Apparently, Michael Egnor just can't get enough of making himself look like an idiot. His latest screed is an attack on me, for criticizing his dismissal of evolution as a tautology.
My observation that "Natural Selection" is a tautology, and therefore useless to modern medicine, seems to have set off quite a few Darwinists. Prominent Darwinist blogger Mark Chu-Carroll took me to task here, and comes up with an approach that he believes gets "Natural Selection" off the tautological hook: he asserts that all scientific theories are reducible to tautologies! Mark writes:
And this brings us to Egnor's idiocy. It's a common tactic among idiots to criticize various scientific theories as tautological... [but] you can derive a tautological statement from any scientific theory. The theory of gravity? If you let go of something, it will fall - therefore, if you let go of something, it will fall. Relativity? Light bends when it passed through a gravitational field - therefore, if I shine a light through a gravitational field, it will bend. Evolution? The things that survive to reproduce are the things that survive to reproduce.
Mark errs. A tautology is a statement that is true by its logical structure. 'A is A' is a tautology, and 'survivors survive' is a tautology. It's logically true, and it cannot be false. Scientific theories generally cannot be reduced to tautologies. Newton's law cannot be reduced to 'If you let go of something, it will fall-- therefore, if you let go of something, it will fall'. Newton's law of gravitation, in its most 'reduced' form, states that the gravitational force acting between two masses is proportional to the product of the masses and inversely proportional to the square of the distance between the centers of the masses, and that the constant of proportionality is the gravitational constant. Newton's law is not a tautology, and it can't be reduced to a tautology. It is not logically true. It's empirically true, but it could have been false. The force of gravity could have been proportional to the inverse cube, not the inverse square. Neither is Einstein's theory of relativity a tautology. The curvature of space-time is given by Einstein's tensor equations. They've been confirmed experimentally, and they're not the least bit tautological.
As usual, Egnor is being a typical IDist idiot - playing games with selective quoting, definition shifting, and general bullshitting.
Let's start with the game he's playing with the definition of tautology, and what I actually said. Here's the relevant quote from my original post:
3. "(A⇒B)∧A⇒B" - A implies B and A implies B. This is just a basic statement of one of the fundamental inference rules of logic. Once again, it doesn't matter what A means, or what B means; and it doesn't matter whether A or B are true or false. No matter what, by virtue of the structure of the statement, it must be true.
That third tautology is particularly important - because it's an example of a fundamental principle of logic. If you take any proof - any sequence of statements and valid inferences from those statements - and you combine all of the statements of the proof together, the resulting statement is, by definition, a tautology. The obvious implication of this is that you can take any statement which is provably true, and present it as a tautology.
Notice the important thing that Egnor chose to omit?
He quotes me as saying that any scientific theory can be reformulated into a tautology - in fact, any statement provable by inference can be reformulated into a tautology by simply folding the inferences into the statement. Any theory can
be stated as a tautology by folding the observations and inferences into the statement of the theory.
A tautology is a statement which is true by its logical structure. What's a proof of a statement in logic? It's a set of basic statements (axioms) which can, by using the inference rules of the logic derive the proven statement. Translate that down a bit and - what's a proof? It's a series of statements that demonstrates that a final statement is inevitably true by the logical structure of the proof. So form the proof up into a single statement (by joining steps with appropriate "and" and "or"s), and the proof is a tautology.
The same holds for scientific theories - except that in general, scientific theories add observations to the set of basic statements.
Egnor tries to use Newton's law of gravitation as a constrast against evolution: an example of a theory that cannot be reduced to a tautology. He does that by trying to throw a lot of terminology in to make the law of gravity look as impressive and complicated as possible, while simultaneously presenting the most silly reductionist statement of evolution that he can. But even so - let's try playing his game. What does Newton's law of gravitation say?
- When we observe the behavior of physical bodies, we observe that
all physical bodies appear to be attracted to one another. - If we carefully measure the masses of bodies, we observe that the force of attraction
between two bodies separated by a specific distance appears to be proportional
to the product of their masses. - If we carefully measure the distance between two bodies of known mass, we observe
that the force of attraction between them appears to diminish as the inverse square
of the distance between them. - Therefore, we conclude that two bodies of known mass m1 and m2, separated by a distance r will attracted towards each other with
a force proportional to (m1×m2/r2).
Strip away the excess verbiage, and that reduces to: we observe that two bodies are attracted by a force proportional to (m1×m2/r2), therefore, we we assume that for all pairs of bodies, the bodies will be attracted by a force proportional to (m1×m2/r2). Quite tautological when you structure the statement as a tautology, right?
Or to reduce it to the silly level of statement that Egnor insists on using for
evolution: if we let go of something, it will fall towards the ground; therefore, if
we let go of something, it will fall towards the ground.
That reduction that I just made - that silly, ridiculous reduction of gravity to "things fall, therefore things fall" - is almost exactly on the level of Egnor's reduction of evolution. It strips away nearly all of the relevant parts of the theory in order to reduce it to a true, accurate, by trite statement that's easily dismissed as a silly tautology.
Egnor's full-fledged argument is that the theory of evolution is irrelevant to the development of antibiotic resistance, because it reduces to "bacteria that don't die when exposed to antibiotics don't die when exposed to antibiotics." But that statement leaves
out a crucially important part of the phenomenon. Just like my reductionist statement
of gravity doesn't say anything about how fast things will fall or how
things are attracted to bodies other than the earth, or anything else that the
theory of gravity could tell us, Egnor's reductionist statement leaves out the crucial
explanation of why there are bacteria that don't die in the first place.
As I keep saying: we can, experimentally, take a single bacteria, and produce a clone-line culture from it. If we then expose it to antibiotics, some of the bacteria won't die. Dr. Egnor's tautology applies: the bacteria that don't die, don't die. But the original cell was not antibiotic resistant! So why are there any bacteria in the culture that don't die?
Egnor can't answer that. So he plays games to avoid the central issue - to avoid the question that he can't answer.
One more quick example of Egnor's idiocy, and I'll stop. Later in his post, he says:
A salient characteristic of a strong scientific theory is the combination of its logical improbability and its empirical verification.
"Logical improbability" is a meaningless term. Egnor is playing the "Mr. Spock" game of logic - trying to use the credibility of formal mathematical logic to give weight to something that has nothing to do with logic.
I've also never heard of a scientist who says that to be a good theory,
a theory must be improbable. In fact, I'm not even sure of what it means for a theory to be improbable in a formal sense, and I'm pretty sure that no real scientist discards theories because they don't seem improbable enough.
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I have a theory about the cause of Dr. Egnor's complete lack of apparent rationality, and it's something that you haven't quite said.
When you write a tautology like "A implies A", the less obvious A is, the less tautological the tautology sounds. So, when you say "organisms survive, therefore organisms survive", it sounds a lot more obvious than "F=Gm1Ãm2/r2, therefore F=Gm1Ãm2/r2", because of course organisms survive. The second one has this complicated equation that makes it look like it's really saying something, even though "organisms survive" is actually just as much an assumption that needs observational backing as "F=Gm1Ãm2/r2" is.
I think that is fundamentally why Dr. Egnor seems to think that physical law tautologies are so different from evolutionary tautologies. I'm not saying this is okay, but it may be the source of his confusion.
I thought that it was interesting that Newton was brought up as an example.
It happens to be that Newton's laws of motion - in particular, F=ma - are often said to be tautologies. The claim is that F (force) and m (mass) are defined so as to make that true. That force has a meaning only as it can be detected by the acceleration of mass. And mass can be measured only by measuring its acceleration when subject to a force.
I just mention this argument, not to argue in favor of it (or against it, for that matter), just as an interesting comment.
Not if you extend logic so that truth-values can be in the range [0, 1]. Then you get... BAYESIAN PROBABILITY!
(sung to the tune of Monty Python's "Spam")
Bayes Bayes Bayes Bayes
Bayes Bayes Bayes Bayes
Bayes Bayes Bayes Bayes...
Of course, Egnor gets the rest of it wrong. To be considered strong, an initially "logically improbable" scientific theory needs more extensive empirical verification. (Here, I'm thinking of the Occam's razor effect in the Bayesian analysis of models with different degrees of freedom.)
TomS: in other words, a definition is a tautology... which is pretty much what Mark is getting at.
I think when Egnor refers to a theory's "improbability", he really means it's "non-obvious-ness". Which you could formulate as a statistical probability, I suppose. How many physicists were there in 1903 trying to figure out what the Michaelson-Morley null result meant to the "obvious" assumption of the ether? How many abandoned the ether in favor of the constancy of the speed of light? Divide one by the other and there is your "probability" of discovering the Special Theory of Relativity :-)
TomS: Any equation is a tautology only if you know the values of all the components. When I was a child my father tried to teach me Ohm's Law. At that time I couldn't yet grasp the utility of the eqn when you know only two of the 3 quantities and the eqn lets you determine what the unknown must be.
This is the wrong approach IMO. The criticism ignores the fact that Egnor's description has removed the most important part of the theory of evolution via natural selection: heredity. Evolution doesn't happen simply because some bacteria die and some don't. Evolution only happens if dieing or not is inheritable, otherwise the next generation of bacteria will not have a gene pool that is different from the first.
Let me put it another way. Resistance is only a medical problem if it is inheritable, which is explained by the theory of evolution via natural selection. Egnor has yet to understand this.
Congratulations, this is high praise indeed! You should use the opportunity to pick up a FCD (Friends of Charles Darwin) suffix for yourself.
Mustafa Mond, FCD:
Do you have to be called a "Darwinist" by a Paleyist in order to pick up an FCD suffix? I'm not particularly desperate for more letters — having an OM for myself which I appreciate very much! — but extra noodles in the alphabet soup have a certain savory appeal. . . .
Ah, the FCD page finally loaded. Question answered, many thanks.
He also leaves out the fact that bacteria mutate, and some mutations have better antibiotic resistance. Not all mutations are harmful or neutral. Some incoporate information (correlation, i.e. mutual information between the bacterium's metabolism and the antibiotic's interaction with the metabolism) about the new environment. Whee! Where did that new information come from? Or did the Designer also create the entire pharmacopia, past, present, and future? And did the Designer create 10^1000000 possible bacteria, each of which is already present on Earth? Seems to me that Earth isn't big enough. Nor is the entire universe.
Perhaps I should have signed my name:
Jonathan Vos Post, FBIS.
That's Fellow of the British Interplanetary Society. Haven't thought of that in years. I still have the lovely BIS silk necktie, with little Hugo-award-like rockets on it.
Should there be a FCD necktie, maybe with Darwin's Finches?
To get this right, you need the other part:(sung to the tune of Monty Python's "Spam")
Bayes Bayes Bayes Bayes
Bayes Bayes Bayes Bayes shut up!
Bayes Bayes Bayes Bayes... SHUT UP! Bloody Vikings...
Keep up the fight Mark. He may learn something yet! (I doubt it, but he may.)
Scientific theories generally cannot be reduced to tautologies.
Egnor is trivially wrong of course. We can always claim that a model is a tautology. If we include the process of observations, it may or may not be true. If we have tested all possible predictions, a theory can again be interpreted as a tautology, at least in its domain of validity.
And we can easily compare evolution theory with Newton's gravitation theory on a quantifiable level. We just need to include the heretability of characters that variation and selection works on, which Egnor has omitted. Then we can observe frequencies of characters or alleles in populations.
A salient characteristic of a strong scientific theory is the combination of its logical improbability and its empirical verification. A strong scientific theory is not tautological but is true empirically. It is, logically speaking, highly improbable that Einstein's tensor equations would describe space-time accurately, but, empirically, they do.
This is meaningless at face value.
One generous interpretation is that Egnor is discussing the explanatory power of theories. That is an interesting discussion, but it is a general one.
Another generous interpretation is that Egnor is discussing how constraints picks a certain theory.
First, in that case he is pulling the rug beneath ID. Egnor himself seems to think evolutionary theory is logically improbable. So he should consider evolution strong, if it is empirically verified. Even if Egnor wont admit that, IDists like Behe admits that evolution works on some scales.
Second, it is still a questionable measure. For example, string theory seems to be a unique quantization for extended objects that preserves lorentz invariance. ( http://cosmicvariance.com/2007/03/31/string-theory-is-losing-the-public… ) At the same time, it admits a large number of different constructions and vacua. Not infinitely many as field theories, but still enough to make a brute search impractical.
Now, does the first sets of constraints, the quantization bottleneck, mean the theory was improbable (because it was unique) or that it was probable (because it was forced)? And what about the second sets of constraints, does that mean the theory is improbable (because the landscape has uniform probability) or probable (because it will be forced or environmentally choosen)?
I don't think anyone knows, and neither does Egnor. What I do think is apparent is that Egnor is alluding to the usual creationist mistake of confusing probability with outcome. Creationists have a limited set of ideas and key quotes, and Egnor has already tried them all.
Uups. I mangled the blockquotes:
Egnor is trivially wrong of course.
[...]
This is meaningless at face value.
[...]
And "probable" should be replaced with "obvious" in my second to last paragraph. Probably.
Over at Respectful Insolence, a commenter noticed that the line Egnor quotes is no longer in Wikipedia. To be precise, at 17:52 UTC today, a physics undergraduate from U. Waterloo deleted this paragraph:
"DrLeebot" said of this that "the analog is too abstract to be worth mentioning", with which I happen to agree.
Oops, I busted my HTML. Try this link instead.
Your demonstration that Newton's law of gravity can be reduced to a tautology is good. I also agree with TomS about the tautological nature of F=ma (certainly that was covered in my basic Physics books). I would also say that there is a flip side to this argument of tautologies in that natural selection can easily be restated to be distinctly not a tautology. My personal choice (although it might well be improved on) is to say that natural selection says: "Heritable characteristics, interacting with a local environment are a primary determinant as to how many offspring an organism leaves behind." This captures heritability and describes a feedback loop.
When a neurosurgeon is reduced to pulling references out of Wikipedia to bolster his appalingly stupid assertions; you realise that the DI has plumbed yet another new low. As usual this scietific pretender doesn't know where his off stump is. The design inference is the ultimate tautology. Design (the appearance of being designed) is the best explanation of complexity of nature because nature exhibits the signs of being desined (i.e., the appearance of being designed).
If it is so friggin' impossible in this Universe to do something so "complex" as changing the genome of a bacterium, how exactly, in *scientific* terms, does one explain the existence of an incomprehensibly complex and powerful Designer?
Maybe it's a garbled version of Popper's statement that good theories "forbid things to happen".
Just guessing.
This doesn't strike me as a tautology but rather a logical fallacy. Going from a single example (or even multiple examples) to a universal statement is not a valid logical inference.In science we do this all the time which is partly why we don't have proofs in science but rather evidence in support of a theory.
You misunderstand. F = ma (e.g.) is PHYSICAL relation of identity, not a logical one. You can't derive F = ma from the rules of language and logic alone. To derive F = ma you have to know physics: what IS F, what IS m, and what IS a. F = ma is not an analytic proposition. Survival of the fittest is an analytic proposition (claims Engor), specifically, a tautology, because survival and fittest are related circularly by DEFINITION. You don't have to know anything at all about biology to reason: if natural selection is survival of the fittest; and since fittest is those that survive, by definition; then natural selection is a tautology. All you have to do is manipulate logically the meaning of words. You can't derive a physical identity relation for F by manipulating logically the meaning of words! Do you understand now? Now, natural selection is indeed not a tautology, but not for the "reasons" you give. If the vulgar solecism "natural selection is survival of the fittest" were scientifically true then Engor would be correct. But it's not true. It's a vulgar solecism.
Jeff:
As I said in the original post: in science, we tend to take observations and treat them as "almost-axioms". So when we follow an inference trail, what we get at the end is a theory, not a theorem. The inference rules that we use for science are weaker than what we'd use in pure math.
For example, in standard mathematical logic, you can never reason from a set of selected examples to a universally quantified statement. In science, if we have many consistent observations, we allow a *tentative* inference of the universal. (tentative until/unless we can show otherwise)
So, for example, we know that Newton's law of gravity isn't actually true. It's damned close to true, accurate to within the range of measurement error for pretty much all observations at the time it was proposed, and for most observations today. So we accept it, and use the equation to figure out gravitational forces all the time, even though we can't prove that gravity does act exactly the same on all bodies.
Likewise, we now have Einstein's gravity equation from relativity. It's accurate to within the range of measurement error for every observation we're able to make today. So we accept the inference that it's universally true - and that inference from the evidence is the "proof" that makes relativity a theory, rather than just a hypothesis.
Torbjörn Larsson's comments here and at EvolutionBlog gave me an idea for a fun project to try.
I couldn't help noticing that Egnor's "Evolution News" site doesn't provide the reader an easy way to comment. I wanted to ask him directly what he meant by his logical improbability argument. It sounds like something from the Hitchhikers Guide to the Galaxy.
"F = ma" is not a tautology.
There are hidden assumptions in the domain and range of the variables.
I've published several papers, which themselves cite a wide and weird literature, on what one should expect to observe if, for instance, we had a particle for which m is an imaginary number (in the co-moving frame of the experiment).
In what direction is the gravitational force between that mass and the Earth? In what direction is the acceleration? I hypothesized that such a particle would leave the brane, perhaps in a Planck-time, and the experiment would observe what appears to be a violation of conservation of momentum.
I calculated the energy to produce a particle-antiparticle pair with masses i and - i times the mass of a proton, and it came out in the range of hypothesized Higgs bosons. So I'm very sad abou the LHC catastrophe. For all we know, LHC would also have seen substructure to electrons (contrary to QED), sparticles, any number of crazy things.
But my crazy idea can still be tested cosmologically -- the universe should cool faster soon after the big bang -- and it the neighborhood of supernova, GRBs, black holes in an observable way.
Jonathan Vos Post, Andrew Carmichael Post, and Christine Carmichael, "Imaginary Mass, Momentum, and Acceleration: Physical or Nonphysical?", Proc. CASOS, 2004.
Jonathan Vos Post, Professor Christine M. Carmichael, Andrew Carmichael Post, "Imaginary Mass, Force, Acceleration, And Momentum", Proceedings of the Fifth International Conference on Complex Systems, 2004.
Now, if "F = ma" is not a tautology, please tell me what happens if m is a quaternion?
Heh heh heh.
I wonder if Egnor instructs his patients to complete the entire course of antibiotics he prescribes, even if the patient feels better? As I understand it, the intent of that is to prevent or at least reduce the "breeding" of resistant bacteria. Doctors want you to complete the course of antibiotics because they hope any initial mutation will provide only partial resistance and they hope to kill off those mutated individuals before a further mutation will make the resistance too good. Leaving the partially resistant population to breed increases the likelihood of the further mutation.
As Reed Cartwright points out, heredity is a key to why Egnor is wrong. If his version of natural selection is right, taking the antibiotics away will only return the population to what it was before, at its earlier stage of pre-antibiotic fitness. *That* would be the result of his "tautological" natural selection. Once the environment changes back, the old fitness standards should apply in his simplistic version.
The reason we think past that is because of evolutionary theory and the recognition that the organisms have been changed by their environment.
I agree that pulling out the Wikipedia for your intense assault on evilution is a pretty junior high school move. Webster's dictionary defines "Moron" as "Any of several edible fungi having a conical cap with a highly pitted surface."
Oh, that's morel. Stupid dictionary.
Stop wasting time on these people and post more about maths.
They're clearly idiots and you're preaching to the choir. If anything you're making yourself look a bit silly by letting them wind you up so much.
If you say so, but I think you added some actual intelligence into the mix.
Btw, nice blog. But it is funny how ingroup terms looks from the outside. xkcd would personally not get me thinking "wobosphere". In fact, it makes me wanna see "wobots", which of course takes me to the Dredded future of a fascist state. (Walter the wobot will have problems with "w", poow guy.)
But I may get used to it. Just don't hit me with that baton meanwhile.
f(n) = f(n-1) + f(f-2) is a tautology.
So there!
Note that there has been work where the flux of substrates through a biochemical pathway was characterized under different conditions and used to explain differences in fitness (Actually, biochemical throughput was used to predict relative growth rates which in turn predicted relative fitness).
Essentially, researchers have tunneled down to the level of specific mutations and created a tight, causal explanation for how well strains perform in different environments. That work doesn't sound like a tautology to me...
"... flux of substrates through a biochemical pathway was characterized under different conditions and used to explain differences in fitness..."
That's at the heart of my PhD Dissertation research 1975-1977. I got me my first publications in Biology. My work in Mathematical Biology has led to many more publications, most highly technical, but some for general scientific multidisciplinary conference audiences. But no job offer ever came to me from this.
It's actually a very nice area, where pure Math, computer science, genetics, molecular biology, and other fields come together in unexpected ways. And someone alive today will be the Bill Gates of biomedicine, by commercially increasing human fitness by reverse engineering human metabolism and retrofitting it.
In fact, Bill Gates himself might be the Bill Gates of biomedicine, since he hired away Leroy Hood and his staff from Caltech... Google the name to see what I mean.
Me, I'd prefer Linux-based immortality to Windows immortality, but I'd settle for Apple immortality.
For a scientific theory to be stated in its tautological form, it's premises must be folded into the theory, yielding an IF...THEN proposition. So the tautological form of the theory of natural selection is something along the line of "If organisms exhibit heritable variation, then the frequency of a variation that enhances the reproduction of organisms bearing that particular variation (relative to other variations) will tend to increase over time."
A tautological form of the theory of gravity might be something along the lines of "If masses are attracted by a force that varies as the inverse square of distance from their centers of mass, then planetary orbits will be elliptical."
It is, of course, possible that the premises (the IF part) are wrong, in which case the theory remains logically true, but does not describe the real world. If gravity did not follow an inverse square law, then the tautological form of the Law of Gravity would still be a correct statement, but it would not describe planetary orbits in the real world.
If there were no heritable variability among organisms, then the tautological formulation of natural selection would still be a true statement, but it would not relate to anything in the real world.
"If gravity did not follow an inverse square law, then the tautological form of the Law of Gravity would still be a correct statement, but it would not describe planetary orbits in the real world."
This applies if the number of dimensions of Space is more than 3. My head has been in 4-D for a while lately, as I've mentioned on other threads.
So I'm thinking that maybe Egnor is from (cue theremin music) ANOTHER DIMENSION! Where he comes from, before he manifested a 3-D simulacrum at Stoneybrook, does NOT have Kepler's Law for planetary motion, and does NOT have evolution by natural selection!
Maybe he's a friend of Mister Mxyzptlk, the name used on Earth by a devilish being from another plane of reality, different from our own, which he calls the "Fifth Dimension."
As Wikipedia summarizes:
"... Originally, Mxyzptlk had designs on conquering the planet for himself, but soon settled for tormenting Superman whenever he got the opportunity...."
So maybe Egnor had designs on conquering Earth for himself, but soon settled for tormenting Darwinists and sciencebloggers whenever he got the opportunity?
Jonathan Vos Post: "In fact, Bill Gates himself might be the Bill Gates of biomedicine, since he hired away Leroy Hood and his staff from Caltech... Google the name to see what I mean."
I know of Dr. Hood. He's one of a class of ambitious mega-lab directors and true gear-head in the best sense of the word. Sometimes those things pan out. But in all cases they chew through buckets of money. Great places for post-docs in any case.
Leroy Hood is very likely to win aq Nobel Prize, any year now...
As to tautology:
arXiv:0704.1001 [ps, pdf, other] :
Title: Tautological relations in Hodge field theory
Authors: A. Losev, S. Shadrin, I. Shneiberg
Comments: 35 pages
We propose a Hodge field theory construction that captures algebraic properties of the reduction of Zwiebach invariants to Gromov-Witten invariants. It generalizes the Barannikov-Kontsevich construction to the case of higher genera correlators with gravitational descendants.
We prove the main theorem stating that algebraically defined Hodge field theory correlators satisfy all tautological relations. From this perspective the statement that Barannikov-Kontsevich construction provides a solution of the WDVV equation looks as the simplest particular case of our theorem. Also it generalizes the particular cases of other low-genera tautological relations proven in our earlier works; we replace the old technical proofs by a novel conceptual proof.
arXiv:0801.4024 [pdf]
Title: Set-based complexity and biological information
Authors: David J. Galas, Matti Nykter, Gregory W. Carter, Nathan D. Price, Ilya Shmulevich
Subjects: Information Theory (cs.IT); Computational Complexity (cs.CC); Quantitative Methods (q-bio.QM)
It is not obvious what fraction of all the potential information residing in the molecules and structures of living systems is significant or meaningful to the system. Sets of random sequences or identically repeated sequences, for example, would be expected to contribute little or no useful information to a cell. This issue of quantitation of information is important since the ebb and flow of biologically significant information is essential to our quantitative understanding of biological function and evolution. Motivated specifically by these problems of biological information, we propose here a class of measures to quantify the contextual nature of the information in sets of objects, based on Kolmogorov's intrinsic complexity. Such measures discount both random and redundant information and are inherent in that they do not require a defined state space to quantify the information. The maximization of this new measure, which can be formulated in terms of the universal information distance, appears to have several useful and interesting properties, some of which we illustrate with examples.