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"Six of One, Half a Dozen of the Other" (Frank Ramsey, "Truth and Probability", 1926)
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Jeffrey Rubard
Dec 11, 2021, 1:08:31 AM12/11/21
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TRUTH AND PROBABILITY
& "Further Considerations" and "Probability and Partial Belief"
by
Frank Plumpton Ramsey, M.A.
Fellow and Director of Studies in Mathematics at King's College,
Lecturer in Mathematics in the University of Cambridge
"Truth and Probability" written 1926. Published 1931 in Foundations of Mathematics
and other Logical Essays, Ch. VII, p.156-198. Edited by R.B. Braithwaite. London:
Kegan, Paul, Trench, Trubner & Co. Ltd. New York: Harcourt, Brace and Company
"Further Considerations" written 1928. Published 1931 op. cit., Ch. VIII, p.199-211.
"Probability and Partial Belief" written 1929. Published 1931, op cit., Ch. IX, p.256-57.
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[p.156]
TRUTH AND PROBABILITY
(1926)
To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is and of what is not that it is not is true.
-- Aristotle.
When several hypotheses are presented to our mind which we believe to be mutually exclusive and exhaustive, but about which we know nothing further, we distribute our belief equally among them .... This being admitted as an account of the way in which we actually do distribute our belief in simple cases, the whole of the subsequent theory follows as a deduction of the way in which we must distribute it in complex cases if we would be consistent.
-- W. F. Donkits.
The object of reasoning is to find out, from the consideration of what we already know, something else which we do not know. Consequently, reasoning is good if it be such as to give a true conclusion from true premises, and not otherwise.
-- C. S. Peirce.
Truth can never be told so as to be understood, and not be believed.
-- W. Blake.
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[p.157]
FOREWORD
In this essay the Theory of Probability is taken as a branch of logic, the logic of partial belief and inconclusive argument; but there is no intention of implying that this is the only or even the most important aspect of the subject. Probability is of fundamental importance not only in logic but also
in statistical and physical science, and we cannot be sure beforehand that the most useful interpretation of it in logic will be appropriate in physics also. Indeed the general difference of opinion between statisticians who for the most part adopt the frequency theory of probability and logicians who mostly reject it renders it likely that the two schools are really discussing different things, and that the word 'probability' is used by logicians in one sense and by statisticians in another. The conclusions we shall come to as to the meaning of probability in logic must not, therefore, be taken as prejudging its meaning in physics.1
1
[p.157] [A final chapter, on probability in science, was designed but not written. -- ED.]
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[p.158]
CONTENTS
(1) The Frequency Theory
(2) Mr Keynes' Theory
(3) Degrees of Belief
(4) The Logic of Consistency
(5) The Logic of Truth
(1) THE FREQUENCY THEORY
In the hope of avoiding some purely verbal controversies, I propose to begin by making some admissions in favour of the frequency theory. In the first place this theory must be conceded to have a firm basis in ordinary language, which often uses 'probability' practically as a synonym for
proportion; for example, if we say that the probability of recovery from smallpox is three-quarters, we mean, I think, simply that that is the proportion of smallpox cases which recover. Secondly, if we start with what is called the calculus of probabilities, regarding it first as a branch of pure
mathematics, and then looking round for some interpretation of the formulae which shall show that our axioms are consistent and our subject not entirely useless, then much the simplest and least controversial interpretation of the calculus is one in terms of frequencies. This is true not only of the ordinary mathematics of probability, but also of the symbolic calculus developed by Mr. Keynes; for if in his a/h, a and h are taken to be not propositions but propositional functions or class-concepts which define finite classes, and a/h is taken to mean the proportion of members of h
which are also members of a, then all his propositions become arithmetical truisms.
[p.159]
Besides these two inevitable admissions, there is a third and more important one, which I am prepared to make temporarily although it does not express my real opinion. It is this. Suppose we start with the mathematical calculus, and ask, not as before what interpretation of it is most convenient to the pure mathematicism, but what interpretation gives results of greatest value to science in general, then it may be that the answer is again an interpretation in terms of frequency; that probability as it is used in statistical theories, especially in statistical mechanics -- the kind of
probability whose logarithm is the entropy -- is really a ratio between the numbers, of two classes, or the limit of such a ratio. I do not myself believe this, but I am willing for the present to concede to the frequency theory that probability as used in modern science is really the same as frequency.
But, supposing all this admitted, it still remains the case that we have the authority both of ordinary language and of many great thinkers for discussing under the heading of probability what appears to be quite a different subject, the logic of partial belief. It may be that, as some supporters of the frequency theory have maintained, the logic of partial belief will be found in the end to be merely the study of frequencies, either because partial belief is definable as, or by reference to, some sort of
frequency, or because it can only be the subject of logical treatment when it is grounded on experienced frequencies. Whether these contentions are valid can, however, only be decided as a
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result of our investigation into partial belief, so that I propose to ignore the frequency theory for the present and begin an inquiry into the logic of partial belief. In this, I think, it will be most convenient if, instead of straight away developing my own theory, I begin by examining the views of Mr Keynes, which are so well known and in essentials so widely accepted that readers probably feel [p.160] that there is no ground for re-opening the subject de novo until they have been disposed of.
(2) MR KEYNES' THEORY
Mr Keynes1
starts from the supposition that we make probable inferences for which we claim objective validity; we proceed from full belief in one proposition to partial belief in another, and we claim that this procedure is objectively right, so that if another man in similar circumstances entertained a different degree of belief, he would be wrong in doing so. Mr Keynes accounts for this
by supposing that between any two propositions, taken as premiss and conclusion, there holds one and only one relation of a certain sort called probability relations; and that if, in any given case, the relation is that of degree α, from full belief in the premiss, we should, if we were rational, proceed to a belief of degree α in the conclusion.
Before criticising this view, I may perhaps be allowed to point out an obvious and easily corrected defect in the statement of it. When it is said that the degree of the probability relation is the same as the degree of belief which it justifies, it seems to be presupposed that both probability relations, on
the one hand, and degrees of belief on the other can be naturally expressed in terms of numbers, and then that the number expressing or measuring the probability relation is the same as that expressing the appropriate degree of belief. But if, as Mr. Keynes holds, these things are not always expressible
by numbers, then we cannot give his statement that the degree of the one is the same as the degree of the other such a simple interpretation, but must suppose him to mean only that there is a one-one correspondence between probability relations and the degrees of belief which [p.161] they justify.
This correspondence must clearly preserve the relations of greater and less, and so make the manifold of probability relations and that of degrees of belief similar in Mr Russell's sense. I think it is a pity that Mr Keynes did not see this clearly, because the exactitude of this correspondence
would have provided quite as worthy material scepticism as did the numerical measurement of pro bability relations. Indeed some of his arguments against their numerical measurement appear to
apply quite equally well against their exact correspondence with degrees of belief; for instance, he argues that if rates of insurance correspond to subjective, i.e. actual, degrees of belief, these are not rationally determined, and we cannot infer that probability relations can be similarly measured. It
might be argued that the true conclusion in such a case was not that, as Mr Keynes thinks, to the non-numerical probability relation corresponds a non-numerical degree of rational belief, but that degrees of belief, which were always numerical, did not correspond one to one with the probability
relations justifying them. For it is, I suppose, conceivable that degrees of belief could be measured by a psychogalvanometer or some such instrument, and Mr Keynes would hardly wish it to follow that probability relations could all be derivatively measured with the measures of the beliefs which they justify.
1
[p.160] J.M. Keynes, A Treatise on Probability (1921).
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But let us now return to a more fundamental criticism of Mr Keynes' views, which is the obvious one that there really do not seem to be any such things as the probability relations he describes. He supposes that, at any rate in certain cases, they can be perceived; but speaking for myself I feel con fident that this is not true. I do not perceive them, and if I am to be persuaded that they exist it must be by argument; moreover I shrewdly suspect that others do not perceive them either, because they
are able to come to so very little agreement as to which of them relates any two given propositions.
[p.162]
All we appear to know about them are certain general propositions, the laws of addition and multiplication; it is as if everyone knew the laws of geometry but no one could tell whether any given object were round or square; and I find it hard to imagine how so large a body of general knowledge can be combined with so slender a stock of particular facts. It is true that about some particular cases there is agreement, but these somehow paradoxically are always immensely complicated; we all agree that the probability of a coin coming down heads is 2 1 , but we can none of us say exactly what is the evidence which forms the other term for the probability relation about
which we are then judging. If, on the other hand, we take the simplest possible pairs of propositions such as 'This is red' and 'That is blue' or 'This is red' and 'That is red', whose logical relations should surely be easiest to see, no one, I think, pretends to be sure what is the probability relation which
connects them. Or, perhaps, they may claim to see the relation but they will not be able to say anything about it with certainty, to state if it is more or less than 31 , or so on. They may, of course, say that it is incomparable with any numerical relation, but a relation about which so little can be
truly said will be of little scientific use and it will be hard to convince a sceptic of its existence.
Besides this view is really rather paradoxical; for any believer in induction must admit that between 'This is red ' as conclusion and ' This is round ', together with a billion propositions of the form 'a is round and red' as evidence, there is a finite probability relation; and it is hard to suppose that as we accumulate instances there is suddenly a point, say after 233 instances, at which the probability relation becomes finite and so comparable with some numerical relations. It seems to me that if we take the two propositions 'a is red', 'b is red', we cannot really discern more
than four [p.163] simple logical relations between them; namely identity of form, identity of predicate, diversity of subject, and logical independence of import. If anyone were to ask me what probability one gave to the other, I should not try to answer by contemplating the propositions and
trying to discern a logical relation between them, I should, rather, try to imagine that one of them was all that I knew, and to guess what degree of confidence I should then have in the other. If I were able to do this, I might no doubt still not be content with it, but might say ' This is what I
should think, but, of course, I am only a fool' and proceed to consider what a wise man would think and call that the degree of probability. This kind of self-criticism I shall discuss later when developing my own theory; all that I want to remark here is that no one estimating a degree of probability simply contemplates the two propositions supposed to be related by it; he always
considers inter alia his own actual or hypothetical degree of belief. This remark seems to me to be borne out by observation of my own behaviour; and to be the only way of accounting for the fact that we can all give estimates of probability in cases taken from actual life, but are quite unable to
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do so in the logically simplest cases in which, were probability a logical relation, it would be easiest to discern. Another argument against Mr Keynes' theory can, I think, be drawn from his inability to adhere to it consistently even in discussing first principles. There is a passage in his chapter on the
measurement of probabilities which reads as follows: -- "Probability is, vide Chapter 11 (§12), relative in a sense to the principles of human reason. The
degree of probability, which it is rational for us to entertain, does not presume perfect logical insight, and is relative in part to the secondary propositions which we in fact know; and it is not dependent upon whether more perfect logical insight [p.164] is or is not conceivable. It is the
degree of probability to which those logical processes lead, of which our minds are capable; or, in the language of Chapter II, which those secondary propositions justify, which we in fact know. If we do not take this view of probability, if we do not limit it in this way and make it, to this extent,
relative to human powers, we are altogether adrift in the unknown; for we cannot ever know what degree of probability would be justified by the perception of logical relations which we are, and must always be, incapable of comprehending." 1
This passage seems to me quite unreconcilable with the view which Mr Keynes adopts everywhere except in this and another similar passage. For he generally holds that the degree of belief which we are justified in placing in the conclusion of an argument is determined by what relation of
probability unites that conclusion to our premisses, There is only one such relation and consequently only one relevant true secondary proposition, which, of course, we may or may not know, but which is necessarily independent of the human mind. If we do not know it, we do not
know it and cannot tell how far we ought to believe the conclusion. But often, he supposes, we do know it; probability relations are not ones which we are incapable of comprehending. But on this view of the matter the passage quoted above has no meaning: the relations which justify probable
beliefs are probability relations, and it is nonsense to speak of them being justified by logical relations which we are, and must always be, incapable of comprehending. The significance of the passage for our present purpose lies in the fact that it seems to presuppose a different view of
probability, in which indefinable probability relations play no part, but in which the degree of rational belief depends on a variety of logical relations. For instance, there might be between the premiss and conclusion the relation [p.165] that the premiss was the logical product of a thousand
instances of a generalization of which the conclusion was one other instance, and this relation, which is not an indefinable probability relation but definable in terms of ordinary logic and so easily recognizable, might justify a certain degree of belief in the conclusion on the part of one who
believed the premiss. We should thus have a variety of ordinary logical relations justifying the same or different degrees of belief. To say that the probability of a given h was such-and-such would mean that between a and h was some relation justifying such-and-such a degree of belief. And on
this view it would be a real point that the relation in question must not be one which the human mind is incapable of comprehending.
1
[p.164] p. 32, his italics.
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This second view of probability as depending on logical relations but not itself a new logical relation seems to me more plausible than Mr Keynes' usual theory; but this does not mean that I feel at all inclined to agree with it. It requires the somewhat obscure idea of a logical relation justifying a
degree of belief, which I should not like to accept as indefinable because it does not seem to be at all a clear or simple notion. Also it is hard to say what logical relations justify what degrees of belief, and why; any decision as to this would be arbitrary, and would lead to a logic of probability
consisting of a host of so-called 'necessary' facts, like formal logic on Mr Chadwick's view of logical constants.1
Whereas I think it far better to seek an explanation of this 'necessity' after the model of the work of Mr Wittgenstein, which enables us to see clearly in what precise sense and why logical propositions are necessary, and in a general way why the system of formal logic consists of the propositions it does consist of, and what is their common characteristic. Just as
natural science tries to explain and [p.166] account for the facts of nature, so philosophy should try, in a sense, to explain and account for the facts of logic; a task ignored by the philosophy which dismisses these facts as being unaccountably and in an indefinable sense ' necessary'.
Here I propose to conclude this criticism of Mr Keynes' theory, not because there are not other respects in which it seems open to objection, but because I hope that what I have already said is enough to show that it is not so completely satisfactory as to render futile any attempt to treat the
subject from a rather different point of view.
(3) DEGREES OF BELIEF
The subject of our inquiry is the logic of partial belief, and I do not think we can carry it far unless we have at least an approximate notion of what partial belief is, and how, if at all, it can be measured. It will not be very enlightening to be told that in such circumstances it would be rational to believe a proposition to the extent of 2/3, unless we know what sort of a belief in it that means. We must therefore try to develop a purely psychological method of measuring belief. It is not enough to measure probability; in order to apportion correctly our belief to the probability we must also be able to measure our belief.
It is a common view that belief and other psychological variables are not measurable, and if this is true our inquiry will be vain ; and so will the whole theory of probability conceived as a logic of partial belief; for if the phrase 'a belief two-thirds of certainty ' is meaningless, a calculus whose sole object is to enjoin such beliefs will be meaningless also. Therefore unless we are prepared to give up the whole thing as a bad job we are bound to hold that beliefs can to some extent be measured. If we were to follow the analogy [p.167] of Mr Keynes' treatment of probabilities we should say that some beliefs were measurable and some not; but this does not seem to me likely to be a correct account of the matter: I do not see how we can sharply divide beliefs into those which have a position in the numerical scale and those which have not. But I think beliefs do differ in measurability in the following two ways. First, some beliefs can be measured more accurately than others; and, secondly, the measurement of beliefs is almost certainly an ambiguous process leading to a variable answer depending on how exactly the measurement is conducted. The degree of a belief is in this respect like the time interval between two events; before Einstein it was supposed
1
[p.165] "Logical Constants", Mind, 1927.
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that all the ordinary ways of measuring a time interval would lead to the same result if properly performed. Einstein showed that this was not the case; and time interval can no longer be regarded as an exact notion, but must be discarded in all precise investigations. Nevertheless, time interval
and the Newtonian system are sufficiently accurate for many purposes and easier to apply. I shall try to argue later that the degree of a belief is just like a time interval; it has no precise meaning unless we specify more exactly how it is to be measured. But for many purposes we can assume that the alternative ways of measuring it lead to the same result, although this is only
approximately true. The resulting discrepancies are more glaring in connection with some beliefs than with others, and these therefore appear less measurable. Both these types of deficiency in measurability, due respectively to the difficulty in getting an exact enough measurement and to an important ambiguity in the definition of the measurement process, occur also in physics and so are not difficulties peculiar to our problem; what is peculiar is that it is difficult to form any idea of how the measurement is to be conducted, how a unit is to be obtained, and so on.
[p.168]
Let us then consider what is implied in the measurement of beliefs. A satisfactory system must in the first place assign to any belief a magnitude or degree having a definite position in an order of magnitudes; beliefs which are of the same degree as the same belief must be of the same degree as
one another, and so on. Of course this cannot be accomplished without introducing a certain amount of hypothesis or fiction. Even in physics we cannot maintain that things that are equal to the same thing are equal to one another unless we take 'equal' not as meaning 'sensibly equal' but a
fictitious or hypothetical relation. I do not want to discuss the metaphysics or epistemology of this process, but merely to remark that if it is allowable in physics it is allowable in psychology also. The logical simplicity characteristic of the relations dealt with in a science is never attained by
nature alone without any admixture of fiction. But to construct such an ordered series of degrees is not the whole of our task; we have also to
assign numbers to these degrees in some intelligible manner. We can of course easily explain that we denote full belief by 1, full belief in the contradictory by 0, and equal beliefs in the proposition and its contradictory by 2 1 . But it is not so easy to say what is meant by a belief 3 2 of certainty, or a belief in the proposition being twice as strong as that in its contradictory. This is the harder part of the task, but it is absolutely necessary; for we do calculate numerical probabilities, and if they are to
correspond to degrees of belief we must discover some definite way of attaching numbers to degrees of belief. In physics we often attach numbers by discovering a physical process of addition1: the measure-numbers of lengths are not assigned arbitrarily subject only to the proviso that the
greater length shall have the greater measure; we determine them further by deciding on a [p.169] physical meaning for addition ; the length got by putting together two given lengths must have for its measure the sum of their measures. A system of measurement in which there is nothing
corresponding to this is immediately recognized as arbitrary, for instance Mohs' scale of hardness1
1
[p.168] See N. Campbell, Physics The Elements (1920), p.277. 1
[p.169] Ibid., p.271.
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in which 10 is arbitrarily assigned to diamond, the hardest known material, 9 to the next hardest, and so on. We have therefore to find a process of addition for degrees of belief, or some substitute for this which will be equally adequate to determine a numerical scale. Such is our problem; how are we to solve it? There are, I think, two ways in which we can begin.
We can, in the first place, suppose that the degree of a belief is something perceptible by its owner; for instance that beliefs differ in the intensity of a feeling by which they are accompanied, which might be called a belief-feeling or feeling of conviction, and that by the degree of belief we mean
the intensity of this feeling. This view would be very inconvenient, for it is not easy to ascribe numbers to the intensities of feelings; but apart from this it seems to me observably false, for the beliefs which we hold most strongly are often accompanied by practically no feeling at all; no one
feels strongly about things he takes for granted.
We are driven therefore to the second supposition that the degree of a belief is a causal property of it, which we can express vaguely as the extent to which we are prepared to act on it. This is a generalization of the well-known view, that the differentia of belief lies in its causal efficacy, which is discussed by Mr Russell in his Analysis of Mind. He there dismisses it for two reasons, one of which seems entirely to miss the point. He argues that in the course of trains of thought we believe many things which do not lead to action. This objection is however beside the mark, because
[p.170] it is not asserted that a belief is an idea which does actually lead to action, but one which would lead to action in suitable circumstances; just as a lump of arsenic is called poisonous not because it actually has killed or will kill anyone, but because it would kill anyone if he ate it. Mr Russell's second argument is, however, more formidable. He points out that it is not possible to suppose that beliefs differ from other ideas only in their effects, for if they were otherwise identical their effects would be identical also. This is perfectly true, but it may still remain the case that the nature of the difference between the causes is entirely unknown or very vaguely known, and that what we want to talk about is the difference between the effects, which is readily observable and important.
As soon as we regard belief quantitatively, this seems to me the only view we can take of it. It could well be held that the difference between believing and not believing lies in the presence or absence of introspectible feelings. But when we seek to know what is the difference between believing more
firmly and believing less firmly, we can no longer regard it as consisting in having more or less of certain observable feelings; at least I personally cannot recognize any such feelings. The difference seems to me to lie in how far we should act on these beliefs: this may depend on the degree of some feeling or feelings, but I do not know exactly what feelings and I do not see that it is indispensable that we should know. Just the same thing is found in physics; men found that a wire connecting plates of zinc and copper standing in acid deflected a magnetic needle in its neighbourhood.
Accordingly as the needle was more or less deflected the wire was said to carry a larger or a smaller current. The nature of this 'current' could only be conjectured: what were observed and measured were simply its effects. It will no doubt be objected that we know how strongly [p.171] we believe
things, and that we can only know this if we can measure our belief by introspection. This does not seem to me necessarily true; in many cases, I think, our judgment about the strength of our belief is really about how we should act in hypothetical circumstances. It will be answered that we can only tell how we should act by observing the present belief-feeling which determines how we should act;
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but again I doubt the cogency of the argument. It is possible that what determines how we should act determines us also directly or indirectly to have a correct opinion as to how we should act, without its ever coming into consciousness. Suppose, however, I am wrong about this and that we can decide by introspection the nature of belief, and measure its degree; still, I shall argue, the kind of measurement of belief with which probability is concerned is not this kind but is a measurement of belief qua basis of action. This can I think be shown in two ways. First, by considering the scale of probabilities between 0 and 1, and the sort of way we use it, we shall find that it is very appropriate to the measurement of belief as a basis of action, but in no way related to the measurement of an introspected feeling. For the units in terms of which such feelings or sensations are measured are always, I think, differences which are just perceptible: there is no other way of obtaining units. But I see no ground for supposing that the
interval between a belief of degree 3 1 and one of degree 2 1 consists of as many just perceptible changes as does that between one of 3
2 and one of 6 5 , or that a scale based on just perceptible differences would have any simple relation to the theory of probability. On the other hand the
probability of 31 is clearly related to the kind of belief which would lead to a bet of 2 to 1, and it will be shown below how to generalize this relation so as to apply to action in general. Secondly, the quantitative aspects of beliefs as the basis of action are evidently more important than the intensities of belief-feelings. [p.170] The latter are no doubt interesting, but may be very variable from individual to individual, and their practical interest is entirely due to their position as the hypothetical causes of beliefs qua bases of action. It is possible that some one will say that the extent to which we should act on a belief in suitable circumstances is a hypothetical thing, and therefore not capable of measurement. But to say this is merely to reveal ignorance of the physical sciences which constantly deal with and measure
hypothetical quantities; for instance, the electric intensity at a given point is the force which would act on a unit charge if it were placed at the point.
Let us now try to find a method of measuring beliefs as bases of possible actions. It is clear that we are concerned with dispositional rather than with actualized beliefs; that is to say, not with beliefs at
the moment when we are thinking of them, but with beliefs like my belief that the earth is round, which I rarely think of, but which would guide my action in any case to which it was relevant. The old-established way of measuring a person's belief is to propose a bet, and see what are the
lowest odds which he will accept. This method I regard as fundamentally sound; but it suffers from being insufficiently general, and from being necessarily inexact. It is inexact partly because of the diminishing marginal utility of money, partly because the person may have a special eagerness or
reluctance to bet, because he either enjoys or dislikes excitement or for any other reason, e.g. to make a book. The difficulty is like that of separating two different co-operating forces. Besides, the proposal of a bet may inevitably alter his state of opinion; just as we could not always measure
electric intensity by actually introducing a charge and seeing what force it was subject to, because the introduction of the charge would change the distribution to be measured.
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[p.173]
In order therefore to construct a theory of quantities of belief which shall be both general and more exact, I propose to take as a basis a general psychological theory, which is now universally discarded, but nevertheless comes, I think, fairly close to the truth in the sort of cases with which
we are most concerned. I mean the theory that we act in the way we think most likely to realize the objects of our desires, so that a person's actions are completely determined by his desires and opinions. This theory cannot be made adequate to all the facts, but it seems to me a useful
approximation to the truth particularly in the case of our self-conscious orprofessional life, and it is presupposed in a great deal of our thought. It is a simple theory and one which many psychologists would obviously like to preserve by introducing unconscious desires and unconscious opinions in
order to bring it more into harmony with the facts. How far such fictions can achieve the required result I do not attempt to judge: I only claim for what follows approximate truth, or truth in relation to this artificial system of psychology, which like Newtonian mechanics can, I think, still be
profitably used even though it is known to be false. It must be observed that this theory is not to be identified with the psychology of the Utilitarians, in
which pleasure had a dominating position. The theory I propose to adopt is that we seek things which we want, which may be our own or other people's pleasure, or anything else whatever, and our actions are such as we think most likely to realize these goods. But this is not a precise statement, for a precise statement of the theory can only be made after we have introduced the notion of quantity of belief.
Let us call the things a person ultimately desires 'goods', and let us at first assume that they are numerically measurable and additive. That is to say that if he prefers for its own sake an hour's swimming to an hour's reading, he will prefer [p.174] two hours' swimming to one hour's swimming and one hour's reading. This is of course absurd in the given case but this may only be because swimming and reading are not ultimate goods, and because we cannot imagine a second hour's swimming precisely similar to the first, owing to fatigue, etc. Let us begin by supposing that our subject has no doubts about anything, but certain opinions about all propositions. Then we can say that he will always choose the course of action which will lead in
his opinion to the greatest sum of good. It should be emphasized that in this essay good and bad are never to be understood in any ethical sense but simply as denoting that to which a given person feels desire and aversion.
The question then arises how we are to modify this simple system to take account of varying degrees of certainty in his beliefs. I suggest that we introduce as a law of psychology that his behaviour is governed by what is called the mathematical expectation; that is to say that, if p is a
proposition about which he is doubtful, any goods or bads for whose realization p is in his view a necessary and sufficient condition enter into his calculations multiplied by the same fraction, which is called the 'degree of his belief in p'. We thus define degree of belief in a way which presupposes
the use of the mathematical expectation.
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We can put this in a different way. Suppose his degree of belief in p is n
m ; then his action is such as he would choose it to be if he had to repeat it exactly n times, in m of which p was true, and in the others false. [Here it may be necessary to suppose that in each of the n times he had no memory of the previous ones.] This can also be taken as a definition of the degree of belief, and can easily be seen to be equivalent to the previous definition. Let us give an instance of the sort of case which might occur. I am at a
cross-roads and do not know the way; but I rather think one of the two ways is right. I propose therefore [p.175] to go that way but keep my eyes open for someone to ask; if now I see someone half a mile away over the fields, whether I turn aside to ask him will depend on the relative inconvenience of going out of my way to cross the fields or of continuing on the wrong road if it is the wrong road. But it will also depend on how confident I am that I am right; and clearly the more confident I am of this the less distance I should be willing to go from the road to check my opinion. I propose therefore to use the distance I would be prepared to go to ask, as a measure of the
confidence of my opinion; and what I have said above explains how this is to be done. We can set it out as follows: suppose the disadvantage of going x yards to ask is ƒ(x), the advantage of arriving at the right destination is r, that of arriving at the wrong one w. Then if I should just be willing to go a distance d to ask, the degree of my belief that I am on the right road is given by r wf dp − = − ( ) 1For such an action is one it would just pay me to take, if I had to act in the same way n times, in npof which I was on the right way but in the others not. For the total good resulting from not asking each time
= npr + n(1-p)w = nw + np(r - w) that resulting from asking at distance x each time = nr - nÆ’(x) [I now always go right.] This is greater than the preceding expression, provided Æ’(x) < (r - w)(1-p),
∴the critical distance d is connected with the degree of belief, by the relation ƒ(d) = (r - w)(1-p) 18 or r wf dp − = − ( ) 1 as asserted above. [p.176]
It is easy to see that this way of measuring belief gives results agreeing with ordinary ideas; at any rate to the extent that full belief is denoted by 1, full belief in the contradictory by 0, and equal belief in the two by 2 1 . Further, it allows validity to betting as means of measuring beliefs. By proposing a bet on p we give the subject a possible course of action from which so much extra good will result to him if p is true and so much extra bad if p is false. Supposing, the bet to be in goods and bads instead of in money, he will take a bet at any better odds than those corresponding to his state of belief; in fact his state of belief is measured by the odds he will just take; but this is vitiated, as already explained, by love or hatred of excitement, and by the fact that the bet is in money and not in goods and bads. Since it is universally agreed that money has a diminishing marginal utility,
if money bets are to be used, it is evident that they should be for as small stakes as possible. But then again the measurement is spoiled by introducing the new factor of reluctance to bother about
trifles.
Let us now discard the assumption that goods are additive and immediately measurable, and try to work out a system with as few assumptions as possible. To begin with we shall suppose, as before, that our subject has certain beliefs about everything; then he will act so that what he believes to be the total consequences of his action will be the best possible. If then we had the power of the Almighty, and could persuade our subject of our power, we could, by offering him options, discover how he placed in order of merit all possible courses of the world. In this way all possible worlds would be put in an order of value, but we should have no definite way of representing them by numbers. There would be no meaning in the assertion that the difference in value between α and β was equal to that between γ and δ. [Here and elsewhere we use Greek letters to represent the
different possible totalities [p.177] of events between which our subject chooses -- the ultimate organic unities.] Suppose next that the subject is capable of doubt; then we could test his degree of belief in different
propositions by making him offers of the following kind. Would you rather have world α in any event; or world β if p is true, and world γ if p is false? If, then, he were certain that p was true, simply compare α and β and choose between them as if no conditions were attached; but if he were
doubtful his choice would not be decided so simply. I propose to lay down axioms and definitions concerning the principles governing choices of this kind. This is, of course, a very schematic version of the situation in real life, but it is, I think, easier to consider it in this form.
There is first a difficulty which must be dealt with; the propositions like p in the above case which are used as conditions in the options offered may be such that their truth or falsity is an object of desire to the subject. This willbe found to complicate the problem, and we have to assume that there are propositions for which this is not the case, which we shall call ethically neutral. More precisely an atomic proposition p is called ethically neutral if two possible worlds differing only in 19 regard to the truth of p are always of equal value; and a non-atomic proposition p is called ethically
neutral if all its atomic truth-arguments1
are ethically neutral.
We begin by defining belief of degree 2 1 in an ethically neutral proposition. The subject is said to have belief of degree 2 1 in such a proposition p if he has no preference between the options (1) α if p is true, β if p is false, and (2) α if p is false, β if p is true, but has a preference between α and β
simply. We suppose by an axiom that if this is true of any [p.178] one pair α, β, it is true of all such pairs.1 This comes roughly to defining belief of degree
2 1 as such a degree of belief as leads to indifference between betting one way and betting the other for the same stakes. Belief of degree 2
1 as thus defined can be used to measure values numerically in the following way. We have to explain what is meant by the difference in value between α and β being equal to that between γ and δ; and we define this to mean that, if p is an ethically neutral proposition believed to degree 2
1 , the subject has no preference between the options (1) α if p is true, δ if p is false, and (2) β if p is true, γ if p is false.
This definition can form the basis of a system of measuring values in the following way:-- Let us call any set of all worlds equally preferable to a given world a value: we suppose that if world α is preferable to β any world with the same value as α is preferable to any world with the same value as β and shall say that the value of α is greater than that of β. This relation 'greater than' orders values in a series. We shall use α henceforth both for the world and its value.
[...]
& "Further Considerations" and "Probability and Partial Belief"
by
Frank Plumpton Ramsey, M.A.
Fellow and Director of Studies in Mathematics at King's College,
Lecturer in Mathematics in the University of Cambridge
"Truth and Probability" written 1926. Published 1931 in Foundations of Mathematics
and other Logical Essays, Ch. VII, p.156-198. Edited by R.B. Braithwaite. London:
Kegan, Paul, Trench, Trubner & Co. Ltd. New York: Harcourt, Brace and Company
"Further Considerations" written 1928. Published 1931 op. cit., Ch. VIII, p.199-211.
"Probability and Partial Belief" written 1929. Published 1931, op cit., Ch. IX, p.256-57.
6
[p.156]
TRUTH AND PROBABILITY
(1926)
To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is and of what is not that it is not is true.
-- Aristotle.
When several hypotheses are presented to our mind which we believe to be mutually exclusive and exhaustive, but about which we know nothing further, we distribute our belief equally among them .... This being admitted as an account of the way in which we actually do distribute our belief in simple cases, the whole of the subsequent theory follows as a deduction of the way in which we must distribute it in complex cases if we would be consistent.
-- W. F. Donkits.
The object of reasoning is to find out, from the consideration of what we already know, something else which we do not know. Consequently, reasoning is good if it be such as to give a true conclusion from true premises, and not otherwise.
-- C. S. Peirce.
Truth can never be told so as to be understood, and not be believed.
-- W. Blake.
7
[p.157]
FOREWORD
In this essay the Theory of Probability is taken as a branch of logic, the logic of partial belief and inconclusive argument; but there is no intention of implying that this is the only or even the most important aspect of the subject. Probability is of fundamental importance not only in logic but also
in statistical and physical science, and we cannot be sure beforehand that the most useful interpretation of it in logic will be appropriate in physics also. Indeed the general difference of opinion between statisticians who for the most part adopt the frequency theory of probability and logicians who mostly reject it renders it likely that the two schools are really discussing different things, and that the word 'probability' is used by logicians in one sense and by statisticians in another. The conclusions we shall come to as to the meaning of probability in logic must not, therefore, be taken as prejudging its meaning in physics.1
1
[p.157] [A final chapter, on probability in science, was designed but not written. -- ED.]
8
[p.158]
CONTENTS
(1) The Frequency Theory
(2) Mr Keynes' Theory
(3) Degrees of Belief
(4) The Logic of Consistency
(5) The Logic of Truth
(1) THE FREQUENCY THEORY
In the hope of avoiding some purely verbal controversies, I propose to begin by making some admissions in favour of the frequency theory. In the first place this theory must be conceded to have a firm basis in ordinary language, which often uses 'probability' practically as a synonym for
proportion; for example, if we say that the probability of recovery from smallpox is three-quarters, we mean, I think, simply that that is the proportion of smallpox cases which recover. Secondly, if we start with what is called the calculus of probabilities, regarding it first as a branch of pure
mathematics, and then looking round for some interpretation of the formulae which shall show that our axioms are consistent and our subject not entirely useless, then much the simplest and least controversial interpretation of the calculus is one in terms of frequencies. This is true not only of the ordinary mathematics of probability, but also of the symbolic calculus developed by Mr. Keynes; for if in his a/h, a and h are taken to be not propositions but propositional functions or class-concepts which define finite classes, and a/h is taken to mean the proportion of members of h
which are also members of a, then all his propositions become arithmetical truisms.
[p.159]
Besides these two inevitable admissions, there is a third and more important one, which I am prepared to make temporarily although it does not express my real opinion. It is this. Suppose we start with the mathematical calculus, and ask, not as before what interpretation of it is most convenient to the pure mathematicism, but what interpretation gives results of greatest value to science in general, then it may be that the answer is again an interpretation in terms of frequency; that probability as it is used in statistical theories, especially in statistical mechanics -- the kind of
probability whose logarithm is the entropy -- is really a ratio between the numbers, of two classes, or the limit of such a ratio. I do not myself believe this, but I am willing for the present to concede to the frequency theory that probability as used in modern science is really the same as frequency.
But, supposing all this admitted, it still remains the case that we have the authority both of ordinary language and of many great thinkers for discussing under the heading of probability what appears to be quite a different subject, the logic of partial belief. It may be that, as some supporters of the frequency theory have maintained, the logic of partial belief will be found in the end to be merely the study of frequencies, either because partial belief is definable as, or by reference to, some sort of
frequency, or because it can only be the subject of logical treatment when it is grounded on experienced frequencies. Whether these contentions are valid can, however, only be decided as a
9
result of our investigation into partial belief, so that I propose to ignore the frequency theory for the present and begin an inquiry into the logic of partial belief. In this, I think, it will be most convenient if, instead of straight away developing my own theory, I begin by examining the views of Mr Keynes, which are so well known and in essentials so widely accepted that readers probably feel [p.160] that there is no ground for re-opening the subject de novo until they have been disposed of.
(2) MR KEYNES' THEORY
Mr Keynes1
starts from the supposition that we make probable inferences for which we claim objective validity; we proceed from full belief in one proposition to partial belief in another, and we claim that this procedure is objectively right, so that if another man in similar circumstances entertained a different degree of belief, he would be wrong in doing so. Mr Keynes accounts for this
by supposing that between any two propositions, taken as premiss and conclusion, there holds one and only one relation of a certain sort called probability relations; and that if, in any given case, the relation is that of degree α, from full belief in the premiss, we should, if we were rational, proceed to a belief of degree α in the conclusion.
Before criticising this view, I may perhaps be allowed to point out an obvious and easily corrected defect in the statement of it. When it is said that the degree of the probability relation is the same as the degree of belief which it justifies, it seems to be presupposed that both probability relations, on
the one hand, and degrees of belief on the other can be naturally expressed in terms of numbers, and then that the number expressing or measuring the probability relation is the same as that expressing the appropriate degree of belief. But if, as Mr. Keynes holds, these things are not always expressible
by numbers, then we cannot give his statement that the degree of the one is the same as the degree of the other such a simple interpretation, but must suppose him to mean only that there is a one-one correspondence between probability relations and the degrees of belief which [p.161] they justify.
This correspondence must clearly preserve the relations of greater and less, and so make the manifold of probability relations and that of degrees of belief similar in Mr Russell's sense. I think it is a pity that Mr Keynes did not see this clearly, because the exactitude of this correspondence
would have provided quite as worthy material scepticism as did the numerical measurement of pro bability relations. Indeed some of his arguments against their numerical measurement appear to
apply quite equally well against their exact correspondence with degrees of belief; for instance, he argues that if rates of insurance correspond to subjective, i.e. actual, degrees of belief, these are not rationally determined, and we cannot infer that probability relations can be similarly measured. It
might be argued that the true conclusion in such a case was not that, as Mr Keynes thinks, to the non-numerical probability relation corresponds a non-numerical degree of rational belief, but that degrees of belief, which were always numerical, did not correspond one to one with the probability
relations justifying them. For it is, I suppose, conceivable that degrees of belief could be measured by a psychogalvanometer or some such instrument, and Mr Keynes would hardly wish it to follow that probability relations could all be derivatively measured with the measures of the beliefs which they justify.
1
[p.160] J.M. Keynes, A Treatise on Probability (1921).
10
But let us now return to a more fundamental criticism of Mr Keynes' views, which is the obvious one that there really do not seem to be any such things as the probability relations he describes. He supposes that, at any rate in certain cases, they can be perceived; but speaking for myself I feel con fident that this is not true. I do not perceive them, and if I am to be persuaded that they exist it must be by argument; moreover I shrewdly suspect that others do not perceive them either, because they
are able to come to so very little agreement as to which of them relates any two given propositions.
[p.162]
All we appear to know about them are certain general propositions, the laws of addition and multiplication; it is as if everyone knew the laws of geometry but no one could tell whether any given object were round or square; and I find it hard to imagine how so large a body of general knowledge can be combined with so slender a stock of particular facts. It is true that about some particular cases there is agreement, but these somehow paradoxically are always immensely complicated; we all agree that the probability of a coin coming down heads is 2 1 , but we can none of us say exactly what is the evidence which forms the other term for the probability relation about
which we are then judging. If, on the other hand, we take the simplest possible pairs of propositions such as 'This is red' and 'That is blue' or 'This is red' and 'That is red', whose logical relations should surely be easiest to see, no one, I think, pretends to be sure what is the probability relation which
connects them. Or, perhaps, they may claim to see the relation but they will not be able to say anything about it with certainty, to state if it is more or less than 31 , or so on. They may, of course, say that it is incomparable with any numerical relation, but a relation about which so little can be
truly said will be of little scientific use and it will be hard to convince a sceptic of its existence.
Besides this view is really rather paradoxical; for any believer in induction must admit that between 'This is red ' as conclusion and ' This is round ', together with a billion propositions of the form 'a is round and red' as evidence, there is a finite probability relation; and it is hard to suppose that as we accumulate instances there is suddenly a point, say after 233 instances, at which the probability relation becomes finite and so comparable with some numerical relations. It seems to me that if we take the two propositions 'a is red', 'b is red', we cannot really discern more
than four [p.163] simple logical relations between them; namely identity of form, identity of predicate, diversity of subject, and logical independence of import. If anyone were to ask me what probability one gave to the other, I should not try to answer by contemplating the propositions and
trying to discern a logical relation between them, I should, rather, try to imagine that one of them was all that I knew, and to guess what degree of confidence I should then have in the other. If I were able to do this, I might no doubt still not be content with it, but might say ' This is what I
should think, but, of course, I am only a fool' and proceed to consider what a wise man would think and call that the degree of probability. This kind of self-criticism I shall discuss later when developing my own theory; all that I want to remark here is that no one estimating a degree of probability simply contemplates the two propositions supposed to be related by it; he always
considers inter alia his own actual or hypothetical degree of belief. This remark seems to me to be borne out by observation of my own behaviour; and to be the only way of accounting for the fact that we can all give estimates of probability in cases taken from actual life, but are quite unable to
11
do so in the logically simplest cases in which, were probability a logical relation, it would be easiest to discern. Another argument against Mr Keynes' theory can, I think, be drawn from his inability to adhere to it consistently even in discussing first principles. There is a passage in his chapter on the
measurement of probabilities which reads as follows: -- "Probability is, vide Chapter 11 (§12), relative in a sense to the principles of human reason. The
degree of probability, which it is rational for us to entertain, does not presume perfect logical insight, and is relative in part to the secondary propositions which we in fact know; and it is not dependent upon whether more perfect logical insight [p.164] is or is not conceivable. It is the
degree of probability to which those logical processes lead, of which our minds are capable; or, in the language of Chapter II, which those secondary propositions justify, which we in fact know. If we do not take this view of probability, if we do not limit it in this way and make it, to this extent,
relative to human powers, we are altogether adrift in the unknown; for we cannot ever know what degree of probability would be justified by the perception of logical relations which we are, and must always be, incapable of comprehending." 1
This passage seems to me quite unreconcilable with the view which Mr Keynes adopts everywhere except in this and another similar passage. For he generally holds that the degree of belief which we are justified in placing in the conclusion of an argument is determined by what relation of
probability unites that conclusion to our premisses, There is only one such relation and consequently only one relevant true secondary proposition, which, of course, we may or may not know, but which is necessarily independent of the human mind. If we do not know it, we do not
know it and cannot tell how far we ought to believe the conclusion. But often, he supposes, we do know it; probability relations are not ones which we are incapable of comprehending. But on this view of the matter the passage quoted above has no meaning: the relations which justify probable
beliefs are probability relations, and it is nonsense to speak of them being justified by logical relations which we are, and must always be, incapable of comprehending. The significance of the passage for our present purpose lies in the fact that it seems to presuppose a different view of
probability, in which indefinable probability relations play no part, but in which the degree of rational belief depends on a variety of logical relations. For instance, there might be between the premiss and conclusion the relation [p.165] that the premiss was the logical product of a thousand
instances of a generalization of which the conclusion was one other instance, and this relation, which is not an indefinable probability relation but definable in terms of ordinary logic and so easily recognizable, might justify a certain degree of belief in the conclusion on the part of one who
believed the premiss. We should thus have a variety of ordinary logical relations justifying the same or different degrees of belief. To say that the probability of a given h was such-and-such would mean that between a and h was some relation justifying such-and-such a degree of belief. And on
this view it would be a real point that the relation in question must not be one which the human mind is incapable of comprehending.
1
[p.164] p. 32, his italics.
12
This second view of probability as depending on logical relations but not itself a new logical relation seems to me more plausible than Mr Keynes' usual theory; but this does not mean that I feel at all inclined to agree with it. It requires the somewhat obscure idea of a logical relation justifying a
degree of belief, which I should not like to accept as indefinable because it does not seem to be at all a clear or simple notion. Also it is hard to say what logical relations justify what degrees of belief, and why; any decision as to this would be arbitrary, and would lead to a logic of probability
consisting of a host of so-called 'necessary' facts, like formal logic on Mr Chadwick's view of logical constants.1
Whereas I think it far better to seek an explanation of this 'necessity' after the model of the work of Mr Wittgenstein, which enables us to see clearly in what precise sense and why logical propositions are necessary, and in a general way why the system of formal logic consists of the propositions it does consist of, and what is their common characteristic. Just as
natural science tries to explain and [p.166] account for the facts of nature, so philosophy should try, in a sense, to explain and account for the facts of logic; a task ignored by the philosophy which dismisses these facts as being unaccountably and in an indefinable sense ' necessary'.
Here I propose to conclude this criticism of Mr Keynes' theory, not because there are not other respects in which it seems open to objection, but because I hope that what I have already said is enough to show that it is not so completely satisfactory as to render futile any attempt to treat the
subject from a rather different point of view.
(3) DEGREES OF BELIEF
The subject of our inquiry is the logic of partial belief, and I do not think we can carry it far unless we have at least an approximate notion of what partial belief is, and how, if at all, it can be measured. It will not be very enlightening to be told that in such circumstances it would be rational to believe a proposition to the extent of 2/3, unless we know what sort of a belief in it that means. We must therefore try to develop a purely psychological method of measuring belief. It is not enough to measure probability; in order to apportion correctly our belief to the probability we must also be able to measure our belief.
It is a common view that belief and other psychological variables are not measurable, and if this is true our inquiry will be vain ; and so will the whole theory of probability conceived as a logic of partial belief; for if the phrase 'a belief two-thirds of certainty ' is meaningless, a calculus whose sole object is to enjoin such beliefs will be meaningless also. Therefore unless we are prepared to give up the whole thing as a bad job we are bound to hold that beliefs can to some extent be measured. If we were to follow the analogy [p.167] of Mr Keynes' treatment of probabilities we should say that some beliefs were measurable and some not; but this does not seem to me likely to be a correct account of the matter: I do not see how we can sharply divide beliefs into those which have a position in the numerical scale and those which have not. But I think beliefs do differ in measurability in the following two ways. First, some beliefs can be measured more accurately than others; and, secondly, the measurement of beliefs is almost certainly an ambiguous process leading to a variable answer depending on how exactly the measurement is conducted. The degree of a belief is in this respect like the time interval between two events; before Einstein it was supposed
1
[p.165] "Logical Constants", Mind, 1927.
13
that all the ordinary ways of measuring a time interval would lead to the same result if properly performed. Einstein showed that this was not the case; and time interval can no longer be regarded as an exact notion, but must be discarded in all precise investigations. Nevertheless, time interval
and the Newtonian system are sufficiently accurate for many purposes and easier to apply. I shall try to argue later that the degree of a belief is just like a time interval; it has no precise meaning unless we specify more exactly how it is to be measured. But for many purposes we can assume that the alternative ways of measuring it lead to the same result, although this is only
approximately true. The resulting discrepancies are more glaring in connection with some beliefs than with others, and these therefore appear less measurable. Both these types of deficiency in measurability, due respectively to the difficulty in getting an exact enough measurement and to an important ambiguity in the definition of the measurement process, occur also in physics and so are not difficulties peculiar to our problem; what is peculiar is that it is difficult to form any idea of how the measurement is to be conducted, how a unit is to be obtained, and so on.
[p.168]
Let us then consider what is implied in the measurement of beliefs. A satisfactory system must in the first place assign to any belief a magnitude or degree having a definite position in an order of magnitudes; beliefs which are of the same degree as the same belief must be of the same degree as
one another, and so on. Of course this cannot be accomplished without introducing a certain amount of hypothesis or fiction. Even in physics we cannot maintain that things that are equal to the same thing are equal to one another unless we take 'equal' not as meaning 'sensibly equal' but a
fictitious or hypothetical relation. I do not want to discuss the metaphysics or epistemology of this process, but merely to remark that if it is allowable in physics it is allowable in psychology also. The logical simplicity characteristic of the relations dealt with in a science is never attained by
nature alone without any admixture of fiction. But to construct such an ordered series of degrees is not the whole of our task; we have also to
assign numbers to these degrees in some intelligible manner. We can of course easily explain that we denote full belief by 1, full belief in the contradictory by 0, and equal beliefs in the proposition and its contradictory by 2 1 . But it is not so easy to say what is meant by a belief 3 2 of certainty, or a belief in the proposition being twice as strong as that in its contradictory. This is the harder part of the task, but it is absolutely necessary; for we do calculate numerical probabilities, and if they are to
correspond to degrees of belief we must discover some definite way of attaching numbers to degrees of belief. In physics we often attach numbers by discovering a physical process of addition1: the measure-numbers of lengths are not assigned arbitrarily subject only to the proviso that the
greater length shall have the greater measure; we determine them further by deciding on a [p.169] physical meaning for addition ; the length got by putting together two given lengths must have for its measure the sum of their measures. A system of measurement in which there is nothing
corresponding to this is immediately recognized as arbitrary, for instance Mohs' scale of hardness1
1
[p.168] See N. Campbell, Physics The Elements (1920), p.277. 1
[p.169] Ibid., p.271.
14
in which 10 is arbitrarily assigned to diamond, the hardest known material, 9 to the next hardest, and so on. We have therefore to find a process of addition for degrees of belief, or some substitute for this which will be equally adequate to determine a numerical scale. Such is our problem; how are we to solve it? There are, I think, two ways in which we can begin.
We can, in the first place, suppose that the degree of a belief is something perceptible by its owner; for instance that beliefs differ in the intensity of a feeling by which they are accompanied, which might be called a belief-feeling or feeling of conviction, and that by the degree of belief we mean
the intensity of this feeling. This view would be very inconvenient, for it is not easy to ascribe numbers to the intensities of feelings; but apart from this it seems to me observably false, for the beliefs which we hold most strongly are often accompanied by practically no feeling at all; no one
feels strongly about things he takes for granted.
We are driven therefore to the second supposition that the degree of a belief is a causal property of it, which we can express vaguely as the extent to which we are prepared to act on it. This is a generalization of the well-known view, that the differentia of belief lies in its causal efficacy, which is discussed by Mr Russell in his Analysis of Mind. He there dismisses it for two reasons, one of which seems entirely to miss the point. He argues that in the course of trains of thought we believe many things which do not lead to action. This objection is however beside the mark, because
[p.170] it is not asserted that a belief is an idea which does actually lead to action, but one which would lead to action in suitable circumstances; just as a lump of arsenic is called poisonous not because it actually has killed or will kill anyone, but because it would kill anyone if he ate it. Mr Russell's second argument is, however, more formidable. He points out that it is not possible to suppose that beliefs differ from other ideas only in their effects, for if they were otherwise identical their effects would be identical also. This is perfectly true, but it may still remain the case that the nature of the difference between the causes is entirely unknown or very vaguely known, and that what we want to talk about is the difference between the effects, which is readily observable and important.
As soon as we regard belief quantitatively, this seems to me the only view we can take of it. It could well be held that the difference between believing and not believing lies in the presence or absence of introspectible feelings. But when we seek to know what is the difference between believing more
firmly and believing less firmly, we can no longer regard it as consisting in having more or less of certain observable feelings; at least I personally cannot recognize any such feelings. The difference seems to me to lie in how far we should act on these beliefs: this may depend on the degree of some feeling or feelings, but I do not know exactly what feelings and I do not see that it is indispensable that we should know. Just the same thing is found in physics; men found that a wire connecting plates of zinc and copper standing in acid deflected a magnetic needle in its neighbourhood.
Accordingly as the needle was more or less deflected the wire was said to carry a larger or a smaller current. The nature of this 'current' could only be conjectured: what were observed and measured were simply its effects. It will no doubt be objected that we know how strongly [p.171] we believe
things, and that we can only know this if we can measure our belief by introspection. This does not seem to me necessarily true; in many cases, I think, our judgment about the strength of our belief is really about how we should act in hypothetical circumstances. It will be answered that we can only tell how we should act by observing the present belief-feeling which determines how we should act;
15
but again I doubt the cogency of the argument. It is possible that what determines how we should act determines us also directly or indirectly to have a correct opinion as to how we should act, without its ever coming into consciousness. Suppose, however, I am wrong about this and that we can decide by introspection the nature of belief, and measure its degree; still, I shall argue, the kind of measurement of belief with which probability is concerned is not this kind but is a measurement of belief qua basis of action. This can I think be shown in two ways. First, by considering the scale of probabilities between 0 and 1, and the sort of way we use it, we shall find that it is very appropriate to the measurement of belief as a basis of action, but in no way related to the measurement of an introspected feeling. For the units in terms of which such feelings or sensations are measured are always, I think, differences which are just perceptible: there is no other way of obtaining units. But I see no ground for supposing that the
interval between a belief of degree 3 1 and one of degree 2 1 consists of as many just perceptible changes as does that between one of 3
2 and one of 6 5 , or that a scale based on just perceptible differences would have any simple relation to the theory of probability. On the other hand the
probability of 31 is clearly related to the kind of belief which would lead to a bet of 2 to 1, and it will be shown below how to generalize this relation so as to apply to action in general. Secondly, the quantitative aspects of beliefs as the basis of action are evidently more important than the intensities of belief-feelings. [p.170] The latter are no doubt interesting, but may be very variable from individual to individual, and their practical interest is entirely due to their position as the hypothetical causes of beliefs qua bases of action. It is possible that some one will say that the extent to which we should act on a belief in suitable circumstances is a hypothetical thing, and therefore not capable of measurement. But to say this is merely to reveal ignorance of the physical sciences which constantly deal with and measure
hypothetical quantities; for instance, the electric intensity at a given point is the force which would act on a unit charge if it were placed at the point.
Let us now try to find a method of measuring beliefs as bases of possible actions. It is clear that we are concerned with dispositional rather than with actualized beliefs; that is to say, not with beliefs at
the moment when we are thinking of them, but with beliefs like my belief that the earth is round, which I rarely think of, but which would guide my action in any case to which it was relevant. The old-established way of measuring a person's belief is to propose a bet, and see what are the
lowest odds which he will accept. This method I regard as fundamentally sound; but it suffers from being insufficiently general, and from being necessarily inexact. It is inexact partly because of the diminishing marginal utility of money, partly because the person may have a special eagerness or
reluctance to bet, because he either enjoys or dislikes excitement or for any other reason, e.g. to make a book. The difficulty is like that of separating two different co-operating forces. Besides, the proposal of a bet may inevitably alter his state of opinion; just as we could not always measure
electric intensity by actually introducing a charge and seeing what force it was subject to, because the introduction of the charge would change the distribution to be measured.
16
[p.173]
In order therefore to construct a theory of quantities of belief which shall be both general and more exact, I propose to take as a basis a general psychological theory, which is now universally discarded, but nevertheless comes, I think, fairly close to the truth in the sort of cases with which
we are most concerned. I mean the theory that we act in the way we think most likely to realize the objects of our desires, so that a person's actions are completely determined by his desires and opinions. This theory cannot be made adequate to all the facts, but it seems to me a useful
approximation to the truth particularly in the case of our self-conscious orprofessional life, and it is presupposed in a great deal of our thought. It is a simple theory and one which many psychologists would obviously like to preserve by introducing unconscious desires and unconscious opinions in
order to bring it more into harmony with the facts. How far such fictions can achieve the required result I do not attempt to judge: I only claim for what follows approximate truth, or truth in relation to this artificial system of psychology, which like Newtonian mechanics can, I think, still be
profitably used even though it is known to be false. It must be observed that this theory is not to be identified with the psychology of the Utilitarians, in
which pleasure had a dominating position. The theory I propose to adopt is that we seek things which we want, which may be our own or other people's pleasure, or anything else whatever, and our actions are such as we think most likely to realize these goods. But this is not a precise statement, for a precise statement of the theory can only be made after we have introduced the notion of quantity of belief.
Let us call the things a person ultimately desires 'goods', and let us at first assume that they are numerically measurable and additive. That is to say that if he prefers for its own sake an hour's swimming to an hour's reading, he will prefer [p.174] two hours' swimming to one hour's swimming and one hour's reading. This is of course absurd in the given case but this may only be because swimming and reading are not ultimate goods, and because we cannot imagine a second hour's swimming precisely similar to the first, owing to fatigue, etc. Let us begin by supposing that our subject has no doubts about anything, but certain opinions about all propositions. Then we can say that he will always choose the course of action which will lead in
his opinion to the greatest sum of good. It should be emphasized that in this essay good and bad are never to be understood in any ethical sense but simply as denoting that to which a given person feels desire and aversion.
The question then arises how we are to modify this simple system to take account of varying degrees of certainty in his beliefs. I suggest that we introduce as a law of psychology that his behaviour is governed by what is called the mathematical expectation; that is to say that, if p is a
proposition about which he is doubtful, any goods or bads for whose realization p is in his view a necessary and sufficient condition enter into his calculations multiplied by the same fraction, which is called the 'degree of his belief in p'. We thus define degree of belief in a way which presupposes
the use of the mathematical expectation.
17
We can put this in a different way. Suppose his degree of belief in p is n
m ; then his action is such as he would choose it to be if he had to repeat it exactly n times, in m of which p was true, and in the others false. [Here it may be necessary to suppose that in each of the n times he had no memory of the previous ones.] This can also be taken as a definition of the degree of belief, and can easily be seen to be equivalent to the previous definition. Let us give an instance of the sort of case which might occur. I am at a
cross-roads and do not know the way; but I rather think one of the two ways is right. I propose therefore [p.175] to go that way but keep my eyes open for someone to ask; if now I see someone half a mile away over the fields, whether I turn aside to ask him will depend on the relative inconvenience of going out of my way to cross the fields or of continuing on the wrong road if it is the wrong road. But it will also depend on how confident I am that I am right; and clearly the more confident I am of this the less distance I should be willing to go from the road to check my opinion. I propose therefore to use the distance I would be prepared to go to ask, as a measure of the
confidence of my opinion; and what I have said above explains how this is to be done. We can set it out as follows: suppose the disadvantage of going x yards to ask is ƒ(x), the advantage of arriving at the right destination is r, that of arriving at the wrong one w. Then if I should just be willing to go a distance d to ask, the degree of my belief that I am on the right road is given by r wf dp − = − ( ) 1For such an action is one it would just pay me to take, if I had to act in the same way n times, in npof which I was on the right way but in the others not. For the total good resulting from not asking each time
= npr + n(1-p)w = nw + np(r - w) that resulting from asking at distance x each time = nr - nÆ’(x) [I now always go right.] This is greater than the preceding expression, provided Æ’(x) < (r - w)(1-p),
∴the critical distance d is connected with the degree of belief, by the relation ƒ(d) = (r - w)(1-p) 18 or r wf dp − = − ( ) 1 as asserted above. [p.176]
It is easy to see that this way of measuring belief gives results agreeing with ordinary ideas; at any rate to the extent that full belief is denoted by 1, full belief in the contradictory by 0, and equal belief in the two by 2 1 . Further, it allows validity to betting as means of measuring beliefs. By proposing a bet on p we give the subject a possible course of action from which so much extra good will result to him if p is true and so much extra bad if p is false. Supposing, the bet to be in goods and bads instead of in money, he will take a bet at any better odds than those corresponding to his state of belief; in fact his state of belief is measured by the odds he will just take; but this is vitiated, as already explained, by love or hatred of excitement, and by the fact that the bet is in money and not in goods and bads. Since it is universally agreed that money has a diminishing marginal utility,
if money bets are to be used, it is evident that they should be for as small stakes as possible. But then again the measurement is spoiled by introducing the new factor of reluctance to bother about
trifles.
Let us now discard the assumption that goods are additive and immediately measurable, and try to work out a system with as few assumptions as possible. To begin with we shall suppose, as before, that our subject has certain beliefs about everything; then he will act so that what he believes to be the total consequences of his action will be the best possible. If then we had the power of the Almighty, and could persuade our subject of our power, we could, by offering him options, discover how he placed in order of merit all possible courses of the world. In this way all possible worlds would be put in an order of value, but we should have no definite way of representing them by numbers. There would be no meaning in the assertion that the difference in value between α and β was equal to that between γ and δ. [Here and elsewhere we use Greek letters to represent the
different possible totalities [p.177] of events between which our subject chooses -- the ultimate organic unities.] Suppose next that the subject is capable of doubt; then we could test his degree of belief in different
propositions by making him offers of the following kind. Would you rather have world α in any event; or world β if p is true, and world γ if p is false? If, then, he were certain that p was true, simply compare α and β and choose between them as if no conditions were attached; but if he were
doubtful his choice would not be decided so simply. I propose to lay down axioms and definitions concerning the principles governing choices of this kind. This is, of course, a very schematic version of the situation in real life, but it is, I think, easier to consider it in this form.
There is first a difficulty which must be dealt with; the propositions like p in the above case which are used as conditions in the options offered may be such that their truth or falsity is an object of desire to the subject. This willbe found to complicate the problem, and we have to assume that there are propositions for which this is not the case, which we shall call ethically neutral. More precisely an atomic proposition p is called ethically neutral if two possible worlds differing only in 19 regard to the truth of p are always of equal value; and a non-atomic proposition p is called ethically
neutral if all its atomic truth-arguments1
are ethically neutral.
We begin by defining belief of degree 2 1 in an ethically neutral proposition. The subject is said to have belief of degree 2 1 in such a proposition p if he has no preference between the options (1) α if p is true, β if p is false, and (2) α if p is false, β if p is true, but has a preference between α and β
simply. We suppose by an axiom that if this is true of any [p.178] one pair α, β, it is true of all such pairs.1 This comes roughly to defining belief of degree
2 1 as such a degree of belief as leads to indifference between betting one way and betting the other for the same stakes. Belief of degree 2
1 as thus defined can be used to measure values numerically in the following way. We have to explain what is meant by the difference in value between α and β being equal to that between γ and δ; and we define this to mean that, if p is an ethically neutral proposition believed to degree 2
1 , the subject has no preference between the options (1) α if p is true, δ if p is false, and (2) β if p is true, γ if p is false.
This definition can form the basis of a system of measuring values in the following way:-- Let us call any set of all worlds equally preferable to a given world a value: we suppose that if world α is preferable to β any world with the same value as α is preferable to any world with the same value as β and shall say that the value of α is greater than that of β. This relation 'greater than' orders values in a series. We shall use α henceforth both for the world and its value.
[...]
Jeffrey Rubard
Dec 17, 2021, 2:09:37 AM12/17/21
to
On Friday, December 10, 2021 at 10:08:31 PM UTC-8, Jeffrey Rubard wrote:
> TRUTH AND PROBABILITY
> & "Further Considerations" and "Probability and Partial Belief"
> by
> Frank Plumpton Ramsey, M.A.
> Fellow and Director of Studies in Mathematics at King's College,
> Lecturer in Mathematics in the University of Cambridge
Personal opinion: Ramsey was a great thinker; people who ignore
> TRUTH AND PROBABILITY
> & "Further Considerations" and "Probability and Partial Belief"
> by
> Frank Plumpton Ramsey, M.A.
> Fellow and Director of Studies in Mathematics at King's College,
> Lecturer in Mathematics in the University of Cambridge
him (unlike his friend Wittgenstein) do poorly by doing so. An
opinion which is "cheap", in its own way.
Jeffrey Rubard
Dec 19, 2021, 1:33:26 AM12/19/21
to
to an event's happening, under certain axioms it can be a *possible probability* to a
standard of Kolmogorovian "frequentist" statistics. If you can conceive it, *maybe*
you could achieve it: maybe you could skip it, too. It also just isn't the case that
"what happens is what would usually happen".
As for everyone else, every 'probability metric' is somehow skewed in favor of
certain 'theoretical biases' we may not even be explicitly aware of: it just seemed
impossible to have a novel world-wide epidemic of importance at this point in
history (it seems impossible to me even today, but maybe *somehow* my
view isn't authoritative).
Jeffrey Rubard
Dec 19, 2021, 12:41:59 PM12/19/21
to
have to remember, for self-interested reasons, that *what usually happens
usually happens*: haters gonna hate, tax forms gonna mail etc.
Jeffrey Rubard
Dec 19, 2021, 12:42:55 PM12/19/21
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legal name -- perhaps it has not changed, etc. Good times.
Jeffrey Rubard
Dec 20, 2021, 2:18:19 AM12/20/21
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Ketchup is not a vegetable, but one does pour water on plants.
Jeffrey Rubard
Dec 20, 2021, 10:20:50 AM12/20/21
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Jeffrey Rubard
Dec 21, 2021, 11:57:05 AM12/21/21
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(Rigorized.)
Jeffrey Rubard
Dec 22, 2021, 12:04:57 PM12/22/21
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is a little "unfortunate" at the present moment, but then hey, *it sure was*.
Jeffrey Rubard
Dec 23, 2021, 3:05:51 AM12/23/21
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So there's a 'role for the amateur' here.
Jeffrey Rubard
Jeffrey Rubard
Dec 26, 2021, 12:25:27 AM12/26/21
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Jeffrey Rubard
Dec 26, 2021, 11:04:14 AM12/26/21
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Jeffrey Rubard
Dec 27, 2021, 1:57:37 AM12/27/21
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Jeffrey Rubard
Dec 28, 2021, 2:02:06 AM12/28/21
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(How easy would it be to say that to everything, and then to 'say it back' to it?)
Jeffrey Rubard
Dec 28, 2021, 5:11:37 PM12/28/21
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(Sure, but you have a weird theory of "certainties" twice over.)
Jeffrey Rubard
Dec 30, 2021, 8:17:50 PM12/30/21
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You don't say.
(Literally. People will find fault with this "schematic" at the oddest of times.)
Jeffrey Rubard
Dec 31, 2021, 1:02:13 AM12/31/21
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"What is necessary must be instantiated."
Or:
"What is actual must have been in some way possible."
(The relationship of "T" to Ramseyan decision theory is highly underexplored.)
Jeffrey Rubard
Dec 31, 2021, 11:31:09 AM12/31/21
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"If today is Dec 31st, tomorrow will be Jan 1st."
Other assessments, though, are highly "theory-laden" -- you might
well think something else, couched in some other terms.
Jeffrey Rubard
Jan 1, 2022, 8:30:10 PM1/1/22
to
(What's that mean? Roughly, that any "hidden evidence" of a thing or state of affairs existing
is the reason everyone else but you is somehow not resistant to that to which you say,
self-importantly, "I'm not sure...")
Jeffrey Rubard
Jan 2, 2022, 11:03:22 PM1/2/22
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last year? I'd have to say "no".
Jeffrey Rubard
Jan 3, 2022, 12:02:13 PM1/3/22
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it's not here-for-you-to-stare-at" true?)
Jeffrey Rubard
Jan 4, 2022, 2:52:06 AM1/4/22
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did much else besides.
Jeffrey Rubard
Jan 4, 2022, 2:52:38 PM1/4/22
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Jeffrey Rubard
Jan 5, 2022, 1:22:20 AM1/5/22
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That makes sense. Can you take other things as "granted"? Good for you.
Jeffrey Rubard
Jan 8, 2022, 3:08:36 AM1/8/22
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Well, you either figure it as a "thing known" or you don't. You also either figure that you have reliable information about it, or you don't.
Jeffrey Rubard
Jan 8, 2022, 10:57:18 AM1/8/22
to
Thing not known / no reliable information: "unknown likelihood"
Thing not known / reliable information: "highly unlikely"
Thing known / no reliable information (about the related matter): "probability uncertain"
Thing known / reliable information: "Of course, is there evidence to the contrary?"
Jeffrey Rubard
Jan 9, 2022, 11:17:31 PM1/9/22
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(Just somehow not a high-quality invitation to candid discourse.)
Jeffrey Rubard
Jan 10, 2022, 7:14:37 AM1/10/22
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as a species of game theory. It's a rational model of "decision making under uncertainty"
(calculating probabilities), just not a model of how we *communicate* about those probabilities.
Jeffrey Rubard
Jan 18, 2022, 9:52:47 AM1/18/22
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Jeffrey Rubard
Jan 18, 2022, 9:47:30 PM1/18/22
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Jeffrey Rubard
Jan 19, 2022, 5:53:42 PM1/19/22
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Perhaps that this "could be"?
Jeffrey Rubard
Jan 19, 2022, 10:53:06 PM1/19/22
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Jeffrey Rubard
Jan 22, 2022, 9:50:18 PM1/22/22
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(Thanks for sleeping here, guys!)
Jeffrey Rubard
Jan 23, 2022, 6:45:35 PM1/23/22
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(You, obviously. But you might be wrong, or just "right for you".)
Jeffrey Rubard
Dec 15, 2022, 9:35:42 PM12/15/22
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Jeffrey Rubard
Dec 16, 2022, 11:48:19 AM12/16/22
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Sure, but maybe you know less about them than you think.
"That seems unlikely."
You're a Ramseyan already!
Jeffrey Rubard
Dec 16, 2022, 3:26:21 PM12/16/22
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I think we all feel that way, sometimes. You do understand that's a little "speculative", though?
Jeffrey Rubard
Dec 18, 2022, 11:08:34 AM12/18/22
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I once thought more like this. However, maybe your perspective is 'foreshortened', you've been in circumstances where you seem more genuinely 'prescient' than against a larger temporal or spatial background.
"I guess that could be."
It basically could, yes.
Jeffrey Rubard
Dec 18, 2022, 2:11:13 PM12/18/22
to
Maybe the second sentence kind of "governs" the first, it's more that you're strong-arming people with your views of what "could be" than that you have exceptional insight.
Jeffrey Rubard
Dec 18, 2022, 9:07:40 PM12/18/22
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That... maybe you believe a little bit too much... in contrived social arrangements in a small area under consideration.
(I've heard of at least 5,000 years of history, and I think I'm not too "expert" on the topic. Which would mean...)
Jeffrey Rubard
Jan 2, 2023, 4:53:55 PM1/2/23
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That's perfectly normal and sensible, and can even help construct a probability metric.
(Just not an "ultra-objective" one.)
Jeffrey Rubard
Jan 3, 2023, 7:55:27 PM1/3/23
to
(That makes sense, but it makes the sense it makes.)
Jeffrey Rubard
Jan 4, 2023, 6:37:36 PM1/4/23
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Life's just not that obvious, no.
Jeffrey Rubard
Jan 8, 2023, 6:45:05 PM1/8/23
to
That makes sense, too.
Jeffrey Rubard
Jan 9, 2023, 12:24:34 PM1/9/23
to
It determines the things you "look into", so it rationally constrains what you *know* is real.
"Hmm... got it."
Maybe not, I find it a difficult concept too.
Jeffrey Rubard
Jan 21, 2023, 5:52:42 PM1/21/23
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And in your mind's eye, too! Without "graphical illustrations"!
Jeffrey Rubard
Jan 27, 2023, 4:21:50 PM1/27/23
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Sure.
Jeffrey Rubard
Feb 6, 2023, 7:41:36 PM2/6/23
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Jeffrey Rubard
Feb 8, 2023, 6:36:36 PM2/8/23
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Jeffrey Rubard
Jul 7, 2023, 11:36:37 AM7/7/23
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"It could be."
Jeffrey Rubard
Jul 7, 2023, 4:55:30 PM7/7/23
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I'm not omniscient, no.
Jeffrey Rubard
Jul 9, 2023, 11:29:54 AM7/9/23
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Aren't we all reckoned "Kolmogorovians" by default, really?
Jeffrey Rubard
Jul 14, 2023, 11:26:04 AM7/14/23
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You don't *think* so?
Jeffrey Rubard
Jul 15, 2023, 11:45:38 AM7/15/23
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Sure, I guess?
Jeffrey Rubard
Jul 17, 2023, 11:17:59 AM7/17/23
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No, that's "leading argumentation". You could have made a correct judgment that some state of affairs obtained, though.
Jeffrey Rubard
Jul 17, 2023, 11:28:07 AM7/17/23
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What's not obvious, or trivial, is when we *don't* know something we could know.
Jeffrey Rubard
Jul 18, 2023, 11:47:50 AM7/18/23
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You don't know what you don't know. It's "axiomatic", even.
Jeffrey Rubard
Jul 19, 2023, 2:23:41 PM7/19/23
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Yeah... only we all never see our "blindspots" like this. (Wittgenstein once drew a diagram to illustrate it.)
Jeffrey Rubard
Jul 20, 2023, 11:32:37 AM7/20/23
to
Would this then be a principle for new further self-growth, then?
"I don't know how."
Hmm, maybe it still would be.
Jeffrey Rubard
Aug 9, 2023, 4:21:36 PM8/9/23
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Jeffrey Rubard
Aug 12, 2023, 11:45:10 AM8/12/23
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Reconcile evidence and your "choice preferences" better, dudes.
Jeffrey Rubard
Aug 13, 2023, 11:46:40 AM8/13/23
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No, I, uh, didn't.
Jeffrey Rubard
Aug 14, 2023, 11:30:21 AM8/14/23
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I didn't.
Jeffrey Rubard
Aug 23, 2023, 2:23:45 PM8/23/23
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Jeffrey Rubard
Aug 25, 2023, 3:07:15 PM8/25/23
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Wittgenstein: "Thought can be of what is *not* the case."
Jeffrey Rubard
Aug 26, 2023, 11:31:47 AM8/26/23
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That sounds a little bit like "wishful thinking".
Jeffrey Rubard
Aug 27, 2023, 5:41:09 PM8/27/23
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That's an interesting question.
Jeffrey Rubard
Aug 28, 2023, 4:23:24 PM8/28/23
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Jeffrey Rubard
Aug 29, 2023, 11:31:12 AM8/29/23
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I read *The Logic of Decision*? (And Ramsey, too.)
Jeffrey Rubard
Aug 30, 2023, 2:28:16 PM8/30/23
to
"It's a book about Ramseyan decision theory by Richard Jeffrey."
Jeffrey Rubard
Sep 1, 2023, 11:25:48 AM9/1/23
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What we were talking about.
Jeffrey Rubard
Sep 3, 2023, 4:51:13 PM9/3/23
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It's what we were talking about. It's a theory of probability.
Jeffrey Rubard
Sep 4, 2023, 2:44:39 PM9/4/23
to
By correctly calculating your "choice preferences" relative to empirical data, "evidential probability".
"Hmm."
Yeah, it's a form of Bayeseanism.
Jeffrey Rubard
Sep 5, 2023, 11:37:55 AM9/5/23
to
A theory of scientifically calculating probability 'relative to' something or other.
Jeffrey Rubard
Sep 8, 2023, 11:38:31 AM9/8/23
to
A set of preferences that "order" empirical data according to their relevance for the choice function?
Jeffrey Rubard
Sep 8, 2023, 2:15:25 PM9/8/23
to
Sure, and we all think like that a lot. It's the details that are the problem.
Jeffrey Rubard
Sep 15, 2023, 2:00:33 PM9/15/23
to
Sure, as with 'Kolmogorovian' views on the topic. What you then know about their 'incidences', though, is...
Jeffrey Rubard
Sep 21, 2023, 6:28:38 PM9/21/23
to
Wrong theme, really. That's more of a rhetorical concept.
Jeffrey Rubard
Sep 22, 2023, 11:25:00 AM9/22/23
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Then it would be the 'Ramseyan'. The Ramseyan would be the one to say.
Jeffrey Rubard
Sep 24, 2023, 4:43:10 PM9/24/23
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No, it would be someone who had an adequate theory of 'subjective probability'.
Jeffrey Rubard
Sep 25, 2023, 11:29:30 AM9/25/23
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Jeffrey Rubard
Sep 26, 2023, 11:19:39 AM9/26/23
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Jeffrey Rubard
Sep 30, 2023, 2:27:52 PM9/30/23
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Jeffrey Rubard
Sep 30, 2023, 5:04:16 PM9/30/23
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Jeffrey Rubard
Oct 1, 2023, 11:17:07 AM10/1/23
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"Were you ever like that?"
As a kid, I guess. Not for a long time, really.
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