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?
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.
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).
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.
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?
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.
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.
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!
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).
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?
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?
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.
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.
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!
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.”
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.
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.
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.
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 “¬”.
And bolding, too. I let no formatting slow me down…