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I have seen some creationists arguing that when evolutionary biologists use homologies (anatomical or genetic) as evidence for evolution, they are committing the fallacy of affirming the consequent. So both evolutionists and creationists would agree that if evolution is/were true, we would expect homologies. We do see homologies, therefore evolution is true. Now, of course this is an example of the fallacy of affirming the consequent. A way that i see of solving this problem is to use Bayes' theorem. So if we have two mutually exclusive and jointly exhaustive hypothesis, with equal prior probabilities, say, evolution and creation ("E" and "C"), and a observation, say, homologies (H) if one of the hypothesis entails the observation, that hypothesis is more likely given the observation, than the hypothesis which does not entail the observation. More formally: 1) P(E or C)=1

2) p(E)=P(C)

3) (E->H)->P(H|E)=1 (since E implies H), (C-/->H)->P(H|C)=x;0

4) P(E|H)=p(H|E)P(E)/P(H), P(C|H)=P(H|C)P(C)/P(H) (Bayes Theorem)

5) Since P(H|E)=1, P(E|H)=P(E)/P(H), and since P(H|C) is a fraction, and P(H|C) is being multiplied by P(C), and for any numbers x and y, where y is a fraction, x multiplied by y is less than x, it follows that P(H|E)P(E)>P(H|C)P(C).

Hence our conclusion: P(E|H)>P(C|H)

So my questions are: 1. Is this proof correct? 2. Does this solve the problem of affirming the consequent, not only in the case of evolution, but also when dealing with other theories/hypothesis?

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What a fun question to answer. Thanks. –  AndrewC Aug 20 at 3:18

5 Answers 5

Affirming the consequent

Scientists distinguish between the merit of explanations on the basis of (a) how accurately and (b) how widely they make experimentally-verified predictions.

This means empiricism is fundamentally based on affirming the consequent (and uses inductive reasoning), so you could argue empiricism is rather weak logically. However, you should note that having used A => B and B, science does not assert A is true, it labels A a good model for B.

Sadly, the media and popular opinion tend to conclude "scientists discovered that A is true". This common fallacy ("science is all true") leads to disillusionment when more data suggests a different conclusion.


Science is fundamentally not about truth, it is fundamentally about degree of statistical agreement.

This is why Newton is held as an eminent scientist, despite his theory of mechanics being demonstrably incorrect. Science doesn't care about correctness, it cares about degree of accuracy of prediction, and Newton was very accurate.

Reasoning statistically

Yes your derivation is correct. I would have presented (5) as:

P(H|E)=1, so by Bayes' theorem P(E|H) = P(H|E)P(E)/P(H), we have
P(E|H) = P(E)/P(H), but similarly, P(H|C)=x<1, so
P(C|H) = x P(C)/P(H), but P(C)=P(E) by prior assumption, so
P(C|H) = x P(E)/P(H) < P(E)/P(H) = P(E|H). Thus we have shown
P(C|H) < P(E|H).

but yours has the merit of being (a) briefer and (b) more explanatory.

Models that are validated empirically

This argument from Bayes' theorem explains why simple explanations making firm predictions are validated over ones which make more equivocal predictions, given similar accuracy. More generally, empiricism's statistical approach favours theories which readily make (preferably numerical) specific and universal deterministic predictions from measurable inputs over theories which make non-deterministic predictions or where it is complicated to find what the theory predicts.

This is absolutely the right approach to take in a scientific paper, since peer-reviewed empiricism is the validation tool for science - not logic, not "the truth", but statistical predictive accuracy.

That's why "children prefer to leave their toys messy than tidy" isn't part of any major scientific model, whereas "the Higgs boson exists" is. The empirical evidence for the toys is overwhelming but hard to quantify and only nearly universal, whereas the evidence for the Higgs Boson is very localised and of comparatively incredibly rare frequency, but easy to analyse numerically.

Thus another too-common belief "all truth is science" is also fallacy.
It is one which, amusingly, would tend to be confirmed by empirical study (because non-numerical, multiple-option and generally hard-to-statistically-analyse truths already evaded being part of the scientific model)!

Can we use this method to decide between theories?

You have a lovely statistical argument in favour of a scientific theory (evolution) over a competing theory (young earth creationism). However, as always with science and statistics, you should be cautious about overgeneralising or misstating your conclusions.

It should come as no surprise that a statistical analysis favours a deterministic theory, as we just proved using no assumption other than E->H and C-/->H that P(E|H)>P(C|H). Don't allow your belief in evolution to encourage you to accept this kind of reasoning as a proof, since you'd be making the fallacy of affirming the consequent, about affirming the consequent!

To dissuade you, I'll give an example the other way around. I could compare evidence E of non-zero credit card bill with theory G: "the nasty green goblins will always cause the numbers to be non-zero on my credit card bill" and theory C:"I prefer to put purchases made at non-small businesses on my credit card to defer payment with "cashback" deducted, maintain a higher current account balance generally, and put the resulting spare money in an interest-earning savings account". Every month, using Bayes' theorem, P(E|C)<1 (albeit slightly) and P(move money into savings)<1 (significantly less slightly!), but P(E|G)=1, so I find that after a year or two, P(G|EEEEEEEEEEEE)>P(C|EEEEEEEEEEEE) by a wide enough margin to be statistically significant, so to behave empirically, I should accept the goblins model.

What can we conclude?

Don't let the green goblins near your credit card.

No, no, that's not it!

To reason this way has logical gaps (as you spotted, plus general inductive reasoning including assumptions about the immutability and uniformity of the universe), favours certain kinds of statements over others (Higgs vs toys), which can lead to misdiagnosis of the truth (credit card).

However, it's perfectly valid to use empirical statistical reasoning to decide between two scientific models. This way you deliberately favour the most statistically predictive theory, but remember, that means you have a good model, not necessarily the truth. (Your school education should have exposed you to a good few models that were the best explanation we had but were later surpassed. Be particularly sceptical about "most fundamental particle"!)

It's against the spirit of empiricism to go around believing in your models, in particular because you're less likely to come up with a new model if you do. However, we're all human, and believing things we like the sound of incredibly readily is what we do.

How to invalidate your Bayesian argument

We've touched on how testing the consequent is intrinsically flawed logically (which doesn't make it bad science), so that we're never testing truth, we're testing accuracy, and at some examples where those are at variance. Now let's look at some popular other ways to logically invalidate your conclusions.

The numerical Bayesian conclusion after one application depends heavily on the assumed probabilities for your competing theories. This is why Bayesian statisticians like to refine their model time and time again, incorporating new evidence (whether inconsistent with previous data or not) into the calculation. You can come to radically erroneous conclusions by making radically erroneous assumptions at the start.

Even slight or subtle errors can flip your conclusions. In real life, an erroneous initial assumption that a black box would be transmitting normally wasted lots of time and money finding a plane crash, because whilst later searches had been thorough, the model hadn't recorded that the first pass over nearby areas had just checked for the signal to speed things up. Correcting this altered the probability map substantially, and the wreck was quickly found in one of the new Bayesian hot spots.

Sadly, many people attempting to argue from evidence erroneously make drastic initial assumptions like P(Y)=0. (eg "I refuse to entertain that as a possibility without first seeing direct evidence" sounds reasonable to many.) This is both unnecessary and counterproductive, since it invalidates the very statistical argument they're trying to make and makes their reasoning circular, boiling it down to "no, that's nonsense" as both assumption and conclusion.

Another pitfall is to fail to consider possible overlaps and hence probabilistic interactions between your theories, credit to Chris Lively for this answer pointing that out.

Summary:

Science is not about truth, it's about accuracy. Empirical reasoning makes great science and concludes science is usually most accurate, but that's circular reasoning.

...which doesn't necessarily mean the conclusion is false! ((A-/->B)-/->not B) Note that empiricism concludes that the current scientific model is likely to be proven demonstrably false at some point in the possibly distant future, and replaced by something more accurate, not necessarily closer to the truth.

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"Science is fundamentally not about truth, it is fundamentally about degree of statistical agreement." <- This. Science is about making the simplest (Occam's razor) matching models of reality based on the data you have, it has nothing to do with truth whatsoever. If people would start realizing this, what a great day that would be. –  David Mulder Aug 20 at 7:54
    
@DavidMulder - That would be a very annoying day, since what we in practice mean by "true" is "statistically overwhelmingly likely according to mental models". Everyone "realizing" what you stated would be forced into nihilism, which is not a fun (or wise) place to be. –  Rex Kerr Aug 20 at 8:06
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@RexKerr No, you can have two radically different models, one true, one false, where the results of both agree to a good degree with experimental data, but the false one comes out on top in the statistical analysis. It's OK to conclude empirically that the green goblins are a better model for predicting whether my credit balance is non-zero, but it's not OK to conclude that it's the truth. Conflating truth and accuracy is bad science. Keeping them separate doesn't imply nihilism. I've edited my answer two try and emphasise this central distinction more in the summary, sorry if I obscured it. –  AndrewC Aug 20 at 12:10
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@RexKerr Now you're arguing from nihilism! Of course I know the other theory is true. I'm me, and it was all about my habits and reasons for them. I didn't acquire evidence, I was there, being me! You may only have my plausible assertions, but I have knowledge. If you refuse to admit any knowledge, even knowledge about oneself, why are you so keen to throw the word "truth" around so freely? –  AndrewC Aug 20 at 18:47
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Please consider taking longer discussions to chat! :) –  Joseph Weissman Aug 29 at 0:57

AndrewC had a very good answer; however I wanted to add a few things.

When beginning to compare theories it's best to begin by looking at your assumptions. In this case you are assuming that both creation and evolutionary theories are mutually exclusive. The second major assumption is that they have equal probabilities.

Covering the first point, there are creationist theories that do not preclude evolution. There are also evolutionary theories that do not throw out the possibility of initial creation. In other words there is a middle ground between the two which is not accounted for in your application of the Bayes theorem because you are starting from a highly over simplified viewpoint.

Ignoring that, the second issue has to do with assuming that each option has equal probability. This is simply impossible to determine. On the evolutionary side, we have a certain amount of physical evidence which is interpreted to have a certain meaning. This meaning might be correct, it might not be. We've certainly had many instances in history where a scientific "discovery" was made which turned out to be completely wrong as new information surfaced, and we've had many instances where the initial hypothesis has turned out to be as close to "settled" as it gets. However in this particular case, science is still learning about our past: and there are gaps. The gaps don't mean that evolution is any less valid, but it certainly means there are points that need further discovery.

On the creationist side, what "proof" could there possibly be other than the creator coming down and demonstrating in front of a world wide audience how man was formed? Maybe finding a barcode written at an atomic level? Even if either of those happened, it would still be contested. In other words, we have no real way of identifying that which was created versus that which may have "naturally" formed. However, absence of proof is meaningless. Now there are irreducible complexity arguments that, frankly, I'm not even remotely qualified to make an opinion on.

What is the likelihood that evolution is the sole means by which life arose on Earth? Is it still solely "evolution" if it was helped by some external source - even something as innocuous as an asteroid hitting the planet? What if that asteroid was hurled by a super intelligent race (or being) which knew it would be a catalyst to start life? Where do we even draw the line? There are just too many questions here.

The point is, we can't assume that either option has equal probability initially because there are in fact more than two options and those grey areas are not mutually exclusive. We also can't assume the probabilities are the same because there is a tremendous amount we don't know; which is further ambiguated by the extremes of both sides attempting to narrow the arguments of their opponents.


  1. Is this proof correct?

I'd have to say no as the comparison you are attempting to make is starting from false assumptions.

  1. Does this solve the problem of affirming the consequent not only in the case of evolution, but also when dealing with other theories/hypothesis?

Again, no. If "other theories" include starting from false assumptions then Bayes won't help you. Honestly we may as well substitute "my big toe" for 'E' and a "can of spam" for 'C' and be able to derive the exact same amount of meaning from the equation.

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+1 Indeed, yes I completely glossed over any kind of created/guided evolutionary process, a(nother) hole in the argument. –  AndrewC Aug 20 at 5:55
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The use of initial probabilities which come from nowhere when using Bayesian methods is one of the key criticisms the Frequentist statisticians have of the Bayesian statisticians, another argument that shows no sign of being resolved, but which I could/should have raised. –  AndrewC Aug 20 at 6:40
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When I have time, I'd like to take the point you made I alluded to in the previous comment and list it among more common errors, explained in the context of the Bayesian method as erroneously assuming P(E and C)=0, along with invalidly starting with P(E)=0 or P(C)=0. I guess you could say they're only invalid if you're reasoning statistically, so to be fair this is more of a problem for someone trying to prove not-C, particularly since it seems presentations of that argument use these assumptions more often anyway. –  AndrewC Aug 20 at 6:50
    
Prior probabilities are determined by simplicity. In the case you mentioned, of creationist theories that involve evolution; these hypothesis have the disadvantage that they are less simple than just creation and just evolution. We assign equal prior probabilities to both hypothesis simply because we have no reason to think of one hypothesis being intrinsically more likely than the other. –  user8083 Aug 20 at 21:20
    
You're missing the point with regards to the proof. I only used evolution and creation as an example. But, the idea was to provide a statistical model for determining which hypothesis has a greater likelihood in a total disjunction of hypothesis. This requires that: 1. We have a total disjuction of hypothesis for some observations. 2. The more likely hypothesis is the one which inplies the observation, and has a probability greater than or equal to the probability of all other hypothesis. –  user8083 Aug 20 at 21:24

Taking 'intelligent design' and 'evolution' as philosophical hypotheses about the origin of life; then homologies in nature are evidence for both hypotheses; but of course this is not the only evidence used.

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Why are homologies evidence for evolution instead of common design?

Better question: Why are homologies stronger evidence for evolution than common design?

Short answer: Evolution (descent with modification) actively predicts and explains homologous structures, and a lack of homologous structures among related organisms would be evidence against evolution. At best, a theory of common design is compatible with homologies but neither predicts nor requires their existence or lack thereof (as a designer is free to either re-use patterns or start over). Therefore, the extensive homologies that we find in nature are considered to be evidence in favour of evolution and are neutral with respect to a common designer.

Longer answer:

There have been a LOT of species on our planet over the past few billion years.[citation needed] Several hundred years ago, humans studying plants and animals discovered something truly bizarre: a nested hierarchy of traits that suggested there was some objective way that animals and plants could be grouped.

For example: Did you know that if you look at an animal with hair or fur, that animal's skull will have one temporal fenestra - but if the animal has scales or feathers the skull will have either two or (in the special case of turtles & tortoises) none at all?

Evolution depends on heredity - the passing on of characteristics from one generation to the next. Each generation of a species will be mostly like the one before, with only slight differences. A novel feature that arises in one population will only be seen in that population and its descendants - it will not and cannot arise in the exact same way in a completely separate lineage, even where convergence would encourage superficial similarity.

For example: Bats are mammals (specifically rodents), and have inherited the basic number and arrangement of bones that all other mammals have. Each individual bone might be bigger or smaller than the corresponding bone in another mammal's skeleton, but we should find far more similarities than differences, and the similarities should be strongest between species that are more closely related. This is indeed what we find - the bones bat wings are homologous to the ones in our arms and hands.

The basic body plan that bats, rats and humans share is derived from the earliest of tetrapods, and looking around us we can see that the basic body plan has been largely conserved. However, if we were to look at a trait that arose independently in different lineages - say, wings/flight - evolution predicts:

  1. Each group that evolved flight (bats, birds and the now-extinct pterosaurs) will have skeletal structure made up of bones that are homologous to the other groups - bats, birds and pterosaurs all inherited the same basic tetrapod skeletal toolkit.
  2. There will be significant differences in how each lineage honed in on flight - each lineage will have inherited its own legacy of mutations and selection pressures to arrive their current wing design - there's no 'going back', 'redo' or 'starting from scratch' option available if evolution is the only force at play.

Examining the different wing structures, this is indeed exactly what we see! Even though all 3 lineages have wings, the devil is, as they say, in the details:

  • Pterosaurs have wings built from an extension of the 5th digit ('pinky finger') to create a wing.
  • The 3rd and 4th digits in a bird's 'hand' have fused together and elongated to create their wing frame, and birds no longer express the 1st and 5th digits.
  • Bats retain all 5 'distinct' digits, with digits 2 through 5 elongated and used to make their wing.

This prediction of homology is rather unique to evolution. If biological creatures were designed & created in some kind of workshop instead, there is no reason why the designer would be forced to work within the constraints of heredity.

As a comparison, cars are, quite clearly, designed and manufactured objects. Designers can borrow ideas from other models (which would be called 'horizontal transfer' in biology) or radically reinvent just about anything about the vehicle. When the engineers sit down to design the 2016 Volkswagon Beetle, there's nothing to stop them from moving the engine from the front to the back. If cars were subject to evolution instead, such a radical change would be impossible.

Coming back to the idea of a common designer as an alternate hypothesis, our knowledge of design from other fields makes the observed strict reliance on homologous structures and simultaneous lack of horizontal transfer and back-to-the-drawing-board redesigns combines to be weak evidence against this hypothesis. This 'common designer' is not behaving in any way like any designers any of us know.

Homology is only one of the many lines of evidence that lead us to the conclusion that the theory of evolution is a mostly correct description of biological diversity.

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Some notes on your question.

(1) You assign probabilities to theories, which doesn't make sense. Probabilities are specific numbers that obey the calculus of probabilities. They are made in the light of explanations that give an account of why the probabilities have the specific values they have. As a result explanations do not have probabilities. An explanation is either true or false. An explanation may say that some specific event E has some probability p(E) and if the relative frequency of E differs from p(E) that is a problem for the theory.

(2) Many evolution supporters may think that structural and genetic homologies support their theories. They are wrong because support doesn't exist. In reality, you can't prove any position or show it is probable, i.e. - support it. Any argument requires premises and rules of inference and it doesn't prove (or make probable) those premises or rules of inference. If you're going to say they're self evident then you are acting in a dogmatic manner that will prevent you from spotting some mistakes. If you don't say they are self evident then you would have to prove those premises and rules of inference by another argument that would bring up a similar problem with respect to its premises and rules of inference.

(3) Knowledge is created by spotting a problem with some current idea, guessing solutions to that problem, criticising the guesses until only one is left and then looking for a new problem with that solution. In the light of this, homologies are relevant to judging whether you should adopt evolution or intelligent design.

If biological evolution is true then there should be structural and genetic homologies and if such homologies don't exist the theory is false. So looking for such homologies tests evolution and evolution has passed that test. The existence of such homologies is a problem that evolution that evolution has solved.

What about design theories? Design theories don't predict anything at all about structural or genetic homologies. The designer could have designed every species with a different genetic code, some species might use DNA, others could use information stored in a flash drive in the form of a text file, still others could store the information in the form of a program written in Clojure, or common lisp, or Ruby, or Python, or C++. He could have made every specimen of every species with blueprints he holds so that no species would have a genetic code. A designer could have constructed roads and made every species with wheels. And that's just a designer acting according to the laws of physics. If we postulate that god was the designer the situation gets even worse. God could have designed the laws of physics to be anything he liked. So there is no constraint on what god could do. Design theories don't pick out any particular state of affairs as the only possible state of affairs and so they explain nothing. As a result of this, they also predict nothing and no prediction can be considered a test of any design theory.

For more on epistemology see "Realism and the Aim of Science" by Karl Popper. For more on many of the issues in this question see "The Beginning of Infinity" by David Deutsch, especially chapters 1,4 and 13.

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"You assign probabilities to theories, which doesn't make sense. Probabilities are specific numbers that obey the calculus of probabilities. They are made in the light of explanations that give an account of why the probabilities have the specific values they have." No. Im assigning probabilities to hypothesis. –  user8083 Aug 27 at 20:12
    
I am using the term theory as synonymous with hypothesis. You can substitute hypothesis if you prefer. –  alanf Aug 28 at 8:46
    
Okay. I dont see why we cant assign probabilities to hypothesis. Each hypothesis has a prior plausability (probability). You've defined what a probability is, but i dont see how that would make assigning probabilities to hypothesis non-sensical. –  user8083 Aug 28 at 14:17
    
@alanf You appear to be objecting in principle to the Bayesian iterative method for refining probabilities. The frequentist movement (including Fisher) object for similar reasons. The frequentists may be more "right", but the Bayesians are more emprical. –  AndrewC Aug 29 at 0:21
    
Bayes' theorem is a correct statement about probabilities of events, not of theories. If you don't know the probability of some specific kind of event and want to find it Bayes' theorem may be useful if you have the appropriate explanation. But it is impossible to assign probabilities to explanations because an explanation is an account of how reality works and such an account is necessary to find probabilities. I'm not a frequentist. See "The Beginning of Infinity" by David Deutsch, Chapter 8, or this vimeo.com/5490979 which has a part about probability. –  alanf Aug 29 at 9:17

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