WE are now in a position to distinguish the principal senses in which we commonly speak of the operations of chance.

(1) A chance event, as we have seen, may be a member of a series which exemplifies a game of chance. What is required of such a series is that it should conform to the a priori calculus of chances. If we take the simple example of a series of tosses with a coin, this would imply that in a reasonably long run there should be an approximate equality of heads and tails. It is, however, doubtful if this condition is sufficient. For instance, it would be satisfied by a series of tosses in which heads and tails regularly alternated: yet a series of this kind would not normally be regarded as typifying chance. What seems also to be required is that the series should satisfy a condition of randomness. I do not think that this condition need be so strong as the principle of indifference to place selection which has been adopted by some modern proponents of the frequency theory of probability. It might be enough that every possible sequence of some arbitrarily determined length should occur with approximately equal frequency.

To speak of a chance event, in this sense, is not to imply that the event is not caused, or even that it is not designed. The results of particular tosses of a coin, or throws of a die, or spins of a roulette wheel, are commonly not designed, but it is very often due to design that the series in which they occur conforms as a whole to the a priori calculus.

It is in accordance with this usage that when events of a certain kind occur with a frequency which deviates significantly from the a priori probabilities, they are said not to occur by chance. I have, however, tried to show that such a deviation does not, in itself, call for any special explanation. It is only when we have empirical evidence that the events are of a kind to which the a priori calculus normally does apply that their discrepancy becomes significant.

(2) Just as we look for a cause when we come upon an 'improbable' deviation in a series of a type which normally conforms to the calculus of chances, so conversely there are cases in which a deviation from an established law-like pattern is put down to chance. It is in this sense, for example, that biologists speak of chance mutations. What is implied here is not necessarily that the event in question is not susceptible of any causal explanation. It is enough that its occurrence should not be predictable in terms of the scientific theory with which we are operating. So, in our example, the theory of evolution provides for the occurrences of mutations only in a general way: it does not enable us to predict when and in what form they will occur. One could say that the theory made provision for them as chance events.

Examples of this usage occur also in historical discourse. 'For want of a nail the shoe was lost, for want of a shoe the horse was lost, for want of the horse the rider was lost, for want of the rider the battle was lost, for want of the battle the kingdom was lost, and all for the want of a horse shoe nail.' We say that the kingdom was lost by mischance, not for lack of a causal explanation but because its being lost in this manner is not something that any historical theory could have enabled us to foresee. It is not part of any recognized historical pattern that so trivial an event as a nail's falling out of a horse's shoe should have such far-reaching consequences.

(3) A third sense of chance is that in which it is contrasted with design. It applies to events which are brought about by human beings, or by other animals in so far as they can be said to have intentions. To attribute an event to chance, in this sense, is just to say that it was not intended by the agent in question. Here again, it is not implied that the event lacked a cause.

(4) We speak of a chance collocation of events when their concurrence is not designed and when, though we may be able to account for them severally, we have not established any law-like proposition which links them together. The ascription of such concurrences to chance is most often made in cases where something of especial interest to us follows from them, or in cases where the concurrence would normally be the result of design. Thus, if in the course of a journey I keep running into friends whom I had not arranged to meet, I am struck by the coincidence, though in fact it is no more of a coincidence than my meeting anybody else. This is on the assumption that the frequency of these encounters does not greatly exceed the average: otherwise, as we have seen, I am justified in suspecting design. If design is ruled out, our speaking of coincidence implies no more than that the events in question are not connected by any law-like generalization which figures in our accepted system of beliefs. It does not commit us to holding that no law which would connect them could ever be discovered.

(5) In one of the senses in which 'chance' is a synonym for 'probability', the chance that an event of such and such a sort will have a given character is equated with the frequency with which the character is actually distributed throughout the class of events in question. There are different ways in which these frequencies may be estimated: they may be extrapolated from recorded statistics, or they may be deduced from a scientific theory. In cases where they are deduced from a theory, it may or may not be assumed that the statistical laws which figure in the theory are derivable, at least in principle, from underlying causal laws. Thus the assumption in classical physics was that everything depended on the state of individual particles the behaviour of which was rigorously governed by Newtonian laws. If, as in the kinetic theory of gases, one was content to rely upon statistical laws, it was because of the practical difficulty of tracing the movements of the individual particles. On the other hand, in contemporary quantum physics, the laws are fundamentally statistical; the individual particles are not represented as obeying causal laws: the states of the system are statistically defined. This does not exclude the possibility, in which some physicists believe, of finding a deterministic theory which would account for the same phenomena: but it would have to be a radically different sort of theory. (6) Let us suppose that no such theory is forthcoming. It can then be said that these are chance events, in a stronger sense of the term than any that we have yet considered. A chance event, in this sense, would be one that was not subject to any causal law. If we are going to maintain that there are chance events of this kind, we must, however, be careful to formulate our position in a way that prevents it from being trivially refutable. The difficulty is that if we set no limit to the form of our hypotheses, then so long as we are dealing with a closed set of events, we shall always be able to find some generalizations which they satisfy. We, therefore, need to place some restriction on what is to count as a causal law. Perhaps the best course would be to stipulate that for a generalization to be a causal law it was necessary that it should apply to events which were not included in the set which it was already known to cover. In other words, one mark of a causal law would be that it was actually used to make successful extrapolations. To deny that phenomena of a given type were subject to causal laws would then have the force of predicting that however far our researches are pressed we shall never succeed in bringing them under 'workable' generalizations of a causal sort.

(7) Following C. S. Peirce,¹ I think that there is another way in which the course of nature may be held to exhibit an irreducible factor of chance. Even in a domain in which causal laws are well established, there is often a certain looseness in their grasp upon the observed facts. The phenomena which are taken as verifying them cover a certain range: if they are quantitative, the values which are actually recorded may be scattered around the values which the laws prescribe. These slight deviations are not held to be significant: they are put down to errors of observation. But 'errors of observation' is here a term of art. Apart from the existence of the deviation, there is usually no reason to suppose that any error has occurred. Now it seems possible that this looseness of fit cannot be wholly eliminated or, in other words, that there are limits to the precision with which observable events can be forecast. If this were so, it might be said that anything which fell outside these limits remained in the hands of chance.

1 See The Collected Papers of Charles Sanders Peirce, VI (1936), p. 46 et al., and my The Origins of Pragmatism, pp. 103-12 (Macmillan, 1968), or 91-9 (Freeman, Cooper, 1968).

Admittedly, this cannot be demonstrated. Whatever limits are set, there can be no a priori reason for assuming that they will never be overstepped. The person who believes in chance, in any such absolute sense, can properly do no more than issue a challenge. He points to certain features of the world and defies anyone to show that they fall entirely, in every detail, within the grasp of causal laws. In the sense in which to speak of chance is to express what I have called a judgement of credibility, I think that there is a good chance that someone who takes this position will be able to maintain it. There is, however, a sense in which it can be said of anything not known to be logically or causally impossible that there is a chance that it will happen: and in this sense, however long the champion of absolute chance has remained in possession of the field, there must always remain the chance that his challenge will eventually be met.