Critical Rationalism vs. Inductive and Subjective Interpretations of Probability

So I am trying to understand the critical rationalist arguments against the inductive and subjective interpretations of probability. I am not all that familiar with the matter, and so I have likely made some elementary mistakes — feedback is appreciated. Although the views expressed here are based on the arguments of Popper and Miller, I have only been exposed to their relevant work secondhand. I thank Kenneth Hopf for originally explaining the basic argument to me, though he shouldn’t be blamed for what are probably my errors.

A ROUGH HISTORY

Repeated failure to justify a principle of induction once caused much frustration among philosophers of science. If no finite set of evidence entailed the truth of any scientific hypothesis, then one could not be justified in believing that any such hypotheses were true. Science had a crisis of identity.

The enlightenment had elevated science to rival traditional political and religious authorities, but to fulfil this role science had to offer its own genus of justified true belief. Sense experience and the experimental method, buttressed with logic and mathematics, emerged as new authorities to challenge custom, might, and religious superstition. But the problem of induction threatened to undermine this edifice; it cast doubt upon the fundamental principles of the scientific method and challenged the integrity of practicing scientists.

Critics accused scientists of hypocrisy: their declarations required a leap of faith no less than religious dogma. Going beyond the evidence, the truth of neither could be guaranteed by rational authorities. But while the devout believer could then appeal to religious authorities for answers, the scientist’s decision to prefer one hypothesis over another was arbitrary. Further investigation revealed that even our most mundane beliefs could not be justified, e.g. other minds exist, the sun will rise tomorrow, and the future will resemble the past.

The problem of induction had been identified, but all attempts to discover a logical solution failed. Positing a special inductive rule of inference or principle of natural uniformity shifted the problem to justifying said principles, but all attempts to do that were demonstrably invalid or question begging.

The emerging field of mathematical probability suggested an alternative solution to the problem of induction: perhaps one could not be justified in believing that any hypothesis is true, but one could be justified in believing that some hypotheses are more probable than others. The theory of probability seemed to offer a way to justify preferences between competing hypotheses without the need to justify the belief that any hypothesis was actually true. Scientists could now answer charges of hypocrisy while remaining loyal to their rational authorities; the only concession was that establishing any hypothesis with absolute certainty would be impossible.

To make the transition from believing in the truth or falsity of hypotheses to believing in their probability, philosophers needed to figure out how to assign and interpret the probability of a proposition. In logic, one would normally assign truth-values to propositional variables, but truth and falsity were now just limiting cases represented by 1 and 0, and the full range of numbers between them could now be assigned.

Two rival interpretations of probability emerged to make sense of statements such as “the probability of P is .75.” The frequentists wanted to interpret probabilities similarly to the idea of truth: probabilities were descriptions of objective facts or, specifically, classes of events. The subjectivists wished to interpret probabilities as the subjective degree of confidence that one has (or should have) given the evidence. With the ascension of Bayes’s theorem and difficulties with the frequentist interpretation, the subjectivists eventually came to dominate.

During all this, the problem of induction was not abandoned; the inductive nature of science could be preserved if induction was probabilistic. While no finite set of evidence entails the truth of any scientific hypothesis, the future events predicted by a hypothesis may be more probable given supporting evidence of past events. The problem of induction had apparently been solved, albeit in a rather more pragmatic and qualified sense than originally desired.

THE CRITIQUE

The critical rationalist critique of induction has many facets; not all of the objections will be covered here. However, before moving onto the central argument of this post, one point is worth emphasising. Parted from the presuppositions which led to the problem of justifying beliefs with sense experience, induction just doesn’t do anything. For critical rationalists, induction has no useful role in the critical evaluation of rival conjectures; it is reduced to merely another invalid inference: a logical fallacy. The rejection of induction runs far deeper than just the observation that induction is invalid or question begging, and the following arguments should be understood in this context.

[EDIT: I have attempted to restate the following argument using a different approach here.][EDIT: Well, there was another argument. I accidently deleted it; I’ll rewrite it again soon.]

Let A and B be propositional variables. Suppose the probability of A is less than probability of A given B. In such a case, B is said to support or partially confirm A by probabilistic inference. For example, suppose

p(A) = .6
p(B) = .4
p(B|A) = .5

From Bayes’s theorem, one can derive

p(A|B) = .75

By subtracting p(A) from p(A|B), one can now calculate precisely how much A is supported by B:

p(A|B) – p(A) = .15

The result seems straightforward: A is not a logical consequence of B, but B increases the probability of A by 15 percent. B appears to be amplified by probabilistic inference. While the induction falls short of implying the truth of A, it nonetheless increases the probability of A being true. Therefore, some say, one should invest more confidence in the truth of A given B than one would have before accepting B.

The counterargument to this view depends on a simple logical equivalence:

A =||= (A v B) & (A v ~B)

It follows that both have the same probability. That is, anything that changes the probability of one side of the logical equivalence must equally change the probability of the other side. The point of decomposing the equivalence on the right is to reveal how the probability of A relates to the probabilities of A v B and A v ~B. In fact, the degree to which B supports A can be calculated by adding the degree to which B supports A v B and A v ~B, respectively:

p(A v B|B) – p(A v B) + p(A v ~B|B) – p(A v ~B) = .15

On the inductive view of probabilistic inference, B is amplified to imply that A is more probably true. This would would mean the logical consequences of A which are not also logical consequences of B should be more probable given B. However, since the probabilities of both A and B are greater than 0 and less than 1, it follows that

p(A v B|B) > p(A v B)

and

p(A v ~B|B) < p(A v ~B)

In other words, while B increases the probability of A v B, it actually reduces the probability of A v ~B. From this we get,

if p(A|B) > p(A), then |p(A v B|B) – p(A v B)| > |p(A v ~B|B) – p(A v ~B)|

That is, to the extent that B increases the probability of A, it does so by increasing the probability of A v B more than it decreases the probability of A v ~B. However, since A v B is a logical consequence of B to begin with, the increase in probability is a purely deductive inference.

The inductive view of probabilistic inference rests on the fallacy of decomposition, i.e. assuming that what is true for the whole must be true for its parts. Not only do logical consequences of A which are independent of B not increase in probability, they may actually decrease in probability. This concludes the refutation of inductive probability.

The subjective interpretation of probability might be retained even if probabilistic inference is not inductive. While it may be true that probabilistic inference cannot amplify B, it can still be used to select among alternative propositions. Probabilities can help us keep score and choose preferences without any presumption of induction. More importantly, probabilities can still be interpreted as the subjective degree of confidence that one has (or should have) in a proposition given some other proposition.

The critique of the subjective interpretation of probability is an extension of the previous argument against inductive probability. Consider that

p(A v ~B|B) < p(A v ~B), therefore p(~(A v ~B)|B) >  p(~(A v ~B))

Since the probabilities of A v ~B and ~(A v ~B) must add to 1, if B decreases the probability of the A v ~B, then it must increase the probability of ~(A v ~B). Therefore, in the subjective interpretation, given B, one should have increased confidence in both A and ~(A v ~B), but that is a flat contradiction with a probability of 0.

While the probability of A v ~B remains greater than ~(A v ~B), what this argument demonstrates is that probabilities do not behave like subjective degrees confidence or belief. When one says that he is more confident in some proposition, he does not mean to suggest that he is also more confident in the contradiction of that proposition, but that is exactly where one may end up with in the subjective interpretation. The concept of subjective degrees of confidence or belief just does not have the same formal structure as the concept of probability, because the former is truly “inductive” in a way that the latter is not.

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74 Responses to Critical Rationalism vs. Inductive and Subjective Interpretations of Probability

  1. Lee Kelly says:

    Nobody biting? Oh well, perhaps I’ll try something else.

  2. Rafe says:

    It is a pity that nobody has got into an argument here, maybe the case against inductive probability is too strong?

    One of my complaints about the obsession with probabilities is that it diverts attention away from scientific problems, the rival theories/programs/tentative solutions etc into technical issues in probability theory. You come away from these arguments feeling that you have learned nothing about science or the world at all. Much the same applies to the massive literature on the philosophy and methodology of economics – both Deirdre McCloskey and I think that this is almost entirely a waste of space because next to none of it is connected to live issues in the field. An exception is Stanley Wong’s critique of Samuelson, which is CR through and through.

  3. Ryan Murphy says:

    My formal education in symbolic logic mostly does not exist (though I still pretty much followed it), but my intuitive priors strongly favor the subjectivist interpretation. I’d have a better grasp of what you were getting at if you tied this in more directly historically with what Keynes and F.P. Ramsay were saying. My opinions on this issue mostly comes from what Nassim Taleb has to say in The Black Swan (especially the added material in the second edition) … I don’t think you would accuse him of backing induction into subjective probability… and also Bryan Caplan’s debate over this issue with Walter Block.

  4. Constantius says:

    I can present argument against the dogmatism of over-confident “Popperians”;
    I, unlike others will be presenting an attempt of justifying induction in a novel & unique way……………..thru Abductive or retroductive inference, thus saving my argument for induction from the charge of question begging.
    My reply to you will be like this.
    Our (sensual) experience of the universe is instantiated by finite number of acts of observations;
    These observations render the same conclusion (provided our empirical & cognitive faculties necessary for comprehending this physical universe is unhindered) when it comes to behavioral regularities of natural objects under given conditions (laws of nature), physical/numerical constants & causal connections between objects & phenomena’
    So all this can best be explained by the (fallible & falsifiable) hypothesis that the universe at least in its observable manifestation is uniform in terms of these features.
    I’m a pragmatists & fallibilist, so I accept that it can be mistaken but until a reliable occurrence of the refutation of it takes place scientifically, it can be regarded as a valid assumption that is necessary & sufficient for the truth of inductive conclusions provided the premises are true.
    p.s. any criticisms will be appreciated. Thank you all………

  5. Constantius says:

    Of course my idea is not to defend inductive fallacies……they are there, but you & other Critical Rationalists expect to much of deduction, Popper even called it the only canon of reason!!!!! You can’t possibly justify deductive reasoning by deduction just as induction can’t be used to justify induction. That is why your “philosophical grandpa'” Hume rejected the possibility of deduction being justified without circularity of reasoning because induction, the only option left that could be used to justify deduction was ruled out by him, since “Induction is bogus” according to him

  6. Constantius says:

    Hey, that’s strange, people seems to be too silent about the post as well as its commentary……………………………people don’t seem to be interested in criticism & complaint any more. Popper’s followers seems to be less vigorous & rigorous now then the Bolsheviks following Marx in the early 20th century. Popper called for criticism & refutation, Marx called for revolution & reconstruction. The latter project failed, Popper’s project also seems to be proving a fad, just like “Logical Positivism”; I mean, which scientist actually follow this doctrine with rigor? May be some of them are impressed by Popper’s fallibilism (even me too), or by his falsifiability criterion as it requires testability & criticism which are antidotes against dogmas. But no one needs Critical Rationalism to be a Fallibilist or to adhere to the idea of testability of scientific propositions.

  7. Lee Kelly says:

    Ryan,

    My argument is that the concepts of probability and confidence behave differently. Perhaps I should explain how I think confidence works.

    In my experience, if someone has 70% confidence in A, then they will have at least 70% confidence to all the non-tautological consequences of A. If an event increases someone’s confidence in A by 10%, then they will be 10% more confident in all the non-tautological consequences of A (while not exceeding 100%). That is, a change in confidence because of some event is transmitted from premises to conclusion similarly to truth.

    If probability and confidence were analogous, in this sense, then the probabilities of A v B and A v ~B given B should both be at least as probable as A given B. However, the argument shows this to be false; A v ~B given B is actually less probable than A given B. Furthermore, if A v ~B is less probable given B, then its negation, ~(A v ~B), must be more probable. It follows that if B increases the probability of A, then it also increases the probability of ~(A v ~B), but ~(A v ~B) actually contradicts A.

    Neither of these results meshes with the idea of confidence or degree of belief: the logical relations do not work out the same.

  8. Rafe says:

    Constantius, it looks as though there are no Popperians in academic philosophy, apart from some who are still active in retirement. This is not due to serious defects in CR, it is all about lazy scholarship and other fads that captured the field. All good scientists are Popperian, that was explicit in the case of an older generation like Einstein, Medawar, Eccles and Monod. It was also obvious in the case of Feynman, although he never discovered Popper and he had contempt for the philosophy of science. Weinberg is regarded by some people as the dean of scientists in the US and he has endorsed Popper although that is not well known.

    I always tell people that a scientist who attacks problems in a critical and imaginative way does not need to take on board a lot of Popper, only if his or her activities are being held back by defective ideas from other schools of thought.

    Because the positivists and logical empiricists were absessed with the definition of science, demarcation and verification, Popper was sucked into these debates for decades, indeed for the rest of his life, long after 1934 when he shifted the advantage line. You need to look past demarcation and see the other advances that Popper made, so he was moving the advantage line time and time again while the rest of the profession was virtually left behind. When some outstanding young philophers find out what has happened and decide that the charade has gone on too long they are going to ask their elders some very awkward questions. Perhaps they will be sidelined like Popper. That would be an intellectual and cultural scandal and a great reflection on the profession that is supposed to have a professional interest in the truth!

    See the Popperian “turns” for a broader view of CR.
    http://www.the-rathouse.com/Pop-Schol/PopperTurns.html

  9. Constantius says:

    I appreciate Popper’s concerns & his solutions, but I think he took some aspects of his philosophy to the extreme in multiple occasions. His commitment to CR and its core ideas (i.e. fallibilism, Critical Rationality, falsification) are well justified……..but he also made mistakes like the method of calculating verisimilitude & the issue of the Quine-Duhem thesis (i.e. how can we justify the refutation of a hypothesis in isolation when it can’t be tested in isolation, but in conjunction with other auxiliary assumption(s)?). It is of course nothing to be ashamed of, since all of us are indeed fallible. The idea of Fact, Conjecture & definition as expressed by him in his writings are confusing to say the least..he said facts are expressed in descriptive statements but his conception of knowledge don’t have any place for them; as he said “All knowledge are conjectural”, then how we know facts, after all “facts ain’t conjectures!!!”

  10. Rafe says:

    How is CR damaged by the failure of Popper’s verisimilitude program?

    What is the problem with his take on the Duhem problem? Adverse observational reports render one or more hypotheses problematic. So…?

    Do you have any comment on the Popperian turns?

    What practical or theoretical problems are you working on which you can share with us to give some background and context for your contribution?

    The reason for that question is that I am frustrated by reading so many philosophy texts where there is no reference to current problems of science or society. The examples that are used tend to be either trivial (knowing that it is raining) or slightly weird (that there is/is not a rhino in the room). The question has to be asked whether they are really interested in the quest for knowledge or whether they have just found a way to make a career by recycling arguments that are more or less clever in the eyes of their peers but don’t really go anywhere or relate to scientific investigation and social reform the way that Popper and the CR people do most of the time.

  11. Constantius says:

    I will be happy to know your take on Quine-Duhem thesis and Popper’s response.
    Now coming to the issue of knowledge & conjectures, see if all knowledge is conjectural then no fact can be knowledge, because they are ontologically different, so according to Popper (I’m not criticizing the principles CR here) we can’t “KNOW” facts yet he admits that facts can be known implicitly as well as explicitly; he does that by stating his criterion of truth; A propositions is said to be true iff corresponds with a fact. Now he says that though we can’t know the truth, we can determine truth-likeness. But to determine that we must presuppose the idea that some truth can be known (at least that portion of truth which is required to establish verisimilitude), so it logically implies that some facts can be known, otherwise we can’t determine the truth-likeness because we won’t know to which degree the proposition corresponds with relevant fact(s) if we don’t know the fact first. If so, then the idea that all knowledge is conjectural/hypothetical is a mere skeptical self-deception, at least some knowledge must be true (at least in principle) without any warranted doubt such as that of facts.
    And you also asked how the failure of Popper with regards to verisimilitude hurts CR. Well it hurts CR quite a lot, if you know what I mean.
    If we don’t have a sound method for that then there will be no way to know which one from the multiple conjectural solutions of scientific problems is the most worthy of attention and use, because we can’t be able to distinguish between them in terms of their performance. Also if there’s no easy, reliable way to tell which hypothesis is better then its competitors then we may have too much disagreement among scientists. Because the mission of CR is not the refutation of everything possible but to show a way how we can detect & correct errors in our knowledge; if we can’t measure the degree of correctness, then how con we be confident that our current solutions is worth the efforts put behind it, after all some other hypothesis might well be able to do better then that. So if we can’t find that out, we be left in the dark regarding our potential ability to enhance our understanding of reality by using science & its method.

  12. Rafe says:

    I don’t know what you mean when you say that problems with the quantification of verisimilitude hurt CR.

    This is my take on the Duhem problem. It is an M Sci thesis that was supervised by Alan Chalmers. http://www.the-rathouse.com/Theses/Duhem-QuineIntroPopperians.html

    On the other issues, can you provide some context, with arguments that relate to specific problems of science or practice.

    Are you sure that your defence of induction is original?

  13. Constantius says:

    I’m sure as about my defense of induction that it is an original contribution of mine, I need no copycat skills to defend Induction, however if you have any reason to doubt me then you can very well present any reliable & credible evidence to undermine my claim.
    I’m a Pragmatist & Fallibilist in my philosophical orientation as I have said earlier. I am not a professional philosopher or logician. I just take up issues in the philosophy of science as a hobby and nothing else.
    Also if you repeatedly fails to understand my reasons to consider Popper’s defective idea of verisimilitude hurting CR then I assume its my failure to communicate with you which is behind this. I have given the reasons about if you fails to take notice. Read them carefully, Hope you will realize my arguments. Tell after that how a defective method of measuring truth-likeness doesn’t hurt CR I’m damn interested to hear from you sir.
    Oh, yeah, don’t forget to come up with a refutation of my justification of induction if you can, thank you.
    The context you are seeking is about the general theoretical & technical criticisms Popper’s ideas have been subjected to over the years. I think you know them more then I do. Alan Sokal’s “Fashionable Nonsense” & Nicholas Dykes “Debunking Popper” (available in the internet) can give you the overview of my position because I agree to most of what they say, if not all.

  14. Hi Constantius,
    If your position is the same as Nicholas Dykes’ “Debunking Popper” you’re in trouble because Dykes’ essay is hopeless.

    Example: In what he obviously considers a bit of a dramatic opening, Dykes tells us that the Law of Identity refutes Hume’s criticism of induction, thus in turn refuting the basic premise of Critical Rationalism. But Dykes, under the unfortunate influence of by Ayn Rand, is, like her, rather overinvested in “A=A”. This line of argument is dimwitted for at least two reasons: 1) Should a hawthorne bush unexpectedly produce grapes one fine day, this would not conflict with the LOI at all as of course it could now be said that it was in the nature of hawthorne bushes to unexpectedly produce grapes!. Far from an iron law keeping philosophy in line as Dykes, aping Rand, believes, it’s a veritable chocolate teapot of a thing. 2) This argument also confuses two quite distinct things: a) reality and b) our knowledge of said reality. Hume’s argument applies to the latter and not the former, whereas Dykes invokes the LOI in regard to the former, not the latter, thus giving us a masterclass demonstration in missing the point. The rest of his essay does not improve.

  15. Constantius says:

    I haven’t subscribed to Dyke’s critique of induction. If you saw the above commentary of mine, there I have presented my own justification for induction. Furthermore it will be really helpful if you can point out the problems of other aspects of that essay. I didn’t say I agree to everything he said, did I? I said that I agree to most (a large portion) of what that person has to say, not each & everything that is there. There are defective things in that essay, like he said that Popper didn’t provided us with a criterion of preferring a hypothesis over competing ones. Popper actually did gave such a criterion; that we should look for the hypothesis with greatest amount of logical content. I acknowledge these mistakes, obviously.

  16. Constantius says:

    I also want to point out a error in your argument against Dyke’s critique of Hume; you said that the LOI is capable of explaining away anomalies like hawthorns producing grapes etc. See, if grapes start growing in hawthorn plants then why should we call it a hawthorn plant. Just because a plant has relative similitude with our prior conceptualization of hawthorns based on their essence, we can’t call them hawthorns. We need to observe whether or not our observational experience of the empirically demonstrable features of that plant (exactly) corresponds with our expectation of what a hawthorn plant is really like (ultimately in terms of the prior conceptualization of hawthorns). If our concept of hawthorn plant implies that it wont produce anything that satisfies the conceptualization of grapes, then anything that produces grapes (unexpectedly or not) will not be eligible to be identified with hawthorn. So, if a plant that unexpectedly produces grapes can’t be called hawthorns as it fails to exactly correspond to our expectation of what a hawthorn must & must not possess or be able to do. Because no being carry identity tags with them to make them known to other observers, their identities are all artificial constructions by we humans. Once it is established that a term (and its conceptual content) exclusively used by any community to refer to a certain being, a individual member of that community, in order to expedite his/her communication, would take up that convention.

  17. d says:

    Constantinus, I hate to intrude, but upon reading your most recent comment, I can only see a word-jumble of concessions to Barnes’s very point.

  18. Constantius, if you’d like to know more about the problems with Dykes’ essay I suggest you look at commenter d’s excellent criticisms here.

  19. Constantius says:

    d says:
    July 11, 2011 at 5:49 am

    Constantinus, I hate to intrude, but upon reading your most recent comment, I can only see a word-jumble of concessions to Barnes’s very point.

    First of all you’ve made a mistake in writing my name!!! It’s Constantius, Not Constantinus.
    And then, What’s actually the point of Mr. Barnes to which I’m conceding?

    I told him to read between the lines. Dyke is a human, I am a human & you are also a human, we are all fallible. But the point I made in my previous comment was that Mr. Barnes confused between Hawthorn plant & something that resembles Hawthorns in its general features but also contains a “Novelty”; grapes produced by what appears (in mind of Mr. Barnes) to be “Hawthorns”. Mr. Barnes told me that this won’t be in conflict with the Law of Identity; “A=A”, as the expression of anything by a being is a part of the identity of that being. Even by producing grapes, the hawthorn plant will retain its identity by allowing the inclusion of the ability to produce grapes as another ‘new’ feature.

  20. Lee Kelly says:

    I had been thinking about writing a post about this very issue. However, the notion that the problem of induction can be solved by appealing to the law of identity is kind of preposterous. I decided that anyone who would accept such an argument probably isn’t going to appreciate any criticism. But, anyway …

    (1) The inductive fallacy is really just a particular form of the fallacy of affirming the consequent: the truth of the consequent does not imply the truth of the antecedent. It is an elementary consequence of how the material conditional is defined. However, this “solution” to the problem of induction effectively asserts that we can validly infer the truth of the antecedent from the truth of the consequent . In other words, there are no alternative antecedents which are consistent with the consequent: if it produces haw-like berries, then it must be a hawthorn bush. For those things we have divined the essence of, all material conditionals become biconditionals.

    (2) Ironically, this “solution” to the problem of induction bears one thing in common with critical rationalism: it denies the existence or usefulness of induction. That is, if one can validly infer a universal from particulars by appealing to the law of identity, then deduction can achieve, by itself, everything one might desire of induction. Therefore, there is no such thing or no need for any kind of inductive logic.

    (3) The practical error of this “solution” is that it confuses questions of logical implication with problems of empirical fact. It amounts to little more than redefining the meaning of terms to “discover” empirical truths. For example, we can validly infer that all swans are white, because a swan is, among other things, a white bird. However, defining swans as white birds doesn’t resolve the empirical issue; it just shifts the matter to whether swans, defined as a species of white-only birds, actually exist. That is, the question we ask changes from being about a universal (“are all swans white?”) to being about an existential (“does a species of swans exist?”), but the empirical problem remains the same. We cannot learn about the facts by subtlety redefining our terms, and the law of identity can’t help with that.

  21. Constantius says:

    I recognize the problems of LOI & recognize them very well, I’m not here to defend the LOI. That’s the point made by Dyke. I expressed my support for a large part of his case against CR, not necessarily all of his arguments. Actually I gave my own defense of Induction above in this page. Just like you Mr.(or Ms./Mrs.) Kelly, I have also pointed out an error in his thesis and I’m sure there are others, which I have admitted. What I contended is that there are practical as well as theoretical criticisms of Popper & not all of them are gibberish, can you deny that? Or do you think Popper is above mistakes? That’s what I’m concerning myself with. The topic here was about inductive probability, actually Popper in conjunction with David Miller worked on this and published their paper on it though I can’t remember the name right at the moment. There they argued that its hopeless to expect anything substantial from this idea of inductive probability. I’m a supporter of Charles S. Peirce in this regard. He thought that Subjectivism in this context is utterly mistaken. Of course there can be difference in the meaning of the term in your & Peirce’s essays, I’ve yet to read the entire extent of the work of Peirce on this.
    However, I’m agreeing with the argument that LOI do really have problems as Lee Kelly pointed out, but I’m still sticking to my previous position regarding what Mr. Barnes said.

  22. d says:

    Excuse me–Constantius, but when you say,

    “… the point I made in my previous comment was that Mr. Barnes confused between Hawthorn plant & something that resembles Hawthorns in its general features but also contains a “Novelty,”

    I don’t know how you could miss it: Barnes is not confusing anything; it is we who are confused. The properties of hawthorns are in fact Goodman-predicates. Are the properties of hawthorns still hawthorns? Of course. They would just have grapes after time t. Our theoretical language is the only thing wrong–not the LOI, for it is a truism.

  23. Constantius says:

    I think you need a better understanding of properties & their function in the identity of objects. Properties in & of themselves not the object, as you tried to say; “Are the properties of hawthorns still hawthorns? Of course.”. How come properties of Hawthorns be Hawthorns? Would you say that the properties of having two hands, the ability to rise up straight & having beard are in and of themselves human being? They are but the features we identify with what is called human beings. Any real object has two dimensions of being; existence & essence. Essence is the bundle of properties that don’t make up the being by their own virtue. They require to be connected with a particular instance of the phenomenon of existence to produce a being.
    Now comes the issue of LOI.
    Does the truism “A=A” resembles “Hawthorns (without grapes, the “Novelty”)= Hawthorns with grapes”?
    It doesn’t seem to be like that, does it? If the term “Hawthorns” is used to refer to a being that don’t produce grapes at time ‘t-1’, then to use that term at time ‘t’ for a being that actually produces it we need to reform the concept to include the grape producing ability. And then the concept of time ‘t’ won’t correspond with that of ‘t-1’ i.e. the prior concept. So the same term is going to be used for beings that are not exactly the same in their respective nature.

  24. d says:

    Constantius,

    I suggest a more charitable reading–I misspoke, and meant to say “…are the properties of hawthorns still the properties of hawthorns.”

    The rest of your comment: you need to take Goodman-predicates seriously; you confuse the word–with all its theoretical assumptions–with the objective properties of the object. If emeralds were to be grue, then emeralds would be grue, no?

  25. Lee Kelly says:

    Constantius,

    You have made many claims. Let’s deal with them one at a time. This comment will be about your solution to the problem of induction.

    You claim that a principle of natural uniformity can be justified using abductive inference. Since abduction is not induction, this avoids accusations of question begging.

    (1) A principle of natural uniformity is not logically strong enough to validate inferences from just any particulars. That is, to infer from “some As are B” that “all As are B” is to assert that nature is uniform in a very particular way. However, a principle of natural uniformity merely asserts that some relations are uniform, otherwise consistent application of such a principle would quickly generate contradictions. Therefore, even assuming a principle of natural uniformity, it is not enough to infer from “some As are B” that “all As are B.”

    (2) The problem with induction is really a problem of logical strength. That is, the conclusions of inductive arguments are logically stronger than their premises. All the issues concerning the invalidity or underdetermination of inductive inference result from this, and it is a feature which induction and abduction have in common. Abductive inference is formally equivalent to the fallacy of affirming the consequent, and induction is just a special case of that fallacy. Therefore, most of the logical objections made against induction can also be made against abduction.

    (3) Abduction is ostensibly an inference to the best explanation, but the “best” explanation for any given abduction is just a deductive result of unstated assumptions. Wikipedia summarises abduction like this:

    For example, the lawn is wet. But if it rained last night, then it would be unsurprising that the lawn is wet. Therefore, by abductive reasoning, the possibility that it rained last night is reasonable.

    Notice that abductive inference depends on all manner of unstated assumptions about rain, clouds, and grass, or the probability of alternative explanations of why the grass is wet. One who does not share these assumptions may abduce an entirely different conclusion. What explanation is “surprising” or plausible is merely a function of what one already believes, and so objecting to alternative beliefs for being more “surprising” is merely to object because they are different.

    (4) For abduction to be valid, one would need to introduce some principle to the effect that all “best” explanations are, in fact, true explanations. Putting aside the matter that such a principle is obviously false, attempting to justify it by claiming it is the “best” explanation would create a problem of abduction analogous to the problem of induction. Since nothing gets justified by invalid or questing begging arguments, it follows that abduction cannot be justified by an appeal to any principle that all (or some) “best” explanations are true.

    (5) Finally, it is logically impossible to justify anything, whether induction, abduction, or deduction. That is, it is a purely logical truth that all attempts to justify the conclusion of an argument with its premises are either invalid or question begging. There are no alternatives. My previous objections are more about why induction and abduction are useless for the critical evaluation of competing ideas, because their unjustifiability is has to do with the impossibility of justification itself.

  26. Lee Kelly says:

    Constantius,

    You claim that deduction cannot be justified using deduction.

    This is true. Critical rationalists do not use deduction because it is justified. Critical rationalists use deduction because otherwise critically evaluating competing ideas is impossible. Criticism proceeds from the logical analysis of ideas and the retransmission of falsity from conclusion to premises. Obviously one cannot be a critical rationalist unless one accepts some minimal logic for the purpose of critical analysis, but nothing is presumed to be justified because of that.

  27. Lee Kelly says:

    Constantius,

    You claim that Popper erred on the Duhem-Quine problem.

    There is a pervasive myth that Popper’s views evolved from “crude falsificationism” to a more nuanced and constrained position in response to prominent critics. However, the main thing to evolve was emphasis; Popper placed less importance in the problem of demarcation in later work. On most technical matters, the developed form of Popper’s work on falsification can be found in the original Logic of Scientific Discovery. This is especially true concerning the Duhem-Quine problem, which could rightly be called, like Rafe Champion argued, the Duhem-Popper-Quine problem.

    Popper addressed the Duhem-Quine problem at length; whole chapters in Logic of Scientific Discovery scarcely make sense except as attempts to address the issue. However, Popper offered no purely logical solution to the Duhem-Quine problem, because there is no such thing. Perhaps this is why so many philosophers act as though he never addressed the issue at all.

    Popper instead proposed a methdological solution: we should voluntarily refrain from ad hoc dismissals of recalcitrant evidence or, what Popper called, “conventionalist stratagems.” Such methodological prescriptions are no guarantee against mistakenly rejecting a true theory and accepting a false criticism, because no such guarantees exist for fallible beings. However, methodological rules of this sort encourage innovation and, supposing our efforts to learn the truth are not entirely futile, gradual improvement to our knowledge.

  28. Constantius says:

    Lee Kelly, you said:
    A principle of natural uniformity is not logically strong enough to validate inferences from just any particulars. That is, to infer from “some As are B” that “all As are B” is to assert that nature is uniform in a very particular way. However, a principle of natural uniformity merely asserts that some relations are uniform, otherwise consistent application of such a principle would quickly generate contradictions. Therefore, even assuming a principle of natural uniformity, it is not enough to infer from “some As are B” that “all As are B.”

    Your contention is indeed true to a certain extent, though it has its own share of problems.
    In science, you can’t help the use of induction to propound tentative & fallible principle, or law statements (with a certain initial degree of verisimilitude, stemming from the premises, as the premises are included in the universalized conclusion).
    There is no other inference i.e. deduction in particular which a scientist can use for this end, without using a finite number of observation reports to give a possibly wrong but still workable tentative solution (in the form of a universal proposition) to the problem in hand i.e. the necessity of a law to explain any given phenomenon.
    They point out behavioral regularities in natural objects under specific given conditions from a given range of data containing the content of finite number of acts of observation and thus identify (possible, but not certain) laws of nature, assuming that this regularity is probably uniform. How can this be denied, even working scientists can’t deny that their works contain straightforward or subtle application of induction in a range of ways.

    Abductive conclusions even if they do not refer to the best possible option, can still be used. The idea of Abduction as equivalent to “affirming the consequent” is still not clear from what you said just earlier. I hope you’ll to give me more insight into this the next time round. See, abduction can be a tool to produce a logically sufficient (but not necessary) hypothesis. C.S. Peirce, who produced was a working chemist & geographer who found value in it, he developed not it to kill his time. The argument of mine, though restrictive, does not seem to involve any fallacy as the two premises can’t render anything better then this. If you look into it you can find out that there are three hypothetical (and even unrealistic) options that can be contemplated:
    1. What I’ve proposed;
    2. The idea that this apparent uniformity of nature in some of its features is a work of a wicked devil who manipulates nature every time we try to observe the universe;
    3. We are deceived by our senses in seeing what is not there.

    Now you tell me what will be your explanation should you find out the premises to be true. Will you agree with me, or say that until a valid deduction can be made with them, every premise is useless, because there is no other inference that is worthy of our high standard?

  29. d says:

    Constantius, in response to your comment to Lee, I can only quote what I said earlier this morning here:

    “Even if enumerative induction is permitted during the context of discovery, it does not help the scientist any more than dreaming next to a raging fire (read: Kekulé’s oroboros), drug use (read: Feynman, Kary Mullis), &c., which is to say that is has no privileged position over even the most arbitrary ‘methods.’ …

    “If ‘induction’ refers to the metaphysical assumption of regularity of systems, which we may approximate if enough inductions of the system are collected, then the inductivist retreats to asserting only that there exists regularities, calling this assumption ‘induction.’ If a proposed regularity should turn out to be false, then this was either a mistaken induction or not induction at all. If it is not [] induction, then this is little more than wordplay: we cannot tell this type of induction apart from a conjecture.”

    In sum, you retreat, you surreptitiously change words in order to save ‘induction’ (whatever it is at the moment).

  30. Constantius says:

    Also Lee Kelly, Where do you get the idea that abduction always seeks the “Best”explanation? Or the idea that Abduction requires a principle to the effect that all “Best” explanations are true.
    You are either ignorant of abduction or dishing out things you yourself don’t understand.
    The aim of abduction is to produce a tentative, & testable hypothesis with empirical content which can be subjected to critical tests for verification, corroboration or falsification.

  31. Constantius says:

    d says:
    Constantius, in response to your comment to Lee, I can only quote what I said earlier this morning here:
    “Even if enumerative induction is permitted during the context of discovery, it does not help the scientist any more than dreaming next to a raging fire (read: Kekulé’s oroboros), drug use (read: Feynman, Kary Mullis), &c., which is to say that is has no privileged position over even the most arbitrary ‘methods.’ …

    Also: “In sum, you retreat, you surreptitiously change words in order to save ‘induction’ (whatever it is at the moment)”.
    I preferred to advocate Induction in a very restrictive way from the very beginning, that’s why I gave my justification for it as applying only in a strict naturalistic sense; about the apparent uniformity of the universe in term of some of its features, and this justification has empirical content a well as verisimilitude, making it falsifiable (& thus scientific). Have any of you ever heard of “the Cosmological Principle” or Einstein’s assumption known as “the Principle of Relativity”, my principle belongs to that class. Because all of them logically imply the same thing; uniformity in one or more features of the nature we live in. The theory of relativity, Hawking’s so called (humorously) “Big Bang theory” all use these assumptions as auxiliary hypothesis while yielding predictions. If they can use it, then what’s the problem if we use it for our purpose.
    David Miller said, all our valid arguments are either circular or actually invalid, Don’t believe me? go find it out in his book “Critical Rationalism: a restatement & defense”. So even deduction is also bogged down with the same problem as induction (and abduction).

  32. d says:

    Constantius,

    Then you were not talking about induction at all; you were talking about conjectures–it’s still wordplay on your part.

    “David Miller said, all our valid arguments are either circular or actually invalid, Don’t believe me? go find it out in his book “Critical Rationalism: a restatement & defense”. So even deduction is also bogged down with the same problem as induction (and abduction).”

    I don’t think you understand what Miller was saying at all.

  33. d says:

    P.S.

    “I preferred to advocate Induction in a very restrictive way from the very beginning, that’s why I gave my justification for it as applying only in a strict naturalistic sense; about the apparent uniformity of the universe in term of some of its features, and this justification has empirical content a well as verisimilitude, making it falsifiable (& thus scientific).”

    So many questions …

    How is this kind of ‘induction’ restrictive when compared to other definitions of induction? Could you explain how your ‘induction’ holds any more epistemic weight than a conjecture? How is this inductive principle of the uniformity of nature falsifiable? How is the metaphysical assumption of naturalism falsifiable?

  34. Constantius says:

    First, The Principle of Induction that I mentioned is falsifiable If & only if someone can prove that some laws of nature (at least one) don’t hold true in the entire continuum of spacetime but are localized in certain restrictive regions within spacetime
    or
    If someone can prove physical/numerical constants like the speed of light or the Planck Constant don’t obtain in certain part of spacetime
    or
    if someone can prove that a phenomenon can be produced by different causes without rendering any conceivable changes in the nature of the phenomenon.

    Secondly, This principle applies only to the cases where laws of nature of other things like constants etc. deemed (tentatively) as capable of holding true irrespective of the observer & the region of spacetime where they are being observed, not each & everything like hawthorns & grapes, man & monkey, cat & donkey. That’s how it’s restrictive, it does not involve itself in all the claims of every Tom, Dick & Harry.

    Thirdly, an Inductive conclusion is an “educated guess” which includes the truth of its premises, rendering it with a initial degree of truth-likeness. On the other hand, your God-forsaken Conjectures are “Wild Guesses” (with nothing to its credit initially) which you Popperians like to call “Creative Imagination” ; which is nothing but a load of hot air. Before hypothesizing, every scientist makes (finite number of ) observations and formulate observation reports, then bases their hypothesis on them. Its far more economical & rational (in a economic sense) to start with something has an initial grain of truth in it then a massive load of “word-jumble”; consisting of a lot of conjectures from which we have to select the best one by some criterion.

  35. Constantius says:

    However, don’t fool yourself with other loads of junk from Popper. He says, All knowledge is conjectural……………how wonderful is it, isn’t it? It’s damn Wrong. Because if we are to assume some degrees of truth in our theories then we need to assume that the facts required to establish this verisimilitude, can be known in practice, because Popper & other CR people adhere to the Correspondence theory of truth, and Popper said that a propositions is said to be true if and only if it corresponds with a fact. We all “know” that the logical content of a universal proposition consists of a range of singular existential propositions. We need to know how much of them are true to measure how much truth is actually there in the hypothesis. For that we need to know the facts with which the propositions must correspond. Otherwise how can someone know that a hypothesis is partially true?
    But Popper with all his brilliance has got it all wrong. He left no room for the facts that is the catalyst for progress of knowledge in Popperian terms as the aim of science is truth according to Popper, and truth can be established by “facts”.

  36. d says:

    Constantius,

    I don’t think you or I will learn much from a continued conversation. You eliminate a priori the possibility of Goodman-predicates, claim this move is permissible because it is falsifiable, call this a principle of induction, and call it a day. What of it?

    Even if we assume that you will not, upon learning that a scientific theory is limited, blame the scientific theory and not your principle of induction, did you not notice that Goodman-predicates are falsifiable as well? So why prefer your falsifiable principle of induction over mine?

  37. Lee Kelly says:

    Constantius,

    I presented an argument that even if nature is assumed to be uniform, then universal propositions still do not follow from any finite set of existentials. You claim that my “contention … has its own share of problems.” However, you fail to provide any criticism of my argument. Instead, you claim that I “can’t help” but use induction, that verisimilitude is transmitted from premises to conclusion in an inductive argument, and that “scientists can’t deny” their work requires the “application of induction.” While all these claims are a mixture of falsehood and confusion, even if one of them were true, then it would still not challenge my argument.

    Let me restate,

    Some As are B
    Therefore,
    All As are B

    This is a basic inductive argument, and it is obviously invalid: the truth of the conclusion does not follow from the premises. Let’s introduce a second premise which asserts that nature has uniformities.

    Some regularities like “all x are y” are true
    Some As are B
    Therefore,
    All As are B

    Even with the assumption that nature has uniformities, the argument is still invalid. Why? Because merely asserting that some universal propositions are true doesn’t entail that any particular universal is true. For example, it is possible that “Some As are B” does not indicate a uniformity without it being false that nature has some uniformities. The truth of the conclusion does not follow from the premises and, therefore, a principle of natural uniformity is not strong enough to validate the inference of universal propositions from existentials.

    The critical rationalist solution to this problem is to not limit our premises to existential propositions. We flip the inductive argument on its head and put the conclusion in place of the premises and vice versa. The argument then becomes:

    All As are B
    Therefore,
    Some As are B

    This is a valid argument: the truth of the premises entails the truth of the conclusion. The premise is a conjecture which doesn’t need to be derived from observation. In consequence:

    Some As are not B
    Therefore,
    Not all As are B.

    This seems a bit like induction, but it is actually a deductive inference. The premises of this argument falsify the premises of our previous argument. Everything is deductive and valid. There is no need for anything like induction. What is more, we get a principle of natural uniformity as a logical consequence of our conjectures:

    All As are B
    Therefore,
    Some regularities like “all x are y” are true

    For critical rationalists, natural uniformity is part of what is being conjectured whenever one tries to comprehend the world. We do not have to assume that nature is uniform before making conjectures, because we have no pretense of inferring universal propositions from existentials. Natural uniformity just comes along for the ride as a metaphysical consequence of empirically testable theories.

  38. Lee Kelly says:

    Should I turn any of these responses to Constantius into full posts on the blog?

  39. @Lee, yes, good idea.

    @Constantius, this seems to be the nub of your criticism:
    “…if we are to assume some degrees of truth in our theories then we need to assume that the facts [are] required to establish this verisimilitude…”

    Have a think about what you’re asking here. It seems to me that you’re really saying that in order to establish the degree of truth-likeness, we first need the truth itself. Do you see the problem with that? If so, you’ll understand why your criticism of Popper unfortunately doesn’t succeed.

    In fact to get around this problem, all we need is the idea of the truth as a regulative principle. We can hope to get nearer to it by cutting away falsehoods, but there is no guarantee that we will succeed – it’s a target to aim at, even if we never hit it. And even if we do eventually have the unvarnished truth in our grasp, which CR says is possible if not very likely, the logical problem of induction means we can never know that we know it.

    This seems to be the situation.

  40. Constantius says:

    Seems that I’m drawing a lot of heat for being a Inductivist heretic!!!!!!
    Oh, well, let’s talk about how much criticisms CR followers are willing to dish out against their wonderful theories…..
    Has anyone the guts to criticize Popper’s principles that underlies CR?
    IF so, then why don’t any of you put up an article explaining the problems of CR?
    As Popper said, we must criticize, not justify ideas, but what I see in both their and my approach is nothing but justification of ones own position. Lets break this “vicious” cycle of justification & lets be critical…….But the problem with CR as I’ve mentioned above about facts & their status as knowable is still to be answered for.
    Daniel Barnes says:
    “Have a think about what you’re asking here. It seems to me that you’re really saying that in order to establish the degree of truth-likeness, we first need the truth itself. Do you see the problem with that? If so, you’ll understand why your criticism of Popper unfortunately doesn’t succeed.

    In fact to get around this problem, all we need is the idea of the truth as a regulative principle. We can hope to get nearer to it by cutting away falsehoods, but there is no guarantee that we will succeed – it’s a target to aim at, even if we never hit it. And even if we do eventually have the unvarnished truth in our grasp, which CR says is possible if not very likely, the logical problem of induction means we can never know that we know it.

    This seems to be the situation.”

    This situation, though appearing as the actual one is nothing but mistaken. I’m not asking about knowing the whole truth first to establish the truth-likeness (partial truth). I never demanded anything as such. What I asked was this: If a hypothesis says :
    All A’s have B; therefore, any (randomly collected) A must have B, then to corroborate the hypothesis we need to know the fact with which the singular proposition (that appears as the conclusion) is supposed to correspond should it be true. If the fact doesn’t correspond with the proposition, then the proposition is refuted. But Popper didn’t appear to recognize the existence of any room for facts to be part of our knowledge. He gave us a conjecture that all knowledge is conjectures. If so, then where are facts? Are they unknowable? If so, then how the hypotheses resists falsification & receives corroboration with the verification of some (if not all) of the singular propositions yielded by the universal one.

    Whether you like it or not, if a singular propositions is falsified (made false), that means its negation has been verified (made true, by facts as Popper said).
    This contra-positive verification is the result of agreement of relevant fact with the negation instead of the original proposition being tested, implying that facts are indeed knowable.

    My question is that how can we possibly start from a premise like “all knowledge is conjectural”?
    Instead Popper could’ve said “all knowledge can have mistakes underlying them and we can find those mistakes out by critically testing them”.

    Throughout this conversation, no body seem to reply to this point of knowledge, conjectures & facts, I asked Mr. Rafe, I asked other, only Mr. Barnes seems to be talking about it, though all what he said are wrong, he thought I demanded knowing the truth of the hypothesis to know how much verisimilitude it has, in fact, it’s the opposite, I asked that how can the logical consequences that are so vital for the corroboration or refutation of the parent hypotheses be tested for their truth values without facts?
    According to Popper, our attempts should be oriented towards demonstrating that at least some of the claims stemming from the universal hypothesis does not obtain. Then, to demonstrate this, we need to know the particular fact that fails the singular existential proposition. That means that if Popper’s true (that in principle, a hypothesis can be falsified by a counter-instance), then facts (that fails the logical consequences of the hypothesis) can be known. But if facts can be “known”, then not all knowledge claims are conjectures, thus Popper’s theory is self-stultifying

  41. d says:

    Constantius, I’m wondering if you need a primer in basic logic.

    “As Popper said, we must criticize, not justify ideas, but what I see in both their and my approach is nothing but justification of ones own position. Lets break this “vicious” cycle of justification & lets be critical…….”

    Do you honestly think that, once this conversation is over, Kelly, Barnes or myself would think critical rationalism is vindicated by us explaining how you have repeatedly misrepresented critical rationalism? Do you think we are stupid?

    “Whether you like it or not, if a singular propositions is falsified (made false), that means its negation has been verified (made true, by facts as Popper said).”

    And what of it? Its negation is little more than the existential statement “There exists at least one X that is ~Y,” which is identical to the universal statement “~All X are Y”. The conclusion’s logical content does not go beyond the logical content of the existential statement! If we should adopt an existential statement as true, provisional that it may be, all things being equal, we must then provisionally reject the universal statement if we wish to retain the logical rule of modus tollens. How is this problematic to critical rationalism in the least?

    And so on, throughout your entire comment. It would take an immense amount of time and effort to respond to everything you’ve just said, and honestly, I just don’t feel up for it, especially after your most recent remarks.

  42. d says:

    Constantius,

    P.S.

    When you said, “Has anyone the guts to criticize Popper’s principles that underlies CR?” that was a low point for you.

    It took me approximately five years after encountering critical rationalism to adopt it–five years of going through all the popular arguments against it, then the more obscure ones, reading up on Quine, Lakatos, Feyerabend, &c., until I adopted Bartley’s comprehensively critical rationalism + (as of last year) a version of van Fraassen’s constructive empiricism.

    Five years. Don’t tell me I didn’t have the guts.

  43. Constantius says:

    Lee Kelly says:
    July 12, 2011 at 3:09 pm

    Constantius,

    I presented an argument that even if nature is assumed to be uniform, then universal propositions still do not follow from any finite set of existentials. You claim that my “contention … has its own share of problems.” However, you fail to provide any criticism of my argument. Instead, you claim that I “can’t help” but use induction, that verisimilitude is transmitted from premises to conclusion in an inductive argument, and that “scientists can’t deny” their work requires the “application of induction.” While all these claims are a mixture of falsehood and confusion, even if one of them were true, then it would still not challenge my argument.

    Your utterly mistaken to say the least. No sane man can ever claim that Universals necessarily “FOLLOW” from particulars, and neither did I.
    What I said was that unlike Popperian conjectures:

    1. an inductive conclusion has a pre-existing degree of truth-likeness’ that stems from the truth of the premises i.e. the particulars.

    2. inductive conclusions are economical, i.e. their acceptance limits that number of hypothesis we’ve to deal with. This is because when an inductive conclusion is drawn from particulars, the commonalities in their properties are the main point of inference.
    If a set observation reports reveals that a group of resembling objects belonging to a certain type A has P, Q, R, X, Y & Z in common, then for optimizing the amount of logical consequence of an inductive conclusion drawn from them, we can possibly have one & only one conclusion that’s the most optimum; that which includes the most number of commonalities (all six of them, in the case of the above set of reports).
    If any observation undermines the claim that any one of the object of the type A does not have, say P, is to assert that the inductive conclusion fails to obtain.

    As such, the number of properties that are included, the more probability of it being refuted because the probability of its being true is inversely related to the number of elements the predicate asserts.

    3. Inductive conclusion has verisimilitude, thus it does not require question begging by assuming (without evidence) that it ‘is/may be’ truth-like.
    Popperian conjectures lack this property, because, there can be three type of hypothesis, a completely true one, a completely false one & a hypothesis with certain degree of verisimilitude.
    While trying to select a hypothesis from a group of rival, a Popperian can’t assume that the selected one is false before testing it, as declaring something false and then proceeding to falsify that makes no sense.
    He/she can’t also say the hypothesis is true, as we “know” that universals can’t be verified.
    The only remaining option is to assume that it (possibly)has verisimilitude, but to do so is nothing but question begging because that’s what he/she is trying to gain.

  44. Constantius says:

    d says:
    “Whether you like it or not, if a singular propositions is falsified (made false), that means its negation has been verified (made true, by facts as Popper said).”

    And what of it? Its negation is little more than the existential statement “There exists at least one X that is ~Y,” which is identical to the universal statement “~All X are Y”. The conclusion’s logical content does not go beyond the logical content of the existential statement! If we should adopt an existential statement as true, provisional that it may be, all things being equal, we must then provisionally reject the universal statement if we wish to retain the logical rule of modus tollens. How is this problematic to critical rationalism in the least?

    Sir, did I told any one of you not to reject the universal proposition even after on of its consequences has been proven false (at least tentatively)?
    Not at all, what I was asking about was how can we believe that all knowledge is conjectural when we also say that a hypothesis can be proven false? Is the proof conjectural, if so, then why it has more weight then the conjecture it’s supposed to refute?
    That was my point, now you have grossly misrepresented my view & taken it to a wrong track. I never meant to say what you make out of my remarks.

    And also try to read your own remarks carefully; “And what of it? Its negation is little more than the existential statement “There exists at least one X that is ~Y,” which is identical to the universal statement “~All X are Y”.”
    I mean, seriously? Are you trying to say that the confirmation of negative existential also means that the corresponding universal is also “VERIFIED!!!!”?

    That’s pretty much what you implied, intentionally or not. Then not all universals are unverifiable after all; at least some are indeed verifiable like what you’ve mentioned above. That’s another mistake of Popper, as he didn’t thought of it when he said that no universal can ever be proven (even provisionally).

  45. Constantius says:

    D, Lee Kelly & others, I didn’t meant to hurt you by personally targeting any of you. You are entitled to your views just as much as I’m entitled to mine. Probably d is extremely upset about asking whether you people have the guts to attack your mentor Popper for his positions. He thought I’m implying that you actually don’t have “GUTS”!!! I didn’t meant anything even near. I just wanted your take on the potential (and actual) problem of adopting what Popper had to say & asked whether any of you can actually free yourself from the theoretical commitment of Popperianism to start criticizing it, of course for constructive (and not destructive) purpose.

  46. d says:

    Constantius,

    “Not at all, what I was asking about was how can we believe that all knowledge is conjectural when we also say that a hypothesis can be proven false?”

    Who here says that hypotheses can be proven false?

    “I mean, seriously? Are you trying to say that the confirmation of negative existential also means that the corresponding universal is also “VERIFIED!!!!”?”

    The ‘~’ mark is understood in logic to be one expression of negation.

    The sentence, “Not all X are Y,” is the same as “~All X are Y” which is logically identical to the sentence, “There exists at least one X that is ~Y.” Do you understand?

    Or should I start giving examples? The statement “there exists at least one wombat that is not yellow,” is the same as saying “Not all wombats are yellow.” Do you see?

  47. Constantius says:

    Yes sir, I see indeed.
    I didn’t argue against what you’re saying’ sir. What I meant was this: “That’s (the idea that all universals are not unverifiable) pretty much what you implied, intentionally or not. Then not all universals are unverifiable after all; at least some are indeed verifiable like what you’ve mentioned above. That’s another mistake of Popper, as he didn’t thought of it when he said that no universal can ever be proven (even provisionally).”

    It’s true what you said:
    “The sentence, “Not all X are Y,” is the same as “~All X are Y” which is logically identical to the sentence, “There exists at least one X that is ~Y.” Do you understand?
    Or should I start giving examples? The statement “there exists at least one wombat that is not yellow,” is the same as saying “Not all wombats are yellow.” Do you see?”

    ‘”Not” all wombat are yellow’ is a universal proposition & it can be “proven/verified/confirmed” as true, at least tentatively. Popper said that no universal can ever be “proven” (and I repeat “Proven”) because they involve an indeterminate quantity of samples required to be tested from past, present & future; which is impossible to obtain. But this does not apply to all universals as you have so wonderfully demonstrated in the above-mentioned proposition; my point here was to prove the mistake that Popper has made in his account of knowledge, not to confuse the singular proposition and its corresponding, identical “universal twin”, so to speak. That was my point, you are either to tired of argument (you & I both can agree to quit if you don’t like the way things are going) or simply have some problem in understanding me.

    Every time I say something, you seem to make something else out of that.
    I told you about one thing, you understand something other then that which is fully irrelevant!!!!!!!

  48. Constantius says:

    d, you asked me: “Who here says that hypotheses can be proven false?”
    Nobody here says that hypotheses can be proven false. That’s what Popper said. And that’s what scientists try to do, according to CR.
    If hypotheses can’t be proven false, then tell me “what we are supposed to do with them?” For us to prove something (true or false), we need to demonstrate their compatibility/incompatibility with facts (according to Popper’s theory of truth). I said Popper failed to reserve a place in his idea of where facts can reside.
    He said, and I quote: “”all knowledge is hypothetical” [Objective Knowledge P. 30] or “All knowledge remains… conjectural” [Realism and the Aim of Science Ed. xxxv]”
    Where are facts then?
    Popper upheld the correspondence theory of truth: “A statement is true if and only if it corresponds to the facts” [Objective Knowledge p. 46].
    Although he reiterated this frequently [e.g. Open Society & its Enemies ed. 2 p.369, Unended Quest p.140], only once did he go into detail about what he meant by ‘fact.’ “Facts are something like a common product of language and reality… they are reality pinned down by descriptive statements…. New linguistic means not only help us to describe new kinds of facts; in a way, they even create new kinds of facts. In a certain sense, these facts obviously existed before the new means were created…. But in another sense we might say that these facts do not exist as facts before they are singled out from the continuum of events and pinned down by statements – the theories which describe them” [Conjectures & Refutations p. 214].
    Now you judge whether there is coherence or contradiction in all these?

  49. Constantius says:

    Is Rafe Champion Australian? Is ‘d’ a follower of CR & of Constructive Empiricism? Is a person calling himself ‘Constantius’ arguing & throwing words against CR in this page?
    What you people think the answers to these questions be like; “Conjectural or Factual?”
    You are all free to respond……..say what you like, and give criticism of alternatives to your ideas (about the question just above) so as to remain true to the critical spirit of CR.

  50. Lee Kelly says:

    Constantius,

    d is right to suggest you need a primer on elementary logic. It’s hard to have a discussion with someone who doesn’t understand the basics, and none of us here is in a position to teach you these things. Some of your comments betray a gross misunderstanding of the matter at hand.

    In any case, this comment thread long ago stopped having anything to do with the original post. I don’t have the time or will to continue this debate with you. I’ll have another post up soon. I suggest you educate yourself about some of the concepts involved before commenting in the future.

  51. My advice at this point is, as always, to recommend Constantius formalise his solution to the problem of induction (or abduction, exduction, obduction, blingduction or whatever), and submit it to a suitably reputable/disreputable journal for publication. If his solution succeeds, he will undoubtedly soon find acclaim as the greatest philosophic genius of the past 100 years, and d, Kelly and I can dine out on the fact that the famous Constantius once deigned to debate us in an internet forum even though we were too stupid to understand what he was talking about. If not well, perhaps Constantius might consider a course in basic logic as d and Kelly suggest.

  52. Constantius says:

    It may just well be true that I’m “in fact” a blockhead as you people seem to imply, but you just seems to be no less dumber then me in understanding me when I was saying things about universals; that boneheaded d (and also other “pseudo-philosophers” occupying the blog as well) seems to think I don’t understand how elements in basic logic like Modus Tollens when I actually said that Popper’s idea that Universal propositions can’t possibly be confirmed (because their logical content talks about an indefinite number of entities from past, present & future) is wrong because negative universals (like what d mentioned) “~All X are Y” are in fact provable.
    I initially expressed my feelings in seeing a follower of CR saying the opposite of what Popper said, that’s all;

    I said:
    “I mean, seriously? Are you trying to say that the confirmation of negative existential also means that the corresponding universal is also “VERIFIED!!!!”?”
    That may be what motivated all of you to view me as an idiot, but what I tried to say actually was this:
    If some universals can be verified, then not all universals are unverifiable.
    And Popper didn’t get that, for he said no universal is verifiable.

    But you people thought that I was trying to deny the role of modus tollens by saying that “The sentence, “Not all X are Y,” is the same as “~All X are Y”” is not true; that they are not logically equivalent; that what d said was wrong etc.

    That’s not what I said, I’m stating that again very clearly……..if I actually implied these then no body could possibly have denied that I’m a simpleton.
    However, I’m accepting Daniel Barnes’s mockery of me quite seriously & vows to do just what you people are making fun of.

  53. Lee Kelly says:

    Constantius,

    Universals cannot be verified.

    “Not all A are B” is not a universal. I know it has the word “all” in there, and that can be confusing, but it is actually just another way of saying the existential “there are some A that are not B.”

    Nobody has ever denied this; it is a trivial result recognised by both Popper and his critics.

  54. d says:

    Constantius,

    I think we might try a better track. Think of it this way: if it were the case that there existed a llama that was not brown, I could say “There exists a llama that is not brown.”

    That existential statement could be rephrased–while keeping the same logical content of the existential statement–as the negation of a universal statement, such as “It is not the case that all llamas are brown.”

    Does this ‘verify’ a universal statement? Yes … but that is a trivial type of ‘verification,’ for the universal statement still has the same logical content of the existential statement. It’s not really ‘verifying’ anything; it’s just translating an existential statement into a universal statement.

    The negated universal statement doesn’t tell us anything about the color of unobserved llamas, does it?

    Discovering llamas that are not brown does not tell us if interesting theories, theories that say more than the color of observed llamas, are true.

    Does this make sense? Is Popper an idiot to make such a gaffe? If so, then by all means, you’re more than free to direct us to a passage where Popper claims that the negation of a universal statement “Not all swans are white” cannot be accepted if one accepts a falsifying existential statement (“There exists a swan that is black”).

  55. Constantius says:

    Agreed, agreed; yeah, thank you all………………….now we can call it quits after all, there’s nothing fruitful here in this discussion now, lets go debate about something else someday.

  56. 1Z says:

    “It is a pity that nobody has got into an argument here, maybe the case against inductive probability is too strong?”

    Maybe someone should deal with the counterarguments to the Popper-Miller argument.

  57. Lee Kelly says:

    1Z,

    I am currently working on some criticisms to the argument presented here. Although the argument seems correct, the conclusion seems less … well, conclusive than I originally suggested.

  58. J says:

    Forgive me if I’m misreading this, but isn’t the probability of (A and ~B|B) zero by definition? If B is a given, there is a 0% probability that ~B is fulfilled.

  59. Lee Kelly says:

    J,

    It’s “A or ~B,” not “A and ~B.”

    You read a conjunction where I wrote a disjunction.

    A disjunction is at least as probable as either of its parts.

    p(A v ~B|B) = p(A|B) + p(~B|B) – p(A & ~B|B)

    Since the probability of B given ~B is zero, and the probability of A & B given ~B is zero (as you rightly state), we get

    p(A v ~B|B) = p(A|B) + 0 – 0

    or

    p(A v ~B|B) = p(A|B)

  60. J says:

    Right, sorry. Confusion was because the equation equaling .15 still works out if you change all the or’s to and’s.

  61. Paul Torek says:

    I’m a latecomer and I breezed through the replies, and I apologize if this has already been covered.

    The result seems straightforward: A is not a logical consequence of B, but B increases the probability of A by 15 percent.

    Sounds good to me. By the way, a more natural comparison, to my mind, is to consider P(A|B) – P(A|~B). If I’ve done my math right, that’s 25 percent. Perhaps the more-natural-ness of this comparison is important, if one wishes to criticize the idea that Bayesian inference is a kind of induction.

    That is, to the extent that B increases the probability of A, it does so by increasing the probability of A v B more than it decreases the probability of A v ~B. However, since A v B is a logical consequence of B to begin with, the increase in probability is a purely deductive inference.

    The second sentence here is either a non sequitur or of dubious relevance. If “the increase in probability” is the net increase, it’s a non sequitur. If it’s the increase of the probability of the “A v B” part of the equation, then, dubious relevance.

    This concludes the refutation of inductive probability.

    I crave first a definition of “inductive probability”, or at least some gesture in the direction of one.

  62. Guy Srinivasan says:

    Can you translate these two lines into math? I’m having trouble unpacking what they mean or are supposed to suggest or imply.

    1. “This would would mean the logical consequences of A which are not also logical consequences of B should be more probable given B.”

    2. “Not only do logical consequences of A which are independent of B not increase in probability, they may actually decrease in probability.” (and in particular, which specific logical consequence of A which is independent of B are you referring to, and why is it ‘independent of B’?)

  63. Lee Kelly says:

    Guy,

    Induction is ampliative: the conclusion says more than the premises. That’s the whole point of induction. If the conclusion of an inference does not say more than the premises, then the inference cannot be an induction. An induction is supposed to amplify the premises so they lend support to logical consequences of the conclusion which cannot be deduced from the premises alone. For example, if induction is not ampliative, then even with an inductive inference, one would have no empirical justification to suppose the future will resemble the past.

    For the logical interpretation of probability, where A and B are propositional variables, if B is to be amplified by the inference, then it must lend support to logical consequences of A which cannot be deduced from B alone. What can be deduced from B alone is deductive (duh!), but what cannot be deduced from B but is still supported by B is inductive.

    So the argument uses the logical equivalence to pry A into two parts: A v B and A v ~B. This division of A is not accidental. A v B is a logical consequence of both A and B independently; it is the shared content. But A v ~B is only a logical consequence of A. Therefore, if the inference is inductive, then B must furnish some kind of support for logical consequences of A which cannot be deduced from B alone. In other words, B must increase the probability of A v ~B to be ampliative.

    But that is impossible; B counter-supports A v ~B. In fact, B only supports A on net if it supports A v B more than it counter-supports A v ~B. In other words, B only supports that part of A which is a deductive consequence of B to begin with, and so the inference is not inductive.

  64. Guy Srinivasan says:

    Good explanation, thanks! Here are my main confusions now.

    1. Clearly (after algebra) p(A v B|B) – p(A v B) + p(A v ~B|B) – p(A v ~B) = p(A|B) – p(A). But I have no intuition about WHY that should be the case, much less why (besides brute algebra) you should be able to break down B’s “support” of A into those two factors and have it work out. I don’t think that it’s relevant to the thrust of the argument, though, which seems instead to rest on B counter-supporting A v ~B. Still, I’m used to thinking of B’s support of A as a function of p(B|A)/p(B|~A) or in other words [p(A|B)/p(~A|B)]/[p(A)/p(~A)], and when I see the term p(A|B) – p(A) outside the context of decision calculations it feels like someone is about to make a units error. Do you have an intuition pump for why this is a natural division for which the p(A|B) – p(A) support works and makes sense?

    2. I am still not entirely sure why B counter-supporting A v ~B is a critique. It feels like you’re saying “every logical consequence of A which is independent of B should be supported by B”, but A v ~B is not independent of B. A v ~B cannot be deduced from B, but that doesn’t mean they’re independent – P(B & (A v ~B)) != P(B)*P(A v ~B). What am I missing here? Why the switch from talking about “not logical consequences of” to “independent”?

    3. Does every logical consequence of A which is not-a-logical-consequence/independent/something of B need to be supported by B for induction to make sense, or simply “most” in some sense? Why does the existence of one logical consequence with these properties constitute a critique, should I correct my intuition to feel like we need all logical consequences with these properties to be supported? (It feels like we’re rapidly approaching asking what “most” means and re-discovering Solomonoff induction.)

  65. Lee Kelly says:

    Guy,

    (1) Perhaps some more context would clear up some confusion. My original intuition was that probabilistic support is not inductive support. That is, “p(A|B) > p(A)” is a purely deductive inference: the appearance of induction is a fallacy of decomposition. I did not expect to learn that some logical consequences of A are actually counter-supported by B; it would be enough for my argument if their probability did not change at all, and so the “thrust” of my argument does not rely on B counter-supporting A v ~B.

    However, the more I think about it, the counter-support it is an important complement to the argument. Consider

    A v ~B =||= B → A

    That is, the disjunction A v ~B is logically equivalent to the conditional B → A. Given B,

    (B → A) & B |= A

    The probability of B → A characterises the strength of the induction from B to A, because A inherits the probability of B → A in the above inference (since B is given). But B → A is less probable given B than without. Therefore, the inductive inference is weakened by B, not supported. Now consider the other half of the argument

    A v B =||= ~B → A

    That is, the disjunction A v B is logically equivalent to the conditional ~B → A. Unlike before, the inference is invalid

    (~B → A) & B |≠ A

    In other words, that part of A which B actually supports is not enough to deduce A, even if we assume a probability of 1.

    (2) By “independent” I meant logically independent, not probabilistically independent. I apologise for not being more precise.

  66. Lee Kelly says:

    I have attempted to restate this argument using a different approach here:

    http://www.criticalrationalism.net/2011/07/20/more-on-inductive-probability/

  67. Lee Kelly says:

    I just accidently deleted the second thread on inductive probability. I apologise to those who commented. They had some good things to say and I had intended to respond with another formulation of the argument. Sorry!

  68. Kenneth Hopf says:

    Lee,

    I believe that which may be countersupported by B, i.e., B → A, is not a consequence of A. The fact that it is not a consequence of A is the reason it is the inductive factor.

    k

  69. Kenneth Hopf says:

    Lee,

    Never mind. I was thinking of B here instead of A. You’re right, that factor is a consequence of A, I just don’t know why you’re talking about the consequences of A in that context since it is the logical relation to B, the evidence, we care about. I’m used to calling these variable ‘h’ and ‘e’ so that we keep clear on which is the hypothesis and which the evidence.

    k

  70. Kenneth Hopf says:

    Lee,

    I think you do a good job of explaining the situation. There is, however, something I would like to add.

    Popper’s discussion of the issue is very deep, at least for me. I don’t pretend to understand it all at the moment. I think Popper and Miller do succeed in showing that probabilistic support is not inductive support, but these days the proof strikes as somewhat a matter of killing a dead horse.

    The problem is that, as Popper explains, probabilistic support is defined in terms of the logical interpretation of probability. But if you adopt the logical interpretation, and you want to talk about physical laws, you have to also become a subjectivist .. because the logical interpretation yields nothing but tautologies. Clearly, physical laws are not tautologies. If they were, there wouldn’t be any point in testing them.

    Of course, Popper has produced an uninterpreted axiom system that works with all interpretations, even though there really would have been no lose of generality if he had stuck with sets. But here’s a subtle point: even though an uninterpreted probability calculus is possible, this says nothing about what the values of the probabilities would be. The same laws hold across all interpretations, but the same values do not! On the contrary, the logical interpretation gives you values that are way way out of line with anything you’d expect with, say, a frequency or propensity interpretation.

    To make a long story short, the subjective interpretation blows up on you long before one reaches the point of showing that probabilistic support is not inductive support. This is why the Popper-Miller proof looks to me a bit like killing a dead horse. I do think it’s possible to use the logical interpretation to attack inductive inference. And Popper does that in _Realism and the Aim of Science_.

  71. Kenneth Hopf says:

    .. that is, I do think it is NEVERTHELESS possible to use the logical interpretation on inductivists as a weapon of mass destruction…. just that you wouldn’t be invoking the Popper Miller proof. You wouldn’t need to.

    k

  72. Kenneth Hopf says:

    Lee,

    OK, now I’ve slept on it, and I see that my last remark does not really get across the point I’m trying to make. Let me try again.

    Inductivism is an unfalsifiable thesis. Therefore, no argument can refute it. What it says is that there is some relation between theory and evidence such that the evidence amplifies the theory. From a logical point of view, this is exactly like saying (to use Popper’s example) ‘there is a Latin phrase which, if chanted in exactly the right way, will cause the Devil to appear’. There is no way you can exhaust the possibilities. However long you try to articulate the logical disposition of this mysterious relation, the fact that you haven’t found it doesn’t imply that it doesn’t exist. Further, it won’t work to simply blast inductive inference as logically invalid, because inductivists needn’t say anything at all about inference. This is actually the appealing thing about probabilistic induction: it simply says that the probability of the hypothesis changes in the face of the evidence. And indeed it does! Critical rationalists cannot deny that — unless they want to be mistaken.

    What the Popper-Miller proof accomplishes is that it debunks one highly esteemed candidate for the throne. In effect, it says ‘whether there is or can be some such relation as you postulate, we do not know .. but however that may be, THIS particular relation doesn’t fill the bill’. It is as if someone said ‘Oh look here .. I have the much sought-after Latin phrase that causes the Devil to appear, and I know how to chant it in exactly the right way’. What Popper and Miller do is to show that the proferred candidate does not work: the s-relation, often touted as the magic phrase, does not cause the Devil to appear. This doesn’t prevent the inductivist, however, from maintaining that, in the words of the famous Ronstadt & Ingram song, somewhere out there beneath the pale moon’s light … there is induction … la la la … This is a transcendental argument.

    So what is an upright critical rationalist to do? The way I see it now .. the only way to effectively attack induction is with reductio arguments. What you have to do is assume that the thing sought after by inductivists actually exists, and then show that absurd consequences follow.

    k

  73. Kenneth Hopf says:

    One minor correction to the above. This does not change the unfalsifiable status of inductivism.

    Inductivism purports to amplify the evidence, not the theory. That is, it purports to yield conclusions that transcend or go beyond the evidence, thereby raising the logical probability of an hypothesis in the process. The idea that there is some such relation between evidence and hypothesis is still an existential assertion, and as such unfalsifiable. Inductivists have defended the view that the relation commonly known as probabilistic support is an instance of inductive support. The Popper-Miller proof debunks this claim. Of course, inductivists are free to continue proposing different candidates, or criticizing the Popper-Miller proof, as they have done. Aside from the Popper-Miller proof, Popper had already argued that an inductive interpretation of probability leads to insuperable failures, contradictions, and absurdities. Chief among these is the realization that supporting evidence, such as it is, points in all directions at once, and therefore points usefully in no direction. Thus an inductive method is bound to be rudderless. One may certainly obtain supporting evidence, but the same evidence invariably supports infinitely many other contrary theories as well. Support is real but useless if the goal is to find some unique truth.

  74. Albert Sachmann says:

    Hi Lee,
    I don’t know if I can bump this comments thread to life, but I give it a try because I want to make sure I understand your argument correctly.

    Is this, my version of your argument, correct:
    Both refutations rest on the fact that the probability for B&~A increases (while the probability for its negation Av~B then of course decreases) when the truth of B is given. In the first case this is a problem because inductive probability should increase for all logical consequences of A when it is receiving additional support (through B). In the subjective case it’s a problem that inductive confidence in B&~A increases when confidence in A increases, because the two propositions contradict each other.

    If I’m mistaken I’d be happy if you could try to explain where I go wrong.

    Albert S.

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