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Classic Texts: Karl Popper - Conjectures and Refutations (1963)
« on: 2003-06-08 19:43:22 »
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Karl Popper

Conjectures and Refutations (1963)



A very important text.
It has already been posted in the BBS, and it is the 3rd post in this thread:

http://virus.lucifer.com/bbs/index.php?board=4;action=display;threadid=25761;start=0





The gist: What is the problem?

<begin quote>

The most characteristic element in this situation seemed to me the incessant stream of confirmations, of observations which "verified" the theories in question; and this point was constantly emphasized by their adherents. A Marxist could not open a newspaper without finding on every page confirming evidence for his interpretation of history; not only in the news, but also in its presentation-which revealed the class bias of the paper-and especially of course in what the paper did not say. The Freudian analysts emphasized that their theories were constantly verified by their "clinical observations." As for Adler, I was much impressed by a personal experience. Once, in 1919, I reported to him a case which to me did not seem particularly Adlerian, but which he found no difficulty in analysing in terms of his theory of inferiority feelings, although he had not even seen the child. Slightly shocked, I asked him how he could be so sure. "Because of my thousandfold experience," he replied; whereupon I could not help saying: "And with this new case, I suppose, your experience has become thousand-and-one-fold."

What I had in mind was that his previous observations may not have been much sounder than this new one; that each in its turn had been interpreted in the light of "previous experience," and at the same time counted as additional confirmation. What, I asked myself, did it confirm? No more than that a case could be interpreted in the light of the theory. But this meant very little, I reflected, since every conceivable case could be interpreted in the light of Adler's theory, or equally of Freud's. I may illustrate this by two very different examples of human behaviour: that of a man who pushes a child into the water with the intention of drowning it; and that of a man who sacrifices his life in an attempt to save the child. Each of these two cases can be explained with equal ease in Freudian and in Adlerian terms. According to Freud the first man suffered from repression (say, of some component of his Oedipus complex), while the second man had achieved sublimation. According to Adler the first man suffered from feelings of inferiority (producing perhaps the need to prove to himself that he dared to commit some crime), and so did the second man (whose need was to prove to himself that he dared to rescue the child). I could not think of any human behaviour which could not be interpreted in terms of either theory. It was precisely this fact -- that they always fitted, that they were always confirmed -- which in the eyes of their admirers constituted the strongest argument in favour of these theories. It began to dawn on me that this apparent strength was in fact their weakness.

<end quote>


On verifiability vs falsifiability:

<begin quote>

(1) It is easy to obtain confirmations, or verifications, for nearly every theory -- if we look for confirmations.

(2) Confirmations should count only if they are the result of risky predictions; that is to say, if, unenlightened by the theory in question, we should have expected an event which was incompatible with the theory -- an event which would have refuted the theory.

(3) Every "good" scientific theory is a prohibition: it forbids certain things to happen. The more a theory forbids, the better it is.

(4) A theory which is not refutable by any conceivable event is nonscientific. Irrefutability is not a virtue of theory (as people often think) but a vice.

(5) Every genuine test of a theory is an attempt to falsify it, or to refute it. Testability is falsifiability; but there are degrees of testability; some theories are more testable, more exposed to refutation, than others; they take, as it were, greater risks.

(6) Confirming evidence should not count except when it is the result of a genuine test of the theory; and this means that it can be presented as a serious but unsuccessful attempt to falsify the theory. (I now speak in such cases of "corroborating evidence.")

(7) Some genuinely testable theories, when found to be false, are still upheld by their admirers -- for example by introducing ad hoc some auxiliary assumption, or by re-interpreting theory ad hoc in such a way that it escapes refutation. Such a procedure is always possible, but it rescues the theory from refutation only at the price of destroying, or at least lowering, its scientific status. (I later described such a rescuing operation as a "conventionalist twist" or a "conventionalist stratagem.")

One can sum up all this by saying that the criterion of the scientific status of a theory is its falsifiability, or refutability, or testability.

<end quote>


On demarcation:

<begin quote>

Thus the problem which I tried to solve by proposing the criterion of falsifiability was neither a problem of meaningfulness or significance, nor a problem of truth or acceptability. It was the problem of drawing a line (as well as this can be done) between the statements, or systems of statements, of the empirical sciences, and all other statements -- whether they are of a religious or of a metaphysical character, or simply pseudo-scientific. Years later -- it must have been in 1928 or 1929 -- I called this first problem of mine the "problem of demarcation". The criterion of falsifiability is a solution to this problem of demarcation, for it says that statements or systems of statements, in order to be ranked as scientific, must be capable of conflicting with possible, or conceivable, observations....

<end quote>


On induction:

<begin quote>

(1) Induction, i.e. inference based on many observations, is a myth. It is neither a psychological fact, nor a fact of ordinary life, nor one of scientific procedure.

(2) The actual procedure of science is to operate with conjectures: to jump to conclusions -- often after one single observation (as noticed for example by Hume and Born).

(3) Repeated observations and experiments function in science as tests of our conjectures or hypotheses, i.e. as attempted refutations.

(4) The mistaken belief in induction is fortified by the need for a criterion of demarcation which, it is traditionally but wrongly believed, only the inductive method can provide.

(5) The conception of such an inductive method, like the criterion of verifiability, implies a faulty demarcation.

(6) None of this is altered in the least if we say that induction makes theories only probable rather than certain.

<end quote>

« Last Edit: 2003-06-13 10:26:18 by rhinoceros » Report to moderator   Logged
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Re:Classic Texts: Karl Popper - Conjectures and Refutations (1963)
« Reply #1 on: 2003-06-12 12:59:05 »
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Quote:

(6) None of this is altered in the least if we say that induction makes theories only probable rather than certain.

I don't see how this can be true. Can anyone explain it?
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Re:Classic Texts: Karl Popper - Conjectures and Refutations (1963)
« Reply #2 on: 2003-06-12 15:41:26 »
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[Lucifer]
<quote from Popper>

(6) None of this is altered in the least if we say that induction makes theories only probable rather than certain.

<end quote>

I don't see how this can be true. Can anyone explain it?



[rhinoceros]
I just added some material on "verifiability vs falsifiability" to the "gist" of Popper's text in this thread.

Now, about the question: What is wrong with this statement? What troubles me is exactly the opposite: It seems so obvious, even unneeded, if we take into account that the truth of all the theories of the empirical sciences is tentative.

This is something which makes me suspect that Popper's statement is somehow vague and does not refer to the possibility or certainty of the truth of a theory, but only to its demarcation -- the possibility or certainty that this theory belongs to the empirical sciences.

Apparently, Popper is not talking about the "mathematical induction" or "perfect induction", which does not belong to the toolbox of the empirical sciences. He is talking about the "empirical" or "imperfect" induction which, according to Hume, cannot logically justify a law of nature. As Popper points out in the same text:


<begin quote>

For a brief formulation of the problem of induction we can turn again to Born, who writes: "... no observation or experiment, however extended can give more than a finite number of repetitions"; therefore, "the statement of a law -- B depends on A -- always transcends experience. Yet this kind of statement is made everywhere and all the time, and sometimes from scanty material.'

In other words, the logical problem of induction arises from (a) Hume's discovery (so well expressed by Born) that it is impossible to justify a law by observation or experiment, since it "transcends experience"; (b) the fact that science proposes and uses laws "everywhere and all the time." (Like Hume, Born is struck by the "scanty material," i.e. the few observed instances upon which the law may be based.) To this we have to add (c) the principle of empiricism which asserts that in science, only observation and experiment may decide upon the acceptance or rejection of scientific statements, including laws and theories.

These three principles, (a), (b), and (c), appear at first sight to clash; and this apparent clash constitutes the logical problem of induction.

Faced with this clash, Born gives up (c), the principle of empiricism (as Kant and may others, including Bertrand Russell, have done before him), in favour of what he calls a "metaphysical principle"; a metaphysical principle which he does not even attempt to formulate; which he vaguely describes as a "code or rule of craft"; and of which I have never seen any formulation which even looked promising and was not clearly untenable.

But in fact the principles (a) to (c) do not clash. We can see this the moment we realize that the acceptance by science of a law or of a theory is tentative only; which is to say that all laws and theories are conjectures, or tentative hypotheses (a position which I have sometimes called "hypotheticism") and that we may reject a law or theory on the basis of new evidence, without necessarily discarding the old evidence which originally led us to accept it.

The principles of empiricism (c) can be fully preserved, since the fate of a theory, its acceptance or rejection, is decided by observation and experiment -- by the result of tests. So long as a theory stands up to the severest tests we can design, it is accepted; if it does not, it is rejected. But it is never inferred, in any sense, from the empirical evidence. There is neither a psychological nor a logical induction. Only the falsity of the theory can be inferred from empirical evidence, and this inference is a purely deductive one.

Hume showed that it is not possible to infer a theory from observation statements; but this does not affect the possibility of refuting a theory by observation statements. The full appreciation of the possibility makes the relation between theories and observations perfectly clear. This solves the problem of the alleged clash between the principles (a), (b), and(c), and with it Hume's problem of induction....

<end quote>


Did I miss the point of the question?

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Re:Classic Texts: Karl Popper - Conjectures and Refutations (1963)
« Reply #3 on: 2003-06-14 12:23:01 »
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Popperian falsification is a special case of Bayes theorem. From that perspective Popper's arguments against verification and induction are misguided.

To illustrate let's look at four related theories of the engimatic monkeybat:
a) All monkeybats are black.
b) At least one monkebat is not black.
c) At least one monkeybat is black.
d) No monkeybats are black.

(a) is a positive universal, (b) is the logical inverse of (a) and is a negative existential, (c) is a positive existential, and (d) is the logical inverse of (c) and is a negative universal.

To translate to logical notation:

a) Ax, Mb(x) -> B(x)
b) Ex, Mb(x) & B(x)
c) Ex, Mb(x) & ~B(x)
d) Ax, Mb(x) -> ~B(x)

where Mb(x) means x is a monkeybat and B(x) means x is black.

Since (a) and (b) are logical inverses the truth values (probability, certainty, confidence level) they should sum up to 1. Same for (c) and (d)

e) P(a) + P(b) = 1
f) P(c) + P(d) = 1

We will start with total ignorance, we have no reason to have any more certainty in (a) or (d) so we assign a value of 0.5 to both. From (e) and (f) we can also assign 0.5 to (b) and (c).

Now say we make an observation of a monkeybat. He isn't black! How do we update our beliefs?

Bayes theorem says that the likelihood of a hypothesis H after observing a datum D should be multiplied by the likelihood ratio P(D|H)/P(D|~H), i.e. the probability of seeing D if H is true divided by the probability of seeing D if H is not true.

In this case the probability of seeing a non-black monkeybat assuming that (a) is true is 0, which is to say that there is no chance of seeing a non-black monkeybat if all monkeybats are black. The probability of seeing a non-black monkeybat assuming that (a) is false is non-zero because that would mean that (b) is true and there is at least one non-black monkeybat. If we had to assign a value it would be 0.5 because that is the limit of our knowledge/ignorance, but it doesn't matter because the numerator of the ratio is 0. So to update our knowledge we convert our current P(a)=0.5 to a likelihood L(a)=1 (because a 50% probability is equivalent to 1:1 odds and likelihood is 1/1=1). Now we multiply it by the likelihood ratio 0 resulting in 0, convert it back to probabilty 0, so now our confidence in (a) has dropped to 0. (a) has been falsified! We can now update P(b) using (e) and P(b)=1, (b) has been verified!

Now what does our observation of the non-black monkeybat do to our confidence in the hypotheses (c) and (d)? We will have to calculate the likelihood ratio for hypothesis (d). The probability of seeing a non-black monkeybat assuming that (d) is 1 (because no monkeybat are black). The probability of seeing a non-black monkeybat assuming that (d) is false (and (c) is true, there is at least one non-black monkeybat) is 0.5 (again because that is the limit of our knowledge/ignorance). The likelihood ratio is 1/0.5 = 2. Again, converting our prior P(d) into a likelihood of 1, we multiply by the ratio 2 to get a likelihood of 2. A likelihood of 2 corresponds to 2:1 odds which translates to a probability of 2/3 so our confidence in (d) increases from 1/2 to 2/3 after observing one non-black monkeybat. Using (f) we see that our confidence in (d) decreases from 1/2 to 1/3.

To sum up, our single observation of a non-black monkeybat affected our knowledge by falsifying (a), verifying (b), decreasing our confidence in (c) and increasing our confidence in (d). We now know that it is not true that all monkeybats are black. We now know that it is true that there is at least one non-black monkeybat. We are less confident that at least one monkeybat is black and we are more confident that no monkeybats are black. The first two changes in knowledge are Popperian, but the latter two represent Bayesian induction and are not acknowledged by Popper.
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Re:Classic Texts: Karl Popper - Conjectures and Refutations (1963)
« Reply #4 on: 2003-06-14 21:57:15 »
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I am not sure that I agree with the opening assertion, "Popperian falsification is a special case of Bayes theorem." Would you care to elaborate on why you see Popper (1963) as being a "special case" of a post-humously published theory ("Essay Towards Solving a Problem in the Doctrine of Chances") propounded two centuries before (in 1763) (source: Encyclopedia Brittanica)?

For my part, I see Bayes' Theorem as speaking to the likelihood of something being true based on probability analysis, while Popper specifically spoke to the fact that the value of any scientific theory, no matter how theoretically elegant or plausible, is ultimately tested by experiment. Conventionally, this crucial element of the scientific process involves extracting a clear and unequivocal prediction from a theory, investigating this prediction experimentally, and assessing the outcome objectively. Certainly Bayes is a valid (and I'd say a more valid technique than the all too common frequency regression) method to analyse the validity of the outcome of an experiment, but it certainly remains "merely" a mathematical method, not an empirical result.

I think that, if truly empirical, as opposed to yet another thought experiment, the "monkeybat experiment" would demonstrate the validity of Popper's assertions. So long as it is recognised as valid by other researchers, the actual method of analysis used to reach the "objective conclusion" is not particularly significant. After all, Popper would, I suspect, ascerbically note that (a) and (d) are useful (testable and falsifiable) theories, while (c) and (d) are merely purported observations, not making useful predictions (unless predicated upon some other unstated assumption e.g. Mendelian characteristics). Or am I missing something here?

Kind Regards

Hermit




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Re:Classic Texts: Karl Popper - Conjectures and Refutations (1963)
« Reply #5 on: 2003-06-15 12:39:54 »
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Quote from: Hermit on 2003-06-14 21:57:15   
I am not sure that I agree with the opening assertion, "Popperian falsification is a special case of Bayes theorem." Would you care to elaborate on why you see Popper (1963) as being a "special case" of a post-humously published theory ("Essay Towards Solving a Problem in the Doctrine of Chances") propounded two centuries before (in 1763) (source: Encyclopedia Brittanica)?


Falsification is the special case of Bayesian induction where the likelihood ratio equals 0. Since the probabilities involved in the ratio can take on any value between 0 and 1, the ratio can take on any value between 0 and infinite so falsification is one of the two limit cases.

I'm not sure what the dates of publication have to do with anything, do you see them as relevant?

Quote:
I think that, if truly empirical, as opposed to yet another thought experiment, the "monkeybat experiment" would demonstrate the validity of Popper's assertions.


The Bayesian approach is not merely for thought experiments. See the reviews of Scientific Reasoning: The Bayesian Approach.

Quote:
So long as it is recognised as valid by other researchers, the actual method of analysis used to reach the "objective conclusion" is not particularly significant. After all, Popper would, I suspect, ascerbically note that (a) and (d) are useful (testable and falsifiable) theories, while (c) and (d) are merely purported observations, not making useful predictions (unless predicated upon some other unstated assumption e.g. Mendelian characteristics). Or am I missing something here?


You are missing the point that Popper would claim that a negative universal such as (d) is not falsifiable and therefore not a scientific hypothesis. That's where Popperians and Bayesians disagree.
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Re:Classic Texts: Karl Popper - Conjectures and Refutations (1963)
« Reply #6 on: 2003-06-15 13:16:45 »
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[Hermit 1] I am not sure that I agree with the opening assertion, "Popperian falsification is a special case of Bayes theorem." Would you care to elaborate on why you see Popper (1963) as being a "special case" of a post-humously published theory ("Essay Towards Solving a Problem in the Doctrine of Chances") propounded two centuries before (in 1763) (source: Encyclopedia Brittanica)?

[David Lucifer 2] Falsification is the special case of Bayesian induction where the likelihood ratio equals 0. Since the probabilities involved in the ratio can take on any value between 0 and 1, the ratio can take on any value between 0 and infinite so falsification is one of the two limit cases.

[Hermit 3] Popperian falsification may be established in any one of a number of ways, from simple observation to careful analysis of minor discrepencies (e.g. Milligan's fudging of the mass of an electron). Bayes' theory addresses some classes of analysis, yet remains a method of establishing probability, not a method of observation. Thus Bayes does not speak to Popperian falsifiability, and thus I cannot see "Popperian falsification" as a "case", "special" or not, of Bayes theorum.

[David Lucifer 2] I'm not sure what the dates of publication have to do with anything, do you see them as relevant?

[Hermit 3] I do, because Bayes theorum was published two centuries earlier and the intervening period was remarkable for the number of brilliant minds attempting to resolve the subjectivity/objective debate - conspicuously unsuccessfully.  This despite the fact that many of these people had access to Bayes and comprehended probability tolerably well even by modern standards. We had to wait for Popper to show that no absolute proof of complex systems was possible, while disproving things through observation and analysis can be done. This has become the method through which scientific knowledge advances.

[Hermit 1] I think that, if truly empirical, as opposed to yet another thought experiment, the "monkeybat experiment" would demonstrate the validity of Popper's assertions.

[David Lucifer 2] The Bayesian approach is not merely for thought experiments. See the reviews of Scientific Reasoning: The Bayesian Approach.

[Hermit 3] I did not suggest that it was. vide:
Quote:
[Hermit 1] Certainly Bayes is a valid (and I'd say a more valid technique than the all too common frequency regression) method to analyse the validity of the outcome of an experiment, but it certainly remains "merely" a mathematical method, not an empirical result.
[Hermit 3] That still does not make Bayes theorum into an observation.

[Hermit 1] I think that, if truly empirical, as opposed to yet another thought experiment, the "monkeybat experiment" would demonstrate the validity of Popper's assertions. So long as it is recognised as valid by other researchers, the actual method of analysis used to reach the "objective conclusion" is not particularly significant. After all, Popper would, I suspect, ascerbically note that (a) and (d) are useful (testable and falsifiable) theories, while (c) and (d) are merely purported observations, not making useful predictions (unless predicated upon some other unstated assumption e.g. Mendelian characteristics). Or am I missing something here?

[David Lucifer 2] You are missing the point that Popper would claim that (d) is not falsifiable and therefore not a scientific hypothesis. That's where Popperians and Bayesians disagree.

[Hermit 3] I remind you:
Quote:
[David Lucifer 1] d) No monkeybats are black.


[Hermit 3] Speaking from a "Popperian perspective", my problems with claiming the status of a "scientific hypothesis" for such an assertion would be more that it does not purport to explain something, does not identify the underlying mechanisms and even when falsified, does not particularly expand our comprehension of monkeybattedness. As such the assertion re what Popper would say appears to me to be an improper projection, particularly as it appears to be untrue.

[Hermit 3] Surely the observation of a single black monkeybat would falsify this hypothesis? Does that not mean that it is indeed a trivially falsifiable statement?  I did some research at http://www.monkeybat.com, discovering that a monkeybat is a "Brussels Griffon- Smooth Variety" (appropriate source for details on monkeybats:-) ) and then at http://www.akc.org/breeds/recbreeds/brusgrif.cfm (appropriately qualified experts) where I discovered that "Color: 4) (...) Black: solid black. Any white hairs are a serious fault, except for "frost" on the muzzle of a mature dog, which is natural." So it appears that black monkeybats are possible, and assertion (d) is not just falsifiable, but falsified.

[Hermit 3] Would a "Popperian" argue with falsification from appropriately reported observation? I know I wouldn't.

Kind Regards

Hermit

PS I greatly respect the work of the LSE and will add Colin Howson (Preface), Peter Urbach (Preface), "Scientific Reasoning: The Bayesian Approach" to my reading list, but with no great expectation of it changing my opinion from the consensus position on the scientific method.
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Re:Classic Texts: Karl Popper - Conjectures and Refutations (1963)
« Reply #7 on: 2003-06-16 11:33:00 »
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Quote from: Hermit on 2003-06-15 13:16:45   

[Hermit 3] Popperian falsification may be established in any one of a number of ways, from simple observation to careful analysis of minor discrepencies (e.g. Milligan's fudging of the mass of an electron). Bayes' theory addresses some classes of analysis, yet remains a method of establishing probability, not a method of observation. Thus Bayes does not speak to Popperian falsifiability, and thus I cannot see "Popperian falsification" as a "case", "special" or not, of Bayes theorum.

Would it matter if Bayes theorem was interpreted to show how to calculate plausability from observation, rather than probability?


Quote:

[Hermit 3] I do, because Bayes theorum was published two centuries earlier and the intervening period was remarkable for the number of brilliant minds attempting to resolve the subjectivity/objective debate - conspicuously unsuccessfully.  This despite the fact that many of these people had access to Bayes and comprehended probability tolerably well even by modern standards. We had to wait for Popper to show that no absolute proof of complex systems was possible, while disproving things through observation and analysis can be done. This has become the method through which scientific knowledge advances.

I read a book recently called Against The Gods about the history of probability theory and risk management. One of the common themes was that advances were a long time coming even though lots of brilliant people who worked on it seemed to have all the right information but failed to put it together in ways that their successors would.

In any case, the scientific method is not at issue. What is at issue is Popper's claim that "None of this is altered in the least if we say that induction makes theories only probable rather than certain." This seems to be inconsistent with Bayes. Confidence in a theory should increase with supporting evidence. Bayes theorem doesn't increase confidence with any supporting evidence, it has to be evidence that is more likely when the theory is true than if the theory is false. This subtle point is often lost on critics.


Quote:

[Hermit 3] I remind you:
Quote:
[David Lucifer 1] d) No monkeybats are black.

Oops, mea culpa. I made an elementary error. "No monkeybats are black" is a positive universal just like (a) and therefore falsifiable. I had meant to illustrate with a negative universal, in this case it should have been "Not all monkeybats are black" and the corresponding positive existential is "There is at least one monkeybat that is not black".


Quote:

[Hermit 3] Speaking from a "Popperian perspective", my problems with claiming the status of a "scientific hypothesis" for such an assertion would be more that it does not purport to explain something, does not identify the underlying mechanisms and even when falsified, does not particularly expand our comprehension of monkeybattedness. As such the assertion re what Popper would say appears to me to be an improper projection, particularly as it appears to be untrue.

You seem to be confusing scientific theories with falsifiable statements here. I didn't help by calling it a scientific hypothesis. I meant falsifiable statement.


Quote:

[Hermit 3] Surely the observation of a single black monkeybat would falsify this hypothesis? Does that not mean that it is indeed a trivially falsifiable statement?  I did some research at http://www.monkeybat.com, discovering that a monkeybat is a "Brussels Griffon- Smooth Variety" (appropriate source for details on monkeybats:-) ) and then at http://www.akc.org/breeds/recbreeds/brusgrif.cfm (appropriately qualified experts) where I discovered that "Color: 4) (...) Black: solid black. Any white hairs are a serious fault, except for "frost" on the muzzle of a mature dog, which is natural." So it appears that black monkeybats are possible, and assertion (d) is not just falsifiable, but falsified.

Not really. It is true that the monkeybat is a Brussels Griffon - smooth variety, but not all smooth griffs are monkeybats. (Any more than all dogs or all mammals are monkeybats).
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Re:Classic Texts: Karl Popper - Conjectures and Refutations (1963)
« Reply #8 on: 2003-06-16 22:48:07 »
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[Hermit 3] Popperian falsification may be established in any one of a number of ways, from simple observation to careful analysis of minor discrepencies (e.g. Milligan's fudging of the mass of an electron). Bayes' theory addresses some classes of analysis, yet remains a method of establishing probability, not a method of observation. Thus Bayes does not speak to Popperian falsifiability, and thus I cannot see "Popperian falsification" as a "case", "special" or not, of Bayes theorum.

[David Lucifer 4] Would it matter if Bayes theorem was interpreted to show how to calculate plausability from observation, rather than probability?

[Hermit 5] Not really. Consider the concensus model steps (Ref: FAQ: The Scientific Method):
    1 Pose a question about nature [Some would say, not necessarily as the result of an observation].
    2 Collect the pertinent, observable evidence.
    3 Formulate an explanatory hypothesis, defining relevant assumptions.
    4 Deduce its implications.
    5 Test all of the implications experimentally.
    6 Accept, reject, or modify the hypothesis based upon the experimental results.
    7 Define its range of applicability.
    8 Peer review
    9 Publish (including methodology, data and analysis)
    10 Evaluation and peers continue to test, extend and challenge the hypothesis.
[Hermit 5] Bayes is applicable when evaluating the force of the experimental results upon the hypothesis (Steps 2, and 6) and the strength of the hypothesis (Step 10)  - and perhaps when designing the experiment (Steps 3 and 4). It also applies during the observation taking (Steps 2 and 6) as observational data can no more be conclusively confirmed or disconfirmed than theoretical conjectures as perception is fallible too. Even so, it doesn't matter what we call the application of Bayes (which should depend perhaps on the application in question), as it is an analytical method used to balance the weight that should be given to observations and hypothesii. Bayes is a tool, a good tool, but still only one of many possible tools with which to approach probability. Not a replacement for the scientific method.

[David Lucifer 4] I read a book recently called Against The Gods about the history of probability theory and risk management. One of the common themes was that advances were a long time coming even though lots of brilliant people who worked on it seemed to have all the right information but failed to put it together in ways that their successors would.

[Hermit 5] Agreed, but the driver, in my opinion, is necessity. When we desperately need some tool, people seem to scrabble through their toolboxes and the detrius of the past, and discover (or rediscover) or invent what is needful. In science, as in mathematics this tends to happen in fairly small, generally unnoticed steps, perhaps because while "giant leaps" can take you a long way, they are unpredictable and difficult to distribute (We prefer, with good reason, to clew to our previously strong theories).

[David Lucifer 4] In any case, the scientific method is not at issue. What is at issue is Popper's claim that "None of this is altered in the least if we say that induction makes theories only probable rather than certain." This seems to be inconsistent with Bayes. Confidence in a theory should increase with supporting evidence. Bayes theorem doesn't increase confidence with any supporting evidence, it has to be evidence that is more likely when the theory is true than if the theory is false. This subtle point is often lost on critics.

[Hermit 5] Please note, I'm neither criticising Bayes theorum nor attempting to limit its applicability, so we may be at cross-purposes here. I do think that perhaps you may not have fully grasped the problems Popper attempted to solve - and the fact that his position was not as inflexible as some proponents of Bayesian theory (amongst others) appear to suggest. Popper's "claim" defines the modern scientific method and the basis of falsifiability which reduces the effects of (but does not eliminate) subjectivity. Popper observed (correctly, but not uniquely, Bacon noticed it first) that the science of the day sought rational theories which produced solid models through a process of induction (which not unexpectedly tended to support the prejudices of the researcher) . Evidence was then sought to substantiate these models (and again, remarkable selectivity was often rationalized, as we tend to select evidence which supports our claims (e.g. the afore mentioned Millikan* ref infra)). Popper's "vast leap" was to recognise that not only was subjectivism skewing the results, but that the method itself was flawed. "Popper sought to solve his two basic problems at one blow with his falsificationist philosophy of science. What demarcates science from non-science (metaphysics, logic, mathematics, and pseudo-science) is not the verifiability but the falsifiability of its theories. The method of science also is not inductive; it does not start out from observations and generalise from them: it starts out from problems, which it attacks with bold conjectures. The latter are unverifiable and unjustifiable but, when well developed, have predictive implications which can be put to the test, the more severe the better. A test will be severe if made with sufficiently discriminating experimental apparatus on predictions which deviate (as Einsteinís did from Newtonís) to a small but detectable extent from unrefuted predictions of the previous theory, or if made on predictions which break new ground. On this view, scientific inferences are all deductive, either from conjectural premises to a falsifiable consequence or from a falsified consequence to the negation of the conjunction of the jointly responsible premises. The problem of induction therefore drops out. (Whether it drops out completely is a question which will come up again.)"  John Watkins, "Obituary of Karl Popper, 1902 - 1994".

[Hermit 5] Popper began with the concept of "naive" falsification, ie Scientific hypotheses generalise on observation reports which can be conclusively verified or falsified enabling the hypotheses to be refuted but never enabling them to be made more likely, as expounded in many ot the early texts we have seen here, and to which I think you are objecting. However, you should take into accunt that he subsequently introduced what is today known as Popperian falsification (which is not congruent with his early position): Only refutation, not confirmation is possible, although we must acknowledge the fallibility of refutation and the legitimacy of dogmatic adherence to favoured theory. Popper used this as a basis to derive the fundamental criterion for deciding which alterations in the face of (presumed) refutation are legitimate: i.e. never make ad hoc revisions, that is revisions which reduce empirical content. Similarly Popper has a distinctive methodological principle: accept the strongest unfalsified theory (It should be noted that today this is applied within the Duhem-Quine guidelines, which states that refutations apply only to total systems of general hypotheses plus auxiliary hypotheses plus boundary conditions, and not necessarily to subsets of the preceding.

[Hermit 5] From the above, it follows that when we apply Bayes theorum to the decisions we are making, we are simply using a tool which sometimes allows us to make better decisions as to what to accept as probable and what to reject as improbable and perhaps to quanity seemingly incomplete or contradictory results. However, these results are then used within the overall framework of the scientific method, and in particular, within the guidelines of Popperian falsification.

[David Lucifer 4.1] Oops, mea culpa. I made an elementary error. "No monkeybats are black" is a positive universal just like (a) and therefore falsifiable. I had meant to illustrate with a negative universal, in this case it should have been "Not all monkeybats are black" and the corresponding positive existential is "There is at least one monkeybat that is not black".

[Hermit 5] This would have changed my answer more than a little :-) But before delving more deeply, please observe that for the two classes of logical statements (i.e. universal generalizations and existential generalizations, the logic switches around, i.e.  universal generalisations can be falsified but not verified by their instances, while  existential generalisations can be verified by an instance (and only for that instance) but can never be disproven. The lack of generality is what condemns the existential generalization to triviality. There is no way to predict anything generally applicable from an existential generalization and thus it is not "useful."

[Hermit 5] So in your instance, finding a tan monkeybat would demonstrate that the hypothesis is true, but tells us nothing about other monkeybats accept that they may appear in non-black colors (which has been confirmed by capturing a tan monkeybat). In otherwords, the hypothesis as formulated in (d) gives you very little to bite on in applying analysis (by Bayes theorem or any other method), as there are an infinite number of colors (E.M. spectrum) that are not black. Thus your postulated hypothesis (whether expressed positively or negatively) only speaks to the monkeybat you have captured and observed (Are there in fact black monkeybats? How about Purple ones?) As it suffices to let the monkeybat (or the observations of it) speak for itself, an hypothesis does not extend our understanding or predictive capability.

[David Lucifer 4] You seem to be confusing scientific theories with falsifiable statements here. I didn't help by calling it a scientific hypothesis. I meant falsifiable statement.

[Hermit 5] Certainly, at this point, I acknowledge confusion (at least in so far as why you imagine I am confused), but suggest that perhaps both of us are confused, seeing as you apparently provided me with erronious formulation of your thoughts [David Lucifer 4.1], which I think I spoke to, that perhaps the confusion originated on your side of the screen! I had observed that your previous formulation met the criteria for a "scientific hypothesis" in so far as it was falsifiable, but suffered from other deficiencies which might tend to cause it to be rejected as a "scientific hypothesis". The troubles with the new formulations have I hope been sufficiently explained above. In a scientific sense, beyond that of scope, there is no difference between a falsifiable statement and a hypothesis (in a scientific sense an hypothesis is usually a series of linked falsifiable statements together with appropriate bounding limits, applicability etc). For completeness, a theory is an hypothesis which has made it through steps 1 to 10 above, and a law is a theory which is regarded as having universal applicability, is coherent with other strong theories and has been through sufficient iterations for it to be unlikely that any new evidence will falsify it. e.g. Ohm's law, Boyle's law and Charles's law).

[Hermit 3]  I did some research at http://www.monkeybat.com, discovering that a monkeybat is a "Brussels Griffon- Smooth Variety" (appropriate source for details on monkeybats:-) )...

[David Lucifer 4] Not really. It is true that the monkeybat is a Brussels Griffon - smooth variety, but not all smooth griffs are monkeybats. (Any more than all dogs or all mammals are monkeybats).

[Hermit 5] I think confusion has crept in again. Not being an expert in monkeybats, I went to what I fondly imagined was the source for information on monkey bats, at the cited site and did not see that I could not project information related to Brussels Griffons onto monkeybats. Could you please provide a reference to this vital information :-o

Kind Regards

Hermit

*Millikan's Fraud (Source: http://lachlan.bluehaze.com.au/popper.html with editing and minor formatting. Interestingly, a site advocating the use of Bayes theorem to avoid such problems!)

Robert Millikan is widely regarded as one of the founders of modern American science, his determination of the charge on the electron winning him the 1923 Nobel Prize for physics. In a now-famous study, the physicist and historian Gerald Holton examined the log-books for Millikanís experiments with the electron, and revealed that he repeatedly rejected data that he deemed "unacceptable" (Holton 1978). The criteria he used were blatantly subjective, as revealed by the comments in the log-books, such as "Very low - something wrong" and "This is almost exactly right". Throughout, Millikan appears to have been driven partly by a desire to get results that were self-consistent, broadly in agreement with other methods, and consistent with his personal view that the electron is the fundamental and indivisible unit of electric charge.

While these criteria may seem reasonable enough, they carry inherent dangers. Even today a fundamental explanation of the precise numerical value of the charge on the electron remains lacking, so Millikan was hardly in a position to decide objectively which values were high and which ones low. Previous results may have been fundamentally flawed, while the demand for self-consistent results may mask the existence of subtle but genuine properties of the electron. Millikan could also have been proved wrong in his belief that the electron was fundamental.

However, it is also clear that Millikan had another powerful motivation for using all means to obtain a convincing determination of the electronic charge: he was in a race against another researcher, Felix Ehrenhaft at the University of Vienna. Ehrenhaft had obtained similar results to those of Millikan, but they were interspersed with much lower values that suggested that the electron was not, in fact, the fundamental unit of charge. Millikan had no such doubts, published his results, and went on to win the Nobel Prize.

Apologists for Millikanís hand-picking of data point out that the numerical result he obtained, -1.592 x10-19 coulombs, is just 0.6% below the modern value of -1.6021892 x 10-19 C (Weinberg 1993 p 99). At first sight, this does indeed seem impressive. However, Millikanís stated result was based on a faulty value for the viscosity of air, which when corrected changes Millikanís result to -1.616 x 10-19 C, increasing the discrepancy with the modern value by over 40 per cent. More importantly, however, it puts the latter well outside the error-bounds of Millikanís central estimate. Indeed, the discrepancy is so large that the probability of generating it by chance alone is less than 1 in 103. Millikanís "remarkable ability" to scent out the correct answer was clearly not as great as his apologists would have us believe. Rather more remarkable is Millikanís ability, almost half a century after his death, to evade recognition as an insouciant scientific fraudster who won the Nobel Prize by deception.

The dangers of the injudicious use of subjective criteria is further highlighted by the aftermath of Millikanís experiments. In the decades following his work and Nobel Prize, other investigators made determinations of the electronic charge. The values they obtained show a curious trend, creeping further and further away from Millikanís "canonical" value, until finally settling down at the modern figure with which, as we have seen, it is wholly incompatible. Why was this figure not reached sooner ? The Nobel Prizewinning physicist Richard Feynman has given the answer in his own inimitable style (Feynman 1988, p 382):

"Itís apparent that people did things like this: when they got a number that was too high above Millikanís, they thought something was wrong - and they would look for and find a reason why something might be wrong. When they got a number closer to Millikanís value they didnít look so hard. And so they eliminated the numbers that were too far off"
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Re:Classic Texts: Karl Popper - Conjectures and Refutations (1963)
« Reply #9 on: 2003-06-17 12:19:35 »
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A nice exchange. Some remarks:


[David Lucifer]
a) All monkeybats are black.
b) At least one monkebat is not black.

[rhinoceros]
True or false, (a) is already a perfectly Popperian universal statement for which an explicit testable falsification condition can be (and has been) formulated. It seems that the universal statements which have been chosen for this example are already "Popperian" meaningful statements of empirical science. It is not always so. There are universal statements for which you can't formulate anything like (b). Would Bayes still have something to say about those?

[David Lucifer]
c) At least one monkeybat is black.
d) No monkeybats are black.

[rhinoceros]
The same goes for (d).

[David Lucifer]
Falsification is the special case of Bayesian induction where the likelihood ratio equals 0. Since the probabilities involved in the ratio can take on any value between 0 and 1, the ratio can take on any value between 0 and infinite so falsification is one of the two limit cases.

[rhinoceros]
Falsification is a technique for finding a truth value, so the above sounds right. But the big thing with Popper is not falsification but falsifiability, and the statements used in the monkeybat example were already perfectly falsifiable.

That said, I just found something related in this online ebook:

E. T. Jaynes, Probability Theory With Applications in Science and Engineering
Lecture 7: Queer Uses For Bayes' Theorem (pdf)
Section 7.3: Testing Scientific Theories


[Lucifer]
In any case, the scientific method is not at issue. What is at issue is Popper's claim that "None of this is altered in the least if we say that induction makes theories only probable rather than certain." This seems to be inconsistent with Bayes. Confidence in a theory should increase with supporting evidence. Bayes theorem doesn't increase confidence with any supporting evidence, it has to be evidence that is more likely when the theory is true than if the theory is false. This subtle point is often lost on critics.

[rhinoceros]
I still can't see the inconsistency. Where is it? Subtle point granted, but confidence in many different alternative theories can increase too with the same available supporting evidence.

It would also be interesting to see whether and how Bayes' theorem would handle an empirically verifiable but not falsifiable inductive theory. Or is it also a prerequisite of Bayes' theorem that universal statements should be falsifiable?


[Hermit]
*Millikan's Fraud (Source: http://lachlan.bluehaze.com.au/popper.html with editing and minor formatting. Interestingly, a site advocating the use of Bayes theorem to avoid such problems!)

Robert Millikan is widely regarded as one of the founders of modern American science, his determination of the charge on the electron winning him the 1923 Nobel Prize for physics. In a now-famous study, the physicist and historian Gerald Holton examined the log-books for Millikanís experiments with the electron, and revealed that he repeatedly rejected data that he deemed "unacceptable" (Holton 1978). The criteria he used were blatantly subjective, as revealed by the comments in the log-books, such as "Very low - something wrong" and "This is almost exactly right". Throughout, Millikan appears to have been driven partly by a desire to get results that were self-consistent, broadly in agreement with other methods, and consistent with his personal view that the electron is the fundamental and indivisible unit of electric charge.


<snip>

[rhinoceros]
This brings us to Imre Lakatos, his concept of "series of theories" or "research programmes" rather than standalone falsifiable theories, and his concepts of "negative" and "positive" heuristics.

Falsification and the Methodology of Scientific Research Programmes
http://virus.lucifer.com/bbs/index.php?board=32;action=display;threadid=28667

As Lakatos' writing is rather heavy with jargon of the field, I added a well written and very interesting critical review of his book which I found last week.

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