It will be convenient to discuss this topic immediately after the above discussion of the notion of Causation. It forms the subject of essays by Dr. Hanson, Professor Nelson, and Professor von Wright.

(A) LAWS OF NATURE.

      I take it that the main point which Dr. Hanson is concerned to make is that "law-sentences" (and in particular those which are said to state the laws of motion and the law of gravitation) have a number of different, though interconnected, uses; that the same law-sentence is often used in different ways by one and the same scientist in the course of a single spell of work or a single bit of scientific writing; and that it is futile for anyone philosophising on the topic of these laws to pick out one sense, and claim that it is the only legitimate one. I certainly do not wish to dispute this, and I will content myself with the following remarks on it.

      (1) The question of fact could be settled only by a careful examination of the writings, the conversations, and the behaviour on relevant occasions of representative scientists from Newton's time to the present day.

      (2) I suspect that it would often be very hard to be sure of the sense in which a given scientist was using a given law-sentence on a given occasion. I should not expect to get much useful information by asking the scientist himself. If he had not had a philosophical training, he would probably not understand the question or see the point of asking it, and he would certainly not have the technical equipment to answer it intelligibly. If he had had a philosophical training, the chances are that he would not be a first-rate working scientist; and, even if he were, he would probably be committed to some particular (and often already exploded) philosophical view, which would bias his answers. In fact, those two modern oracles, "the plain man" and "the working scientist," resemble in one respect their ancient forerunners. The artless prattlings of the former and the sophisticated technicalities of the latter stand in as much need of expert interpretation as did the inspired ravings of the Pythia at Delphi or the Sybil at Cumae.

      (3) If the question of fact can be settled, I agree that it is then the business of the philosopher of science to accept the situation, and not to pretend that there is one and only one legitimate sense in which a law-sentence can be used.

      Dr. Hanson mentions five different uses of a law-sentence. The second third, and fourth of these agree in that in each of them such a sentence expresses something that can significantly be described as "true" or "false." In the first use such a sentence formulates a definition of the technical meaning of a term, e.g., "force," which may already have had a long-standing use in ordinary speech. In that case it can hardly be said to be true or false, but it can be judged by various criteria to be well or ill fitted for its purpose. The fifth use itself covers five very different alternatives, according to Dr. Hanson; but they all agree in making a law-sentence express something which (though not a definition) can hardly be said to be true or false.

      On all these matters I will confine myself to the following comments.

      (1) Dr. Hanson distinguishes, among others, the following uses of a law-sentence, viz.,

1. to express a proposition which it is "psychologically impossible" to think of as false, and
2. to express a proposition, the rejection of which would have extremely upsetting repercussions in departments of science which have for long been regarded as models of complete and detailed explanation and absolutely reliable prediction.

      I should suppose that the state of affairs described under (ii) is an important factor in causing that which is described under (i). The alleged psychological impossibility of contemplating the falsity of what is expressed by a law-sentence would seldom be of much philosophical interest unless it sprang from some such cause.

      (2) In his discussion of the attitudes of scientists toward the law of gravitation, Dr. Hanson makes the following remark. A most important function of such a law is that it unifies a great many empirical facts through being a common premiss from which they all follow. He speaks of the typical situation in modern theoretical physics as "observation statements in search of a premiss." And he alleges that philosophers of science have failed to recognise this, and have concentrated their attention too much on empirical correlations. By these I take him to mean straightforward inductive arguments from all the observed S's having been P to the conclusion that all past, present, and future S's respectively have been, are, or will be P.

      My comments on this are as follows.

1. I agree that the procedure in question is most important in all advanced sciences.
2. I do not agree that it has been ignored or underrated by philosophers of science. I should have thought that it was adequately recognised, e.g., by Mill in his Logic (Cf. Book III, Chapters XI to XIV, both inclusive) and by Jevons in his Principles of Science, to mention only two of the older English writers.

      (iii) As a matter of logic, it seems to me that philosophers of science nevertheless do well to lay great stress on the problem of straightforward inductive generalisation. For all the "facts," which are shown to follow from the supposed law, are themselves propositions which have been accepted as straightforward inductive generalizations. If the original inductive arguments for accepting them cannot be defended, then there is to that extent a doubt whether they are facts; and, unless they are they cannot support the more general law which entails them.

      Of course I am well aware that the evidence for each of these more restricted general propositions is greatly strengthened by its being entailed by a single more general law, which also entails many other such propositions, each of which was originally accepted only as an independent inductive generalization. In this way the whole system becomes comparable to a net, as contrasted with a lot of separate threads. But the net is made of threads, and the ultimate strength or weakness of the threads depends in the last resort on the validity or invalidity of straightforward inductive generalization.

      (3) Among what I may call the "non-informative" uses of law-sentences Dr. Hanson mentions their use

1. as rules of inference, and
2. as principles for constructing instruments.

      This is surely obvious when it is a question of constructing an instrument. A sane person constructs an instrument to perform certain functions, and he designs it in such a way as he believes will ensure its performing those functions efficiently. If he uses a law-sentence prescriptively in giving directions for constructing an instrument, or if he understands it in that sense in receiving and following such directions, it must be because he already accepts as true the propositions about nature which it states when used, not prescriptively, but informatively.

      This is perhaps less obvious in reference to the use of a law-sentence as a rule of inference. If so, that is because it is not altogether clear what that phrase means in the present connexion. But surely this at least can be said. Such a "principle" is admittedly not one of pure logic or of pure mathematics, like the principle of the syllogism or the formula (x + y) (x - y) = x² - y². If, then, the conclusions which one "deduces in ac cordance with it" are to have any relevance to actual or causally possible natural phenomena, the "principle of inference" must surely rest on a general proposition which is held to be true of the relevant department or aspect of nature.

(B) INDUCTION.

      The topics treated in Professor Nelson's and Professor von Wright's essays partly overlap and partly diverge, so in some of the sub-sections which follow I shall be concerned with what is common to both and in others with what is peculiar to one or the other.

(1) The so-called "Problem of Induction."

      This question is treatedby both writers, Professor von Wright quotes a dilemma, in whicn I summed up my position in the paper entitled "Mechanical and Teleological Causation," in Aristotelian Society's Supplementary Volume XIV. He suggests that, instead of pursuing the course which seems inevitably to end in that dilemma, we should begin with the question: What do we mean by calling an inductive belief "grounded" (as opposed to "groundless" or "ill-founded"), and what do we mean by "rationally believing" in reference to an inductive generalisation? He is inclined to think that, if we do this, we shall see that what we call "grounds of rational belief in induction" are just empirical premisses without support of any general principles. He does not attempt to argue his case, but hopes that I may comment on it. So I will take this as the text of my sermon in this sub-section.

      I would suggest that what must presumably have happened in the case of deductive logic may be useful as an analogy and a contrast. Here, I suppose, we could distinguish in theory three stages, though very likely they overlapped historically.

      (1) There would have been a number of particular bits of deductive argument which all or most sane persons accepted in the law-courts, in monetary calculations, in mensuration, and so on, except when under the influence of some strong desire or emotion which was known to distort a person's judgment. There would have been a number of particular bits of deductive argument which all or most sane persons, with similar qualifications, rejected. Finally, there would be a number of particular bits of deductive argument which were accepted by some and rejected by other sane persons when in an emotionally calm state.

      (2) It would be natural, then, to compare and contrast the generally accepted with the generally rejected arguments, in order to see whether there were other features, beside general acceptance, common and peculiar to the former. This stage might be illustrated by the discovery and formulation of the traditional rules of the syllogism. At this stage it might be agreed to be a fair test, in the case of a disputed argument, to note whether it did or did not have the characteristics which had been found to be in fact common and peculiar to arguments commonly accepted by sane men in their calmer moments.

      (3) One might still, however, see no reason why an argument having all the characteristics in question should be valid, and why one which lacked any of them should be invalid. There is nothing, e.g., obviously wrong with a syllogism having a negative conclusion and two affirmative premisses. The next stage, then, would be to try to get behind the empirical tests, and to show that they are consequences of more fundamental principles which are self-evident. That can be done in various alternative ways, which I need not describe here, for the rules of the syllogism.

      Let us now compare and contrast this with the case of inductive arguments. In the case of deductive inference we are all, I suppose, agreed as to what we mean by calling an argument "valid." At any rate there is one condition which would generally be acknowledged to be necessary and sufficient for the validity of a deductive argument. It is this. It must be impossible that the premisses should be true and the conclusion false; and this impossibility must rest, not on the impossibility of the premisses (though they may be impossible, as in a reductio ad absurdum argument in pure mathematics), nor on the necessity of the conclusion (though it may be necessary, as it always is in the case of any valid argument from true premisses in pure mathematics), but on a certain relationship between the logical form of the premisses and the logical form of the conclusion. The task for philosophers of deduction is to classify arguments which answer to this admittedly necessary and sufficient condition of validity; to elicit the formal features common and peculiar to them; and then, if possible, to bring them under one or a few general principles, which all or nearly all sane and competent persons find self-evident. That programme has in the main been accomplished.

      But it is not obvious what we mean by calling an inductive argument "valid"; or, if you prefer it, there is no one condition which is generally acknowledged to be necessary and sufficient for the validity of such an argument. What is quite certain is this. If we use the accepted definition or criterion of "validity" as applied to a deductive argument, and if we take the complete premiss of an inductive argument to be: This, that and the other S (which are all that have so far been observed) have been P, and the conclusion to be: All S's, past, present or future, respectively have been, are, or will be P, then all inductive arguments are invalid. Now we all use inductive arguments, and we all accept the conclusions of many of them and guide important actions by reference to these. So reflective persons cannot but find this situation intellectually disturbing.

      Now at this point there seem to be two alternatives open to us. One is to suppose that the definition or accepted necessary and sufficient condition of "validity," as applied to deductive arguments, applies also to inductive ones. The other is to deny this, and to set out from that point. I will now say something about each of these alternatives in turn.

      (1) If we are going to use the old definition or accepted necessary and sufficient condition of "validity," and yet to admit the possibility that some inductive arguments are valid, we must try to save the situation in one or other or a combination of the following ways. We might suppose either

1. that a valid inductive argument has an additional implicit premiss beside the instantial propositions which are its only explicit premisses, or
2. that the conclusion of a valid inductive argument must take a weaker form than the unqualified All S is P.

      If that were done, it is evident that the principles (as distinct from the premisses) of inductive inference would include, beside those of non-problematic deductive inference, at least the formal principles of the calculus of probability, e.g., the axiom of addition concerning the probability of a disjunctive proposition, and the axiom of multiplication concerning that of a conjunctive one. I do not think that this in itself would be felt to raise any special difficulty.

      Anyone who follows this line will have to deal with the following three questions, which might be described respectively as "logical," "ontological," and "epistemological."

1. What are the minimal universal premisses which, if added to the explicit instantial premisses, would make very highly probable the conclusions of those inductive arguments which are commonly accepted as practically certain by sane and instructed persons?
2. What account of the structure of nature as a whole, or of certain departments of nature, would best fit in with the assumed truth of these universal premisses?
3. How, if at all, do we know that these premisses are true or that they are highly probable?

      Before considering the other alternative, suggested by Professor von Wright, I will make the following comments on the alternative outlined above.

      (a) There is no guarantee that the whole enterprise may not break down at the first stage. In that case we should have to admit that, so far as we can tell up to date, no inductive arguments are valid, in the sense of "validity" supposed, even when their conclusions are stated in terms of probability.

      (b) Even if the logical problem can be solved satisfactorily, the epistemological problem would (as both Professor Nelson and I have emphasised) remain very troublesome. The additional premiss (and still more obviously the propositions about the "structure of nature" which have to be assumed in order that it shall be applicable) must be general, and it cannot be merely analytic. Yet our acceptance of it cannot, without circulatory, be based on induction; and, even if the possibility of necessary synthetic propositions were admitted (which it is not by most contemporary English and American philosophers), no additional premiss which has been plausibly alleged to fulfill the conditions has any trace of self-evidence.

      (c) This leads me to the following two reflexions,

(α) Even if the epistemological difficulties should be insoluble, that would not diminish the value of the analytic and the ontological sections of this line of thought. The justification of induction, where it is thought to be justifiable, would have to be stated conditionally, and not categorically. But even that would be no small gain in insight.

(β) The situation would be remarkably like that which Kant (as I understand him) contemplated in regard to such allegedly synthetic a priori propositions as he held to be capable of "transcendental proof."

People claim to know, or to have good grounds for very strongly believing certain general propositions as a result of inductive reasoning. Suppose we grant their claim. Suppose we can show that it can be valid, if and only if certain propositions about the structure of nature are true. Then we are entitled to accept those propositions, even though they be synthetic and though they have no trace of self-evidence. They would be "synthetic a priori propositions" in precisely the sense in which Kant held that, e.g., the law of universal causation and the conservation of mass are so.

      (2) Let us now consider the other alternative, suggested by Professor von Wright, which is nowadays much the more popular of the two. The contention is that if an inductive argument can properly be described as "valid" or "invalid," those words must be understood in a special sense, appropriate to such arguments. On that supposition, it is of course quite possible that certain inductive arguments may be "valid," in the appropriate sense, without the addition of any implicit general premiss to their explicit instantial premisses, and perhaps without reformulating their conclusions in terms of probability. On this suggestion I would make the following comments.

      (i) Plainly the first task would be to formulate a definition, or generally acceptable necessary and sufficient condition, of what I will call "inductive validity." Here we may compare and contrast this enquiry with Stage (1) of what I supposed above to have happened in the case of deductive arguments. We should have to consider typical inductive arguments, which all or most sane persons in their calmer moments accept and compare and contrast them with typical inductive argumentS which all or most of such persons under such conditions reject. But the difference would be this. In the case of deductive arguments there was from the outset no doubt as to what is meant by "valid" and "invalid" as applied to them. The object of the comparison and the contrast was not to elicit the meaning of "validity," but to discover, and if possible to rationalism tests for its presence or absence in any deductive argument. But in the case of inductive arguments the primary object of this comparison and contrast would be to discover what competent persons, who use and criticise such arguments, mean when they call some of them "valid" and others "invalid."

      Unless it turned out that inductive validity had some fairly close and important analogies to deductive validity, it would be better not to use the word "validity" or "invalidity" of inductive arguments, but to coin some other technical term. I should think that the irreducible minimum of analogy would be that the "validity" of an inductive argument should depend in some assignable way on relationships of logical form between its premisses and its conclusion.

      (ii) However that may be, it might still be worth while, after having elicited an agreed definition of "inductive validity" in this or in some other way, to proceed thenceforth as logicians did with deductive arguments. That procedure would be as follows.

1. To try to discover features, other than those which enter into the definition of "inductive validity," which are common and peculiar to arguments which are inductively valid.
2. If that can be done, to try to show why the presence of all these features entails inductive validity, and the absence of any of them entails inductive invalidity.
3. To try to reduce these features as far as possible to one or a few very general headings.

      (iii) Now it might happen that, when one elicited the meaning of "inductive validity," the consequence which Professor von Wright thinks would follow, viz., that the grounds of rational belief in induction are just empirical premisses without support of any general principles, would be seen to follow. Or it might not. All that I will say in conclusion is this. We must of course distinguish between the premisses of a valid argument, and the principles which the argument exemplifies and which ensure and make evident its validity. In the valid syllogism, e.g., All men are mortal, and all Greeks are men, therefore all Greeks are mortal the only premisses are the two propositions which are stated before the word "therefore." The principles which the argument exemplifies, and which together make evident its validity, are such propositions as the following: --

1. If a class is empty, every sub-class of it is empty; and
2. If every member of an exhaustive set of sub-classes of a class is empty, then that class is empty.

(2) Professor Nelson's account of Inductive Argument.

      I found this of very great interest. I will first try to state it, as I understand it, in my own words, and will then make a few comments on it. To simplify the exposition I will confine myself to inductive generalisation where the instantial premiss is that N instances of S have been observed (say S₁, S₂, . . . . . S_N) and that all of them have been P. With that understanding I would summarise Professor Nelson's theory as follows.

1. If the argument is to be defensible, the conclusion must not take the unqualified non-modal form All S is P. It must take the form It is likely, to such and such a degree, that all S is P.
2. This must be carefully distinguished from any statement of the form: "The proposition All S is P has such and such a degree of probability with respect to the datum q." The following points are very important to notice here.
   1. "Likely," in the sense in which Professor Nelson uses it, is analogous (except in that what it stands for is capable of degree) to "true."
   2. On the other hand, the statement that p has such and such a degree of probability with respect to q is comparable to the statement that p is entailed by q.
   Like it, it is a statement which is necessarily true or necessarily false, as the case may be. And, like it, its truth or falsity depends on certain relations between the forms of p and of q, and not on their individual necessity or impossibility, truth or falsity, likelihood or unlikelihood.
3. Nevertheless, in order to establish inductively the conclusion It is likely to such and such a degree that all S is P, we require such a proposition as is expressed by the sentence: "With respect to the proposition that N instances of S have been observed and all of them have been P, it is probable to such and such a degree that all S is P." I will symbolise the proposition, expressed by the sentence in inverted commas, by Π_N(σ) where σ is the degree of probability in question.
4. The part played by Π_N(σ) in establishing a conclusion inductively may be compared with that which is played, in establishing deductively that all Greeks are mortal from the premiss that all men are mortal and all Greeks are men, by the proposition which is expressed by the sentence: "All men are mortal & All Greeks are men entails All Greeks are mortals." The following points are important to notice here: --
   1. In the deductive argument we use a principle of "Deductive Detachment." Knowing that in fact all men are mortal and all Greeks are men, we are entitled to drop those premisses and to accept as true theproposition that all Greeks are mortal. In the inductive argument we need a comparable principle of "Inductive Detachment." Knowing that in fact N instances of S have been examined and that all of them were P, we are entitled to drop that premiss and to accept as likely to such and such a degree that all S is P.
   2. According to Professor Nelson, the degree of likelihood which it is justifiable to assign to All S is P, under the conditions supposed, is a function of the degree of probability σ, which All S is P has with respect to the premisses in the complex proposition Π_N(σ). As to this function he will say no more than the following. The degree of likelihood of All S is P, given that the premisses in Π_N(σ) are known to be true and can therefore be dropped, increases with σ, the degree of probability of All S is P in respect to those premisses.
5. The last point in the theory is this. Professor Nelson holds that we never have any good reason to accept such a proposition as Π_N(σ) on its own merits, as we have, e.g., to accept the proposition expressed by the sentence: "All Greeks are mortal is entailed by All men are mortal & All Greeks are men." The only ground for accepting such a proposition as Π_N(σ) is that it is entailed by a certain other proposition, which he calls the "Principle of Induction," and that we know this to be true. We will denote this principle by P_I. It is important to note the following points about it.
   1. Professor Nelson does not claim to be able to formulate it satisfactorily. But he thinks that progress has been made towards doing so, and that this is illustrated, e.g., by the substitution of Keynes's "Principle of Limited Variety" for Mill's ''Uniformity of Nature."
   2. He draws a distinction between P_I itself, and the characteristics which we must ascribe to the actual world if P_I is to be true and applicable to natural phenomena. The proposition that nature has these characteristics is ontological, whilst P_I itself is described as "formal."

      Supposing this to be a fair account of Professor Nelson's very inter esting theory, I will make the following comments.

1. I wonder why he uses P_I as a premiss which entails Π_N(σ), instead of modifying Π_N(σ) by introducing P_I into it as an additional premise. The modified proposition, which we will denote by Π'_N(σ), would then be expressed by the sentence: "The proposition All S is P has probability of degree σ with respect to the conjunction of P_I with the premisses of Π_N(σ)." I do not see any obvious objection to this. And, unless there be some objection, I should think it would have one obvious advantage. For Π'_N(σ) would hold in virtue of the form of its premisses and its conclusion, just as a valid syllogism does; whilst Π_N(σ) would not do so (if I understand Professor Nelson aright), though I suppose that the proposition that P_I entails Π_N(σ) would do so.
2. Professor Nelson puts the argument in terms of a definite degree of probability σ, and a definite degree of likelihood, which increases with σ. I take it that he does not suppose that these can be exactly measured in any particular case. It would be enough that in favourable cases one should know that σ was high enough to ensure that the degree of likelihood of All S is P is considerable.
3. As regards P_I itself I have two remarks to make.
   1. Taking it as a "formal" principle, I feel rather uncomfortable about a premiss which it is admitted that no one has so far managed to formulate satisfactorily. In order to "detach" P_I in Professor Nelson's form of the argument, one must know that it is true (or at any rate "highly likely"). But unless one knows what it is, how can one know this about it? I suppose we should have to say that what one knows is that there is some formulable proposition (never as yet satisfactorily formulated), which has the logical properties ascribed to P_I and which is true or highly likely.
   2. I think that the distinction between the "formal" principle and its ontological ground might be rather difficult to define. Would it come to this? The formal principle would state in extremely abstract terms the conditions which must be fulfilled in any possible world in which inductive generalisation would be a valid process leading in favourable cases to highly likely conclusions. The ontological principle would be a much more concrete statement as to the structure of the actual world which ensures that these conditions are fulfilled in it.
4. On the notion of "likelihood" I will make the following comments: --
   1. When a person accepts a proposition (rightly or wrongly, reasonably or unreasonably) as true, he is prepared (so far as he is not hindered by temperamental or occasional defects, intellectual or moral) to apply it without hesitation in practice where it is relevant, to accept without question in theory any consequences which seem to him to follow from it, to use it unhesitatingly as a basis for his further reflexions and investigations, and so on. Now there is undoubtedly an attitude which we often have towards a proposition, where all this holds good with the substitution of "with very considerable confidence" for "unhesitatingly." The latter may fairly be described as accepting a proposition (justifiably or unjustifiably) as more or less likely.
   2. One important way in which a person comes to accept a proposition as true is by noting that it seems to him to be logically entailed by certain other propositions, which he accepts as true. In such cases we may say that he accepts it as "deductively established." One important way in which a person comes to accept a general proposition as more or less likely is by what he takes to be a valid inductive argument from premisses which he accepts as true. These always include at least a number of favourably instantial propositions, together with a proposition to the effect that these are all the relevant instances that have been observed. In such cases we may say that he accepts it as 'inductively supported."
   3. If a person accepts a proposition as true, because deductively established, he cannot hope to strengthen his case through the possible discovery of additional true propositions which entail that conclusion. These will only provide him with alternative lines of proof, all of which could be dispensed with, and each of which could be substituted for his original line of proof They are like a lot of ropes, each attached to a different hook, and each amply sufficient to support a certain weight. But suppose a person accepts a general proposition as likely to at least a certain degree, because inductively supported. Then he can hope to strengthen his case (though he must also fear its complete collapse) by the examination of further relevant instances. The mere addition of further true premisses of the same kind (provided that the proposition that they include all the observed instances remains true) will inductively support the conclusion still more strongly and will justify one in accepting it as likely to a still higher degree. Here the additional true premisses are comparable to additional strands in a single rope, which is always liable suddenly to give way.
5. Lastly, I would like to say how fully I agree with the following contention of Professor Nelson's. It is hopeless to consider the principles of induction in isolation from the other principles and categories which are involved in the notion of a world of persistent things with varying states, co-existing and inter-acting in a single spatio-temporal system. Whatever defects there may be in Kant's discussion of the "Principles of Pure Understanding," he had at least grasped this essential point, which his predecessors had failed to note and which most of his successors seen to have forgotten.

(3) Assumptions about Antecedent Probability.

      Professor von Wright discusses this in connexion with problems in probability concerned with drawing counters from a bag, noting their colours, and thence arguing to the probability of various propositions about the colours of the counters in the bag. I have considered such problems in "Induction and Probability" and in "The Principles of Problematic Induction".

      In the former I assumed that the n+1 alternatives, that a bag containing n counters should contain 0 or 1 or . . . . n counters of an assigned colour (e.g., white), would be equi-probable antecedently to any of them being drawn and looked at. In the latter paper, after having read Keynes's Treatise on Probability, I argued that this assumption leads to a contradiction. I there assumed instead that there are v distinguishable colours (including black and white), and that it is antecedently equiprobable with regard to any counter in the bag that it would have any one of these colours. Professor von Wright mentions a third possible assumption, which I did not consider in either paper, viz., that every possible "constitution" of the contents of the bag with respect to an assigned colour (e.g., white) is antecedently equally probable. He expresses regret that I did not work out the consequences of this.

      Now I think that this third possible assumption can be dismissed quite briefly. I take it to be equivalent to assuming that it is antecedently equi-probable with regard to any counter in the bag that it would either have or not have the assigned colour (e.g., white). For all purposes of mathematical deduction that is equivalent to putting √ = 2 in the calculations in "Principles of Problematic Induction". It seems to me obvious that the assumption as to equi-probability which I made there is more defensible than the assumption of equi-probability of "constitutions." For the latter lumps together under the heading "other-than-white" all the remaining colours, and then counts this disjunction of colours as precisely on a level with the single colour white

      Professor von Wright says that he thinks there is no possibility of proving or of disproving any of these alternative assumptions about equi-probability. I am inclined to agree with him as to the impossibility of proving any of them without making factual assumptions. I think, e.g., that the assumption which I made in P. of P. I. would be reasonable only if one had the following information, or something formally equivalent to it, viz., that the bag had been filled by drawing n counters from another bag, which contained equal large numbers of counters of each of the ν colours, well mixed with each other. But I should have thought that it was possible to refute some assumptions by showing that they lead to consequences which are plainly absurd. I do not see anything wrong with the argument by which I tried to show in P. of P. I. that the assumption made by me in "Induction and Probability" leads to absurdities, if we admit that there is more than one colour (e.g., red and blue) besides the assigned one (e.g., white), which might belong to one or more of the counters in the bag.

(4) The notion of "Loading."

      From problems concerned with drawing counters from bags the transition is natural to problems concerned with throwing dice, spinning roulette-wheels, and so on. The notion of "loading" has its most obvious applications in reference to the latter problems. It is discussed both by Professor von Wright and by Professor Nelson. I will take their remarks in turn.

      In P.P.I. I made the following assertions.

1. "The notion of loading is the notion of a constant cause-factor which operates throughout the whole series of throws, and co-operates with other and variable cause factors to determine the actual result of each throw."
2. "I shall say that the counter is loaded to degree s in favour of red, if and only if the antecedent probability of its turning up red would be s for anyone who knew in detail how it was constructed."

      The answer is that I meant the following. I thought of the load, not as a probability, but as a physical factor (e.g., the location of the centre of gravity at such and such a position in relation to the geometrical centre of the body in question) determining the antecedent probability of a face of such and such a colour coming up. It would strike me as linguistically barbarous to talk of a probability as a cause-factor, and I should not wittingly do so.

      My statement that induction, in such cases, presupposes a reference to causation was therefore intended to mean something different from the minimum which Professor von Wright suggests that I might have meant by it. In the context it was intended to mean something like the following. The fact that the antecedent probability of a loaded die turning up a 6 on any occasion is so-and-so is determined jointly by the following facts.

1. That the position at which it comes to rest on any occasion is causally determined jointly by
   1. the position of its centre of gravity in relation to its geometrical centre,
   2. its geometrical, elastic, and other permanent properties,
   3. the correlative properties of the surface on which it falls, and
   4. the angle at which it hits the surface.
2. That it is antecedently equally likely to hit the surface at any one of the innumerable alternative geometrically possible angles.

      Passing now to Professor Nelson's "roulette-wheel," I would make the following comments: --

1. He contrasts the case of a wheel which is "honest" and one which is not. But ought we not rather to contrast one that is known to the player to be honest, and one which is not known to him to be so or not to be so. In the latter case the possibility that it is biased is admitted from the beginning.

         If the wheel is known to the player to be honest, then no run of a single number, however long, and no sequence of numbers, however often repeated , would give him any rational ground for betting in favour of a repetition of that number or of that sequence. That is almost, if not quite, an analytical proposition. But, if the bare possibility of bias is admitted from the first, then it might be argued that a sufficiently preponderant proportion of a certain number, or of a certain sequence of numbers, would provide a ground for a rational belief that it is biased in a certatn way. That in turn would provide a reasonable ground for betting in a corresponding way on its future behaviour.

         Professor Nelson does in fact consider this kind of argument in connexion with his criticism of the "Precept Theory." The essential point seems to me to be one which he himself makes. A glance at the formula for the application of the principles of inverse probability shows that all that an accumulation of uniformly favourable instances can do for a hypothesis is continually to multiply by a new factor its initial probability. Now, if that is to lead to a final probability whose upper limit is 1, we must have reason to believe beforehand, not merely that the initial probability is greater than 0, but that the lower limit of its possible values is greater than 0. Now that is not secured merely by the negative fact that it is not impossible that the wheel may be loaded in one way or an

2. About artificial cases, such as roulette-wheels, the following points may be worth making: --
   1. No one in practice is in a position to know (even in the popular sense of that word) that a roulette-wheel is honest. At most he may have extremely good reasons to believe that it has been made by a competent and reliable firm in accordance with the accepted methods for making honest roulette-wheels, that it has not become worn or tampered with and so on.
   2. Conversely, in certain circumstances one might have very good reasons for thinking it quite probable antecedently that a certain rouette-wheel would not be honest. In all artificial cases an essential part of one's ground for holding any reasonable opinion on the antecedent probability of the machine being honest or being biased is knowledge of the general laws of human motivation and of the characters and motives of certain particular individuals. Again, an essential part of one's ground or inferring, from the supposed construction of the machine and its observed performance up to date, to any conclusion about its future behaviour in any assigned respect, is one's knowledge of the general laws of physics and of the properties of specific kinds of matter.
   3. It might therefore seem that there is a risk of circularity in taking, as a model for the inductive inference of natural uniformities from observed regularities of co-existence or of sequence, the case of inferring from the past results of spinning a roulette-wheel to the probable results of further spins. I mention this appearance of circularity only in order to say that I do not think it harmful for the purpose for which the analogy is used. That purpose is simply to exhibit the presuppositions of an inductive argument in a case where they are very obvious, and to suggest

      (α) that inductive generalisation everywhere presupposes the finite antecedent probability of something analogous to bias in the case of a roulette-wheel or a die, and
      (β) that this always rests on some view about the "concealed structure and mechanism" (to use those words very widely) of nature as a whole or of a particular department of it.

(5) Induction by Simple Enumeration and the Hypothetical Method.

     Under this heading I will discuss a number of inter-related points raised by Professor von Wright.

     I alleged that induction by simple enumeration (so far as it is exemplified by taking counters out of a bag, noting their colours, and then drawing conclusions with more or less probability as to the original proportion of counters in the bag) is a particular case of the hypothetical method. Professor von Wright objects to this. I think that his objection rests partly on a mere difference in the use of words, and partly on an important matter of principle.

      (i) The matter on which I think there is no real difference is this. Let h₀, h₁, . . . . . . h_n be a set of mutually exclusive and collectively exhaustive alternative propositions, which it is proposed to test by specific experiment or observation. Let f be any relevant data which one may have before undertaking the test, and let Q_N be a summary of the relevant information that has accumulated at the N-th stage of carrying out the test. Then for any typical one of these alternatives h, the probability relative to the conjunction of f with Q_N is given by the equation

	r=n
hr/f&QN = [ (hr/f) X (QN/f&hr) ] ÷ [	∑	(hr/f) X (Qn/f&hr)]
	r=0

where any symbol of the form "p/q" stands for the probability of the proposition p given the proposition q.

      Now in the case of bag-problems the propositions of the form h, are alternative "hypotheses" to the effect that exactly so many of the n counters in the bag are of such and such a colour. The proposition Q_N is a summary, at any given stage of the experiment, of the accumulated information as to the whiteness or non-whiteness of the counters drawn and inspected up to that point.

      In what is commonly called "the hypothetical method" we use what is in principle the same formula, but there are the following important differences in detail.

1. Instead of considering a number of mutually exclusive and collectively exhaustive alternative propositions h₀, h₁, . . . . . . h_n, we consider just a single proposition H and its logical contradictory H.
2. H is such that at every stage Q_N/f&H is either 0 (in which case the hypothesis is refuted and the experiment comes to a natural end), or 1 (in which case there is no reason why the experiment should not be continued).
3. In the bag experiment H is analogous to the single alternative h_n, viz. that all the counters in the bag are white. And Q_N/f&H is either 0 (if Q_N includes the information that at least 1 non-white counter has been drawn), or 1 (if it consists of the information that all the counters drawn up to that stage have been white). The Formula therefore reduces to

   	_		_
   H/f&QN = (H/f) ÷ [ (H/f) + (	H	/f) X (QN/f&	H	) ].

      So what I was trying to say could be more accurately expressed as follows. The reasoning in induction by simple enumeration (so far as this is accurately represented by experiments in drawing counters from a bag), and the reasoning in the hypothetical method, are instances of essentially the same general formula in the calculus of probability. And the latter can fairly be regarded as in certain respects a more restricted case of that formula, since it is by definition subject to the three conditions stated above.

      (ii) The important difference in principle is this. Is the kind of hypothesis which is tested in what is ordinarily called the "hypothetical method" really on all fours with the (n + 1)-th. of the alternative "hypotheses" which are tested in an artificial experiment with counters in a bag? Is All swans are white a proposition of the same logical kind as All the n counters in the bag are white? Professor von Wright objects that the former are propositions about what he calls "open classes" and that the latter are about "closed classes," and that these two are fundamentally dissimilar kinds of proposition.

      I think that he is right to object, and that I was wrong to overlook this distinction, but that his objection hardly goes far enough. It seems to me now that we have to contrast at least three fundamentally different kinds of proposition,

1. "All S is P" might express simply the proposition that S₁ is P & S₂ is P & . . . . . S_n is P, and that these are all the S's that there are.
2. It might express a rather complicated proposition of the following form. Consider a sequence of collections of the following kind, viz., (S₁), (S₁ & S₂), . . . . . . (S₁ & S₂ & . . . . . . S_n) . . . . . . Let the percentage of the members of these collections which are P be respectively P₁, P₂ . . . . . . P_n . . . . . . Then "All S is P" might be taken to mean the same as "p_n tends to the limiting value 100% as n tends to infinity." This latter sentence is itself a highly condensed expression for a rather complicated proposition, but we need not unpack it further here.
3. "All S is P" might be taken to mean that in the actual world (though not in all possible worlds) any instance of S would be an instance of P. I do not know how to analyse such propositions further. But I can perhaps indicate their peculiarity by remarking that one is tempted to say of any such proposition

   (α) that, if it is true, it is necessary, but
   (β) that the fact that it is necessary is contingent. (In contrast with this, one can say of the necessity of a true a priori proposition that its belonging to that proposition is itself a necessary fact.)

      We might call these respectively the "enumerative," the "limiting-frequency," and the "nomic" interpretations of such a sentence as "All S is P." It is immediately obvious that (b) differs from (a). If it is not immediately obvious (as I think it should be) that (c) differs from (b), this becomes evident when one reflects that (b) is compatible with there being any finite number of S's which are not P, whilst (c) is not compatible with there being a single S which is not P.

      Now the "limiting-frequency" interpretation certainly presupposes "open classes," in the sense of classes which contain an infinite number of members. For that is involved in the notion of a limit. For that very reason I doubt whether it has any application outside pure mathematics. The "nomic interpretation" does not presuppose "open classes" in that sense. For the proposition that any instance of S would be an instance of P in the actual world is consistent with the number of actual instances in the whole course of the world's history being finite or even zero. What it does presuppose is the notion of classes determined by intension as distinct from by enumeration of their members.

      It seems to me that what we commonly try to test by the so-called "hypothetical method" is universal propositions in the nomic sense. If so, they are fundamentally different from such propositions as All the n counters in the bag are white. But the difference is even more fundamental than would be suggested by the contrast between "open" and "closed" classes.

(6) The Theory of "Generators."

      I have very little to object to in Professor von Wright's comments in what I said about this in P. P. I.

      (i) He is correct in saying that the argument on p. 27 of that paper does not presuppose that the number of generated characteristics is finite. It presupposes only that n, the number of generating characteristics, is finite. The further argument, in the section entitled Effect of the Relative Values of n and N certainly assumes N to be finite when considering the alternatives that N is less than or equal to n, since n is assumed throughout to be finite. In discussing the alternative that N is greater than n, I certainly did assume in my own mind that N is finite; and, although the mere supposition that N is greater than n does not entail this, there are many steps in the argument which presuppose it.

      (ii) He is correct also in saying that I have nowhere shown that the factor (μ_r&ν_s)/h in the formula on p. 27 is greater than 0. This is the probability (relative to the general assumption of the theory of generators, and to the special assumption that each generated characteristic is generated by only one set of generators) of the proposition expressed by the sentence: "In a generalization, whose subject is a conjunction of μ generated characters, and whose predicate is a conjunction of ν generated characters, the former require exactly r, and the latter exactly s, generating factors respectively to generate them."

      (iii) He says, rightly, that all my arguments presuppose that the antecedent probability of a generalization "can be linked with a ratio of true generalisations among a class of generalizations." But he complains that the nature and justification of this link are not made clear. I do not see exactly what the difficulty is here. If it could be shown that at least a certain proportion of possible generalizations of a certain kind must be true, e.g., at least p% of generalizations with a μ-fold subject and a ν-fold predicate, surely the antecedent probability of any generalization of that kind would be at least

      (iv) He mentions my remarks on p. 41 of P. of P. I., that the generating factors must be supposed to be determinable characters, and that it would follow that the generated characters must be so too. He finds the notion of "determinables" and "determinates" obscure, and asks me to try to clarify it.

      I regret that it is impossible for me to go into this very large question here. The following very sketchy and therefore rather obscure remarks must suffice.

1. My account of generating factors explicitly assumes that no conjunction of such factors is either logically necessary or logically impossible. The statement about generating factors having to be determinable characters is bound up with this.
2. That is because of the following properties of determinable characters and of determinate dharacters. Supreme determinables are all logically independent of each other. But it is logically necessary that any thing which possesses a determinate character should possess all the determinables, of whatever order, under which this falls. And it is logically impossible that any thing should possess two determinate characters of the same order which fall under one and the same determinable.

     I think that a more accurate statement of what I had in mind would run as follows. A complete collection of generating factors would have either

1. to contain nothing but supreme determinables; or
2. to contain nothing but determinates, each of which falls under a different supreme determinable; or
3. to be a mixture of (α) supreme determinables, and (β) determinates, none of which fall under any of these determinables, and each of which falls under a different supreme determinable.

(7) Necessary Conditions and Sufficient Conditions.

      I take some credit for seeing by 1930, when I published my two papers on "Demonstrative Induction," that these are the essential concepts involved in demonstrative induction, and for having worked out the formal logic of them in some detail and without serious mistakes, though not without one very serious omission. But all that I have written on this topic has now been superseded by Professor von Wright's more thorough and more accurate work.

      I agree with him that it sounds odd to say: "The ground becoming wet is a necessary condition of rain having fallen in the neighbourhood," and I agree that both he and I are committed by our definitions to saying such things. I agree too that the verbal paradox is bound up with the conviction that a causal condition must be fulfilled before that which it conditions begins. That is why, in the present essay, I have introduced the terms "necessary precursor" and "sufficient precursor," when discussing, in Section IV, C, 2 above, Professor Russell's comments on my account of Causation in the EMcP.

      I should be inclined to say that we must distinguish between a "condition," in the sense of a ground for inference, and a "condition" in the sense of a factor in causation. Given a knowledge of causal laws, one can often infer from knowledge of a later event to the conclusion that such and such an earlier event must have happened. (Unless one is a prophet, one cannot of course infer from knowledge of a future event to the occurrence of such and such an event in the present or the past, since one cannot be in possession of such knowledge.) But, when a person makes such an inference from a later to an earlier event, he does so because he has reason to believe that the later state of affairs (e.g., the ground being wet) would have come into being only if it had been preceded by a state of affairs containing such and such an event (e.g., a fall of rain in the neighbourhood) as a cause-factor.