[ Acknowledgements | Introduction | Part One | Part Two | Part Three | Appendix ]
The theory of probability, on any interpretation of the word `probability', contains propositions of two different kinds which it is most important to distinguish. One of these kinds is that which Carnap calls `elementary statements of probability' [1], and which I will call simply `statements of probability'. The other kind I will call `the principles' of probability.
A statement of probability is any categorical proposition which expresses a measurement, or more generally an assessment, not necessarily a correct one, of some particular probability or other. The principles of probability, on the other hand, do not themselves assess any probabilities, correctly or otherwise, but are general conditional propositions from which a statement of probability can be deduced when, but not before, other statements of probability have been asserted.
The relation which exists between statements, and the principles, of probability can best be made clear by an analogy with two kinds of propositions in geometry. When we consider any particular right-angled triangle, whether actual or imaginary, the Theorem of Pythagoras cannot itself, of course, tell us the length of any of the sides. Conjoined, however, with measurements of any two of the sides, the Theorem enables us to deduce the length of the third. In just the same way, the principles of probability, without statements of probability, are (to paraphrase Kant) empty; and statements of probability, without principles, are blind.
I speak of `statements', and by contrast, of `the principles' of probability, for the following simple reason. The correct assessment of particular probabilities, and especially their numerical assessment, often presents great or even insuperable difficulty, and as a result people often disagree in their statements of probability. At least some of the statements of probability which are believed must, therefore, be false. But the general principles by which further probabilities are to be calculated, once some probabilities are supposed to have been correctly assessed to begin with, are, on the other hand, extremely well known and nowhere in dispute. They must, in fact, have been known, at least in an implicit way, for a very long time indeed; and in the present century, in particular, they have been reduced many times over, and in many different ways, to a system of axioms and theorems. These `axioms and theorems of the calculus' of probability are what I call principles of probability. And I speak of the principles, because I intend to take for granted the truth of familiar ones (and of course of no others). When I call a proposition a statement of probability, on the other hand, I will always intend to leave open the question whether or not it is a true one.
Along with Carnap and many other recent philosophical writers on probability, I think that there are two different senses of `probability', a factual one and a logical one [2].
If this is so, then there are two theories of probability, and there are four kinds of propositions about probability which need to be distinguished. There are the principles of factual probability, and statements of factual probability; and again, there are the principles of logical probability, and statements of logical probability. What is factual about the theory of factual probability, and what is logical about the theory of logical probability, will not show forth, however, in connection with their respective principles; for the principles of factual probability and the principles of logical probability are alike in both being propositions of a non-factual kind. It is only in connection with statements of probability, therefore, that the difference between the theory of factual probability and the theory of logical probability shows itself.
A statement of factual probability is being made when and only when we assess the probability of one attribute in relation to a second attribute, which picks out a class of individuals in which the first attribute may occur. Thus we may say, for example, that the probability of a radium atom decaying in a certain interval of time is = 1/2; that the probability of throwing a `four' with this die is > 1/6; that the probability of a genetic mutation being beneficial is small; that the probability of a woman over thirty giving birth to twins is greater than the probability of a woman under thirty doing so; and so on. Such propositions clearly are assessments of probability, and equally clearly, they are propositions of a factual, contingent kind. Experience is needed, and is able, to confirm or disconfirm them.
What factual probability is, need not detain us here. We need not even pause to inquire whether Carnap and many others are right in supposing that factual probability is to be analyzed in terms of the relative frequency with which one attribute occurs in the population of individuals picked out by the other. For my concern in this book is exclusively with the other, logical, theory of probability.
A statement of logical probability is being made when and only when we assess the probability of one proposition in relation to a second proposition, which picks out possible evidence for or against the first. Thus we may say, for example, that in relation to the evidence that the factual probability of throwing a `four' with this die is = 1/6, the probability of the hypothesis, that John at his first trial with it will throw `four', is = 1/6; that the probability that Socrates is mortal, in relation to the evidence that Socrates is a man and all men are mortal, is > 0; that relative to the knowledge existing in the physical sciences in April 1970, the steady-state cosmological theory is less probable than the big-bang theory; that in relation to what has so far been observed of comets, the probability of the hypothesis that the next comet observed will have a low-density tail, is high, and so on.
Propositions such as these (whether true or false) clearly assess probability, but they are not propositions of a factual kind. Any one of them mentions two propositions, viz. the evidence and the hypothesis, the `primary propositions', as Keynes called them [3], of a statement of logical probability; and these, indeed (our interests being what they are), will usually both be of a factual kind. But the assessment itself, of the probability of the hypothesis relative to the evidence, is a proposition of a different kind. What determines its truth or falsity is not the truth or falsity of its primary propositions (and still less that of any other factual proposition), but just the relation between the two primary propositions. That relation is fully fixed once the content of each of the primary propositions is fixed; and what those contents are, is not a question of fact, but a question of meaning. Statements of logical as distinct from factual probability, therefore, are propositions of a kind, the truth or falsity of which experience is not able [4], and not needed, to inform us.
So far I have followed Carnap in speaking of the subject of a statement of logical probability as being a certain ordered pair of propositions, and of these propositions respectively as `evidence' and `hypothesis'. In what follows, however, I intend to follow the usage of Keynes and many others [5], including Carnap himself at times [6], and to speak instead of the primary propositions as respectively `premiss' and `conclusion', and of the subject of the whole statement, correspondingly, as being the `inference' or `argument' from the former to the latter. Also, where Carnap speaks of the whole statement as `assigning' a certain degree of logical probability `to' its subject, I will speak of it always as `assessing' the logical probability `of' its subject. Clearly, the latter way of speaking accords much better than the former with the possibility which, as I have said, I intend to always leave open when calling a proposition a statement of logical probability, that the proposition is a false one.
But what is that magnitude, the degree of which in a particular case a statement of logical probability assesses? What is logical probability?
According to Carnap, it is the degree of confirmation of the hypothesis h by the evidence e; and nothing in this book would need to be radically different from what it will be, if one were to adopt this answer. But a different answer is in fact suggested if we think of the primary propositions as the premiss and conclusion of an argument; and this answer is preferable because, among other things, it is closer to the conception which most of the writers in the tradition of Johnson, Keynes, and Carnap have had in mind, and also because it testifies to the continuity which exists between these writers and the classical theorists of probability. This is, that logical probability is the degree of conclusiveness of the argument from e to h. That is, the degree of belief which a completely rational inferrer, who knew the premiss of the argument, and was influenced by nothing else, would have in its conclusion.
Some arguments, it is evident, have this property in the highest possible degree. A completely rational inferrer, that is to say, if he knew the premiss and were influenced by nothing else, would have in the conclusion the same degree of belief as he has in the premiss. All valid arguments, for example, have the highest possible degree of conclusiveness. (By calling an argument `valid', I will always mean just that its premiss logically implies its conclusion). It is equally evident that some other arguments do not have the highest possible degree of conclusiveness. A completely rational inferrer who knew the premiss of such an argument would have not the same but at most a lower degree of belief in its conclusion. (We are again assuming the influence of any other information he might have, beyond what the premiss contains, to be excluded). No invalid arguments, for example, are of the highest degree of conclusiveness.
With these assertions, I think, everyone would agree, and they suffice to establish that conclusiveness is a property of arguments which is a magnitude at least in the minimal sense that some arguments have it in the highest possible degree and others do not. Deductive logic stops here, and is concerned to discriminate among arguments only according as they are, or are not, of the highest degree of conclusiveness. The theory of logical probability, on the other hand (or `inductive logic' in Carnap's sense, but here see section (ix) below), is distinguished by the assertion that the conclusiveness of arguments is a magnitude in the further sense that degrees of it are ordered, at least to the extent that some of them lie between others. In other words, the fundamental thesis of the theory of logical probability (since some arguments certainly are valid and hence have the highest degree of conclusiveness) is that two arguments may be of unequal degrees of conclusiveness, even though both are invalid.
One must freely admit, or rather hasten to affirm, that `degree of conclusiveness of an argument', even if a correct answer to `What is logical probability?', is a far from ultimate answer to that question. Arguments certainly are not ultimate entities, and no magnitude which is characteristic of them can be ultimate either. Answers to this question which are more ultimate than the one I have given---more objective and `logical', less epistemic---must exist; and indeed some already do exist, notably the answer which is given by Carnap in terms of the relation between the `ranges' of two propositions. It is to be observed, however, that when writers attempt a more ultimate kind of answer than the one above to `What is logical probability?', it is by reference to just such a non-ultimate conception of logical probability as I have given, and no other, that they test their own answers. Besides, no answer to this question, which is in terms of propositions and the relations between them, can hope to be an ultimate one. For propositions, although no doubt more ultimate than arguments, are certainly not ultimate entities. This reflection may be salutary, as reminding us how far we are from knowing what logical probability really is at bottom. On the other hand, it ought not to prevent us from admitting that there is a great deal which we already do know about logical probability.
One must distinguish, of course, between the fundamental idea---degree of conclusiveness of arguments---and the fundamental thesis of logical probability. Whether that thesis is true---whether, that is, there are unequal degrees of conclusiveness among invalid arguments---certainly can be doubted (and of course has been). That question will be under discussion, by implication, in Chapter 6 section (iv) below. But without the idea of degree of conclusiveness of arguments, it is not possible to understand the theory of logical probability, or indeed the classical theory of probability, at all.
Within the theory of logical probability it is important to distinguish not only principles from statements of logical probability, but also different kinds of propositions within the latter class.
The statements of logical probability which generally receive most attention from writers on probability are what I will call `numerical equalities'. That is, statements of the kind usually abbreviated P(h,e) = r, where e and h are premiss- and conclusion-propositions respectively, and r is a number between 0 and 1 (limits included): which assert, of course, that the probability of the argument from e to h is equal to r.
Statements of logical probability are sometimes spoken of as `measuring' the logical probability of an argument (or ordered pair of propositions) which is their subject. This way of speaking is rather too apt to suggest, what need by no means be true, `correctly measuring'; but otherwise it is appropriate enough, when the statement in question is a numerical equality. But numerical equalities, while they are the strongest and therefore the most interesting, are far from being the only kind of statements of logical probability; and there are other, weaker, kinds which, except in an intolerably extended sense of `measure', could not be said to measure logical probability at all.
First, there are numerical inequalities, i.e. statements of the kind: P(A,B) > 1/5, P(C,D) < 1, etc. (The capital letters are used here, for brevity, in place of concrete propositions). Statements of this kind are weaker than numerical equalities, since, while any numerical equality entails infinitely many numerical inequalities, no conjunction of inequalities entails even one equality. And, it will be of importance later to note, among numerical inequalities, those which say of the logical probability of a certain argument only that it is less than the maximum (P(C,D) < 1), or that it is more than the minimum (P(F,E) > 0), are the weakest of all. For each of these excludes only one numerical equality, while every other inequality excludes infinitely many equalities.
There are other weaker kinds of statements of logical probability, however, which are not numerical at all, but purely comparative. That is, they say, of the logical probability of a certain argument, only that it is equal to the logical probability of a certain other argument: P(A,B) = P(C,D) for example. Obviously, there are also inequalities of this purely comparative kind. The relative weakness of comparative, as compared with numerical, statements of logical probability appears from the fact that while a conjunction of numerical inequalities will often entail a comparative statement, no conjunction of comparativeness will suffice to entail even one numerical inequality.
The importance, despite their weakness, of comparative assessments of logical probability was first stressed by Keynes, and he rightly attached a special importance to a certain sub-class of comparative statements to which he gave the name of `judgements of (ir)relevance' [7]. A judgement of irrelevance is a comparative equality in which the two arguments mentioned have the same conclusion, while the premiss of one of them entails, without being entailed by, the premiss of the other. Thus `P(A,B) = P(A,B.C)' (or `The argument from C and B to A has the same probability as the argument to A from B alone') asserts the `irrelevance' of C to A (relative to B); the contrary inequalities assert the `favorable' and `unfavorable' relevance, respectively, of C to A (relative to B).
Besides numerical and comparative equalities and inequalities, there are vague, but common and useful, `classificatory' statements of logical probability. As when, for example, we say of a certain logical probability, only that it is `low', or `high', or `considerable', or `negligible', etc.
Statements of all these kinds are certainly used to assess the conclusiveness of arguments; although numerical equalities alone of them could properly be said to measure it. And assessments of all these kinds---numerical, comparative, and classificatory---are of course made of factual, as well as of logical, probability.
Statements of logical probability, whether numerical or of any other kind, can be less or more general. One will be less general than another, if, in making it, we assess the conclusiveness of a single argument, or of a certain class of arguments; while in making the other, we assess the conclusiveness of another class of arguments which properly includes the first class, or includes the single argument as a member.
Thus, for example, `P(Socrates is a mortal, Socrates is a man and all men are mortal) = 1' is a less general statement of logical probability than `P(x is G, x is F and all F are G) = 1'. In fact the first assesses the logical probability of only a single argument. Any such statement of logical probability I will call `singular'; any other, `general'.
Any general statement of logical probability will require the employment of universally quantified variable expressions of one of more kinds. (The above example of a general statement employs both individual and predicate variables). In writing general statements of logical probability, universal quantification of variables is usually left tacit (as it was, again, in the above example).
Many statements of logical probability are so very general that propositional variables are required for their expression; and these, too, have to be understood as tacitly universally quantified. But a statement which made the same assessment, P(h,e) = r, for every value of h and for every value of e, would of course be ludicrously false. Consequently, a statement of logical probability employing propositional variables (unless it is a ludicrously false one) will always be accompanied by an indication of some limitation on the range of values which at least one the variables is intended to take. For example, `P(h,e) = 1 for any contingent e and tautological h'; `If h is a universal empirical proposition and e is observational, P(h,e) < 1', etc.
The contemporary theory of logical probability, it must be admitted, is something of a specialism even within the specialism which is philosophy. The standard abbreviations adopted for statements of logical probability tend especially to invest those statements with an air of technicality. For this reason among others, it deserves to be stressed that statements of logical probability are not an out-of-the-way kind of proposition, or of interest only to a few contemporary specialists. They are in fact as common as daylight, and of interest to everyone.
Certainly, philosophers and logicians at all periods have been much engaged in making assessments of the degree of conclusiveness of arguments, and even in making numerical assessments. For they constantly make, concerning arguments, judgements of validity and judgements of invalidity; and these are numerical equalities and inequalities respectively. To judge the argument from B to A valid is to assess its conclusiveness as being of the maximum degree, i.e. P(A,B) = 1; while to judge it invalid is to affirm that P(A,B) < 1.
Far more common still are the weaker kinds of statement of logical probability which were distinguished from the numerical ones in section (iii) above; and the making of these is not even more characteristic of philosophers than of non-philosophers. The scientist, the judge, the detective, and indeed any intelligent person, is constantly being called on the decide whether comparative statements of logical probability are true: whether, that is, P(h1,e1) = P(h2,e2), or not. Even more commonly than other kinds of comparative assessment, we all make many judgements of (ir)relevance. In fact, any one who, with his inferential capacity in working order, has new information constantly flowing in upon him (as we, for example, have from perception) must make countless judgements of (ir)relevance. For such an inferrer has constantly to consider whether the degree of conclusiveness of his `old' inference from e1 to h is the same as that from e1.e2 to h, where e2 is his new information; i.e. whether P(h,e1) = P(h,e1.e2). For most of his hypotheses h, and most of his old pieces of information e1 and most of his new pieces of information e2, he will in fact, presumably, make the judgement of irrelevance. But this itself suffices to show that assessments of logical probability constitute an immense fabric of belief, even with the least reflective among us.
Where the reflective and the educated differ from those who are neither is not in making assessments of logical probability at all, or in making more of them, but in the greater generality of many of the assessments that they do make. The ordinary man may be unfailingly accurate in the assessments he makes of the conclusiveness of single arguments, but it is the mark of the reflective mind to assess whole classes of arguments at once: to judge (truly) that every syllogism in Barbara is valid; to judge (falsely) that every case of `affirming the consequent' is invalid; to judge (truly) that P(h,e1) = P(h,e1.e2) for every contingent h and e1 and every tautological e2; and so on. But whether true or false, singular or general, assessments of conclusiveness are constantly being made, by the learned and the vulgar alike.
Even the kinds of statement of logical probability which were enumerated in section (iii) above, however, are far from exhausting the language which we use to assess conclusiveness. They are, rather, only the tip of the iceberg. Far more common than all of those kinds of statement put together are assessments of the conclusiveness of inferences which are expressed in the utterly untechnical terminology of one proposition's being `some grounds', or `slight foundation', or `no reason', etc., for belief in another. It is in such language as this that most of the assessing of the conclusiveness of inferences is carried on, at the present time as at earlier times, and by philosophers as well as by non-philosophers.
Such language is very vague, of course. Yet it can be used so as to convey the same thing as one of the more definite kinds of statements of logical probability which were discussed above. For example, if a writer, to express his assessment of the conclusiveness of two arguments, repeatedly used the same phrase, then, however untechnical and vague it may be, we will clearly be entitled to ascribe to him belief in a comparative equality concerning those two arguments.
We will meet a remarkable instance of just this sort in Part Two below. It will involve me in attributing to an eighteenth-century writer certain statements of logical probability. Such an attribution is apt at first to seem absurdly anachronistic. But there can be no anachronism about such an attribution, if what has been said in this section, and the two preceding ones, is true. The attribution may be erroneous, of course, but that is another matter, to be decided on the basis of the texts. And even the chance of error will be somewhat diminished by the fact that the eighteenth-century writer in question was a philosopher, and consequently was engaged even more often than most men in assessing the conclusiveness of arguments.
It will be important later in this book to make assessments of the degree of conclusiveness of certain arguments from tautological premisses. Such an assessment I will call, following Carnap, a statement of `initial' [8] logical probability; and it will be found convenient to follow Carnap's practice of letting the propositional variable `t' take only tautological values when it occurs as a primary proposition in a statement of logical probability. Thus, a statement of initial logical probability will, when abbreviated, always begin P(h,t) [...]; and conversely, whatever begins so will be a statement of initial logical probability.
Since most of the arguments in which we are actually interested are, as I have observed earlier, from (as well as to) a factual proposition, we are apt to find the idea of statements of initial logical probability strange at first. Yet initial logical probabilities do exist. Indeed, unless the principles of logical probability are supposed to have an application much more restricted than has ever, so far as I know, been ascribed to them, then for every h and e, P(h,e) = P(h.e,t)/P(e,t) (by the conjunction (`multiplication') principle, and the principle that logically equivalent propositions can always be substituted for one another salva probabilitate in statements and principles of logical probability; along with the equivalence of `e' and `e.t' for any e). Every logical probability, in other words, requires the existence of at least two initial logical probabilities.
But there is no need for us thus to infer that initial logical probabilities exist. For there are a good many assessments of the logical probability of arguments from tautological premisses which we know the truth of directly, as directly as we know any statements of logical probability whatever. Some of them, moreover, are of very great generality. For example, `P(h,t) = 0 for every self-contradictory h'. Or again, `For any contingent h, P(h,t) < 1'.
This last proposition is essentially Carnap's requirement of `regularity' for adequate measures of logical probability [9]. It is, clearly, such a `requirement', though not in the sense of being in the least an arbitrary stipulation; only in the sense of being very general, not usually inferred from anything else, and true. For to deny it would be to affirm that some contingent propositions can be inferred with the highest possible degree of conclusiveness from a tautology.
Statements of logical probability asserting P(h,e) < 1 for some limited range of values of h and e, I have earlier called `judgements of invalidity'. I propose to call judgements of invalidity P(h,t) < 1 where h is contingent, `judgements of regularity'. They will prove especially important in Part Three below.
The logical probability of an argument is its degree of conclusiveness; that is, the degree of belief which would attend its conclusion in a completely rational inferrer who knew the premiss and was influenced by nothing else. What this degree is, in any particular case, will depend on the content of the premiss and conclusion, and on nothing else; and what their contents are, is a question of meaning, not of fact. Statements of logical probability, consequently, are not propositions of a factual, or more particularly, of an empirical kind. Consequently neither their truth or their falsity can be discovered by experience.
All this was said in section (ii) above. We have now to notice, with respect to the final conclusion just drawn, first, some exceptions (essentially just one exception) to it; and second, an important consequence of its being, with this exception, true.
Within the class of all statements of logical probability, consider the sub-class consisting of numerical equalities, P(h,e) = r. Within this sub-class, consider just the further sub-class in which the number r is 1, or is 0. Then all the statements which we are considering are, or are equivalent to, judgements of validity P(h,e) = 1. Within this class, consider just the further sub-class which consists of false judgements of validity. Now, the proposition ~h.e, consisting of the premiss of the argument which is being assessed, conjoined with the negation of the conclusion, is certainly inconsistent with the false judgement of validity P(h,e) = 1. This proposition, moreover, may happen to be an observational one, in virtue of the content of e and ~h, if P(h,e) = 1 is singular. And if P(h,e) = 1 is general, it will usually be possible to choose values of the predicate or propositional variables in h and e is such a way as to construct an observational counter-example, ~h.e, to the judgement of validity. Sometimes at least, therefore, false judgements of validity are inconsistent with some observation-statement; i.e. are falsifiable. Some statements of logical probability at least, therefore, are such that their falsity can in principle be learned from experience.
For precisely the same reasons, some statements of logical probability are such that their truth can in principle be learned from experience: viz. the contradictories of the statements just mentioned. True judgements of invalidity (that is to say, P(h,e) < 1 or equivalent statements) are at least sometimes deducible from some observation-statements; i.e. are verifiable. (This `second' exception to the generalization which I made above is of course only apparently different from the first: for the falsifiability of a false judgement of validity differs only verbally from the verifiability of a true judgement of invalidity).
But the restrictions which were laid down in the last paragraph but one were all necessary in order to make possible even this (essentially single) exception to my generalization. Where the number r is not extreme (1 or 0), P(h,e) = r is never falsifiable: for then it is not inconsistent with ~h.e even if that proposition is observational. And if the judgement of validity P(h,e) = 1 is not false, but true, then of course ~h.e is self-contradictory and consequently not an observation-statement. All other numerical equalities or inequalities, therefore (i.e. where the condition of extremeness or the condition of falsity fails), are neither falsifiable nor verifiable. The same is true a fortiriori of all those statements of logical probability which are not even numerical, but merely comparative or classificatory.
Since the vast majority of statements of logical probability cannot have their truth or their falsity determined by experience, how is their truth-value to be determined?
One can prove the truth of a statement of logical probability, by validly deriving it, via the principles of logical probability, from other statement(s) of logical probability of which the truth is intuitively obvious. And one can prove the falsity of a statement of logical probability by validly deriving from it, via the principles, some other statement(s) of logical probability of which the falsity is intuitively obvious. These are the only ways in which any statement of logical probability (other than the exceptions just discussed) can have its truth-value determined indirectly, i.e. by inference. And it will be obvious that the possibility of such indirect inference determinations of truth-value depends on there being some statements of logical probability, the truth or falsity of which can be known directly.
In this sense, almost every application of the theory of logical probability depends in the end on intuitive assessments of logical probability [10]. This is not at all to admit that statements of logical probability are, in any important sense whatever, `subjective'. It is not at all to admit, in particular, that hardly any false statement of logical probability can `really' be proved false. For it would certainly prove the falsity of a given statement of logical probability if it were possible to derive from it, via the principles, P(x is G, x is F and all F are G) = 1/2, for example; or P(x is G, x is F and all F are G) = P(x is F, x is G and all F are G); or P(x is not G, x is F and all F are G) > P(x is not G, x is F); or a violation of regularity P(h,t) = 1 for some contingent h; or any one of very many other equally obviously false consequences. Yet our knowledge that these consequences are false rests on no factual (and in particular no empirical) foundation whatever.
Reliance on intuition, in the sense in which it is admitted here, is not even peculiar to the theory of logical probability, but is equally characteristic of deductive logic. For deductive logic is concerned principally with true judgements of validity; and as we have seen, the truth of a judgement of validity, like the truth of most other statements of logical probability, is not discoverable empirically [11].
Statements of logical probability of which the truth is discoverable empirically, we have seen, are confined to the small minority which consists of true judgements of invalidity. But of course, even when a statements belongs to this minority, its truth, even if known and capable of being discovered empirically, need not have been in fact discovered empirically. The truth, for example, of P(x is F, x is G and all F are G) < 1 (the invalidity of the `undistributed middle' for logically independent predicates) may become known to us by the aid of a counter-example to the contrary false judgement of validity. But it need not be inferred from such an actual case. Nor need it be inferred from anything else; for its truth may be intuitively obvious without inference. Most true judgements of invalidity, at least when they concern simple arguments very obviously invalid, are in fact discovered directly.
It will be worthwhile to emphasize that, in addition, even among true judgements of invalidity, there are indefinitely many the truth of which can only be discovered non-empirically. These are the true judgements of invalidity which are singular, and which concern an argument of which either the conclusion h is true, or the premiss e false in fact. Clearly, in such cases the possibility is logically excluded of learning the truth of P(h,e) < 1 by finding that in fact ~h.e. The statement of logical probability, if known, must be known directly by intuition; or, if it is not known directly but inferred, it must be inferred from other statements of logical probability some at least of which are known intuitively. As an interesting special case it may be observed that, since h is contingent if and only if its negation is contingent, half at least of all singular judgements of regularity can be discovered to be true only non-empirically.
The two kinds of propositions---statements, and the principles, of logical probability---as well as excluding one another, exhaust the theory of logical probability. (Purely mathematical propositions `belong' to the theory of logical probability only in the external and auxiliary sense in which they also belong to, say, physics). Consequently, what the theory of logical probability enables us to do is just what the principles, conjoined with some statements of logical probability, enable us to do. And that is, to derive further statements of logical probability.
This does not sound as thought it could be very instructive; and under some circumstances, it would not be. Suppose, for example, that a man made just one statement of logical probability, and never made another. Or suppose that a man made many such statements, but all of them singular, and about arguments of utterly unrelated subject-matter, so that the statements of logical probability to which he committed himself had no primary propositions in common. Then, in either of these cases, application of the principles, in order to derive other statements of logical probability, could not furnish the materials for any criticism of his original statements; for all of the consequences derivable from them are statements which, by the hypotheses, he neither affirms nor denies.
Our actual case is far otherwise, however, if what I have said earlier in this chapter is true: viz. that we make very many, and many of these very general, assessments of logical probability. Under these circumstances there is the possibility that the principles will bring to light an inconsistency in our assessments which would never have been disclosed except by tracing those assessments into their remoter consequences. And the likelihood of there being such inconsistencies is the greater, of course, the more numerous, and the more general, our assessments are to begin with.
More generally, however, the theory of logical probability is instructive because it can bring to light consequences of our assessments of logical probability which are unforeseen, and which, even if not inconsistent with any of our other assessments, are nevertheless surprising and unwelcome. Let us call an assessment of the conclusiveness of a certain class of arguments `the natural' assessment, if it is that which is or would be made by almost all men; and let us call an assessment a `sceptical' one, if it ascribes, to a certain class of arguments, less conclusiveness than the natural assessment does; if it ascribes more, a `credulous' assessment. Natural assessments exist for many classes of arguments, though of course (since no one could consider all possible arguments) not for all. Now, an assessment of certain arguments which is either sceptical or credulous will entail, in virtue of the principles of logical probability, further assessments, either credulous or sceptical, of other arguments (so long as the natural assessments are themselves consistent). To take the simplest sort of illustration: if for a certain class of arguments, i.e. for a certain limited range of values of h and e, a man makes a sceptical assessment of P(h,e) = r, then the negation principle commits him to making a credulous assessment, P(~h,e) = 1-r, of the class of arguments from e to not-h.
This kind of service which the principles of logical probability, combined with our statements, can perform for us, is particularly important. For philosophers are rather too apt to think that they can depart from the natural assessments of inferences, in the direction of scepticism, in a more piecemeal fashion than is really possible. The principles of probability can show us that our scepticism about one class of arguments must be extended to classes of arguments about which we had never dreamed of being sceptical, or again, must be compensated for by an embarrassing credulity concerning some other arguments. (A striking instance of the latter kind will be discussed in Chapter 5 below). This, rather more often than the disclosure of an outright inconsistency in our assessments, is the valuable kind of instruction which the theory of logical probability can afford us.
As the word `inductive' has almost always been used, inductive inference, whatever else it may be, is at any rate inference from experience. An inference, that is, is not called `inductive' unless its premiss is observational: consisting of reports of (actual or possible) past or present observations. Thus, the class of inferences of which the following is a paradigm (`Bernoullian' inferences as I will call them), is not inductive. `The factual probability of a human birth being male is = 0.51, and there were a large number of human births in Australia between 1960 and 1970, so the relative frequency of male births in Australia at that time was close to 0.51' [12]. The inference, on the other hand, which has the major premiss of the above inference as its conclusion, and the above conclusion and minor premisses as its premisses, is inductive; or at any rate, it satisfies the condition stated above as being necessary for an inference to be inductive. For the statement of factual probability is not observational, whereas the statements about the number of births in Australia between 1960 and 1970, and about the observed relative frequency of males among them, are observational.
What class of inferences is to be understood in this book by the phrase `inductive inference' will be specified more narrowly in the next chapter. But the restriction already mentioned---that the premiss is observational---is sufficient to ensure that inductive inferences constitute a quite special class of inferences, and as such have no more intimate a connection with the theory of logical probability than any other special class of inferences. No more, for example, than Bernoullian inferences have. The theory of logical probability is no more especially about inferences from observational premisses than it is especially about inferences, for example, from a premiss which includes a statement of factual probability. The principles of logical probability are perfectly general [13]. Not only do they not assess the conclusiveness of, they in no way mention, any particular class of inferences such as the inductive ones. Of course, once some assessment has been made of the conclusiveness of some or all inferences from experience, though not before, the principles will enable us to derive therefrom other assessments of other inferences; not all of which will have observational premisses, and not all of which, therefore, can be inductive. But of course that is neither more nor less than the kind of thing which the principles will do for us in connection with any other particular class of inferences.
It seems advisable to insist on this, in view of the great amount of confusion which, in about the last 100 years, has come to surround the relation between the two topics, induction and probability. (What has brought these two topics in close connection, why the confusion about their relation should have arisen at all, and why it should have arisen just when it did, will, I hope, be explained by Chapter 8 below).
This is also the place to comment on Carnap's highly idiosyncratic usage of the word `inductive'. This departs from the mainstream usage of the word, once by omitting something, and once by adding something not present before. First, Carnap omits from the sense of `inductive' the descriptive element, remarked on above, which confines it to inferences from observational premisses. And second, he imports into the meaning of the word an evaluative element which was lacking before. During most of the last 300 years, it was no part of the function of the word `inductive' to convey any assessment whatever of the conclusiveness of any argument from observational premisses. But as Carnap uses `inductive', in contrast with `deductive', it is at least part of the meaning of calling an inference `inductive', that it is invalid. One result of this second departure from the mainstream usage is, as we will see in Chapters 7 and 8, the trivialization of an important truth. The result of the two departures combined is that Carnap regularly calls the great enterprise, of which he is himself the chief architect, by a seriously misleading name. What he calls the construction of a system of `inductive logic' is in reality the construction of a system of non-demonstrative logic in general, or simply, the theory of logical probability [14].
In speaking, for most of this chapter, about logical probability, therefore, I have been speaking about `inductive logic' in Carnap's sense. But if we take `inductive' in its usual sense---in which, applied to inferences, the word does convey a limitation on the nature of the premisses, and does not convey any assessment of the conclusiveness of the inference---then it will be evident that I have so far said or assumed nothing about inductive inference. In particular I have not said or implied anything about the degree of conclusiveness of any arguments from experience; any more than I have, say, about the degree of conclusiveness of Bernoullian inferences. The degree of conclusiveness of inductive inferences is, in fact, the subject of most of the remainder of this book; but it is a question that is entirely untouched by anything said so far.
[1] Logical Foundations of Probability (Chicago University Press, 2nd edition, 1962), pp.29--36. (Referred to hereafter as Foundations).
[2] Foundations, Chapter II.
[3] Treatise, p.II.
[4] In section (vii) below, some minor exceptions will be noticed to what is said here.
[5] e.g. W.E.Johnson, Logic (Cambridge U.P., 1921--4), Part III, Chapter IV and Appendix; Keynes, Treatise, Parts I and II especially; H.Jeffreys, Scientific Inference (Cambridge U.P., 2nd edition, 1957), Chapter I; R.T.Cox The Algebra of Probable Inference (Johns Hopkins Press, Baltimore, 1961), esp. Chapter I.
[6] Foundations, pp.567 ff.
[7] Treatise. See its index under `Irrelevance'.
[8] Foundations. Section 57.
[9] Cf. Foundations, pp.294 ff. Strictly, of course, for Carnap this requirement is confined to measures of logical probability for languages containing a finite number of individual constants.
[10] Cf. Carnap, `Inductive Logic and Inductive Intuition', in Lakatos, ed., The Problem of Inductive Logic (North-Holland, Amsterdam, 1968). Popper, in his comments on this essay in the same volume, points out in effect that (unlike other statements of logical probability, some) judgements of validity can have their falsity discovered (not only intuitively but) by empirical counter-example. See ibid., pp.286, 296--7.
[11] This, I suggest, is the important element of truth in the thesis of symmetry, as between deductive logic and `inductive logic' in Carnap's sense, which is maintained by Carnap, and criticized by Popper, in the volume referred to in the preceding footnote.
[12] Inferences of this kind become statistical inferences, of course, as soon as the probability referred to in the major premiss is given an interpretation in terms of relative frequency. They become, in particular, inferences from the relative frequency of an attribute in a certain population to its relative frequency in a large sample from that population. For this reason the name `Bernoullian inference' will be applied later in this book to inferences of this latter kind as well, and also to the closely related kind of inference in which both the minor premiss and the conclusion are singular.
[13] The only inferences which the theory of probability is prevented by its principles from taking account of, are those from self-contradictory premisses. Cf. Carnap Foundations, pp.295 ff.
[14] Carnap's usage of `inductive', however idiosyncratic, is clear, and it ought to have been possible for him to adhere to it consistently. But as so often happens, normal usage reasserts its rights, and Carnap is sometimes led to say something which is true only in the normal sense of `inductive', quite false in his own sense of that word. Thus in his article `On Inductive Logic', Carnap says that the evidence-sentence e, mentioned in any statement of logical probability, `is usually a report on the results of our observations'. (Philosophy of Science, 1945, p.72. My italics). But taking `inductive logic' in his broad sense, this remark is simply baseless and untrue. It can only be regarded as an inconsistent concession to the mainstream of usage of `inductive'.
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