left out from either: "All raven are black" or "Everything that isn't black, isn't a raven."

 

[Yazata]

The Hempel’s Ravens paradox uses the principle to prove the absurd conclusion that an observation of a green parrot is evidence that ravens are black. The only way of avoiding this clearly unacceptable conclusion is to reject the principle above.

I don't entirely agree. I can imagine situations where observing a green parrot might be better evidence that all ravens are black than observing a black raven would be.

The paradox goes like this:

Consider the two statements:

(1) “All ravens are black.”
(2) “Everything that isn’t black, isn’t a raven.”

These two statements say exactly the same thing.

I don't agree with that.

The first statement says that everything of a particular kind has a certain property. The second statement says that everything that lacks that property isn’t of that kind.

The two statements are therefore logically equivalent; they are true and false in exactly the same circumstances.

Right. That doesn't mean that the two statements mean the same thing or that they are interchangeable. It just means that whenever one of them is true (or false) that the other will be too.

If there is anything that is a raven but isn’t black then both (1) and (2) are false; oherwise, they are both true. As the two statements are logically equivalent, any observation that supports one will also support the other.

And that delivers us to the crux of the difficulty. The issue is how strong that support will be in particular cases.

Suppose, then, that I observe a green parrot. This observation confirms (2), “Everything that isn’t black isn’t a raven”. A green parrot isn’t black and isn’t a raven. The observation is evidence that (2) is true.

Given what has been said so far, my observation of a green parrot must also confirm (1). (1) and (2) are logically equivalent, so any evidence for one is evidence for the other. My observation of a green parrot, then, is evidence for the statement, “All ravens are black”. In fact, any observation of something that isn’t black and isn’t a raven is evidence that ravens are black.

Right. But it needn't be very strong evidence.

This, though, is absurd; there is no way that we can discover what colour a raven is without looking at a raven.

Sure there is. It all depends on the nature of the universe of discourse.

Suppose that a hypothetical universe contains only one object that isn't black. And suppose that the universe has billions upon billions of black ravens. Wouldn't it make sense to examine that one single non-black object to see whether or not it's a raven (it turns out to be a green parrot in this example) than it would be to examine every one of the billions of ravens, one after another, to make sure they are all black? Simply ascertaining that the one non-black thing in the universe isn't a raven would prove that all the ravens must be black, since everything else in the universe must be black (including all the ravens).

In our universe, where the vast majority of objects aren't black and aren't ravens, it wouldn't make very much sense to approach things that way, since examining all non-black things would be a vastly larger task than examining all the ravens. Each instance of a non-black non-raven would presumably have less inductive force than a black raven would, because it would be a much smaller sample of a much larger set.

In other words, what seems at first to be a logical problem might actually be more empirical, resulting from the nature of the universe in which we seek to apply the logic.

 

[C C]

But what about the two statements about ravens being black was ambigious? In your example the statement "fingers of a hand" IS ambigious in that it leaves out the number of fingers a hand has. I can't see anything similarly left out from either: "All raven are black" or "Everything that isn't black, isn't a raven."

Yet, despite designating that ravens were black, this was rejected: "This, though, is absurd; there is no way that we can discover what colour a raven is without looking at a raven." So why include "five" with the hand at the outset of my departing analogy? Since the whole point carried over seems to be that items like "Hands have ___ fingers" and "Ravens are ____ color" have to be left so blank because of this kind of bewildering criticism. Which then ironically helps illustrate that reliable templates or "formulas" often (if ever) cannot be "blown-up" by the empirical content or choices plugged into their abstract placeholders. I mean, is 5+5=10 really bunk because someone might complain that "A hand's five fingers and Sue's five marbles combined equals ten" is unacceptable because the sentence is unable to "see" that a hand really has five fingers? That is, this seems to be the actual gist of what the author is saying in the quote above, that the sentences "cannot see", rather than us. An AI could be in a similar fix: Asserting that it has never seen a hand.

 

[Magical Realist]

 

[Fednis48]

This is a fascinating paradox! I think Yazata largely nailed the solution, but I'll add my two cents to the discussion, and reply to C C while I'm at it. A big problem with focusing just on the hypothesis "all ravens are black" is that finding evidence to support one hypothesis, in a vacuum, can't help us refine our picture of the world. Only when we get evidence that supports one hypothesis but not others can we make meaningful inferences. In the example of ravens, if I were to spend a long time randomly sampling non-black birds in my town and never finding a raven, this would indeed support the hypothesis that all ravens are black. But it would also support, for instance, the hypothesis that ravens don't live in my town, or that my "random sampling" is actually somehow biased against finding ravens. If for other reasons I am confident that I have no sampling bias and do in fact have a fair chance of spotting any non-black ravens, the absence of any such spottings would be fair evidence that they don't exist.

Yet, despite designating that ravens were black, this was rejected: "This, though, is absurd; there is no way that we can discover what colour a raven is without looking at a raven." So why include "five" with the hand at the outset of my departing analogy? Since the whole point carried over seems to be that items like "Hands have ___ fingers" and "Ravens are ____ color" have to be left so blank because of this kind of bewildering criticism. Which then ironically helps illustrate that reliable templates or "formulas" often (if ever) cannot be "blown-up" by the empirical content or choices plugged into their abstract placeholders. I mean, is 5+5=10 really bunk because someone might complain that "A hand's five fingers and Sue's five marbles combined equals ten" is unacceptable because the sentence is unable to "see" that a hand really has five fingers? That is, this seems to be the actual gist of what the author is saying in the quote above, that the sentences "cannot see", rather than us. An AI could be in a similar fix: Asserting that it has never seen a hand.

The problem here is that "5 + (# of fingers on each hand) = 10" is not at all equivalent to the principle posted at the start of this thread. If you knew nothing about hands at all, you couldn't plug # of fingers into 5 + 5 = 10, because you would have no reason to think # of fingers = 5. The raven example, by contrast, is logically sound even if you make no assumptions about the nature of ravens. This implies that even if you know nothing at all about ravens, you can learn about them just by studying things that aren't ravens. Hence the apparent paradox.

 

[C C]

The problem here is that "5 + (# of fingers on each hand) = 10" is not at all equivalent to the principle posted at the start of this thread.

It's not supposed to be equivalent to that principle. OR: How many times do I have to re-quote the last sentence of the quote in MR's original post to indicate that's what the departing analogy of the hand was directed at? It's all that was ever quoted/referenced in either of my posts (from the OP message).

 

[Fednis48]

It's not supposed to be equivalent to that principle. OR: How many times do I have to re-quote the last sentence of the quote in MR's original post to indicate that's what the departing analogy of the hand was directed at? It's all that was ever quoted/referenced in either of my posts (from the OP message).

Well, ok then. The last sentence of the original post is "This, though, is absurd; there is no way that we can discover what colour a raven is without looking at a raven." Maybe it would be better to say "This, though, is absurd; there is no way that we can discover what colour a raven is without looking at a raven or already knowing what its color is from previous experience." The (apparent) paradox is still there, because it just involves us postulating a hypothesis about ravens and then observing non-ravens to see if it's true, without invoking any prior knowledge. In your hand analogy, you wouldn't know "5 + # of fingers = 10" unless you already knew "# of fingers = 5". When you accept "5 + # of fingers = 10" as true, you're bringing in prior knowledge. If, on the other hand, you treat "5 + # of fingers = 10" as just a hypothesis (like "all ravens are black" in the original argument), you really can't infer that "# of fingers = 5", because you don't know whether your hypothesis is true.

 

[C C]

The (apparent) paradox is still there, because it just involves us postulating a hypothesis about ravens and then observing non-ravens to see if it's true, without invoking any prior knowledge.

It's not much of an hypothesis and h-tester. "All X are/have ___"; If it lacks ___ it isn't X" really just seems to express a property or characteristic that X must have in order to be classified or identified as X. IOW, there are features or conditions that distinguish X from non-X, what makes entities or their groups distinct from each other to begin with (rats are not water pipes and vice versa, for yata-yata-yata reasons / differences). So when an empirical item (sparrow, oak tree, washing machine, etc) is plugged into the abstract placeholders, one is dealing with entities of experience or perception that are already known. When we plug "ravens" into the template we already know what they are and some if not all their characteristics, like black color (or at least that they are recorded in the whole of human knowledge, if "Zoldar The Extremely Isolated Person" happens to have never even saw a picture of the standard raven before).

Plugging "raven" and "black" into that kind of template is to use things already known, and then conjuring this pretense that "black" wasn't already known for ravens. Same with plugging the empirical content of a "hand" into another abstraction like 5+5=10. An artificial intelligence which for some peculiar reason was never downloaded with a description of a human hand or lacks the capacity of having experiences in association with image data can complain it doesn't know "hand" represents "5". But the average human can't get away with feigning ignorance. S/he's dabbling in matters s/he already has knowledge about.

Imaginary entities can be plugged in, too: Like "behorpha" have "jognalix"...."; but those become empty placeholders themselves replacing "X"and "___". They don't refer to anything that will be found in the world. Or even in fiction, if no author has ever utilized them to name a fictional entity or circumstance (Sherlock Holmes wasn't real but Doyle created / provided assorted literary "facts" about him that can be referenced). Science might propose things that are possible but which there is no evidence for yet: Like intelligent life on other worlds. We might plug in some characteristic about ourselves that supposedly qualifies us as "intelligent" ("All intelligent ETs will have ___"), but most everything else, such as asserting that Drogons from planet Porgesh must have a triple-peaked dorsal fin on their backs to qualify as Drogons is just made-up fiction or speculation. Meaningful plug-ins to insert in the template must come from what we already know about, or fall out of some principle-guided process that sometimes provides information / inferences about things that can be demonstrated consistent with its own system axioms later, or literally have the potential to be found in nature, etc.

 

[Fednis48]

It's not much of an hypothesis and h-tester. "All X are/have ___"; If it lacks ___ it isn't X" really just seems to express a property or characteristic that X must have in order to be classified or identified as X. IOW, there are features or conditions that distinguish X from non-X, what makes entities or their groups distinct from each other to begin with (rats are not water pipes and vice versa, for yata-yata-yata reasons / differences). So when an empirical item (sparrow, oak tree, washing machine, etc) is plugged into the abstract placeholders, one is dealing with entities of experience or perception that are already known. When we plug "ravens" into the template we already know what they are and some if not all their characteristics, like black color (or at least that they are recorded in the whole of human knowledge, if "Zoldar The Extremely Isolated Person" happens to have never even saw a picture of the standard raven before).

There are enough properties that define a raven without including their black color (proof: see MR's picture of a white raven). You say that when we plug "ravens" into a template, we already know some if not all of their characteristics. Is it really so crazy to say for argument's sake that we might know enough about ravens to identify them on sight but not know what color they are?

 

[Captain Kremmen]

To prove that all ravens are black you need to see all ravens, not just one.
Before Australia was discovered, you could have said that all swans are white,
and been supported by every observation.
But, as we know now, Australia has black swans.

Seeing one bird is equivalent for both instances.
One black raven does not prove that all ravens are black.
Neither does one green parrot prove that anything that isn't black isn't a raven.

What about albino ravens?

 

[Fednis48]

To prove that all ravens are black you need to see all ravens, not just one.
Before Australia was discovered, you could have said that all swans are white,
and been supported by every observation.
But, as we know now, Australia has black swans.

Seeing one bird is equivalent for both instances.
One black raven does not prove that all ravens are black.
Neither does one green parrot prove that anything that isn't black isn't a raven.

What about albino ravens?

Now we're getting into more fundamental questions. I totally agree that to prove all ravens are black, you need to look at all ravens. Since there is no way to know for certain how many ravens are in the universe, one can never be sure that one has looked at all ravens, therefore it is impossible to prove that all ravens are black. In fact, except for a few, almost tautological cases (e.g. all ravens are birds because the category "birds" includes all ravens by definition), one can never prove a general statement. What we can do is gain evidence in favor of hypothesis. My question to you is, how do we do that? If a statement like "all ravens are black" cannot be proven true, what does it even mean to find evidence in favor of such a statement?

According to verificationism, of which Bayesian statistics is a refined form, gaining evidence for a hypothesis means making observations that are in agreement with that hypothesis. In the more nuanced Bayesian form, we can examine the likelihood of different observations under different hypotheses to quantify just how much a given observations supports each, but the principle is the same. The raven paradox is that, according to verificationism, we can gain evidence in favor of a hypothesis about ravens just by observing non-ravens, which runs counter to the intuition that we have to study something to gain information about it.

 

[rcscwc]

There are white ravens too.

 

[rcscwc]

Like ravens, take humans.

Examine the statement rather premise.

All humans are Mortal

Logical problem is that the testator really does not, cannot know that all humans are REALLY mortal. There might be some non mortal human who has not come light.

But hold it. How do determine that your subject Mr. X is really a mortal? If an observer Mr. Y is appointed how will he know unless he too is immortal if indeed X turns out to immortal? How will he report back that X is mortal? If after a billion years X does die how will the original testator know?

Fact is that this statement CANNOT be proved true or false.

 

[Captain Kremmen]

Now we're getting into more fundamental questions. I totally agree that to prove all ravens are black, you need to look at all ravens. Since there is no way to know for certain how many ravens are in the universe, one can never be sure that one has looked at all ravens, therefore it is impossible to prove that all ravens are black. In fact, except for a few, almost tautological cases (e.g. all ravens are birds because the category "birds" includes all ravens by definition), one can never prove a general statement. What we can do is gain evidence in favor of hypothesis. My question to you is, how do we do that? If a statement like "all ravens are black" cannot be proven true, what does it even mean to find evidence in favor of such a statement?

According to verificationism, of which Bayesian statistics is a refined form, gaining evidence for a hypothesis means making observations that are in agreement with that hypothesis. In the more nuanced Bayesian form, we can examine the likelihood of different observations under different hypotheses to quantify just how much a given observations supports each, but the principle is the same. The raven paradox is that, according to verificationism, we can gain evidence in favor of a hypothesis about ravens just by observing non-ravens, which runs counter to the intuition that we have to study something to gain information about it.

I don't think that you have actually disagreed with me.
I'm not sure whether you were intending to do so.

It's like one of those diagrams with the overlapping circles. Venn diagrams.
If you made a Venn diagram of this situation, it would look like this:
You would have a large circle A, birds.
Somewhere fully inside that circle, you could place a second circle, B, black birds.
Then, if you wanted to place a third circle, C, ravens, the question for you would be whether it overlaps with B, or is wholly inside it.

As humans, we prefer to define things with certainty, but there is as much information in the first instance, "overlaps",
as in the second, "is wholly inside".

 

[Fednis48]

Like ravens, take humans.

Examine the statement rather premise.

All humans are Mortal

Logical problem is that the testator really does not, cannot know that all humans are REALLY mortal. There might be some non mortal human who has not come light.

But hold it. How do determine that your subject Mr. X is really a mortal? If an observer Mr. Y is appointed how will he know unless he too is immortal if indeed X turns out to immortal? How will he report back that X is mortal? If after a billion years X does die how will the original testator know?

Fact is that this statement CANNOT be proved true or false.

You're right, but you've identified a separate question. The raven example, and verificationism in general, have to do with how we treat categorical statements. Since it's usually impossible to be sure we've examined every instance of a category, categorical statements are typically unprovable. This is the problem you note when you say "There might be some non mortal human who has not come to light," and it's the problem I talk about in my previous post.

But the question "How do [you] determine that your subject Mr. X is really a mortal?" is a different sort of question. "Mr. X is a mortal" is not a categorical statement but a specific one, and we could prove or disprove it simply by applying some "mortality test" to Mr. X. The fact that such a test might not exist is interesting, and there's probably a whole other thread to be started on the question of whether things can have properties that in principle cannot be tested for. But that has nothing to do with verificationism or the raven paradox.

I don't think that you have actually disagreed with me.
I'm not sure whether you were intending to do so.

It's like one of those diagrams with the overlapping circles. Venn diagrams.
If you made a Venn diagram of this situation, it would look like this:
You would have a large circle A, birds.
Somewhere fully inside that circle, you could place a second circle, B, black birds.
Then, if you wanted to place a third circle, C, ravens, the question for you would be whether it overlaps with B, or is wholly inside it.

As humans, we prefer to define things with certainty, but there is as much information in the first instance, "overlaps",
as in the second, "is wholly inside".

True. In some sense, both "C is wholly inside B" and "C overlaps with B but is not wholly inside it" are equally strong statements. But what if you can't make either statement with certainty? Say you want to decide whether the hypothesis "C is wholly inside B" (ie. "all ravens are black) is true. So you go out and look at some ravens, and you see that they all are black. Your observations are consistent with your hypothesis, but they certainly don't prove it. In fact, no matter how many black ravens you find, there might be some exceptions to the rule hiding somewhere you haven't looked yet. If there is no number of observations that would prove the hypothesis true, what does it mean to gather evidence in favor of it?

 

[wynn]

But the two statements aren't premises by which to conclude if something is or isn't a raven.

Then of what use are they? Why state them?

If they are not intended to be appplied in some real-world situation, even though they clearly seem to be about things of the real world, then what's their use?