.Probability theory or the calculus of chances or probabilities is the mathematical theory underlying probability arguments and most (though not all) theories of induction with a mathematical basis. (See CONFIRMATION, where the relation between confirmation and probability is also discussed.) This calculus contains elementary rules governing results to be expected from tossing dice or drawing marbles from a bag. It also covers BAYES’S THEOREM, and topics like the law of large numbers (see NUMBERS (LAW OF LARGE)). Results within this theory are purely mathematical, and are not predictions about what actual dice, etc. will do. Use of the theory simply draws out the logical implications of assumptions already made.
Various theories have been offered about the nature of probability. The classical theory defines an event’s probability as the proportion of alternatives, among all those possible in a given situation, that include the event in question. There are 36 possible results of tossing two dice, of which 11 include at least one six, so the probability of getting at least one six in a throw of two dice is 11/36. But the alternatives must be equiprobable (equally probable)—or equispecific, if ‘equiprobable’ seems question-begging in an analysis of ‘probable’. This is hard to ensure. Attempts to ensure it have often used the principle of INDIFFERENCE. Other difficulties concern the probability of theories, such as Darwinism, and cases where the alternatives are not obviously finite and definite in number, e.g. the probability that all swans including future ones are white; since we can breed swans the number of future swans could depend on our very probability calculations. Also BERTRAND’S PARADOX and BERTRAND’S BOX PARADOX become relevant here. Kneale’s ‘range’ theory attempts to answer some of these difficulties. Range is used elsewhere too. For Carnap a proposition’s range is the set of state-descriptions compatible with it. See CONFIRMATION.
The frequency theory defines probability in terms of the ratio of times something happens to times it might happen. If the proportion of smokers who die of cancer remains steady at 10 per cent then the probability of smokers dying of cancer is 10 per cent. Since most of the classes we are concerned with are open classes, the probability is defined as the limit, in the mathematical sense, to which the frequency tends in the long run. We often talk of the probability of single events, e.g. that Smith will die of cancer, and it is disputed how, if at all, the frequency theory can account for this. Also the notion of a limiting frequency raises a problem because in an infinite or open-ended series, such as tosses of a coin, any limiting frequency is compatible with any result in a finite run. If a penny falls heads a million times running the limiting frequency of heads could still be a half it it fell tails the next million, or indeed if it merely behaved normally the next million times. Therefore in applying the theory we seem to have to say things like ‘Probably the limiting frequency is this’ or ‘Probably present trends of cancer among smokers will continue’, where ‘probably’ is unexplained.
The propensity theory, substituted by Popper for the frequency theory, defines probability as a propensity of objects themselves, e.g. of a die to show a six. Popper claims propensities are no more ‘mysterious’ than gravitational fields, but one can still ask just what propensities are and how wide an area the theory covers. The word ‘chance’ can also be used for ‘propensity’, and for objective probability when this is distinguished from subjective degrees of belief (see below).
The logical relation theory makes probability a logical relation between evidence and a conclusion, rather like entailment (see IMPLICATION) only weaker (cf. CONFIRMATION). Probability is therefore always relative to evidence. Apart from the difficulty of finding such a relation, one defect of this theory as an analysis of ‘probably’ is that if we know a true proposition, p, which entails another, q, we can ‘detach’ q, i.e. assert it on its own, but if p only makes q probable, we can at best say ‘Probably q’, which leaves ‘probably’ unanalysed—and even that we cannot say if we know there is another true proposition which makes q improbable.
The subjectivist theory analyses probability in terms of degrees of belief. A crude version would simply identify the statement that something is probable with the statement that the speaker is more inclined to believe it than to disbelieve it. Degrees of belief may be measured in terms of the bets the believer would be willing to place, and more refined versions of the theory say one is only entitled to use ‘probably’ if one’s bets are ‘coherent’, in the sense that one does not bet on contradictory propositions in such a way that one is bound to lose whatever happens, which can be expressed by saying that one does not let oneself have a ‘Dutch book’ made against one. This, however, still bases probability on the attitudes of the believer. Because ‘coherence’ is required, subjectivism is sometimes described as the view that probability is the degree of the rational man’s belief. However, when this means that calling something probable is saying that it is rational to believe it, it is not subjectivist, since it no longer analyses probability in terms of beliefs actually held. It then has no special name.
Another version of the subjectivist theory is the speech act theory. To call something probable is not to describe one’s belief but to express it. To say war is probable is to say, but only tentatively, that war will occur. Like other SPEECH ACT analyses (e.g. of ‘good’, ‘true’) this faces the objection at least prime facie (cf. GOOD), that it ignores cases like ‘If war were probable we would emigrate’, where it is not being even tentatively asserted that war will occur. It has never in fact been popular, and perhaps applies better to sentences like ‘It may happen’ than to ‘It is probable that it will happen’, or even ‘It will probably happen’.
Between them these theories try to account for the ideas that probability is objective and not of our choosing, and yet is somehow relative to our knowledge, since things in the world are either so or not so, and not probably so (though quantum physics may provide an exception to this). Problems also arise over when to say something was probable, especially if eventually it never happened.
Many recent writers think that there is more than one kind of probability. They often distinguish between probability as a logical relation, where probability statements are true or false as a matter of logic (Carnap’s probability1), and probability as relative frequency, where probability statements are empirical statistical statements which form the material for the mathematical calculus of chances (Carnap’s probability2). Some, like the frequency theorist Reichenbach, hold the identity conception of probability, saying that these two kinds are really one. Surely, however, one should distinguish at least three kinds of statement: empirical statistical statements, like ‘The probability of an Englishman being a Catholic is 10 per cent’, where this just means that 10 per cent are so; purely mathematical statements, like ‘The probability of a double six with two throws of a true die is 1/36’, where this makes no prediction about any actual dice; and ‘ordinary’ probability statements, like ‘The probability of Smith being a Catholic is high’, ‘Smith is probably a Catholic’, ‘The probability of a six with this die is low’, ‘This die will probably not show six’, ‘The probability of rain tomorrow is high’, ‘The probability of Darwinism being true is high’. These ‘ordinary’ statements may of course themselves be of various kinds, and may rest on statistical or mathematical statements.
Probabilities are called absolute, a priori or prior if they are considered as relative either to nothing or to the general background of knowledge rather than to some real or assumed set of evidence statements; otherwise they are relative or conditional. When a certain probability is assumed, e.g. the probability of an a which is b being c, the probability of an a which is c being b is called the inverse probability (cf. BAYES’S THEOREM). This notion raises no problems itself, but has been put to controversial uses, as a result of which LIKELIHOOD has been introduced.
Probabilism is the view that scientists can and should seek to attach probabilities to their hypotheses, i.e. to confirm them. Pop-per’s opposing view that this is impossible and that the scientist should seek the most improbable, i.e. the most easily falsifiable (though not yet falsified), hypothesis is sometimes called improbabilism. See also BAYES’S THEOREM, CONFIRMATION.
A.J.Ayer, The Concept of a Person, Macmillan, 1963. Miscellaneous Essays. (Chapter 7 discusses logical relation theory, and single events. Criticized by C.G.Hempel, Aspects of Scientific Explanation, Free Press, 1965, pp. 65–6).
R.Carnap, ‘The two concepts of probability’, Philosophy and Phenomenological Research, 1945, reprinted in H.Feigl and W. Sellars (eds), Readings in Philosophical Analysis, Appleton-Century-Crofts, 1949, and in H.Feigl and M.Brodbeck (eds) Readings in the Philosophy of Science, Appleton-Century-Crofts, 1953. (Cf. also his book Logical Foundations of Probability, 1950, and, for another version, J.O.Urmson, ‘Two of the senses of “probable”’, Analysis, vol. 8, 1947, reprinted in M.Macdonald (ed.), Philosophy and Analysis, Blackwell, 1954).
B.de Finetti, ‘Foresight: its logical laws, its subjective sources’, translated from 1937 French original in H.E.Kyburg and H.E. Smokler (eds), Studies in Subjective Probability, Wiley, 1964. (De Finetti is main representative of subjectivist theory.)
I.Hacking, The Emergence of Probability, Oxford UP, 1975. (Historical.)
J.M.Keynes, A Treatise on Probability, Macmillan, 1921. (Chapter 1 defends logical relation theory. See also chapter 4 for principle of indifference.)
*H.E.Kyburg, Probability and Inductive Logic, Macmillan, 1970. (Part I gives basis of calculus of chances and discusses various theories of probability. Includes exercises and bibliographies. Other elementary accounts of the calculus of chances include B. Mates, Elementary Logic, 2nd edn, Oxford UP, 1972, chapter 2, §5, H.C.Levinson, The Science of Chance: From Probability to Statistics, Faber and Faber, 1952, I.Copi, Introduction to Logic, Macmillan, 6th edn, 1982, chapter 14, R.Fogelin, Understanding Arguments, Harcourt, Brace, Jovanovich, 1978, 4th, revised, edn (with W.Sinnott-Armstrong), 1991, chapter 10.)
K.R.Popper, ‘The propensity interpretation of probability’, British Journal for Philosophy of Science, 1959. (Propensity theory. Cf. D.H.Mellor, The Matter of Chance, Cambridge UP, 1971, chapter 4, and A.O’Hear’s review of Popper in Mind, 1985, pp. 463–9.)
H.Reichenbach, Experience and Prediction, Chicago UP, 1938, (§ 33 advocates identity conception, and discusses single events.)
R.Swinburne, An Introduction to Confirmation Theory, Methuen, 1973. (First two chapters discuss kinds of probability.)
*R.von Mises, Probability, Statistics and Truth, William Hodge, 1939, revised (Allen and Unwin/Macmillan), 1957 (German original, 1928). Non-technical account of one version of frequency theory (first chapter), followed by discussion and applications.
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