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Old 07-26-2008, 10:57 PM
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The Raven Paradox - Can anyone explain this in layman's terms?

I followed it for a bit then it kind of slid into some sort of symbolic logic meta wankery. Why is this paradox imporant? Can you 'splain me?
Old 07-28-2008, 11:49 AM
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Personally, I've never seen this presented as a paradox until this morning.

There is the set of all objects O, the set of non-black objects N, and the set of ravens R,with both of the last sets being subsets of O. Exhaustively searching either of those two subsets will determine the truth of the inital proposition "All ravens are black." Just because we know intuitively that the set R is much smaller than N doesn't make it logically more rigorous: the two searches are logically equivalent.
Old 07-28-2008, 11:55 AM
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William Poundstone dwells on that paradox quite a bit in his book Labyrinths of Reason, which is a very recommendable read.

The problem is that the application of the accepted rules of inductive logic yields the conclusion that a certain hypothesis is confirmed by an observation which is obviously irrelevant to the hypothesis.

Contrary to deductive reasoning, induction is not based on strictly logical conclusions and syllogisms drawn from given premises; instead, it relies on observations which confirm a hypothesis by giving a little bit more credence to it. In the Raven paradox, the hypothesis which is examined is: "All ravens are black."
If you find only one raven which is not black, the hypothesis is debunked. The rules of inductive logic state that every observation of another black raven confirms the hypothesis; even after seein a million black ravens, the observation of the 1,000,001st black raven will give a tiny little bit more credence to the hypothesis. It cannot actually prove it, because there might be white ravens somewhere which we haven't found yet, but clearly the more black ravens we see the more justified are we in our belief that all ravens are black.

The problem arises from the fact that you can rephrase every hypothesis using negators. The statement "All ravens are black" is logically identical to the statement "All non-black things are non-ravens." It seems as if this theory is confirmed by the observation of any thing which is not black and not a raven. My mousepad to the right of the laptop I'm using right now, for example, is blue. It is not black, and it is not a raven. The existence of my blue mousepad seems to prve the theory that all ravens are black, which is obviously nonsense (especially if you consider that the same mousepad would also confirm the hypothesis that all ravens are white).
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Old 07-28-2008, 12:15 PM
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I concur with the last two, especially the idea that it is not a paradox. The crux of the matter is that one example of a black raven or non-black non-raven is not proof. Think of it in terms of supporting your statement all ravens are black. The more ravens I see that are black without seeing a non-black raven supports my statement a little better. If I were to show you EVERY raven and they are all black then the statement would be proven.

The procedure would be similar to supporting the non-black = non-raven statement. Every non-black object I show you without seeing a raven supports my statment a little more. Finally if I show you all non-black object in the universe and they are all non-raven, then finally my statement would be proven.
Old 07-28-2008, 12:53 PM
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I wrote this last night and it wouldn't post b/c of the database problem, so I'll just ignore everybody else and go ahead and post it.

"Every raven is black" and "Everything that is non-black is not a raven" are logically identical statements; they are contrapositives of each other. That is, each entails the other--they have the exact same truth conditions the same thing. So if two sentences are true under precisely the same conditions, then intuitively if something is evidence for the truth of one sentence, then it should be evidence for the truth of the other. In fact, from the perspective of formal logic, these two sentences are actually synonymous--they mean exactly the same thing, merely stated differently. So it is highly intuitive that evidence for one should count as evidence for the other.

But "This green thing is an apple" is not evidence for "Every raven is black", even though it is evidence for its logical equivalent. That doesn't make any sense,
because "Every raven is black" <--> "Everything that is non-black is not a raven." They are either both true or both false; they necessarily have the same
truth value (and, as I said above, they are synonymous from the perspective of formal logic). So evidence for one should count as evidence for the other. But that seems not to be the case. So the question is, what gives?
Old 07-28-2008, 01:06 PM
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Quote:
Originally Posted by Sophistry and Illusion
But "This green thing is an apple" is not evidence for "Every raven is black", even though it is evidence for its logical equivalent.
Yes it is, though it may not be very strong evidence. In order to test the theory, "Every raven is black," I can choose to look for ravens, and see if any of them are not black, or I can choose to look for non-black things, and see if any of them is a raven. So if I see a green thing, I can check whether it's a raven: if it's an apple, then since apples are not ravens, it confirms the theory.

However, it's not very strong confirmation, because there are a lot more not-black things than there are ravens. For example, looking around from where I am now, I can see there are a lot of not-black things, and there are no ravens. In fact, I see not-black things all the time, and I rarely see ravens.
Old 07-28-2008, 01:09 PM
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Originally Posted by Sophistry and Illusion
I wrote this last night and it wouldn't post b/c of the database problem, so I'll just ignore everybody else and go ahead and post it.
Same thing for me:

One way to put the gist of it: Suppose you were to hypothesize that "All Ts have property P". What could be evidence for this? You might intuitively feel that, each time you observe a T which has property P [without ever seeing any counterexamples], this serves as further evidence of the hypothesized universal claim. For example, you might hypothesize "All ravens are black", and feel that each new instance of observing a black raven (without ever seeing any non-black ravens) is further evidence for the claim (giving you even stronger reason to believe it, etc.).

However, a hypothesis like "All ravens are black" is logically equivalent to a hypothesis like "All non-black things are non-ravens". Applying the principle from above, evidence for the latter hypothesis would be any observation of a non-black non-raven (for example, observation of a white cow). If one takes the reasonable position that evidence for X is also evidence for anything logically equivalent to X, it would follow that every observation of a white cow is further evidence for the claim "All ravens are black". But this is, to many, very counterintuitive, perhaps even unacceptably so; what should observations of cows have to do with claims about properties of ravens? Would anyone really try to establish "All ravens are black" by going out and finding a bunch of non-black non-ravens?

The paradox then leads one to re-examine one's beliefs on the nature of evidence, inductive argument, etc. Different resolutions avail themselves at that point, depending on what you find your beliefs to be, when put under such strict scrutiny and possible need for re-evaluation to avoid inconsistency.
Old 07-28-2008, 01:30 PM
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Let's look at a similar hypothesis, but a different universe. Suppose you have a school where every student but two is black, and every class has about 25 students. Someone asserts, ":Every student in class 1A is black." Now, you could check the hypothesis by going through the 25 students in class 1A, or you could check it by going through the two non-black students. So you go that second route: John Jones is a white student in class 1C, and Wendy Lee is a Chinese student in class 2B -- and checking the non-black students was a faster route to go that checking the students in class 1A. It was all a matter of relative numbers!
Old 07-28-2008, 01:47 PM
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Originally Posted by Sophistry and Illusion
So evidence for one should count as evidence for the other. But that seems not to be the case. So the question is, what gives?
My own opinion: Evidence for one is, automatically, evidence for the other. But not all evidence has equal weight—a very common phenomenon. If you're investigating a murder, some forms of evidence are stronger or more compelling than others. If you find evidence that Suspect A is innocent, you could consider it, indirectly, to be evidence that it's Suspect B who's guilty. But it's probably not as strong a piece of evidence as evidence that ties B directly to the crime (of which, some such pieces would be more conclusive than others).
Old 07-28-2008, 01:51 PM
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Another aspect that is counter-intuitive (not sure it rises to the level of paradox) is that one observation can support an infinite number of hypotheses.

I observe that that my hair is brown.

This is evidence that:
a) All Ravens are black.
b) All pizzas are blue.
c) All dogs are black.
d) Tom Cruise is white.
e) Spike Lee is white.
f) Cecil Adams is bright pink.
etc..

It can also support mutually exclusive hypotheses. For instance, it is evidence that:
a) All Ravens are black.
b) All Ravens are white.

How can one observation support two completely incompatible hypotheses?
Old 07-28-2008, 02:38 PM
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I think muttrox's point has the potential to reply to what Giles and Thudlow Boink are saying. Since "This apple is green" is equally evidence for "All ravens are black", "All ravens are white," "All ravens are transparent," etc., then it in fact is evidence for none of them, since it provides equal evidence for incompatible claims. It's like saying, "My analog clock reads 12:00" is evidence for both "It is day" and "It is night." But in fact, it is evidence for neither, precisely because it is equally compatible with both (incompatible) claims.

But even if Giles and Thudlow Boink are right, it's still odd. Why should a sentence provide a good piece of evidence for A, and a very weak piece of evidence for B, if A and B are logically equivalent?

Last edited by Sophistry and Illusion; 07-28-2008 at 02:39 PM. Reason: wrong logical connective--in a logic thread!
Old 07-28-2008, 02:39 PM
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This is sloppy and artificial. It's a forced error. The first premise chooses a non-unique identifier (the color black) which is bound to lead to an error in the contrpositive.

All ravens are black

is not an entirely inclusive/exclusive statement, it merely delimits ravens in one aspect. You might as well substitute the identifiers tall, happy, smart, etc. It means nothing. To be accurate, the first premise must be phrased:

All (read: Only) ravens are black

which would then allow for the contrapositive

Everything which is not a black is not a raven


I could be generous and allow that's the intent of the argument, but if so, it's a slouch of an example. A poorly worded hypothesis is worthless as a test of logic.
Old 07-28-2008, 02:43 PM
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Maybe we are approching this all wrong. Rather than looking at evidence as supporting a statement, let's look at counter evidence.

You make a statement that all ravens are black. Let's split up everything in the universe into two piles, ravens and non-black objects. Notice that already we have said that any black objects that are not ravens don't matter.

According to your statement, every raven in your pile should be black and every object in my pile is not a raven. You could prove your statement by showing me that everything in your pile is black and I can DISPROVE your statement by finding ONE raven in my pile.

Now support is a tricky concept. It is a mutually agreed-to criteria that when met, I will ASSUME your statement is true unless later evidence disproves your statement, specifically I see a non-black raven which is the counterexample I was looking for. It is true that brown hair does support the statement all onions are blue, but I doubt you will find many to accept that as meeting the criteria of agreement. This is not really a paradox since it is not self-contradictory; it is more of a fallacy.
Old 07-28-2008, 02:46 PM
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Quote:
Originally Posted by muttrox
It can also support mutually exclusive hypotheses. For instance, it is evidence that:
a) All Ravens are black.
b) All Ravens are white.

How can one observation support two completely incompatible hypotheses?
The propositions a and b are incompatible only if we know that there exists at least one raven. For example, I assert:
c) All unicorns are black.
d) All unicorns are white.

Now, I observe a brown animal in a field, which from a distance looks like it could be a unicorn. But then I go closer, and see that it is, in fact, a horse. So, it confirms both proposition c and proposition d!

Last edited by Giles; 07-28-2008 at 02:46 PM.
Old 07-28-2008, 02:48 PM
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Originally Posted by muttrox
Another aspect that is counter-intuitive (not sure it rises to the level of paradox) is that one observation can support an infinite number of hypotheses.

I observe that that my hair is brown.

This is evidence that:
a) All Ravens are black.
b) All pizzas are blue.
c) All dogs are black.
d) Tom Cruise is white.
e) Spike Lee is white.
f) Cecil Adams is bright pink.
etc..

It can also support mutually exclusive hypotheses. For instance, it is evidence that:
a) All Ravens are black.
b) All Ravens are white.

How can one observation support two completely incompatible hypotheses?
Suppose there are three suspects in the murder case: A, B, and C. If I can prove that A is innocent, or even find evidence making it more likely that A is innocent, that simultaneously supports the hypotheses "B is the murderer" and "C is the murderer."

ETA: The way it does this is that by ruling out one possibility, you in a sense make the remaining possibilities more likely.

When you say, "My hair is brown," you are telling us, "This brown stuff on my head is not a raven." And so you have ruled out the possibility that "The set of ravens contains this particular brown object."

Last edited by Thudlow Boink; 07-28-2008 at 02:51 PM.
Old 07-28-2008, 02:58 PM
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Let's forget about ravens for a minute.

The hypothesis is that all A's are B. The logical equivalent of this statment is that all not-B's are not-A.

So how do we go about providing evidence for the hypothesis? We could gather A's and examine them to see if they are B. If any A's are not-B, then the hypothesis is disproved, and every A that is B strengthens the hypothesis. We could also gather up not-B's and examine them to see if they are A. If any not-B turns out to be A, then we disprove the hypothesis, and every not-B that is not-A strengthens the hypothesis.

But the question then becomes, how MUCH does it strengthen the hypothesis? If we examine 1000 ravens, and find that all 1000 are black that is much stronger evidence for the statement that all ravens are black than if we examined 1000 non-black things and found zero ravens.

Since the number of non-black things in the universe is many many many orders of magnitude large than the number of ravens in the universe, you'd have to examine many many many orders of magnitude more non-black things without finding a raven to arrive at the same confidence in the statement, compared to examining ravens without finding non-black ones.

The statements all A's are B and all not-B's are not-A are logically equivalent. There's no paradox there. But if we want to efficiently allocate our research efforts, you're much better off looking for ravens and seeing if they are black than looking for non-black things and seeing if they aren't ravens. And this is only because we know there are trillions and trillions of times more non-black things in the world than there are ravens.
Old 07-28-2008, 04:56 PM
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Since "This apple is green" is equally evidence for "All ravens are black", "All ravens are white," "All ravens are transparent," etc., then it in fact is evidence for none of them, since it provides equal evidence for incompatible claims.
Not quite. The observation "This apple is green" is not evidence for the claim "all ravens are green". So it isn't evidence for every statement one can make about raven color.

One also has to be careful that one doesn't include anything about ravens in one's selection criteria for "non-black objects" to examine. If I'm truly looking for all non-black objects, and giving all of them equal weight, then there's still a chance that one of the non-black objects I observe will be a white bird standing in the middle of the street, which turns out to be an albino raven. As long as there is that chance, then every time that chance doesn't come up is a (very weak) piece of evidence. But if I refuse to examine anything but apples, then my observations of non-black non-raven apples don't mean anything.
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Old 07-28-2008, 04:57 PM
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Originally Posted by NaSultainne
This is sloppy and artificial. It's a forced error. The first premise chooses a non-unique identifier (the color black) which is bound to lead to an error in the contrpositive.

All ravens are black

is not an entirely inclusive/exclusive statement, it merely delimits ravens in one aspect. You might as well substitute the identifiers tall, happy, smart, etc. It means nothing. To be accurate, the first premise must be phrased:

All (read: Only) ravens are black

which would then allow for the contrapositive

Everything which is not a black is not a raven


I could be generous and allow that's the intent of the argument, but if so, it's a slouch of an example. A poorly worded hypothesis is worthless as a test of logic.
The contrapositive of "All ravens are black" is indeed "Everything which is not black is not a raven"; however, these are logically equivalent even without restricting the first statement by inserting "(and only)" before "ravens". Perhaps you are thinking of the converse, which would be "All black things are ravens"; it's true that "All ravens are black" does not imply "All black things are ravens", and that "All (and only) ravens are black" would imply it. However, this is entirely irrelevant to the raven paradox.

I don't think you've quite grasped the "paradox" correctly (or perhaps I've not quite grasped what you're saying correctly). The paradox is that "All Ts are Ps" and "All non-Ps are non-Ts" are logically equivalent; if one endorses the position that (observations of) instances of Xes which are Zs are evidence for "All Xs are Zs", and also that evidence is blind to distinctions between logically equivalent propositions, one is forced to conclude that instances of non-P non-Ts are themselves evidence for "All Ts are Ps", which seems counterintuitive.

Of course, there are a number of fine posts in this thread explaining why one might want to accept that counterintuitive result, and even why it's not that counterintuitive after all.

Last edited by Indistinguishable; 07-28-2008 at 05:00 PM.
Old 07-28-2008, 05:03 PM
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Quote:
Originally Posted by NaSultainne
This is sloppy and artificial. It's a forced error. The first premise chooses a non-unique identifier (the color black) which is bound to lead to an error in the contrpositive.

All ravens are black

is not an entirely inclusive/exclusive statement, it merely delimits ravens in one aspect. You might as well substitute the identifiers tall, happy, smart, etc. It means nothing. To be accurate, the first premise must be phrased:

All (read: Only) ravens are black

which would then allow for the contrapositive

Everything which is not a black is not a raven


I could be generous and allow that's the intent of the argument, but if so, it's a slouch of an example. A poorly worded hypothesis is worthless as a test of logic.
I believe this is incorrect. The statement "All ravens are black" can be broken down logically as follows:

if IS-RAVEN then IS-BLACK (1)

The contrapositive of (1) is (note: the symbol "¬" means "not"),

if ¬IS-BLACK then ¬IS-RAVEN (2)

or, in layman's terms, no non-black things are ravens. (1) and (2) are logically equivalent. Note that your statement, "Only ravens are black" is actually the converse of (1), logically

if IS-BLACK then IS-RAVEN (3)

The contrapositive of (3) is, as one would expect,

if ¬IS-RAVEN then ¬IS-BLACK (4)

which is logically equivalent to (3). However, the truth of (1) does not speak to the truth of (3) or (4). This is obvious from the statement, "only ravens are black." If that were the case, I could show you a white raven without disproving the statement that "only ravens are black." However, it would disprove the statement, "all non-black things are non-ravens."

Edit: Darn you, Indistinguishable, for making exactly the same point as me, but earlier. On the other hand, my post used fancy symbols like "¬".

Last edited by MilTan; 07-28-2008 at 05:04 PM.
Old 07-28-2008, 05:07 PM
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And bolding, too. I let no formatting slow me down...
Old 07-28-2008, 05:17 PM
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I drew up a detailed proof last night.

On my writing-desk.
Old 07-28-2008, 05:46 PM
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Originally Posted by Indistinguishable
(clip)I don't think you've quite grasped the "paradox" correctly (or perhaps I've not quite grasped what you're saying correctly). The paradox is that "All Ts are Ps" and "All non-Ps are non-Ts" are logically equivalent; if one endorses the position that (observations of) instances of Xes which are Zs are evidence for "All Xs are Zs", and also that evidence is blind to distinctions between logically equivalent propositions, one is forced to conclude that instances of non-P non-Ts are themselves evidence for "All Ts are Ps", which seems counterintuitive.
Ah, but that's the rub, isn't it? All choices are not logically coherent when put to the test. My position is that this is akin to the 'can God create a rock so big even He can't lift it' logic, which is to say, poor logic indeed. By choosing a simple statement as representative of 'All A are B', to wit: all ravens are black, there can be nothing but paradox because you and I and we all know that there exist many things which are black and not ravens, hence this is a contrived - I would argue, even nonsensical - statement intended to artificially create paradox. If I speak gibberish, it remains gibberish even when put into formal logic. Why, then, isn't the obvious answer to remove the issue of paradox through a corrected hypothesis?
Old 07-28-2008, 05:55 PM
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I drew up a detailed proof last night.

On my writing-desk.
But we shall see it....

nevermore.


NaSultainne, in what way is it gibberish to say that "all ravens are black"? IANAOrnithologist, but so far as I know, that's true.
Old 07-28-2008, 06:06 PM
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"All Ravens are black" is NOT the same statement as "All black things are ravens".

Different statments entirely. If all ravens are black you could still have plenty of black things that aren't ravens, and if all black things are ravens you could have plenty of ravens that aren't black.

And it is not true that all ravens are black: http://images.google.com/images?hl=e...no+raven&gbv=2
Old 07-28-2008, 06:14 PM
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Originally Posted by Lemur866
And it is not true that all ravens are black: http://images.google.com/images?hl=e...no+raven&gbv=2
I'm so glad that particular search doesn't return porn.
Old 07-28-2008, 06:35 PM
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Retroactively, I guess now I am too!
Old 07-28-2008, 06:38 PM
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A is not B. So not B is A.
Old 07-28-2008, 06:44 PM
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Originally Posted by Bryan Ekers
I drew up a detailed proof last night.

On my writing-desk.
Why?
Old 07-28-2008, 06:45 PM
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I'd argue that the paradox is solved by neither the existence of the apple nor the existence of any individual or many black ravens being evidence. The existence of 9 black ravens out of 10 total doesn't say anything about the colour of the tenth raven. A premise which requires an absolute can't be backed up by anything less than that absolute.
Old 07-28-2008, 06:49 PM
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Child asks:
What is a rose?

Parent says:
See that rock there, that is not a rose.
See that dog there, that is not a rose.
See that house there, that is not a rose.
See that cloud there, that is not a rose.
See that potato there, that is not a rose.
See that ice cream cone there, that is not a rose.
See that truck there, that is not a rose.
See that shoe there, that is not a rose.
See that girl there, that is not a rose.
See that shadow there, that is not a rose.
See that n-1 there, that is not a rose.
See that flower there, that is a rose.

Last edited by ouryL; 07-28-2008 at 06:51 PM.
Old 07-28-2008, 07:00 PM
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Originally Posted by Revenant Threshold
I'd argue that the paradox is solved by neither the existence of the apple nor the existence of any individual or many black ravens being evidence. The existence of 9 black ravens out of 10 total doesn't say anything about the colour of the tenth raven. A premise which requires an absolute can't be backed up by anything less than that absolute.
Well, the whole point of inductive inference is to be looser than deductive inference, allowing confirmation of universal hypotheses from such less than conclusive evidence. Science, and indeed daily reasoning, is thoroughly infused with this kind of inference. Every ice cube ever observed has melted when heated to sufficient temperature; is this no evidence at all that the same holds true of all ice cubes, including those not tested yet? [Sorry, couldn't think of a better example, but you get the idea]

Last edited by Indistinguishable; 07-28-2008 at 07:01 PM.
Old 07-28-2008, 07:01 PM
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Originally Posted by Lemur866
"All Ravens are black" is NOT the same statement as "All black things are ravens".
That's not actually the problem. As I understand it:

Premise: (1) All ravens are black.

No problem so far.

Corollary premise: (2) If something is not black, it is therefore not a raven.

Also clear enough.

Observation: I have a green apple.

Alleged Paradox: The apple is not black and is not a raven, which tends to support (2). But if (2) is equivalent to (1), my green apple must also tend to support (1). Therefore having a green apple proves all ravens are black.

The problem as I see it is that "non-contradiction" is being treated as equivalent to "support". Sure, the green apple does not contradict (2), but I don't see that non-contradiction as providing evidence of the correctness of (2). It would be like claiming "whatever does not disprove God, proves God".

Last edited by Bryan Ekers; 07-28-2008 at 07:03 PM. Reason: clarification
Old 07-28-2008, 07:02 PM
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Lemur properly understands the problem; he was pointing out someone else's confusion.

Also, nothing so glib as "Non-contradiction is evidence" is being appealed to (in any direct way). Rather, the intuitive plausibility of Nicod's condition is being used (that is, the idea that observations of Ps which are Qs should be evidence for "All Ps are Qs"), plus the idea that evidence for a claim is also evidence for all logically equivalent claims. From the Nicod condition, we get that non-black non-ravens are evidence for "All non-black things are non-ravens"; then, by logical equivalence, it becomes evidence for "All ravens are black" as well.

Last edited by Indistinguishable; 07-28-2008 at 07:05 PM.
Old 07-28-2008, 07:10 PM
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Quote:
Originally Posted by Indistinguishable
Well, the whole point of inductive inference is to be looser than deductive inference, allowing confirmation of universal hypotheses from such less than conclusive evidence. Science, and indeed daily reasoning, is thoroughly infused with this kind of inference. Every ice cube ever observed has melted when heated to sufficient temperature; is this no evidence at all that the same holds true of all ice cubes, including those not tested yet? [Sorry, couldn't think of a better example, but you get the idea]
Yes, I would argue that it is no evidence at all. The smallest possible point of data which would not only be negative evidence but disprove the premise is one ice cube that doesn't melt. No matter how many ice cubes you melt, unless you can do so for them all there remains the possibility that that smallest point of data exists. I'd agree that you could use all those observations as evidence for "ice cubes tend to melt", but they're no evidence for the premise that none do.
Old 07-28-2008, 07:57 PM
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Er, I'm not talking about negative evidence, I'm talking about positive evidence. Is there anything which, to you, would count as positive evidence for the claim that all ice cubes will melt when heated to sufficient temperature? If you say no, I'm going to say you're fixated on deductive, indefeasible inference. That is a very nice kind of reasoning, but it is not the only kind. Surely you are familiar with a different standard of reasoning, and even employ it regularly for many purposes, under which certain observations could serve as positive evidence for such a claim (or that electrical attraction follows an inverse square law or that monkeys do not speak English or what have you), even though the possibility of fallibility remains.

Last edited by Indistinguishable; 07-28-2008 at 08:02 PM.
Old 07-28-2008, 08:12 PM
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The problem as I see it is that "non-contradiction" is being treated as equivalent to "support". Sure, the green apple does not contradict (2), but I don't see that non-contradiction as providing evidence of the correctness of (2). It would be like claiming "whatever does not disprove God, proves God".
It is support, as long as you accept the premises that at least one raven exists and that there are a finite number of objects in the universe.

Look at the big picture. Suppose you were able to look at every single non-black object in the universe. And when you were done, you hadn't seen a raven. Logically, if you know ravens exist, you must conclude that ravens are black even though you haven't seen a single raven. So every non-black non-raven that you saw supported the theory that all ravens are black.

Or bring it down to a smaller scale. Let's say you go to shopping at a used car dealership. You know from their advertisments that they have at least one Toyota on the lot and when you get there you see that they're having a special sale on red cars and all of the cars inside their showroom are red. But you find that the showroom is closed and you can only look at the cars parked outside. You drive around and look at every car that's parked outside and don't see any Toyotas.

What color are the Toyotas they have for sale? You can correctly assume that they are red because you've seen all the non-red cars on the lot and they're all non-Toyotas.
Old 07-28-2008, 08:13 PM
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Why do you impose the conditions that at least one raven exists and that there are a finite number of objects in the universe?
Old 07-28-2008, 08:51 PM
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Originally Posted by Indistinguishable
Er, I'm not talking about negative evidence, I'm talking about positive evidence. Is there anything which, to you, would count as positive evidence for the claim that all ice cubes will melt when heated to sufficient temperature?
Yup, all ice cubes melting when heated to sufficient temperature.
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If you say no, I'm going to say you're fixated on deductive, indefeasible inference. That is a very nice kind of reasoning, but it is not the only kind. Surely you are familiar with a different standard of reasoning, and even employ it regularly for many purposes, under which certain observations could serve as positive evidence for such a claim (or that electrical attraction follows an inverse square law or that monkeys do not speak English or what have you), even though the possibility of fallibility remains.
Sure, but not when absolutes are used. I don't have any problem with such observations being used as evidence for claims that there are many cases where electrical attraction works in that way or where monkeys don't speak English. But were you to say that electrical attraction always works that way, or there are zero monkeys who speak English, I wouldn't consider specific examples of instances following those premises to be evidence.
Old 07-28-2008, 09:24 PM
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Alright. It would appear to me that you use the word "evidence" in an unusual way, then; would you say that most of the claims of physicists, chemists, etc., as they would naturally make them, are made without any evidence, and should be reformulated? And if so, what would you say is wrong with the notion of evidence they appear to be working with?

Last edited by Indistinguishable; 07-28-2008 at 09:28 PM.
Old 07-28-2008, 09:39 PM
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Why do you impose the conditions that at least one raven exists and that there are a finite number of objects in the universe?
I don't impose them. They're premises.
Old 07-28-2008, 09:40 PM
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What I mean is, why do you restrict your claim about the nature of evidence to the situations where those premises hold? Why do you not make the same claim more generally?

(To put it another way, where, in such reasoning as you may have in mind from those premises to the stated conclusion ("It is support"), do you see those premises being invoked?)

Last edited by Indistinguishable; 07-28-2008 at 09:44 PM.
Old 07-28-2008, 09:52 PM
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Another thing I might ask about, in line with the symmetry-breaking of Hempel's paradox: since "All ravens are black" and "All non-black things are non-ravens" are logically equivalent, why have a premise "At least one raven exists" but not a premise "At least one non-black thing exists"?
Old 07-28-2008, 10:24 PM
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Well, it's definitely the case that if no ravens existed, then "All ravens are black" would be trivially true. In fact, any statement of the logical form:

if IS-RAVEN then FOO

would be trivially true.

----

For those who want the logical reasoning:

if A then B

is defined as:

¬A or B

So if no object satisfies A, then the statement trivially holds.

Edit: As a homework assignment, it should be obvious from the logical definition of implication as ¬A or B why:

if A then B

is equivalent to

if ¬B then ¬A

Last edited by MilTan; 07-28-2008 at 10:27 PM.
Old 07-28-2008, 10:36 PM
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Originally Posted by Bryan Ekers
The problem as I see it is that "non-contradiction" is being treated as equivalent to "support". Sure, the green apple does not contradict (2), but I don't see that non-contradiction as providing evidence of the correctness of (2).
That depends on how the green apple was "selected," which I think is, or at least relates to, the important point that Chronos made in Post #17.

Imagine the world as a very large (finite or infinite) urn and all of the objects in the world as balls in the urn. So, we have an urn filled with balls of different colors and labeled according to what they are: there are black balls labeled "RAVEN," green balls labeled "APPLE," red balls labeled "APPLE," red balls labeled "STRAWBERRY," and so on and so on and so on. Then, the statement that "All ravens are black" would mean every ball labeled "RAVEN" is a black ball.

The only way to know for sure whether this is true would be either to (1) look at all the balls in the urn that were labeled "RAVEN" and make sure they were all black, or (2) look at all the non-black balls and make sure none of them were labeled "RAVEN." If you could only look at some of the balls—say, a randomly-selected statistical sample—you wouldn't have conclusive proof but you could have evidence. Pulling out a "RAVEN" ball and seeing that it was black would be evidence, and so would pulling out a green ball and seeing that it was labeled "APPLE" rather than "RAVEN."
Old 07-28-2008, 11:20 PM
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Originally Posted by MilTan
Well, it's definitely the case that if no ravens existed, then "All ravens are black" would be trivially true.
Aye, on the conventional formalization. Also, though, the same would hold if no non-black things existed.
Old 07-28-2008, 11:26 PM
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Originally Posted by Indistinguishable
Aye, on the conventional formalization. Also, though, the same would hold if no non-black things existed.
Right. Although people tend to have an easier time believing that "if A then B" is true if B is always true than they do believing that it's true if A is always false.

Last edited by MilTan; 07-28-2008 at 11:26 PM.
Old 07-28-2008, 11:27 PM
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Originally Posted by Thudlow Boink
The only way to know for sure whether this is true would be either to (1) look at all the balls in the urn that were labeled "RAVEN" and make sure they were all black, or (2) look at all the non-black balls and make sure none of them were labeled "RAVEN." If you could only look at some of the balls—say, a randomly-selected statistical sample—you wouldn't have conclusive proof but you could have evidence. Pulling out a "RAVEN" ball and seeing that it was black would be evidence, and so would pulling out a green ball and seeing that it was labeled "APPLE" rather than "RAVEN."
It's not enough merely to have seen all the balls labelled "RAVEN" and observed them to be black. One also needs to know that those were all the balls labelled "RAVEN'. Similarly with looking at all the non-black balls.

Also, there's nothing particularly special about your two ways; one could also look at all the balls which were both non-black and ravens, and make sure that there actually aren't any of them, or whatever (after all, "All Ts are Ps" and "All non-Ps are non-Ts" are also both equivalent to "All (T and non-P)s are non-existent"). In a sense, that's the minimal set to look at, the two sets you propose looking at being sufficient because they contain it.

Last edited by Indistinguishable; 07-28-2008 at 11:28 PM.
Old 07-28-2008, 11:33 PM
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Originally Posted by MilTan
Right. Although people tend to have an easier time believing that "if A then B" is true if B is always true than they do believing that it's true if A is always false.
Indeed. Though perhaps this is because much of ordinary-language implication isn't material implication; it's important to remember that it isn't an error to speak in a manner which works differently, just a situation that calls for a different analysis. [Of course, typically, this will require implication to be an intensional (i.e., non-truth-functional) connective]

Not to deny that people do screw up and hold inconsistent beliefs about the nature of implication or have difficulty bringing themselves to understand the conventional mathematical formalization or so on...

Last edited by Indistinguishable; 07-28-2008 at 11:36 PM.
Old 07-28-2008, 11:36 PM
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Originally Posted by Indistinguishable
It's not enough merely to have seen all the balls labelled "RAVEN" and observed them to be black. One also needs to know that those were all the balls labelled "RAVEN'.
That was what I meant by "look at all..."; but you're probably right to make it explicit.
Old 07-28-2008, 11:45 PM
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Originally Posted by Indistinguishable
Alright. It would appear to me that you use the word "evidence" in an unusual way, then; would you say that most of the claims of physicists, chemists, etc., as they would naturally make them, are made without any evidence, and should be reformulated? And if so, what would you say is wrong with the notion of evidence they appear to be working with?
I wouldn't say that they were wrong. Generally the claims made by various scientists in specific are in the form of particular observations, and in the general by theories which are supportable by the evidence they have. I'd say my idea with what evidence is really isn't all that different - if your premise involves A, or which B, C and D are subsets, that B might be observed to be accurate is not (in my eyes) evidence for A, only for B. But if you do have B, C and D, then you do have evidence for A. And even if you don't, you're still alright with B.
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