REPLIES TO DIANE: POPPER ON LEARNING FROM EXPERIENCE
Sir Karl Popper has claimed repeatedly that different events are independent of each other in the sense of “independence” used in the theory of probability. That is to say, the probability of a conjunction of two different events is the product of their probabilities.
From this it follows that the probability of any event given any set of different events is the same as it was prior to those events having been given. For example, the chance of the next swan being white is not affected by our having seen many swans before, all of which are white. Similarly, Sir Karl has insisted that the initial probability of any universal law is zero. From this it follows that the probability of any universal law is always zero, regardless of how much empirical evidence supports or backs it.1This should lead to a complete breakdown of rational science. If Sir Karl is right, then, it seems, there is no mode of rational choice of scientific theories; and with this the hopes that we do - or at least may - learn from experience must evaporate.
All this worried Diane, and she has asked for a clarification of his view on this point: how does Sir Karl view learning from experience? I have set my clarification in a few preliminary paragraphs which break into an imaginary dialogue between Diane and Sir Karl. Though I do think that the dialogue is faithful in content, and though I have chosen the atmosphere with as much care as I could, I am afraid I have taken a literary license here and there and have made no attempt to make the dialogue faithful in every respect to Sir Karl’s characteristic style of conversation. Hence, this work has no claim for great biographical accuracy. Also, the style of the dialogue is less conversational than originally intended, because editors and referees exercise not only philosophical judgments but also artistic and stylistic ones.
Customarily, philosophers of science discuss at length the problem of choice of scientific theories. The words “choose”, or “choice,” and the frequently used expression “we choose,” are ambiguous. By “we” a philosopher often means Einstein and himself. By “choose” he means, believe to be true. Thus, “we choose general relativity” usually means Einstein and the philosopher believe that general relativity is true. This, of course, is utterly false: Einstein disbelieved general relativity on account of some metaphysical arguments, and the philosopher all too often does not believe general relativity since he cannot believe a thesis he does not know and all too often he does not know what general relativity says. So there - refuse to understand “we choose” and maybe you will have nothing more to understand or fail to understand.
To set the problem a bit more cautiously. Men of science are often faced with alternative hypotheses, and they try - sometimes successfully - to make an experiment help them choose one, i.e., take one to be the one which they tell their students and lay-audiences to choose, or at least the one which they talk more about. Also, some say, it is the one hypothesis which they train engineers and navigators and their likes to use. This is ridiculous because engineers and navigators use Newton, not Einstein - excepting some nuclear engineers and some space navigators (and even then, they are more likely to use special relativity, hardly ever general relativity).
Never mind. Suppose we have a set of competing hypotheses. (What Makes Newton and Einstein, Lamarck and Darwin, but not Lamarck and Einstein, competitors? If you find the answer keep it in mind.) Suppose under some conditions experience helps us choose. Which conditions? What rules of choice? The process of choice may be called inductive. The rules may be called inductive logic.
Now, the Carnapian, or current, theory may be characterized simply as follows. Inductive logic follows the rules of the calculus of probabilities.
The conditions of choice are two: (1) We must set a measure of a priori probabilities and dependences. These are not specified by the calculus, and must be added. As we shall see, this is easier said than done. (2) We may now bring relevant evidence to tip probability in favor of one candidate against the rest.By contrast, and the accent is on contrast, the Popperian theory is this. In science the important process is not at all choice or endorsement but rather criticism or rejection, namely the conclusion that a given theory is unsatisfactory in view of this, that or another specific criticism - usually specific empirical data which (seem to) conflict with the theory.
Q. Do scientists choose?
A. Usually, yes.
Q. Do they have to?
A. Not really.
Q. Is their actual choice rational?
A. Yes.
Q. How so?
A. They do reject unsatisfactory theories, and so what they do not reject they have failed to declare unsatisfactory, i.e., they have corroborated.
Q. Is this a logic?
A. To some extent, though not all the way.
Q. Can we call this logic inductive?
A. Call anything by any name.
Q. Sorry. I mean, is there a problem of choice and is the problem soluble by empirical means and a logic and have you not supplied the logic?
A. No.
Q. How so?
A. It is not the case that scientists have a problem of choice and we philosophers of science offer them tools. Rather, it is the case that scientists do choose. Their choice may be extrascientific or it may be anti-scientific, but it cannot be scientific - it is always metaphysical, though more or less in accord with science (more or less in accord with my theory of corroboration).
Q. The distinction is subtle: you simply refuse to include the metaphysics as a part of the logic of science. Why?
A. Because the problem of induction is not that of how we choose a hypothesis, but that of how we learn from experience. To the second question “How do we learn from experience?” the current answer is by choice.
And it is this very answer which leads to the first question “How do we choose?” If you do not ask the second question, or if you do not give the second question the current answer, you need not ever bother with the first question. And, indeed, the current answer is untrue. We learn from experience not by choice, but by rejection. And this answer leads not to the question “how do we choose?” but to the question, “how do we reject?”Q. If so, why did you, Sir Karl, study choice?
A. Choice does occur, and it is not inductive, as I have tried again and again to show.
Q. Why not inductive?
A. Inductive logic is the logic of learning from experience by choice and it follows the rules of the calculus of probability. Corroboration is somewhat a logic of choice, but not a logic of learning from experience and it does not follow the rules of the calculus of probabilities.
Q. So you agree with Carnap: his logic does and yours doesn’t follow the calculus of probabilities, and so they differ?
A. Obviously.
Q. So your views do not compete?
A. Our logics do differ} this is why our views do clash.
Q. How so?
A. In my view scientists choose in accord with my logic. In Carnap’s view scientists choose in accord with his logic. Our logics differ: we both apply them to the same phenomena: therefore, we arrive at a conflict. Q.E.D.
Q. Not so fast. Can you not apply different logics in different ways so as to get the same results?
A. Yes, you can. You are quite right.
Q. In which case there will be no different views?
A. Correct again.
Q. Can this be the case of Popper versus Carnap?
A. No.
Q. Can you prove this?
A. Easily: Carnap’s logic as applied by Carnap, when applied to choices in scientific situations leads to some wrong results: Popper’s only to right results. Hence they differ. (Even if they are both ultimately mistaken, they still differ because the one is now known to be mistaken but not the other.)
Q. Please spell this out.
A. You are tiresome: all you need to do is read my works, they are crystal clear.
Q. I deny that vehemently. Don’t go away! Please explain.
A. As Carnap says in his Continuum of Inductive Methods, if the next event depends in no way on the last few observed ones, our last observations are no guide to the choice of a guess on its outcome (by definition of independence). Even the observation of a million white swans does not tell us, in this case, what is the color of the next swan, and a fortiori, what is the color of all (unobserved) swans. And therefore, as Carnap notes, probabilistic independence prevents experience from helping us choose. Hence we do not choose by Carnap’s rules unless there is dependence. Carnap says we do choose by my rules, hence that there is dependence. Let it be so. What measure of dependence? Should we assume that measure to be high or low? That is to say, at what speed do we learn from past experience? Carnap says that he doesn’t know. He wants to consult experience about this. This is funny: he wants to consult experience about the rules for consulting experience, which is either a vicious circle or infinite regress. Also, we may learn fast in one field and slowly in another field. How will Carnap find this out? By experience? But experience may be balanced by unequal distributions of intellectual energy. Carnap correlates intuitively worlds with higher measures of dependence in them, namely worlds with more order in them, with ideal students learning more rapidly from experience, namely quicker to generalize, and less ordered worlds with ideal slow learners. But learning is supposed to tell us chiefly how and to what measure the world is ordered! Moreover, probabilistic dependence leads under the best conditions merely to a choice of limited forecasts, but in a possibly infinite world (in space or time) even dependence does not lead to choice or universal statements*, even a big measure of dependence fails to let experience rescue universal statements from their initial zero probability.
Q. But if there is dependence not all universal statements have zero probability.
A. You mean I have made a small logical error?
Q. Did you not?
A. No. There can be dependence in an infinite sequence of eventstatements the conjunction of all of which is equivalent to one universal statement. And there can be dependence in an infinite sequence of universal statements; these two possibilities are mutually (logically) independent.
Q. Good Lord!
A. That is right. You try to solve a simple question of choice by a simple calculus of probability, and the calculus rebels and turns up in a really complicated fashion.
Q. Why? Is it as complicated in mathematics?
A. Not in such an annoying way.
Q. Meaning?
A. Questions of dependence and of probability measures are decided in mathematics arbitrarily, just as in the case of arbitrarily different geometries, and their consequences can be studied.
Q. And in statistics, how do we determine which of these arbitrary probability measures to use?
A. Much as we decide which arbitrary geometry to apply.
Q. Namely?
A. We try any of them for size and if it fits -
Q. - we choose them?
A. - we test them -
Q. - and then choose them?
A. - and try to eliminate them -
Q. - and if the tests fail we choose them?
A. - and if the tests fail we may choose them.
Q. - Oh!
A. The probability of the inductive philosophers is that of compulsory choice. But there is no compulsory choice. And so philosophers need no probability, and thus no probability measure, and no dependence measure. Rather, for different problems in probability theory we assume alternating solutions involving different measures of probability and of dependence.
Q. So we need not say that all events are a priori independent and all hypotheses have zero initial probability!
A. You are right.
Q. So why do you say what we need not say?
A. I say what may be said.
Q. On what ground?
A. On the ground that experience and the laws of probability make me choose it.
Q. You must be pulling my leg.
A. Indeed, I am.
Q. Why do you tease me? Do you have no heart?
A. I do. It is because I do, and because you try to make me apply induction to my choice of equiprobability of events and to my choice of zero initial probability of hypotheses, that I say what you wish me to say.
Q. Sorry. What criteria, other than inductive, do you employ for the choice of independence and zero probability?
A. I do not fully know, but perhaps simplicity is one.
Q. But how can you call your choice of zero probability and total independence simple if it makes life so difficult; for does it not prevent any further choice, in particular choice of empirical hypotheses or learning from experience?
A. Take care! Now you are committing a small logical error. My choice is legitimate and excludes inductive logic. But it permits, to begin with, learning from experience by rejection and after this, as an option, the choice of hypotheses by corroboration.
Q. And is this inductive or not? I am at a loss.
A. As you yourself say, my dear, zero initial probability of hypotheses plus equiprobability of events prevents inductive logic, but not the logic of rejection and of choice by corroboration.
Q. Can you prove the latter point - on corroboration, I mean?
A. Certainly; with mathematical precision, even.
Q. And is it not less simple to say that learning from experience is by rejection but that choice is corroboration?
A. Less simple than what?
Q. Than saying that both learning from experience and choice are corroborations?
A. This is an inductive variant of Popperism.
Q. I am delighted. What’s wrong with it?
A. It does not work.
Q. Oh! Why?
A. Infinite regress and all that. Good night.
Q. Wait! You didn’t say yet. Why do you assume events to be independent?
A. I do not. They interdepend in accord with causal laws.
Q. I mean why do you assume events to be a priori independent?
A. I do not. I assume a priori that some depend on others some not, in accord with....
Q. Sorry again; why is the probability of events in your system such that they are probabilistically independent of each other? Is that precise enough for your finicky taste?
A. Almost. You are very acute, if I may compliment you. I say, events are probabilistically independent in the absence of laws.
Q. Thanks. Why?
A. Because laws and only laws are the measures of the mutual dependence of events.
Q. How interesting! And for Carnap both laws of nature and a priori probabilities are measures of dependence of events. Is that why you say your system is simpler than Carnap’s?
A. This is, roughly, the general idea.
Q. Any thrid alternative?
A. I may not be overjoyed to agree with Carnap, but I am afraid I must. No third alternative is logically possible. Only two inductive systems are possible. Either you assume a priori dependence in your system, or you do not. The former is his, the latter is Wittgenstein’s and mine. You see, I even have to agree with Wittgenstein.
Q. And Carnap’s system is defective, whereas in yours no learning from experience is at all possible!
A. Not so fast or you have again a small logical error on your hands. What you should have said is this. In Carnap’s system no kind of learning from experience is possible, and in mine no inductive learning from experience is possible. Good night.
Q. Please wait. But learning from experience by elimination is possible?
A. By elimination of errors, not by confirmation of one view through the elimination of another. Good night.
Q. Please, wait. One last question, please. Just how do we learn from experience by refutation?
A. Good night.
Q. Are you peeved?
A. Yes.
Q. Why?
A. All the time I told you again and again that we learn from experience by refutations and you didn’t bother to try to understand.
Q. But I was bothered with another point. Can I bother about all points at once?
A. No. But why bother with inductive logic for so long before even trying to see what I mean?
Q. What do you mean by learning from experience?
A. I mean our intellectual horizon widens with a refutation. I mean that a refuting observation report is more theoretically loaded the more abstract and general the theory it refutes. I mean that when a theory is refuted we may see better how far it goes, explain it, and so see perhaps why it goes that far: some breakthroughs of great importance are great refutations.
Q. Is this a theory of breakthrough?
A. No. Breakthroughs are unique; get them anyway you like. If you can’t, try refuting an existing theory.
Q. Which one?
A. Good night.
Q. Peeved again?
A. No. Tired. Inductivists think that all repetition is reinforcement. I wonder. Sometimes repetition simply fills one with profound tiredness, if I may make an empirical observation. But I know: philosophers are not supposed to take recourse to experience. Good night.
NOTE
1 Definition of independence: a and b are independent when and only when p (a & b) — p(a)xp®.
Definition of conditional probability: If p(Z>) / 0, then p(a, b) = p (a & b)/p(b).
If a and b are independent, then p(a, b) = p (a & b)/p(b) = [p(a) x p(b)]/p(b) = p(a). If p(a) = 0 then, by the law of monotony, p(a & b) — 0, and p(a, b) = 0.
If p(6) = 0, then p(a, b) is undefined for most systems (including Carnap’s). It is normally assumed in the literature that since b is an observation or observation-report of unique events, p(Z>) is never zero. This seems to me to be very questionable. Assume p(Z>) to be zero, and the computing of p(a, b) may be highly problematic.