Then again, the scientific method is not set in stone. Scientists and philosophers both are still trying to understand just how to identify the best hypothesis or when to discard one. This is not as trivial as it sounds, and this difficulty is well illustrated by the Raven Paradox, which I want to talk about today.
The Raven Paradox was first discussed in the 1940s by the German philosopher Carl Gustav Hempel and it is therefore also known as Hempel’s paradox. Hempel was thinking about what type of evidence counts in favor of a hypothesis. As an example, he used the hypothesis “All ravens are black”. If you see a raven, and the raven is indeed black, then you’d say this counts as evidence in favor of the hypothesis. So far, so good.
Now, the hypothesis that all ravens are black can be expressed as a logical statement in the form “If something is a raven, then it is black.” This statement is then logically equivalent to saying “If something is not black, then it is not a raven.” But once you have reformulated the hypothesis this way, then anything not black that is not a raven counts in favor of your hypothesis. Say, you see a red bus, then that speaks for the hypothesis that ravens are black, because the bus is not black and it not a raven either. If you see a green apple, that’s even more evidence that ravens are black. Yellow post-its? Brown snails? White daisies? They’re all evidence that ravens are black!
To most of you this will sounds somewhat nuts, and that’s what’s paradoxical about it. The argument is logically entirely correct. And yet, it seems intuitively wrong. This is not how we actually go about collecting evidence for hypotheses. So what is going on? Do we maybe not understand how science works after all?
Hempel himself seems to have thought that our intuition is just wrong. But the more commonly accepted explanation is today that our intuition is right, at least in this case. This explanation has it that we think black ravens are better evidence for the hypothesis that ravens are black than non-black non-ravens because there are more non-black non-ravens than there are black ravens, and indeed we have seen a lot of non-black non-ravens in our lives already. So, if we see a green apple, that’s evidence, alright, but it’s not very interesting evidence. It’s not very surprising. It does not tell you much new.
This argument can be made more formal using Bayesian inference. Bayesian inference is a method to update your evaluation of the probability of a hypothesis if you get more information. And indeed, for the raven paradox the calculation seems to be showing that the non-black non-ravens *are evidence in favor of the hypothesis, but black ravens are better evidence. They help you gain more confidence in your hypothesis.
But. The argument from Bayesian inference expects you to know how many non-black non-ravens there are compared to ravens. You might estimate this to be a large number, but where do you get the evidence for that number from? And how have you evaluated it? What do you even mean by a non-black non-raven. Come to think of it, just how do you define “raven”? And what does it mean for something to be “black”? And so on. You can debate this endlessly, if you want.
But you know me, I don’t want to debate this endlessly, I just want to inspire you to think about this paradox for a moment and maybe confuse some other people with it.
The Raven Paradox demonstrates why our brains don't use logic or Bayesian inference to make such judgments. Our intuitive statistics module is more practical.
ReplyDeleteThe exception determines the rule.
ReplyDeleteThere's nothing wrong with the contrapositive statement, "If something is not black, it is not a raven." It's just not very specific. So it's not very useful.
ReplyDeleteIs this really a paradox? You're applying a narrow rule (whether a thing is a raven) to a very wide set of candidates (things that are black); so of course the number of non-ravens will outnumber the ravens. Formulate any hypothesis to be as broad and useless as this, and of course you'll get broad and useless results.
"You're applying a narrow rule (whether a thing is a raven) to a very wide set of candidates (things that are black)" is not entirely correct or complete.
DeleteThere are two categories: entities with the attribute 'black' and a category with entities with the attribute 'non-black'. Of the category black there is a sub-entity identified as 'raven'. There are no ravens in the category non-black. (Leave aside the question how to define a raven,how to define black, and whether one entity can have two color, black and non-black, or not).
If you randomly take one entity and check its color and it is not black, than it is not a raven. If it is black it might be a raven or it might not be a raven.
Is it not better to say: "With one statement you're applying a narrow rule (whether a thing is a raven) to a very wide set of candidates (things that are black, and with the other statement: to a wide set of candidates (things that are not black).
One can deal with this paradox by noting that observing a black raven is NOT evidence supporting the hypothesis that ALL ravens are black, but is merely a fact consistent with it, just as observing non-black non-ravens are more consistent facts. With respect to this hypothesis, these observations are on the same plane as "2+2=4", they are merely consistent. Now, surveying all ravens on earth (which is abundantly possible, though expensive) and finding them all black is evidence to support, just as surveying all non-black things and finding them to be non-ravens. The survey can be done many times, steadily increasing the support for the hypothesis. On that view, it is the intuition for applying the induction in the first place that is wrong.
ReplyDeleteHa, the raven paradox! I always like teaching this topic in my introductory course on philosophy of science. Some reflections:
ReplyDelete- To arrive at the Bayesian conclusion that observing non-black non-ravens does increase the evidential support for the hypothesis that "all ravens are black", no exact numbers are needed. Physicists should be able to deal with ">>", which suffices for the qualitative result.
- Like with the Monty Hall problem (where it is crucial to know how it is decided which door will be opened), the core issue of understanding the Bayesian result might not be in the formulas, but rather in the silent assumptions. Here, the implicit 'observation protocol' is crucial.
- In addition to the detailed Bayesian computation, you could consider the limiting case of inspecting all non-black objects and discovering that none of them are ravens. This may help to see that the evidence increases ever so slightly with each observation.
- Even without going to this limiting case, it helps to imagine that some large but far from exhaustive set of non-black objects will be sampled. If this bizarre collection happens to contain just one raven, the hypothesis is falsified. For some, this may help to see that in the opposite case, actually some partial confirmation has been accumulated.
- "Even a blind pig can sometimes find truffles, but it helps to know that they grow in oak forests." - David Ogilvy. Likewise, our intuitive judgment factors in that we normally _don't_ inspect random objects to find out about ravens. Instead of a uniform probability across possible observations, we sample more strategically.
- Finally, Hempel does make explicit a key assumption (Nicod's criterion), that is usually left implicit in the popular/teaser versions. The criterion says that logically equivalent sentences should be (dis)confirmed by the same observation objects.
Hi Sylvia,
DeleteThanks for the interesting comment, to which I have nothing to add other than that the Germans have an idiom to the same end as the quote you name by Ogilvy, which has it that "even a blind chick sometimes finds a grain".
Its not scientific if you don't test the null hypothesis.
DeleteIn this case the null hypothesis would be for example "not all ravens are black". So the scientific method would not require for you to reinforce the hypothesis by counting all the things that are not black and aren't ravens, but would to specifically look to all ravens and see the color they are.
What is more, scientific method would require you to formulate the question why are all ravens black, and look at the DNA and biology of the Raven to explain why.
Statistics is just a tool
Unfortunately, Bayesian analysis is becoming popular in physics, and not for the better!
ReplyDeleteThis may have been the first time you made a sensible comment.
DeleteAs I listened to this I though Bayes theorem, and you then mentioned that. It is the case that black ravens are more germane as empirical evidence and provide a better Bayesian update for the hypothesis that all ravens are black. The modus tolens, all nonblack objects are not ravens, is logically correct, but provides little information. So there is little utility it provides in building a Bayesian updated prior.
ReplyDeleteSo what might we do? Bayes' rule is subjective; there is no precise analytic approach to getting a prior. It is all subjective estimation. So one approach might be to look up all the nouns in the dictionary, narrow in on those that are not black, say classic bowling balls, tar, and so forth, and get a number. Then one might take a prior estimate for finding a black raven and divide by this number and subtract from one. This will be a number close to one, which will do little as a useful prior, for the posterior probability estimate will not be significantly different. This does little to inform us.
Yes, well of course it's possible to get frequency counts. Just do this as a Frequentist to establish initial priors, then put your Bayesian hat on :-)
DeleteLawrence Crowell wrote: "Bayes' rule is subjective; there is no precise analytic approach to getting a prior."
DeleteI recommend that you have a look at "Probability Theory" by Edwin T. Jaynes. He discusses analytic approaches to getting a prior a great length. (And many paradoxes!)
It is misleading to call Bayesian probabilities subjective -- anybody having the same information should arrive at the same conclusions. In most paradoxes relevant information is left unspecified.
Well, yes there are methods, but they are not exact. I will confess to knowing just the basics on probability and statistics. It is my least favorite area of mathematics to be honest. I learned just enough to get by :-O
DeleteThat ravens are black is not even a hypothesis in any sensible understanding of science. Even in middle school, the notion of a hypothesis as a proposed solution to a problem should make that obvious. That ravens are black is part of a description of ravens. The notion of birds being divided into species in the first place is a generalization from experience, tested by bird breeding, for one thing.*
ReplyDeleteIf you were so foolish as to insist "Ravens are black" is a hypothesis, what it means to falsify it requires saying NOT that non-ravens are non-black, but that this bird is raven, yet it is white. If a population of white ravens were to be discovered, then the description of ravens would change. I suppose you could insist the hypothesis ravens are uniquely defined by their blackness was refuted. But that was never a good "hypothesis" anyhow, as "ravens" as real biological entities are defined by morphology, life history, breeding practices, behavior. Mistaking the proposition for a genuine hypothesis in any valid scientific sense is mistaking consistent use of words as reason. You can't reason correctly without good grammar, aka formal logic, but good grammar, aka formal logic, is not necessarily correct reason.
*The notion that "science" is only a few hundred years old is nonsense. Science includes things like chick embryologist, which means Aristotle was also a scientist. The process where Aristotle's notion the brain was a cooling organ (apparently because he saw blood flow as primarily about carrying heat, and the brain consumed much of it,) was refuted and it was shown the brain was the seat of intellect was an early example of science.
Intuition aside, in saying “If something is not black, then it is not a raven” I don't see a logical connection to observe a green apple, red bus, etc.. as being evidence confirming ravens are black. The relationship of those observations to all ravens being black are innuendo, they are only direct evidence that non-black things can't be a raven, evidence for Ravens being black requires direct observation of ravens.
ReplyDeleteFor what is worth, I endorse your statement. Well, I like it anyway.
DeleteSuppose that we say "All aliens are little green men".
Does that mean that observing white or black tall men (not to mention red buses) must be taken as a proof of the original statement ? So, looking at Donald Trump must now taken as a (though indirect) proof that aliens actually exist ? OK, perhaps I should choose another example but still... you get the point.
No it doesn't. If you could find and observe all non-black things and not find a raven amongst them you could have proven the hypothesis without ever having seen a raven. To prove it by observing ravens you'd have to find all of them.
DeleteGmack:"Suppose that we say "All aliens are little green men".
DeleteDoes that mean that observing white or black tall men (not to mention red buses) must be taken as a proof of the original statement?"
Again, no. No single observation can do that. What it does is add evidence, and so increase the probability of the truth. It is the weight of all such observations that may ultimately provide proof.
"The argument from Bayesian inference expects you to know how many non-black non-ravens there are compared to ravens. "
ReplyDeleteNo, it doesn't. For the hypothesis "not all ravens are black," that proportion can itself be an uncertain value to be inferred, with an appropriate prior on it. You might use data from other bird species for this purpose. Or if you lack such data maybe you'll use a uniform Dirichlet prior for the proportion vector p (where p[i] is the unknown proportion of ravens having color i).
Dr. Hossenfelder
ReplyDeleteI am getting ready to break one of my cardinal rules, bringing up part of my past. However, in this case I believe it can provide a valid related view of things. I am a retired cop/investigator. Investigations are not research and not science. However, investigations still gather evidence that can be, and many times is, just as strong/valid as conclusions based on research/science. Now sometimes science is performed on evidence, but science was a secondary option in the investigation. When a person gets up on the witness stand and says I saw my spouse raise the gun and shoot me, that person was in a hospital with a gunshot wound and there was a history of spousal abuse, people (jurors) tend to reach a formal conclusion about the validity of the information/evidence. No science, no math and no formal probability analysis.
Evidence for conclusions does not always have to have a rigorous mathematical or research based background, science, philosophy or the scientific method in order to be valid.
Last things, very good topic Dr. Hossenfelder, opening up discussions for the exchange of information, not to push a belief on things, and to stimulate thinking and discussion is very important. And, many times I tell people I have two strikes against me in life; I am trained as a cop/investigator but educated as a physicist/mathematician (undergrad degrees). This means I like to dig into things and find facts and evidence for myself, which in turn means I ask a lot of questions that people are not always happy about.
We can never forget there is a big difference between direct and indirect evidence; the raven analogy blurs those lines. Having video footage of a perpetrator shooting someone is direct evidence of the crime. While eye witness accounts, fingerprints on the gun, are indirect evidence. It’s all useful in solving the crime however one can never have 100% certainty using only indirect evidence and that always needs to be taken into account. Witnesses can be wrong; fingerprints don’t conclusively prove who used the gun to shoot the victim, etc.
DeleteIn the raven analogy they go from directly observing ravens to observing non-ravens and don’t account for the fact the latter is indirect evidence (a much lower standard of proof).
The entanglement correlation is non-raven evidence for non-local(causal)ity.
ReplyDeletePink signal haven't seen - never. They are all black-dark non-observable.
Just to complicate things, there are such things as white ravens!
ReplyDeleteI think the real problem here is that traditional logic is very fragile. So for example, if suns orbiting round galactic centres don't follow the the laws of Newton/Einstein, do you throw those laws out, when it may well be that some form of dark matter exists and saves the day (or maybe not).
Without wishing to restart that debate, it is obvious that a wrong theory can usually be retained by making ad hoc hypotheses.
Stars orbiting galactic centres has nothing to do with logic. The fact that glass knocked off a table in Lima shattered on concrete floor, as one would expect, also has nothing to do with logic.
DeleteLawrence Crowell writes: "Bayes' rule is subjective; there is no precise analytic approach to getting a prior. It is all subjective estimation."
ReplyDeleteNot all. Jaynes got some useful results obtaining ignorance priors from symmetry considerations, and the Principle of Maximum Entropy is another way of creating a prior from certain sorts of information. Furthermore, there is a rigorous form of the Principle of Insufficient Reason that can be proven as a theorem rather than assumed as an axiom. It gives a mandatory objective prior in the finite case when all of your prior information can be expressed as constraints on what situations are possible.
We're a long ways away from being able to compute the objective posterior odds of two theories from first principles + observed data -- even if we knew how to construct the necessary objective priors, the computational problem would be daunting, and probably intractable -- but it is at least conceivable in principle.
I think the confusion starts with the supposed equivalence of the two statements:
ReplyDeleteIf something is a raven, then it is black.
and
“If something is not black, then it is not a raven.”
This is of course correct in first order logic. In reality, it is another question. In a model where we quantify over all kinds of birds, it is alright.
But in reality, we don't really know what we quantify over. all that exists? Well ...
Then, in my opinion, the equivalence is, at least not relevant for science.
Groan...
ReplyDeleteThis "paradox" looks like an example of bad reasoning. First, there is a failure to define the terms. What precisely is a "raven"? What precisely is "something"? After carefully defining the terms, it is blindingly obvious that the class of "raven" is very different from the class of "something" -- the class of "something" is obviously far, far larger than the class of "raven". For example, the first statement can be logically correct in an imaginary world where nothing else exists besides ravens and geese.) So the so-called paradox takes a statement about "raven" and then recasts it as an equivalent statement that invokes the new class -- "something" -- which didn't exist in the first statement. This is just restating a somewhat strong claim in a far weaker form. There is not a logical inconsistency though, because the second statement logically includes the first. But the two statements are not logically equivalent: implicitly, the second one additionally contains a whole lot of totally unrelated statements that didn't exist before.
It is really bad reasoning to claim that using observations on the new class of "something" gives "evidence" about the first statement without first restricting the class of "something" to the class of "ravens". If one agrees that ravens are forms of life that require an atmosphere with plenty of oxygen, then it is obvious that making observations of objects on the Moon cannot give evidence for or against the first statement, because those objects aren't going to be ravens unless someone has surreptitiously transported some dead ravens to the Moon...
I agree almost entirely, but your "somethings" can't just be "ravens" but a more general class that includes ravens, such as "birds" or "animals" and so on. Otherwise you're just looking at ravens and precludes a pink flamingo adding evidence for the proposition.
DeleteI think the commentator »Matt H« already asked the “right” question. »Is this really a paradox?« It is more like a childish thought game based on limited perception.
ReplyDeleteJust for the sake of it, if you “treat” mathematics, philosophy or rationality like a person, you can fool “math” and philosophy but you can’t fool rationality.
Math: See for example the »Banach–Tarski paradox« which is non existent in the real world (of objects). The “funny” thing is you have to be a mathematician to be a fool otherwise you won’t be able to construct this (“axiomatic”) nonsense.
Where does this purely mathematical problem-thinking came from?
A hint
Euclid was still looking for a plausible view of mathematical foundations and thus created an interdisciplinary connection that could be rated as »right or wrong«, later the question of »right or wrong« did not arise in modern mathematics. Euclid's definitions are explicit; they refer to non-mathematical objects of "pure intuition" such as points, lines and areas. "A point is what has no width. A line is length without width. A surface is what has only length and width." When David Hilbert (1862 - 1943) axiomatized geometry again in the 20th century, he only used implicit definitions. The objects of geometry were still called "points" and "lines", but they were simply elements of sets that were not further explored. Hilbert is said to have said that instead of points and straight lines, tables and chairs could be used at any time without the purely logical relationship between these objects being disturbed.
The question is to which extent axiomatically based abstractions are linked to real-physical objects. Mathematics is a tool and does not create physical reality, even if theoretical physicists tend to believe – especially within the framework of the standard models of cosmology and particle physics - that mathematics is more than applying a highly formalised language. This reminds me on Sabine Hossenfelder’s book (title) »Lost in math«.
I guess I was a bit hasty in commenting, since it looks like Sylvia's already includes my main points (in particular, Sylvia's 4th and 6th "reflections").
ReplyDelete> To arrive at the Bayesian conclusion that observing
ReplyDelete> non-black non-ravens does increase the evidential
> support for the hypothesis that "all ravens are black",
> no exact numbers are needed.
I think exact numbers is exactly what is needed to justify this position. There are mathematical issues with samples where every trial is a success (every raven sampled so far is black). The first signs of trouble show up when you try to make a confidence interval estimate for a proportion, given a random sample proportion of 1. The naive normal approximation wants to divide by zero, but even the Clopper Pearson method, which is arguably the most appropriate and robust way, leaves this case undefined, as far as I can tell.
What can be statistically justified is a statement of the kind "the proportion of black ravens is in excess of 0.999". In other words, sufficiently large samples of all black ravens will offer convincing evidence that the proportion of non-black ravens cannot exceed a number which depends on the confidence desired. Intuitively, this may seem like we are inching closer to the logical fact "all ravens are black", but this, I think, is where the delusion lies.
There's yet another sign of trouble, and this one is epistemological. What happens when we observe a single green raven? Can our hypothesis be instantly rejected, no further testing needed? Note that legitimate statistical analysis described in the above paragraph would not be affected: as long as we get more and more all black ravens after than, we can still drive the proportion estimate towards 1, and that limit 1 is just as unobtainable as before. But what are we to do with a hypothesis that can be "proven beyond any doubt" with a single observation? We have a very vibrant field where this happens all the time, and the field is called Math. A single valid proof instantly provides all evidence and support we will ever need for a formal statement. But the idea of a natural, physical hypothesis that can be proven or refuted by a single observation does not sit right with me.
All of this leads me to think that we should regard universally quantified statements as formal logical propositions almost completely detached from the reality they describe. Alternatively, we can think of them as an abuse of nomenclature: having the valid statistical meaning, but sounding weird.
And I say "almost detached", because of the tremendously important measurement problem that Sabine brought up recently. There must be a natural, physical connection, however tenuous, between the reality on one hand, and the very words we say, write, hear, and read in order to describe that reality, and this connection cannot be made clear before we fully describe what happens inside the brains. At the same time, in cases like this one, we tend not to believe that mere musings can somehow affect the actual blackness of ravens, so we presume statistical independence.
But what is a raven? It would seem that we need to know what a raven is. I believe that a raven is a bird, which is black and flies. I went out birding today and saw several all-black birds, and one partially black one, with three little bits of red.
ReplyDeleteThe statment does not say "all ravens are all black". Which of my black birds could be ravens? Now Baysian inference says that none were likely to be, because, if I remember correctly, there are supposed to be no ravens around here.
But several of the all-black birds did look a bit like pictures of ravens in the bird book. Others looked like vultures, but they too had a bit of red. What could the truly all-black birds have been? How do we define a raven?
Since I didn't capture any of the birds, I could not sample their DNA. But one more tidbit ... the bird namers are hesitant to demand DNA tests.
More to the point than ravens is Swainson's Thrushes versus Veerys. I saw a bird yesterday that is rare around here, but not unknown. It matches some pictures in the books of either a Veery or a Swainson's Thrush. All the pictures of Veerys match, but only some of the Swainson's Thrushs. Should I again apply Mr. Bayes's rule, noting that the Veery-like pictures were all made in California and I'm in Illinois?
Could the so-called Swainson's Thrushes in California actually be Veerys? I didn't find any DNA studies.
What about Penguins? All Penguins have black on them. Or do they? I have photos of what looked liked a penguin, acted like a penguin, was flirting (its was mating season) with penguins, and was on Antarctica like penguins. But it had no black on it. The light gray parts were light gray, but the supposedly black parts were a very pale buff color. Was it a penguin? How does one define "Penguin" (in this case, Adelie penguin.)
What about benzene molecules? Benzene molecules are supposed
to be six-fold symmetric. That gives them certain spectroscopic properties. One is that if you hit them with pulses of light at a certain frequency around 3 microns, they then fluoresce at that exact wavelength. Yet when I did the experiment I found that some of them fluoresced at 10 microns! I found that if I replaced a carbon 12 with a carbon 13, ALL of them fluoresced at 10 microns. If I replaced a hydrogen with a deuterium, they didn't even absorb the light! Were those molecules still benzene?
(The molecules started out at 0.1 K, isolated in a vacuum.)
Its arguments like this that made us in college say "Philosophy is bullshit".
I think that it is implicit unless stated otherwise that a raven is that which results when two other ravens love each other in a special way and their combined DNA subsequently drives the development of something which then emerges from an egg.
DeleteSomething that is black but not a raven is irrelevant to this paradox,as it adds no evidence either way.
If you consider something to be not black because of markings and it isn't a raven by the above definition, then it does add evidence.
If you have something that is not a raven and is black, then that supports the original form of the hypothesis "If it's a raven, then it's black". Logically that statement evaluates to "true" in such cases. Check the truth table.
Delete"false" implies "true" = True
I'm reminded of an example of tricky confirmatory evidence, which I think is due to David Foster Wallace but I can't find the reference. Your hypothesis is that all humans are less than eight feet tall. You head out the door, notebook in hand, and the first person you see is seven feet eleven inches tall. That's evidence in favor of your hypothesis -- but how do you feel about it now?
ReplyDelete"All Ravens are black." My reaction is, why would anyone make such a claim in the first place? One has not seen all ravens that currently exist, much less all those that existed in the past or will in the future. Given the incentive, one could trap a raven and paint it blue, or even insert modified genes for color into a raven embryo. If that doesn't count, I still think the statement is probably false since there is such a thing as albinoism.
ReplyDeleteIt only makes sense as a definition, that only black ravens are considered to be ravens.
Or perhaps it could be considered a scientific prediction, that the next raven you see will be black, but in science as I understand it there should be a mechanism or phsyical reason put forth as part of the hypothesis before it could be accepted other than provisionally, and maybe not even then. Black of course does have advantages in temperate regions, especially nocturnally.
Even in particle physics, there are outliers above five or six sigma. "Most ravens are black," would be a better hypothesis, and one for which non-black, non-ravens would not present any paradox. Don't ask for 100% certainty in an uncertain world. (That's a variation on another one of Mario's dictums: "V__, don't try to make sense out of a senseless world!"
Here's another one he told his wife soon after they got married: "Whenever you ask me something, I'm gonna tell you whatever I think you want to hear--"Yes, dear, that dress looks great on you!"--but since we're married you have the right to ask me, "Are you lying?", and then I have to tell you the truth--but don't ask unless you really, really want to know!"
We have not examined every possible physical system, yet we claim that any closed system conserves energy and that in every closed system, entropy only increases. This is the physics equivalent of black ravens.
Delete""All Ravens are black." My reaction is, why would anyone make such a claim in the first place? One has not seen all ravens that currently exist, much less all those that existed in the past or will in the future."
DeleteIt's not a claim, it's a proposition or hypothesis. The whole point is that it is made without definitive evidence and requires corroboration. By seeking out and observing all ravens, or all non black things, you can increase the probability that it is true towards 100% when you have completed that task, or false by finding a counterexample.
I tried to argue, apparently unsuccessfully, that hypotheses based purely on successive observations with no theory or mechanism behind them are only held provisionally, not absolutely, as far as I know; and sometimes not even with, as with General Relativity (thought to be incomplete requiring QM completion).
DeleteThat aside, I tend to disagree with both of your examples. General Relativity does not respect Conservation of Energy on the cosmological level (a big closed system), and entropy increasing is a statistical law (similar to "the House always wins") while on the microscopic level (a small closed system) classical physics is time-symmetric allowing for increases in entropy to be randomly reversed--as I understand it. E.g., a colder electron could transfer some of its energy to a hotter electron in a small fraction of collisions (those close to a right angle between trajectories with the hotter electron arriving at the collision region first).
I guess the main point is that short descriptions of one's point of view in Internet comments almost always leave something to quibble with.
Quick! Which kind of physics, classical or quantum, is inextricably dependent on the existence of intelligent observers?
ReplyDeleteIf you incorrectly answered “quantum”, don’t worry, lots of folks get this one wrong.
Here’s the problem. If in quantum mechanics you say you have two objects are exactly the same, the statement has a precise, quantifiable meaning: If you place the two objects together in a wave function, that wave function will show exhibit either fermionic or bosonic dynamics.
Let's try classical counting: How many fingers do you have?
Did you say “ten”?
Are you sure? After all, your fingers aren’t exactly the same, are they? What about those short ones with fewer joints? Are thumbs really fingers, or are they just similar to fingers?
My point is simple: In contrast to the mathematically precise definition of “sameness” in quantum physics, “sameness” in classical physics is always approximate. But if the universe does not provide a classical definition of sameness, who does?
You do.
This is deeply biological. To help you survive in hostile environments, your brain is built to seek “sameness rules” that help it estimate efficiently what could happen next. Knowing that the sun will rise again — knowing what a day is — increases your odds of surviving the next one. But the days themselves are never exactly the same, just similar.
The point is that assessing classical similarly is a complex process that requires an intelligent observer. To meet this challenge, the observer must include sensors, memory, and classification rules that enable it to decide if entities are "the same".
This even the simplest and most fundamental of all math concepts, counting "same" objects, exists only as a symbiotic relationship between a classical world full of never-exact similarities, and primate brains that act as the final arbiters of which similarities to embrace or ignore. Mathematics thus is no more isolated from the role of observers than is classical physics.
Classical decoherence and mathematical decoherence provide convenient labels for these relationships. Both phrases parallel quantum decoherence, the interpretation of quantum mechanics that insists that wave functions cannot be understood or even defined without looking at how they are entangled with their classical environments and observers.
Mathematical decoherence asserts that no matter how perfect a formal proof sounds on paper, its meaning is always entangled with a classical environment in which no two entities can ever be exactly the same. There are no Gödel’s theorems in a mathematically decoherent universe, since the recursive and timeless self-examination steps upon which they depend decohere before they can achieve their asserted perfection.
In classical decoherence the Raven’s Paradox just becomes an unusually inefficient bag of selection rules. For example, if the initial guess (heuristic) was "everything black is a raven", then every sighting of a not-black not-raven would indeed add a tiny bit more weight to the heuristic, but of course would never prove it, since black feathers are not a full definition of raven-ness. When viewed as a bag of heuristic rules, the Raven’s Paradox is not fundamentally different from the definition of finger-ness.
In short, the only things that exist in the universe are subatomic particles that seem to interact in certain ways.
DeleteAnything more than that is merely our brain's attempt to tell us that a whole bunch of these particles have clumped together in a way that it says is a "tree", a "table", a "raven" etc, but in reality has no relevance to the universe at all.
if a non-black raven is found, then the whole argument falls. On the other hand, the assertion would remain by maintaining that a non-black raven is not actually a raven since it doesn’t come under the fact that all ravens are black. And so a scientific hypothesis can be maintained through a self-justifying circular argument.
ReplyDeleteNo, I don't agree with that rebuttal. If its parents were ravens then it's a raven, and not being black doesn't change that fact. I think you're misunderstanding the intention of this by saying that "All ravens are black" is a definition rather than a proposition which requires corroboration.
DeleteI would like to share my view on Raven Paradox. I totally agree with Matt H. that means the Paradox itself involves two groups that refer to a property that applies to theoretically and practically incomparable scales (group scale). Therefore, the Paradox is nothing more than a misconception. However, when the so called Raven Paradox is used properly then it turns to be another version of the law of the excluded middle. A known Paradox as that of Zeno, when is formulated in terms of the Raven Paradox would be written in a general modern form as follow: "A spacetime with infinite small length and time properties implies no motion. This is the hypothesis.". So, an equivalent statement would be "Motion implies/presupposes a spacetime with finite small length and time properties". The law of the excluded middle is so powerful that might even reveal new physics. Unfortunately, QM Probabilistic interpretations hijacked and replaced objectivity/causality with a stochastic reality that has not even to do with the reality itself (or better objectivity).
ReplyDeleteLet's reconsider the Zeno independently from Raven Paradox as we saw above just by using the law of the excluded middle: "According to Zeno, infinite small space (length) and time in general leads to no motion, right? Then why math lead to a Paradox? What is the conclusion?" The conclusion is that Zeno Paradox is not a Paradox but a test proposition aiming to falsify our experience. The result of it is since motion is evident everywhere and in all scales of matter then, spacetime has finite small space and time properties, otherwise we wouldn't experience/measure motion. So, without even writing a mathematical expression, we just discovered new physics (spacetime should be quantised). I can imagine and share many of the misconceptions of modern and classical Physics just by making the "right questions" (isolated fundamental statements that do not lead to misconceptions or contradictions), meaning using just the law of the excluded middle. Another example might be the Casimir or Quantum Tunneling effect where with careful isolated fundamental statements, one may prove in one case the vacuum fluctuations has no role and on the other the probabilistic description is a consequence of the cause (observation further down the line) and not the cause itself.
While I love the discussion of logic paradoxes, I'm not sure ravens are a good example since logic generates: “If something is not black, then it is not a raven.”
ReplyDeleteBut does this statement hold? What about white ravens (actually more cream colored) of Vancouver. White ravens have leucism which results in only partial loss of this pigment and their eyes are typically blue. This is a link to pictures:
https://www.pinterest.com/carrie_d_miller/white-ravens/
It only works if blackness and "ravenness" are uncorrelated. Jaynes gave a nice counterexample in his book Probability Theory. Here's one of mine, a somewhat simpler one:
ReplyDeleteYou have a magic wand with the power to turn every black object in the world into white. You accidentally drop it, but you don't know for sure if the spell has been activated. You go to check the color of your black leather jacket. If it turned white, you know there are no black ravens left in the world. In other words, observing a non-black non-raven *weakened* the proposition that all ravens are black.
Well yes, but can't you do that with any argument? You put the bunny in the hat by using an extra premise, what Douglas Hofstadter called a "defeater" -- a statement not about ravens, but about the first premise itself:
Delete1. "All ravens are black"
2. If my BLJ is white, then "All ravens are black" is false (because magic wand).
3. My BLJ is white.
The "weakening" of premise 1 wasn't due to the kind of empirical evidence that Dr. B was talking about; it was based on reasoning about propositions and evidence about BLJ's that was only relevant because you made it so by premise 2.
I think it illustrates a more fundamental issue. We all make assumptions about reality. The basic assumption is that we all share the same objective reality. We also thinks heuristically. We automatically discard trivial information without a second thought. One of the great breakthroughs of science is the paradox. As we move towards a better model of reality, we uncover the hidden assumptions we made in drawing our conclusions.
ReplyDeleteMy opinion is, Bayesian or not, I think you need a decay curve for the significance of additional observations of the same thing.
ReplyDeleteConsider number theory. No matter how many times we observe that some unproven mathematical relationship holds true, even for billions or trillions of samples produced by computers, a single counter-example will disprove it as a general rule.
No number of samples can raise the certainty to 100%, a single counter-example is enough to drop the certainty to 0%.
Forget about defining a raven or making sure you don't include being black as part of the definition or other bonehead mistakes.
If they hypothesis is "all ravens are black", then no matter how many non-black non-ravens you have, one non-black raven will disprove the hypothesis.
Which means there was still doubt, no matter how many trillions of non-black non-ravens we found. There is a point of diminishing returns that applies.
The same applies to all of science and physics, the counter examples have to prompt us to close examination, accepting when we are wrong, and a revision of what we believe.
"no matter how many non-black non-ravens you have, one non-black raven will disprove the hypothesis."
DeleteThat amounts to tautology. If you have found literally all the non black things that exist and none of them are ravens then you have proved the proposition without seeing a single raven.
As far as the usefulness in science in something like the Raven Paradox; ie observing non-ravens that are not black: maybe we can consider Popper's falsification criterion. Hypotheses become stronger when you legitimately attempt to falsify them, and fail. It's impossible to falsify "All ravens are black" by observing non-ravens. If you observe non-black things you CAN falsify it (if one of them happens to be a raven), but what's the probability of a raven being in the set of non-black things, compared to a non-black raven being in the set of ravens? The relative probability of falsification determines the weight the evidence should have.
ReplyDeleteAlso, the implication in logic (usually written as "if...then..." or "implies") is much weaker than how we normally use those words in natural language (when the first is false, the second can be either true or false). I think most people intuitively read into it "if not the first...then not the second" while the logical symbol doesn't mean that.
"The relative probability of falsification determines the weight the evidence should have."
DeleteThe weight of the evidence is actually irrelevant for the purposes of this thought experiment. That observing a non black non raven adds any weight at all to the proposition that all ravens are black is the essence of the apparent paradox. You appear to agree that it does add evidence, however small you think it is. That is enough.
You are correct that observing non ravens is a flawed strategy. You need to observe either all ravens or all non black things to determine the veracity of the proposition.
If the set of non-black things and/or non-ravens are infinite, as they very well may be (besides all the permutations of how everything in the world can be arranged, numbers?) then can elements in those sets even be used to make statistical calculations?
DeleteAfter thinking about this question, I came up a with a resolution that worked for me. I looked up prior work and found that my own viewpoint was more or less spelled out by W.V.O. Quine around 1969. The search term is "natural kinds".
ReplyDeleteQuine said that the "innate flair we have for natural kinds" may be responsible for the "illogical" success of induction. Quine attributed this to our "psychology". More recent work suggests that all human societies categorize the natural world into similar sorts of what we call Linnaean hierarchies. So perhaps this is built into the biology of our brains.
A kind of "universal illogic", especially about the natural world, may explain why we can so easily communicate about our illogical results.
This has interesting implications for physics. Which was perhaps your point.
this is not a paradox for me because of simple counting argument.
ReplyDeletesuppose 900 red balls and 100 blue blocks in a jar, well mixed.
my hyp is all blocks are blue. the equivalent is all non blue things aren't blocks.
seems like paradox that looking at all those nonblue things gives you no information, because there are 10x as many, BUT.
suppose i want to support first form of hypothesis by saying: ok i'll find 10 blocks and see if they are all blue. in order to find 10 blocks by choosing blocks out of the jar at random, i'm going to have to choose on average, 100 blocks, 90 of which will be my nonblue nonblocks.
so in terms of the AMOUNT of work i have to do... either way, i'm going to have to look at 10x as many nonblue nonblocks. so the alternative form of the hyp is equivalent
Sorry, I totally misread the formula for Clopper Pearson. It is, in fact, up to the task in cases x=0 and x=n, yielding smaller and smaller intervals on the appropriate side as the sample size increases. So there's no mathematical hurdle here, only the need for clarity of the interpretation.
ReplyDeleteTesting the origin of Casimir Force using logical statements (an alternative Raven "Paradox"):
ReplyDelete(A OR B) implies C <-> C implies D
(NOT D) implies (NOT C) <-> NOT (A OR B)
A = Metals
B = Dielectrics
C = Charges
D = Casimir Force (regards its manifestation)
Do the above statements imply Vacuum Fluctuations? Either YES or NO, please justify your answer.
I consider this an 'induced paradox' caused by reasoning out of context.
ReplyDelete“All ravens are black”.
ReplyDeleteA biologist, a physicist, and a mathematician are on a train in Scotland. The biologist looks out the window, sees a black sheep, and says "sheep in Scotland are black". The physicist scoffs and says "some sheep in Scotland are black". The mathematician takes a sip of his tea and says "There is at least one sheep in Scotland, which is black on at least one side". :-)
A logically correct reasoning does not entail that it is relevant pragmatically, that is, in relation to the concrete conditions of an experience.
ReplyDeleteIn the present case, the experience is observation and consists to check the (hypothetical) systematic association of a certain property (black color) with a special object of study (ravens).
From there, you have these two practical possibilities:
1 - you can restrict your observations only to the object of study (or exclude the rest of the universe), checking whether many of these objects exhibit such a property. This will give you a relevant and reliable information, even if you can’t check all the existing ravens. If we assume that you know what a raven is and what the color black is (but don’t know if they are all black), the question about the definition of the categories «ravens» and «black» seems to be of weak relevance. Now, what you would have to do anyway is to diversify your observations in several regions on earth, to avoid an abusive generalization (what the hypothesis may turn out to be) based on local observations.
2 - you can extend your observations to all the creatures populating the universe, to all non-black non-ravens entities, and eventually deduce from these observations that ravens must all be black. But where do you start and where do you stop your observations? Whatever the way you take it, this is a mess, and as Sabine put it, you can debate endlessly about it.
But you also may think, well, it could be wiser (much less time-consuming) to select a class of objects, for example non-black birds, since at the end of the day you are not really interested by non-birds. This gives you very little, very weakly relevant information. But still, there are many species of non-black birds, and then you will eventually end up thinking that, well….you should better check the existence of non-black ravens. This would be much more simple. And indeed, from a pragmatic point of view, the relevant equivalence if you want to check whether all ravens are black, is to check the existence of non-black ravens.
The conclusion for scientists : don’t drop too easily your object of study for logic, pure logical reasoning (and more broadly intellectualism) does not necessarily lead you somewhere. Just define as intelligently as possible the conditions of your experience. No mystery here, all scientists already know that.
I didn’t know it before, and what strikes me is that It reproduces in an indirect and metaphorical way the functioning of sense-perception, for which It turns out that subtraction (or selection) is a much more efficient operation than addition.
If you allow me to drift a little bit, this is probably what the meaning of the measurement problem is about. That is, the conditions through which something (the environment) can be perceived (or felt). This is an anthropic question, if you wish: we always perceive much less, this is the main condition for perceiving something rather than nothing, and that’s why you need several modes of sense-perception.
Bear with me here, because I think I have a way to represent this, but haven't put it into the language of logic yet.
ReplyDelete1) Draw a circle and call it the set of "All Ravens".
2) Then, anything outside that circle is the set of "All Non-Ravens".
3) Draw a larger circle around "All Ravens" and call it the set of "All Black Things".
4) Then, everything outside this larger circle is the set of "All Non-Black Things".
"All ravens are black" tells us that "1)" is a subset of "3".
"All non-black things are non-ravens" tells us that "4)" is a subset of "2)".
This latter statement refers to sets that are for all purposes unbounded.
Sabine said,
ReplyDeleteNow, the hypothesis that all ravens are black can be expressed as a logical statement in the form “If something is a raven, then it is black.” This statement is then logically equivalent to saying “If something is not black, then it is not a raven.”
I beg to differ. To me, the statement “If something is a raven, then it is black” is equivalent to the statement "“If something is a black raven, then it is raven”, which is not logically equivalent to saying “If something is not black, then it is not a raven.” And therefore, not anything not black that is not a raven should count in favor of your hypothesis.
"Logically equivalent" means that whatever the truth values for "It's a raven" (true or false) and "It's black" (true or false), you'll get the same truth value for the two logically equivalent statements "If it's a raven then it's black" and "If it's not black then it's not a raven".
Delete@RV
@ I aM wh. Yes, I am aware of the fact that two negatives resolve to a positive, but I still beg to differ.
Delete- If something is a raven, then it is black (true even for our raven, which was painted in blue before the statement was made, because ravens are naturally black and the current color of our raven is secondary to the absolute truth of the statement. Otherwise, our blue raven would have ceased to be a raven when we painted it, which is not the case).
- If something is not black, then it is not a raven (false for our raven, because it's a blue animal, which happens to be a raven).
Of course, you could perfect both statements by adding "naturally” before “black”, but surely one could still find some gap in the logical equivalence of the two statements.
We start with the statement "All ravens are yellow". Then observations of the red bus, the green apple, the white daisies are all evidence in support of the statement. This is clearly wrong. As with induction, we must start with one observation of one yellow raven, which we can't do. So, for me, the correct posing of the paradox must be preceded by "One raven is black", then we can propose "All ravens are black". This also fixes statements such as "All aliens are little green men", as we have to begin by producing one green alien.
ReplyDeleteMeta-question(s) for some generous expert: If predicate logic is an axiomatic system like arithmetic or geometry, why is a such a (seemingly) simple proposition generating so much disagreement - like 50 or so different opinions in these comments? Is the argument just about the meaning of words? Without pretending that I follow all of the discussion, I'd have expected this issue to have been definitively resolved a century ago. Is this a real paradox (like Russell's) or, as Sean Carroll puts it, just a feature of the world that we're surprised about (the Twins paradox)?
ReplyDeleteYour question is answered in original post (not a true paradox). I don't think anyone has disagreed with that, at least I hope not, but people are giving various opinions about whether or how the issue applies to science, because people like to give opinions on the Internet, or at least many of us do.
DeleteWe should pay for that privilege via the Donate button, of course.
If this was less respectful blog I would simply say: +1
ReplyDeleteSchools need to re-intruduce logic classes, I'm thinking 3-4th grade. While it is true logic on itself isn't really required in everyday's life it is hidden foundation for common sense and that is going the way of the dodo.
Hi! I would like to propose a solution to The raven Paradox.
ReplyDeleteInitially, Green Apple and Red Bus do appear as pieces of evidence in support of "ALL Ravens are black" hypothesis and that applying Bayesian Inference to such objects would incrementally provide ever-increasing support of the hypothesis. What we must consider, however, is that Green Apple is an equally valid quanta of supporting evidence for the mutually exclusive hypotheses of "ALL ravens are BLACK" and "ALL ravens are WHITE", meaning that whatever support adds up, consequently subtracts itself in the end rendering itself NULL. This applies to all non-raven objects.
What must be taken into the consideration is that if we indeed had every "non-black non-raven" object in our possession, we would indeed be able to confirm the hypothesis of "ALL ravens are black". So that's an interesting juxtaposition here. IF X amount of "non-black non-ravens" EQUALS any number other than "ALL" the evidence is NULL.
However IF X amount of "non-black non-ravens" EQUALS ALL, then evidence is TRUE.
Equivalency statements:
If "ALL ravens are black" then "ALL non-black are non-ravens". A = B, therefore A and B are the same.
Therefore IF X amount of "non-black non-ravens" EQUALS NOT ALL is NULL, then X amount of "black ravens" EQUALS NOT ALL is NULL also.
As well as the opposite is true:
IF X amount of "non-black non-ravens" EQUALS ALL is TRUE, then X amount of "black ravens" EQUALS ALL is TRUE also.
TRUE and NULL being designations for validity of evidence as demonstrated above.
What this means is that using Bayesian Inference (counting ravens) is NULL evidence, unless the amount of ravens sampled = ALL, in which case it is TRUE evidence.
And of course as outlined in the firs paragraph the same applies to all "non-black non-raven" objects. If the amount X of "non-black non-raven" objects = ALL then the evidence is TRUE. If it's anything less then the evidence is NULL.
Hi again!
ReplyDeleteWith your permission Sabine, I would like to restate the original propositions with regards to the solution of The Raven Paradox in clearer, more concise terms, as well as to provide further evidence in support of the original propositions. Due to the character count limit I will break it up into 2 parts.
Part 1/2
The solution to the Raven Paradox relies on the hypotheses of "(all color) Ravens are accepted as "TRUE" evidence only when X (all color) ravens = ALL" as well as "Non-black (objects) are accepted as "TRUE" evidence only when X non-black (objects) = ALL", which we will see expressed below as "X (all color) ravens = NULL, unless X = ALL then TRUE" and "X non-black (objects) = NULL, unless X = ALL then TRUE" respectively; where "NULL" represents evidence of "0 weighted value" and "TRUE" represents evidence of "1 weighted value".
Please notice, that as compared to the original propositions, there is a difference in what is considered "TRUE" evidence on the "ALL non-black are non-ravens" side of things as well "ALL ravens are black" side of things. The difference is that we had to drop the term "non-ravens" from the evidence qualifier statement for "ALL non-black are non-raven" side and replace it with "(objects)", to sort of fill the void, make it more palatable for the mind; consequently going back and applying an equivalent change on the "ALL ravens are black" side to keep everything consistent and equivalent. This is due to the fact that when we utilize Bayesian inference to collect evidence in support of "ALL ravens are black", we look at ALL the (all color) ravens that we encounter, because we can't ignore the non-black ravens in the process of evidence collection if we were to come across them. Likewise, when we utilize Bayesian inference to collect evidence in support of "ALL non-black are non-ravens" we look at ALL the non-black (objects) that we encounter, because we can't ignore the non-black ravens in the process of evidence collection if we were to come across them.
End Part 1/2
Part 2/2
ReplyDeleteStarting from the "ALL non-black are not ravens" side of things, a RED BUS, i.e. a member of "non-black (object)" group, initially appears as quanta of evidence in support of both hypotheses. It is however rendered as "NULL" evidence after seeing that it equally applies as quanta of evidence to the mutually exclusive hypotheses of "ALL ravens are black" and "ALL ravens are white". Which is an interesting juxtaposition in light of the fact that if we indeed had ALL "non-black (objects)" in our possession and none of them turned out to be ravens, we would have full support for both hypotheses. Which we express as "X non-black (objects) = NULL, unless X = ALL then TRUE".
Moving onto the "ALL ravens are black" side of things, a BLACK RAVEN, i.e. a member of "(all color) raven" group, initially appears as quanta of evidence in support of both hypotheses. It is however rendered as "NULL" evidence after seeing that it equally applies as quanta of evidence to the mutually exclusive hypotheses of "ALL non-black are non-ravens" and "ALL non-white are ravens". Which is again an interesting juxtaposition in light of the fact that if we indeed had ALL "(all color) ravens" in our possession and none of them turned out to be non-black, we would have full support for both hypotheses. Which we express as "X (all color) ravens = NULL, unless X = ALL then TRUE".
Please note that second set of mutually exclusive hypotheses may appear less straight forward than the first set. This is because our minds are not accustomed to thinking in terms of "non-something", but upon closer examination it becomes clear that both hypotheses cannot be true at the same time.
What the above propositions go to show is that even if we avoid dealing with "non-something" mutually exclusive hypotheses, we can still arrive at the same conclusions by treating "ALL ravens are black" = "ALL non-ravens are non-black" hypotheses as 2 sides of the same equation, such as A = B, where A and B are the same. Such that we consequently derive: IF "X non-black (objects) = NULL, unless X = ALL then TRUE" then "X (all color) ravens = NULL, unless X = ALL then TRUE" (where "NULL" represents evidence of "0 weighted value" and "TRUE" represents evidence of "1 weighted value").
Thank you.
End Part 2/2