Hist-nnal t ic would l ike to thank the edi tor of theProceedings of the nristotel ian society for permission towebpubl i sh Russel I 's paper.

www. ari stotel i ansoci ety. org. uk

HERBERT FEIGL and MAY BRODBECK

EditorsUNIVERSITY OF MINNESOTA

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On the Notion of Cause, with Applicationsto the Free-Will Problem"

BERTRAND RUSSELL

Ix rno FoLLowrNG paper I wish, firstn to maintain that the word "cause"is so inextricably bound up with misleading associarions as to make itscomplete extrusion from the philosophical vocabulary desirable; secondly,to inquire what principle, if any, is employed in science in place of thesupposed "law of causality" which philosophers imagine to be employed;thirdly, to exhibit certain confusions, especially in regard to teleology anddeterminism, which appear to me to be connected with erroneous notionsas to causality.

All philosophers, of every school, imagine that causation is one of thefundamental axioms or postulates of science, yet, oddly enough, in ad-vanced sciences such as gravitational astronomy, the word "cause" neveroccurs. Dr. James Ward, in his Naturalism and Agnosticismo makes rhisa ground of complaint against physics: the business of those who wish toascertain the ultimate truth about the world, he apparently thinks, shouldbe the discovery of causes, yet physics never even seeks them. To me itseems that philosophy ought not to assume such legislative functions, andthat the reason why physics has ceased to look for causes is that, in fact,there are no such things. The law of causaliqy, I believe, like much thatpasses muster among philosophers, is a relic of a bygone age, surviving,Iike the monarchy, only because it is erroneously supposed to do no harm.

In order to find out what philosophers commonly understand by"cause," I consulted Baldwin's Dictionary, and was rewarded beyond myexpectations, for I found the following three mutually incompatible defi-nitions:-, "CAusetnv. ( r ) The necessary connection of events in the time-

series."CAusE (notion of). Whatever may be included in the thought or

perception of a process as taking place in consequence of another

Process. .* Reprinted by kind permission of the author and the publishers from Bertrand

Russell, Mysticiswt and Logic (George Allen & Unrvin, Ltd.), pp. r8o-2o5, and his OzrKnoutledge of the External World (W. W. Norton & Co., London, 1929, New York.rgzc), pp. 247-256.

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388 CAUSALITY, DETERT\{rNrSM, AND pRoBABrLrry

"CAUSE eNp Ernnci. ( r ) Cause and effect . . . are correlative termsdenoting any two distinguishable things, phases, or aspects ofrealiqy, which are so related to each other that whenever the firstceases to exist the second comes into existence immediately after,and whenever the second comes into existence the first has ceasedto exist immediately before."

Let us consider these three definitions in turn. The first, obviously,is unintelligible without a definition of "necessary." Under this head, Bald-win's Dictionary gsves the following:-

"NEcESSARv. That is necessary which not only is true, but would betrue under all circumstances. Something more than brute com-pulsion is, therefore, involved in the conception; there is a gen-eral law under which the thing takes place."

The notion of cause is so intimately connected with that of necessiqythat it will be no digression to linger over the above definition, with a viewto discovering, if possible, some meaning of which it is capable; for, as itstands, it is very far from having any definite signification.

The first point to notice is that, if any meaning is to be given to thephrase "would be true under all circumstances," the subiect of it must bea propositional function,,not e proposition.l A proposition is simply true orfalse, and that ends the matter: there can be no question of "circum-stances." "Charles I's head was cut off" is iust as true in summer as in win-ter, on Sundays as on Mondays. Thus when it is worth saying that some-thing "would be true under all circumstances," the something in questionmust be a propositional function, i.e. an expression contairring a variable,and becoming a proposition when a value is assigned to the variable; thevarying "circumstances" alluded to are then the different values of whichthe variable is capable. Thus if "necessary" means "what is true under allcircumstances," then "if r is a man, r is mortal" is necessary, because it istrue for any possible value of .r. Thus we should be led to the followingdefinition:-

"Nncnssany is a predicate of a propositional function, meaning that itis true for all possible values of its argument or arguments,D

Unfortunately, however, the definition in Baldwin's Dictionary s^ysthat rvhat is necessary is not only "true under all circumstances" but is also"true." Now these two are incompatible. Only propositions can be "true,"and only propositional functions can be "true under all circumstances."Hence the definition as it stands is nonsense. What is meant seems to bethis: "A proposition as necessary when it is a value of a propositionalfunction which is true under all circumstances, i.e. for all values of itsargument or arguments." But if we adopt this definition, the same prop-

1.A propositional funcrion is an.expression containing a. variable,.or undererminedconstituent, and becoming a proposition as soon as a definite value is assigned to thevariable. Examples are: "A is A," "* is a number." The variable is called the argumentof the function'.

ON THE NOTION OF CAUSE l8qosition will be necessary or contingent according as we choose one orother of its terms as the argument to our propositional function. For ex-ample, "if Socrates is a man, Socrates is mortal," is necessary if Socrates ischosen as argument, but not if. man or mortal is chosen. Again, "if Socratesis a man, Plato is mortal," will be necessary if either Socrates or man ischosen as argument, but not if Plato or mortdl is chosen. However, thisdifficulty can be overcome by specifying the constituent which is to beregarded as argument, and we thus arrive at the following definition:

"A proposition is necessary with respect to a given constituent if itremains true when that constituent is altered in any way compatible withthe proposition remaining significant."

We may now apply this definition to the definition of causality quotedabove. It is obvious that the argument must be the time at which the earlierevent occurs. Thus an instance of causality will be such as: "If the eventel occurs at the time f it will be followed by the event e"." This prop-osition is intended to be necessary with respect to f i.e. to remain truehowever tr m y be viewed. Causality, as a universal law, will then be thefollowing: "Given any event e1, there is an event e, such that, whenevere1 occurs, e2 occurs later." But before this can be considered precise, wemust specify how much later eris to occur. Thus the principle becomes:-

"Given any event, el; there is an event e, and a time interval r suchthat, whenever e1 occurs, e, follows after an interval r."

I am not concerned as yet to consider whether this law is true orfalse. For the present, I am merely concerned to discover what the Iaw ofcausaliqy is supposed to be; I pass, therefore, to the other definitions quotedabove.

The second definition need not detain us long, for two reasons. First,because it is psychological: not the "thought or perception" of a process,but the process itself, must be what concerns us in considering causality.Secondly, because it is circular: in speaking of a process as "taking placein consequence of" another process, it introduces the very notion of causewhich was to be defined.

The third definition is by far the most precise; indeed as regards clear-ness it leaves nothing to be desired. But a great difrculty is caised by thetemporal contiguity of cause and effect rvhich the definition asserts. Notwo instants are contiguous, since the time-series is compact; hence eitherthe cause or the effect or both must, if the definition is correct, endurefor a finite time; indeed, by the wording of the definition it is plain thatboth are assumed to endure for a finite time. But then we are faced witha dilemma: if the cause is a process involving change within itself, weshall require (if causalitv is universal) causal relations between its earlierand later parts; moreover, it would seem that only the later parts can berelevant to the effect, since the earlier parts are not contiguous to theeffect, and therefore (b1' ths definition) cannot influence the effect. Thuswe shall be led to dinrinish the duration of the canse u,ithout limit. and

390 CAUSALITY, DETERMINISM, AND PROBABILITY

however much we may diminish it, there will still remain an earlier partwhich might be altered without altering the effect, so that the true cause,as defined, will not have been reached, for it will be observed that thedefinition excludes plurality of causes. If, on the other hand, the cause ispurely static, involving no change within itself, then, in the first place, nosuch cause is to be found in nature, and in the second place, it seems strange-too strange to be accepted, in spite of bare logical possibility-that thecause, after existing placidly for some time, should suddenly explode intothe effect, when it might just as well have done so at any earlier time, orhave gone on unchanged without producing its effect. This dilemma,therefore, is fatal to the view that cause and effect can be contiguous intime; if there are causes and effects, they must be separated by a finitetime-interval r, BS w?s assumed in the above interpretation of the first defi-nition.

What is essentially the same statement of the law of causality as theone elicited above from the first of Baldwin's definitions is given by otherphilosophers. Thus John Stuart l\{ill says:-

"The Law of Causation, the recognition of which is the main pillarof inductive science, is but the familiar truth, that invariability of suc-cession is found by observation to obtain between every fact in nature andsome bther fact which has preceded it."'

And Bergson, who has rightly perceived that the law as stated byphilosophers is worthless, nevertheless continues to suppose that it is usedin science. Thus he savs:-

"Now, it is argued, this law [the law of causalityJ means thar everyphenomenon is determined by its conditions, or, in other words, that thesame causes produce the same effects." 3

And again:-"We perceive physical phenomena, and these phenomena obey laws.

This means: (r) That phenomena a, b, c, d, previously perceived, canoccur again in the same shape; (z) that a certain phenomenon P, whichappeared after the conditions A, b, c, d, and after these conditions only,will not fail to recur as soon as the same conditions are again present." r

A great part of Bergson's attack on science rests on the assumption thatit employs this principle. In fact, it employs no such principle, but phi-losophers-even Bergson-are too apt to take their views on science fromeach other, not from science. As to what the principle is, there is a fairconsensus among philosophers of different schools. There are, however,a number of difficulties which at once arise. I omit the question of pluraliryof causes for the present, since other graver questions have to be consideitd.Two of these, which are forced on our attention by the above statementof the larv, are the following:-

z Logic, Bk. I I I , Chap. V, 52.s Time dnd Free lVill, p. r9g.t Time and Free Will, p. zoz.

ON THE NOTION OF CAUSE 39r(r) What is meant by an "event"l(z) How long may the time-interval be between cause and effect?(r) An "euentr" in the statement of the law, is obviously intended to

be something that is likely to recur, since otherwise the iaw becomestrivial. It follows that an "event" is not a particular, but some universalof which there may be many instances. It follows also that an "event" mustbe something short of the whole state of the universe, since it is highlyimprobable that this will recur. What is meant by an "event" is somethinglike striking a match, or dropping a penny into the slot of an automaticmachine. If such an event is to recur, it must not be defined too narrowly:we must not state with what degree of force the match is to be struck, norwhat is to be the temperature of the penny. For if such considerationswere relevant. our "event" would occur at most once, and the law wouldcease to give information. An "event," then, is a universal defined suf-ficiently widely to admit of many particular occurrences in time beinginstances of it.

(z) The next question concerns the time-interval. Philosophers, nodoubt, think of cause and effect as contiguous in time, but this. for reasonsalready given, is impossible. Hence, sinle there are no infinitesimal time-intervals, there must be some finite lapse of time r between cause andeffect. This, however, at once raises insuperable difficulties. However shortwe make the interval r, something m"y h"pp"n during this interval whichprevents the expected result. I put my penny in the slot, but before I candraw out my ticket there is an earthquake which upsets the machine andmy calculations. In order to be sure of the expected effect, we must knowthat there is nothing in the environment to interfere with it. But this meansthat the supposed cause is not, by itself, adequate to insure the effect. Andas soon as we include the environment, the probability of repetition isdiminished, until at last, when the whole environment is included, theprobability of repetition becomes almost nil.

ln spite of these difficulties, it must, of course, be admitted that manyfairly dependable regularities of sequence occur in daily life. It is theseregularities that have suggested the supposed law of causality; where theyare found to fail, it is thought that a better formulation could have beenfound which would have never failed. I am far from denying that theremay be such sequences which in fact never do fail. Ic may be that therewill never be an exception to the rule that when a stone of more than acertain mass, moving with more than a certain velocity, comes in contactwith a pane of glass of less than a certain thickness, the glass breaks. I alsodo not deny that the observation of such regularities, even when they arenot without exceptions, is useful in the infancy of a science: the observa-tion that unsupported bodies in air usually fall was a stage on the way to

the law of gravitation. What I deny is that science assumes the existenceof invariable uniformities of sequence of this kind, or that it aims at dis-

covering them. All such uniformities, as we saw, depend upon a certain

J92 CAUSALITY, DETERMINISM, AND PROBABILITY

vagueness in the definition of the "events." That bodies fall is a vaguequalitative statement; science wishes to know how fast they fall. This de-pends upon the shape of the bodies and the densiry of the air. It is true thatthere is more nearly uniformity when they fall in a vacuum; so far asGalileo could observe, the uniformity is then complete. But later it ap-peared that even there the latitude made a difference, and the altitude.Theoretically, the position of the sun and moon must make a difierence.In short, every advance in a science takes us farther away from the crudeuniformities which are first observed, into greater differentiation of ante-cedent and consequent, and into a continually wider circle of antecedentsrecognised as relevant.

The principle "same cause, same effect," which philosophen imagineto be vital to science, is therefore utterly otiose. As soon as the antecedentshave been given sufficiently fully to enable the consequent to be calculatedwith some exactitude, the antecedents have become so complicated that it isvery unlikely they will ever recur. Hence, if this \4/ere the principle in-volved, science would remain utterly sterile.

The importance of these considerations lies partly in the fact that theylead'to a more correct account of scientific procedure, partly in the factthat they remove the analogy with human volition which makes'the con-ception of cause such a fruitful source of fallacies. The latter point willbecome clearer by the help of some illustrations. For this purpose I shallconsider a few maxims which have played a great part in the history ofphilosophy.

( r ) "Cause and effect must more or less resemble each other." Thisprinciple was prominent in the philosophy of occasionalism, and is stillby no means extinct. It is still often thought, for example, that mind couldnot have grown up in a universe which previously contained nothingmental, and one ground for this belief is that matter is too dissimilar frommind to have been able to cause it. Or, more particularly, what are termedthe nobler parts of our nature are supposed to be inexplicable, unless theuniverse always contained something at least equally noble which couldcause them. All such views seem to depend upon assuming some undulysimplified law of causality; for, in any legitimate sense of "cause" and"effect," science seems to show that they are usually very widely dissimilar,the "cause" being, in fact, two states of the whole universe, and the "effect"some Pafticular event.

(z) "Cause is analogous to volition, since there must be an intelligiblenexus between cause and effect." This maxim is, I think, ofteru uncon-sciously in the imaginations of philosophers who would reject it whenexplicitly stated. It is probably operative in the view we have iust beenconsidering, that mind could not have resulted from a purely materialworld. I do not profess to know what is meant by "intelligible"; it seems tomean "familiar to imagination." Nothing is less "intelligible," in any othersense, than the connection berween an act of will and its fulfilment. But

ON THE NOTION OF CAUSE 393obviously the sort of nexus desired between cause and effect is such as couldonly hold between the "events" which the supposed law of causality con-templates; the laws which replace causality in such a science as physicsleave no room for any two events between which a nexus could be sought.

(3) "The cause compels the effect in some sense in which the effectdogs not compel the cause." This belief seems largely operative in thedislike of determinism; but, as a matter of fact, it is connected with oursecond maxim, and falls as soon as that is abandoned. We may define "com-pulsion" as follows: "Any set of circumstances is said to compel A whenA desires to do something which the circumstances prevent, or to abstainfrom something which the circumstances cause." This presupposes thatsome meaning has been found for the word "s2s5e"-x point to which Ishall retum later. What I want to make clear at present is that compulsionis a very complex notion, involving thwarted desire. So long as a persondoes what he wishes to do, there is no compulsion, however much hiswishes may be calculable by the help of earlier events. And where desiredoes not come in, there can be no question of compulsion. Hence it is, ingeneral, misleading to regard the cause as compelling the effect.

A vaguer form of the same maxim substitutes the word "determine"for the word "compel"l we are told that the cause determines the efrectin a sense in which the effect does not determine the cause. It is not quiteclear what is meant by "determining"; the only precise sense, so far as Iknow, is that of a funcdon or one-many relation. If we admit plurality ofcauses, but not of effects, that is, if we suppose that, given the cause, theeffect must be such and such, but, given the effect, the cause may havebeen one of many alternatives, then we may say that the cause determinesthe effect, but not the effect the cause. Plurality of causes, however, resultsonly from conceiving the effect vaguely and narrowly and the cause pre-ciselv and widelv. Manv antecedents mav "cause" a man's death. becausehis death is vague and narrow. But if we adopt the opposite course, takingas the "cause" the drin\ing of a dose of arsenic, and as the "effect" thewhole state of the world five minutes later, we shall have plurality of ef-fects instead of plurality of causes. Thus the supposed lact of ryrrr-.ttybetween "cause" and "effect" is illusory.

(4) "A cause cannot operate when it has ceased to exist, because whathas ceased to exist is nothing." This is a common maxim, and a still morecommon unexpressed prejudice. It has, I fancy, a good deal to do with theattractiveness of Bergson's "durde": since the past has effects now, it muststill exist in some sense- The mistake in this maxim consists in the sup-position that causes "operate" at all. A volition "operates" when what itwills takes place; but nothing can operate except a volition. The beliefthat causes "operate" results from assimilating them, consciously or un-consciously; to volitions. We have already seen that, if there are causes atall, they must be separated by a finite interyal of time from their effectsand thus cause their effects after they have ceased to exist.

39+ CAUSALITY, DETERMINISM, AND PROBABILITY

It may be obfected to the above definition of a volition "operating"that it only operates when it "causes" what it wills, not when it merelyhappens to be followed by what it wills. This certainly represens theusual view of what is meant by a volition "operating," but as it involvesthe very view of causation which we are engaged in combating, it is notopen to us as a definition. We may say that a volition "dperates" when thereis some law in virtue of which a similar volition in rather similar cir-cumstances will usually be followed by what it wills. But this is a vagueconception, and introduces ideas which we have not yet considered. Whatis chiefly important to notice is that the usual notion of "operating" is notopen to us if we reject, as I contend that we should, the usual notion ofcausation.

(j) "A cause cannot operate except where it is." This maxim is verywidespread; it was urged against Newton, and has remained a source ofprejudice against "action at a distance." In philosophy it has led to a denialof transient action, and thence to monism or Leibnizian monadism. Likethe analogous maxim concerning temporal contiguity, it rests upon theassumption that causes "operate," i.e. that they are in some obscure wayanalogous-to volitions. And, as in the case of temporal contiguity, theinferences drawn from this maxim are wholly groundless.

I return now to the question, What law oi laws can be found to takethe place of the supposed law of causality?

First, without passing beyond such uniformities of sequence as arecontemplated by the traditional law, we may admit that, if any such se-quence has been observed in a great many cases, and has never been foundtb fail, there is an inductive piobability that it will be found to hold infuture cases. If stones have hitherto been found to break windows, it isprobable that they will continue to do so. This, of course, assumes the in-ductive principle, of which the truth may reasonably be questioned; butas this principle is not our present concern, I shall in this discussion treatit as indubitable. We may then say, in the case of any such frequentlyobserved sequence, that the earlier event is the cause and the later event theeffect.

Several considerations, however, make such special sequences verydifferent from the traditional relation of cause and effect. In the first place,the sequence, in any hitherto unobserved instance, is no more than prob-able, whereas the relation of cause and eftect was supposed to be necessary.I do not mean by this merely that we are not sure of having discovered atrue case of cause and effect; I mean that, even when we have a case bfcause and effect in our present sense, all that is meant is that on groundsof observation, it is probable that when one occurs the other will alsooccur. Thus in our Dresent sense. A mav be the cause of B even if thereactually are cases where B does not follovr A. Striking a match will be thecause of its igniting, in spite of the fact that some matches are damp and failto lgnlte.

ON THE NOTION OF CAUSE 395In the second place, it will not be assumed thrt eaery event has some

antecedent which is its cause in this sense; we shall only believe in causalsequences where we find them, without any presumption that they alwaysare to be found.

In the third place, any c^se of sufficiently frequent sequence will becausal in our present sensel for example, we shall not refuse to say thatnight is the cause of day. Our repugnance to saying this arises from theease with which we can imagine the sequence to fail, but owing to thefact that cause and effect must be separated by a finite interval of. time, nnysuch sequence might fail through the interposition of other circumstancesin the interval. Mill, discussing this instance of night and day, says:-

"It is necessary to our using the word cause, that we should believenot only that the antecedent always Das been followed by the consequent,but that as long as the present consdrution of things endures, it always,uill be so." 5

In this sense, we shall have to give up the hope of finding causal lawssuch as Mill contemplated; any causal sequence which we have observedmay at any moment be falsified without a falsification of any laws of thekind that the more advanced sciences aim at establishing.

In the fourth place, such laws of probable sequence, though usefulin daily life and in the infancy of a science, tend to be displaced by quitedifferent laws as soon as a science is successful. The law of gravitationwill illustrate what occurs in any advanced science. In the motions ofmurually gravitating bodies, there is nothing that can be called a cause,and nothing that can be called an effect; therp is merely a formula. Certaindifferential equations can be found, which hold at every instant for everyperticle of the system, and which, given the configuration and velocitiesat one instant, or the configurations at rwo instants, render the configura-tion at any other earlier or later instant theoretically calculable. That is tosay, the configuration at any instant is a function of that instant and theconfigurations at two given instants. This statement holds throughoutphysics, and not only in the special case of gravitation. But there is nothingthat could be properly called "cause" and nothing that could be properlycalled "effect" in such a system.

No doubt the reason why the old "law of causality" has so long con-tinued to pervade the books of philosophers is simply that the idea of afunction is unfamiliar to most of them, and therefore they seek an undulysimplified statemint. There is no question of repetitions of the "same"'cause producing the "same" effect; it is not in any sameness of causes andeffects that the constancy of scientific law consists. but in sameness ofrelations. And even "sameness of relations" is too simple a phrase; "same-ness of differential equations" is the only correct phrase. It is impossibleto state this accurately in non-mathematical language; the nearest approachwould be as follows: "There is a constant relation between the state of the

6 Loc. cit., $6.

396 CAUSALITY, DETERMINISM, AND PROBABILITY

universe ^t

any instant and. the rate of change in the rate at which any paftof the universe is changing at that instant,

"nd this relation is many-one, i.e.

such that the rate of change in the rate of change is determinate when thestate of the universe is given." If the "law of causality" is to be somethingactually discoverable in the practice of science, the above proposition hasa better right to the name than any "la'iv of causality" to be found in thebooks of philosophers.

In regard to the above principle, several observations must be made-(r) No one can pretend that the above principle is a priori or self-

evident or a "necessiry of thought." Nor is it, in any sense, a premissof science: it is an empirical generalisation from a number of laws whichare themselves empirical generalisations.

(z) The law makes no difference between past and future: the future"determines" the past in exactly the same sense in which the past "deter-mines" the future. The word "determine," here, has a purely logical sig-nificance: a certain number of variables "determine" another variable ifthat other variable is a function of them.

(3) The law will not be empirically verifiable unless.the course ofevents'within some sufficiently small volume will be approximately thesame in any two states of the universe which only differ in regard to whatis at a considerable distance from the small volume in question. For ex-ample, motions of planets in the solar system must be approximately thesame however the fixed stars may be distributed, provided that all thefixed stars are very much farther from the sun than the planets are. Ifgravitation varied directly as the distance, so that the most remote starsmade the most difference to the motions of the planets, the world might bejust as regular and lust as much subject to mathematical laws as it is atpresent, but we could never discover the fact.

(4) Although the old "law of causality" is not assumed by science,something which we may call the "uniformity of nature" is assumed, orrather is accepted on inductive grounds. The uniformity of nature doesnot assert the trivial principle "same cause, same effect," but the principleof the permanence of laws. That is to say, when a law exhibiting, e.g. anacceleration as a function of the configuration has been found to holdthroughout the observable past, it is expected that it will continue tohold in the future, or that, if it does not itself hold, there is some otherlaw, agreeing with the supposed law as regards the past, which will holdfor the future. The ground of this principle is simply the inductive groundthat it has been found to be true in very many instances; hence the prin-ciple cannot be considered certain, but only probable to a degree whichcannot be accurately estimated.

The uniformity of nature, in the above sense, although it is assumedin the practice of science, must not, in its generality, be regarded as a kindof maior premiss, without which all scientific reasoning would be in error.The assumption that all laws of nature are permanent has, of course, less

397probability than the assumption that this or that particular law is permanent;and the assumption that a particular law is permanent for all time has lessprobability than the assumption that it will be valid up to such and such adate. Science, in any given case, will assume what the case requires, but nomore. In constructing the Nautical Alruanac for r9r5 it will assume thatthe law of gravitation will remain true up to the end of that year; but it willinake no assumption as to 1916 until it comes to the next volume of thealmanac. This procedure is, of course, dictated by the fact that the uni-formity of nature is not known a priori, but is an empirical generalisation,like "all men are mortal." In all such cases, it is better to argue immediatelyfrom the given particular instances to the new instance, than to argue byway of a major premiss; the conclusion is only probable in either case, butacquires a higher probability by the former method than by the latter.

In all science we have to distinguish two sorts of laws: first, thosethat are empirically verifiable but probably only approximate; secondly,those that are not verifiable, but may be exact. The law of gravitation, forexample, in its applications to the solar system, is only empirically verifiablewhen it is assumed that matter outside the solar system may be ignored forsuch purposes; we believe this to be only approximately true, but we cannotempirically verify the law of universal gravitation which we believe to beexact. This point is very important in connection with what we may call"relatively isolated systems." These may be defined as follows:-

A system relatively isolated during a given period is one which, withinsome assignable margin of error, will behave in the same way throughoutthat period, however the rest of the universe may be constituted.

A syslsrn may be called "practically isolated" during a given period if,although there might be states of the rest of the universe which wouldproduce more than the assigned margin of error, there is reason to believethat such states do not in fact occur.

Strictly speaking, we ought to specify the respect in which the systemis relatively isolated. For example, the earth is relatively isolated as regardsfalling bodies, but not as regards tides; it is practi.cally isolated as regardseconomic phenomena, although, if Jevons' sunspot theory of commercialcrises had been true, it would not have been even practically isolated inthis respect.

It will be observed that we cannot prove in advance that a system isisolated. This will be inferred from the observed fact that approximateuniformities can be stated for this system alone. If the complete laws forthe whole universe were known, the isolation of a system

"ooid b" deduced

from them; assuming, for example, the law of universal gravitation, the prac-tical isolation of the solar system in this respect can be deduced by the helpof the fact that there is very little matter in its neighbourhood. But it shouldbe observed that isolated systems are only important as providing a possi-bility of discovering scientific laws; they have no theoretical importancein the finished structure of a science.

ON THE NOTION OF CAUSE

398 CAUSALITY, DETERMINISM, AND PROBABILITY

The case where one event A is said to "cause" another event B, whichphilosophers take as fundamental, is really only the most simplified instanceof a practically isolated system. It may happen thet, as a result of generalscientific laws, whenever A occurs throughout a certain period, it is fol-lowed by B; in that case, A and B form a system which is practically isolatedthroughout that period. It is, however, to be regarded as a piece of goodfortune if this occurs; it will always be due to special circumstances, andwould not have been true if the rest of the universe had been differentthough subject to the same laws.

The essential function which causality has been supposed to perform isthe possibiliry of inferring the future from the past, or, more generally,events at any time from events at ceftain assigned times. Any system inwhich such inference is possible may be called a "deterministic" system.We may define a deterministic system as follows:-

A system is said to be "deterministic" when, given certain data, e' e,. . . , en, at tlmes ta, t2, . . . , t* respectively, concerning this system, ifEr is the state of the system at

^ny time f, there is a functional relation of

the form

Et : f (ey tn e2, tz, . . . , en, tn, t). (A)

The system will be "deterministic throughout a given period" if r, in theabove formula, may be any time within that period, though outside thatperiod the formula may be no longer true. If the universe, as a whole, issuch a system, determinism is true of the universe; if not, not. A systemwhich is part of a deterministic system I shall call "determined"; one whichis 4ot part of any such system I shall call "capricious."

The events erl e2, . . . , €n I shall call "determinantst'of the system.It is to be observed that a system which has one set of determinants willin general have many. In the case of the motions of the planets, for example,the configurations of the solar system at

^ny two given times will be de-

terminants.We may take another illustration from the hypothesis of psycho-

physical parallelism. Let us assume, for the purposes of this illustration,that to a given state of brain a given state of mind always corresponds, andvice versa, i.e. that there is a one-one relation between them, so that eachis a function of the other. We may also assume, what is practically certain,that to a given state of a certain brain a given state of the whole materialuniverse corresponds, since it is highly improbable that a given brain is evertwice in exactly the same state. Hence there will be a one-one relation be-tween the state of a given person's mind and the state of the whole materialuniverse. It follows that. if ?t states of the material universe are determinantsof the material universe, then n states of a given man's mind are determi-nants of the whole material and mental universe-assuming, that is to say,that psycho-physical parallelism is true.

The above illustration is important in connection with e certain con-

ON THE NOTION OF CAUSE 399

fusion which seems to have beset those who have philosophised on therelation of mind and matter. It is often thought that, if the state of the,grind is determinate when the state of the brain is given, and if the materialworld forms a deterministic system, then mind is "subject" to matter insome sense in which matter is not "subiect" to mind. But if the state of thebrain is also determinate when the state of the mind is given, it must beexactly as true to regard matter as subject to mind as it would be to re-gard mind as subject to matter. We could, theoretically, work out thehistory of mind without ever mentioning matter, and then, at the end,deduce that matter must meanwhile have gone through the correspondinghistory. It is true that if the relation of brain to mind were many-one, notone-one, there would be a one-sided dependence of mind on brain, whileconversely, if the relation were one-many, as Bergson supposes, therewould be a one-sided dependence of brain on mind. But the dependenceinvolved is, in any case, only logical; it does not mean that we shall becompelled to do things we desire not to do, which is what people instinc-tively imagine it to mean.

As another illustration we may take the case of mechanism and tele-ology. A system may be defined as "mechanical" when it has a set ofdeterminants that are purely material, such as the positions of certainpieces of matter at certain times. It is an open question whether the worldof mind and matter, as we know it, is a mechanical system or not; let ussuppose, for the sake of argument, that it is a mechanical system. Thissupposition---+o I contend-throws no light whatever on-the questionwhether the universe is or is not a "teleological" system. It is difficult todefine accurately what is meant by a "teleoiogical'; system, but the argu-ment is not much affected by the pardcular definition we adopt. Broadly,a teleological system is one in which purposes are realised, i.e. in whichcertain desires-those that are deeper or nobler or more fundamental ormore universal or what not-are follorqed by their realisation. Now thefact-if it be a fact-that the universe is mechanical has no bearing what-ever on the cluestion whether it is teleological in the above sense. Theremight be a mechanical system in which all wishes were realised, and theremight be one in which all wishes were thwarted. The question whether,or how far, our actual world is teleological, cannot, therefore, be settledby proving that it is mechanical, and the desire that it should be teleologicalis no ground foi wishing it to be not mechanical.

There is, in all these questions, a very great difficulty in avoidingconfusion between what we can infer and what is in fact determined. Letus consider, for a moment, the various senses in which the future may be"determined." There is one sense-and a very important one-in which itis determined quite independently of scientific laws, namely, the sensethat it will be what it will be. We all regard the past as determined simplyby the fact that it has happened; but for the accident that memory worksbackward and not forward, we should regard the future as equally de-

4OO CAUSALITY, DETERMTNISM, AND PROBABILITY

termined by the fact that it will happen. "But," we are told, "you cannotalter the past, while you can to some extent alter the future." This viewseems to me to rest upon iust those errors in regard to causation whichit has been my obiect to remove. You cannot make the Past other thanit was-true, but this is a mere application of the law of contradiction. Ifyou already know what the past was, obviously it -is useless to wish itdifferent. But also you cannot make the future other than it will be; thisagain is an application of the law of contradiction. And if you happen toknow the future-e.g. in the case of a forthcoming eclipse-it is just asuseless to wish it different as to wish the past different. "But," it will bereioined, "our wishes c n cause the future, sometimes, to be different fromwhat it would be if they did not exist, and they can have no such effectupon the past." This, again, is a mere tautology. An effect being definedas something subsequent to its cause, obviously we can have no effect uponthe past. But that does not mean that the past would not have been dif-ferent if our present wishes had been difterent. Obviously, our presentwishes are conditioned by the past, and therefore could not have beendifferent unless the past had been different; therefore, if our present wisheswere diffe-rent, the past would be different. Of course, the past cannot bedifferent from what it was, but no more can our present wishes be dif-ferent from what they are; this again is merely the law of contradiction.The facts seem to be merely ( r ) that wishing generally depends uponignorance, and is therefore commoner in regard to the future than inregard to the past; (z) that where a wish concerns the future, it and itsrealisation very often form a "practically independent system," i.e. manywishes regarding the future are realised. But there seems no doubt thatthe main difference in our feelings arises from the accidental fact that thepast but not the future can be known by memory.

Although the sense of "determined" in which the future is determinedby the mere fact that it will be what it will be is sufficient (at least so itseems to me) to refute some opponents of determinism, notably M. Berg-son and the pragmatists, yet it is not what most people have in mind whenthey speak of the future as determined. What they have in mind is a for-mula by means of which the future can be exhibited, and at least theo-retically calculated, as a function of the past. But at this point we meetwith a great dfficulty, which besets what has been said above aboutdeterministic systems, as well as what is said by others.

If formule of any degree of complexity, however great, are admitted,it would seem that any system, whose state at a given moment is a functionof certain measurable quantities, must be a deterministic system. Let usconsider, in illustration, a single material particle, whose co-ordinates attime / xtl !c1, !t, %t.Theq however the particle moves, there must be,theoretically, functions f'fr,fu such that

4: f r ( t ) , yt : f r ( t ) , zt - f " ( t ) .

ON THE NO'TION OF CAUSE 40r

It follows that, theoretically, the whole state of the material universeat time t must be capable of being exhibited as a function of t. Hence ouruniverse will be deterministic in the sense defined above. But if this betrue, no information is conveyed about the universe in stating that it isdeterministic. It is true that the formula involved mav be of strictlv infinitecomplexity, and therefore not practically capabl. of being writtln downor apprehended. But except from the point of view of our knowledge,this might seem to be a detail: in itself, if the above considerations aresound, the material universe must be deterministic I m'ltst be subiect to laws.

This, however, is plainly not what was intended. The difference be-tween this view and the view intended may be seen as follows. Given someformula which fits the facts hitherto-say the law of gravitation-there willbe an infinite number of other formule not empirically distinguishablefrom it in the past, but diverging from it more and more in rhe furure.Hence, even assuming that there are persistent laws, we shall have noreason for assuming that the law of the inverse square will hold in future;it may be some other hitherto indistinguishable law that will hold. Wecannot say that eaery law which has held hitherto must hold in the future,because past facts which obey one law will also obey others, hithertoindistinguishable but diverging in future. Hence there must, at every mo-ment. be laws hitherto unbroken which are now broken for the first time.What science does, in fact, is to select the sirnplest formula that will fit thefacts. But this, quite obviously, is merely a methodological precepr, not alaw of Nature. If the simplest formula ceases, after a time, to be applicable,the simplest formula that remains applicable is selected, and science has nosense that an axiom has been falsified. We are thus left with the brute factthat, in many departments of science, quite simple laws have hirherto beenfound to hold. This fact cannot be regarded as having any a priori ground,nor can it be used to support inductively tl're opinion that the same lawswill continue; for at every moment laws hitherto true are being falsified,though in the advanced sciences these laws are less simple than those thathave remained true. Moreover it would be fallacious to argue inductivelyfrom the state of the advanced sciences to the future state of the others, forit may well be that the advanced sciences are advanced simply because,hitherto, their subiect-matter has obeyed simple and easily ascertainableIaws, while the subiect-matter of other sciences has not done so.

The difficulty we have been considering seems to be met partly, if notwholly, by the principle that the time mrst not enter explicitly into ourformula. All mechanical laws exhibit acceleration as a function of con-figuration, not of configuration and time jointly; and this principle of theirrelevance of the time may be extended to all scientific laws. In fact wemight interpret the "uniformity of nature" as meaning just this, that noscientific law involves the time as an argument, unless, of course, it is givenin an integrated form, in which case lapse of time, though not absolute

time, may appear in our formula. Whether this consideration sufHces to

+o2 CAUSALITE DETERMINISM, AND PROBABILITY

overcome our difficulty completel/, I do not know; but in any case itdoes much to diminish it.

It will serve to illustrate what has been said if we apply it to the ques-tion of free will.

The problem of free will * is so intimately bound up with the analysisof causation that, old as it is, we need not despafu of obtaining new lighton it by the help of new views on the notion of cause. The free-will prob-lem has, at one time or another, stirred men's passions profoundly, and thefear that the will might not be free has been to some men a source of greatunhappiness. I believe that, under the influence of a cool analysis, thedoubtful questions involved will be found to have no such emotional im-portance as is sometimes thought, since the disagreeable consequencessupposed to flow from a denial of free will do not flow from this denialin any form in which there is reason to make it. It is not, however, on thisaccount chiefly that I wish to discuss this problem, but rather because itaffords a good example of the clarifying effect of analysis and of theinterminable controversies which may result from its neglect.

tet us, first try to discover what it is we really desire when we desirefree will. Some of our reasons for desiring free will are profound, sometrivial. To begin with the former: we do not wish to feel ourselves in thehands of fate, so that, however much we may desire to will one thing, wemay nevertheless be compelled by an outside force to will another. Wedo not wish to think that, however much we may desire to act well,heredity and surroundings may force us into acting ill. We wish to feelthat, in cases of doubt, our choice is momentous and lies within our power.Besides these desires, which are worthy of all respect, we have, however,others not so respectable, which equally make us desire free will. We donot like to think that other people, if they knew enough, could predictour actions, though we know that we can often predict those of otherpeople, especially if they are elderly. Much as we esteem the old gentle-man who is our neighbour in the country, we know that when grouse arementioned he will tell the story of the grouse in the gun-room. But weourselves are not so mechanical: we never tell an anecdote to the sameperson twice, or even once unless he is sure to enjoy it; although we oncemet (say) Bismarck, we are quite capable of hearing him mentioned with-out relating the occasion when we met him. In this sense, everybodythinks that he himself has free will, though he knows that no one eise has.'The desire for this kind of free will seems to be no better than a form ofvanity. I do not believe that this desire can be gratified with any certainty;but the other, more respectable desires are, I believe, not inconsistent withany tenable form of determinism.

We have thus two questions to consider: ( r ) Are human actions* The remainder of this selection is reprinted from pp. 247-56 of Our Knou.sledge

of the External World, by Bertrand Russell by permission of W. W. Norton & Com-pany, Inc. Copyright ryz$ by W. W. Norton'&'Company, Inc.

ON THE NOTION OF CAUSE 403theoretically predictable from a sufficient number of antecedents? (z) Arehuman actions subject to an external compulsion? The two questions, as Ishall try to show, are entirely distinct, and we may answer the first inthe affirmative without therefore being forced to give an affirmative answerto the second.

(r) Are hu.man actions theoretically predictable from a sufficientnumber of antecedents? Let us first endeavouf to give precision to thisquestion. We may state the question thus: Is there some constant relationbetween an act and a certain number of earlier events, such that, whenthe earlier events are given, only one act, or at most only acts with somewell-marked character, can have this relation to the earlier events? Ifthis is the case, then, as soon as the earlier events are known, it is theo-retically possible to predict either the precise act, or at least the characternecessary to its fulfilling the constant relation.

To this question, a negative answer has been given by Bergson, ln aform which calls in question the general applicability of the law of cause-tion. He maintains that every event, and more particularly every mentalevent, embodies so much of the past that it could not possibly have oc-curred at any earlier time, and is therefore necessarily quite different fromall previous and subsequent events. If, for example, I read a certain poemmany times, my experience on each occasion is modified by the previousreadings, and my emotions are never repeated exactly. The principle ofcausation, according to him, asserts that the same cause, if repeated, willproduce the same effect. But owing to memory, he contends, this principledoes not apply to mental events. What is apparently the same cause, ifrepeated is modified by the mere fact of repetition, and cannot produce thesame elfect. He infers that every mental event is a genuine novelty, notpredictable from the past, because the past contains nothing exactly likeit by which we could imagine it. And on this ground he regards the free-dom of the will as unassailable.

Bergson's contention has undoubtedly a great deal of truth, and Ihave no wish to deny its importance. But I do not think its consequencesare quite what he believes them to be. It is not necessary for the deterministto maintain that he can foresee the whole particularity of the act whichwill be performed. If he could foresee that A was going to murder B,his foresight would not be invalidated by the fact that he could not knowall the infinite complexiqy of A's state of mind in committing the murder,nor whether the rnurder was to be performed with a knife or with a re-volver. lf the kind of act which will be performed can be foreseen wirhinnarrow limits, it is of little practical interest that there are fine shades whichcannot be foreseen. No doubt every time the story of the grouse in thegun-room is told, there will be slight differences due to increasing habitual-ness, but they do not invalidate the prediction that the story will be told.And there is nothing in Bergson's argument to show that we can neverpredict wha;t kind of act will be performed.

4o.+ CAUSALITY, DETERMINISM, AND PROBABILITY

Again, his statement of the law of causation is inadequate. The lawdoes not state mercly that, if the same cause is repeated, the same effect willresult. It states rather that there is a constant relation between causes ofcertain kinds and effects of certain kinds. For example, if a body falls freely,there is a constant relation between the height through which it falls andthe time it takes in falling. It is not necessary to have a body fall throughthe same height which has been previously observed, in order to be ableto foretell the length of time occupied in falling. If this were necessary,no prediction would be possible, since it would be impossible to make the

height exactly the same on two occasions. Similarly, the attraction whichthe sun will exert on the earth is not only known at distances for whichit has been observed, but at all distances, because it is known to vary asthe inverse square of the distance. In fact, what is found to be repeated isalways the relation of cause and effect, not the cause itself ; all that is neces-sary as regards the cause is that it should be of the same kind (in therelevant respect) as earlier causes whose effects have been observed.

Another respect in which Bergson's statement of causation is inade-quate is in its assumption that the cause must be one event, whereas it maybe two or more events, of even some continuous Process. The substantivequestion ai issue is whether mental events are determined by the past. Nowin such a case as the repeated reading of a poem, it is obvious that ourfeelings in reading the poem are most emphatically dependent upon thepast, but not upon one single event in the past. All our previous readingsof the poem must be includedin the cause. But we easily perceive a certainlaw according to which the effect varies as the previous readings increasein number, and in fact Bergson himself tacitly assumes such a law. Wedecide at last not to read the poem again, because we know that this timethe effect would be boredom. We may not know all the niceties and shadesof the boredom we should feel, but we know enough to guide our decision,and the prophecy of boredom is none the less true for being more or lessgeneral. Thus the kind of cases upon which Bergson relies are insufficientto show the impossibility of prediction in the only sense in which predic-tion has practical or emotional interest. We may therefore leave the con-sideration of his arguments and address ourselves to the problem directly.

The law of causation, according to which later events can theoreticallybe predicted by means of earlier events, has often been held to be a priori,a necessity of thought, a category without which science would be im-possible. These claims seem to me excessive. In certain directions the lawhas been verified empirically, and in other directions there is no positiveevidence against it. But science can use it where it has been found to betrue, without being forced into any assumption as to its truth in otherfields. We cannot, therefore, feel any a priori certainty that causation mustapply to human volitions.

The question how far human volitions are subiect to causal laws is apurely empirical one. Empirically it seems plain that the great majority

ON THE NOTION OF CAUSE +o5of our volitions have causes, but it cannot, on this account, be held neces-sarily certain that all have causes. There are, however, precisely the samekinds of reasons for regarding it as probable that they all have causes asthere are in the case of physical events.

We may suppose-though this is doubtful-that there are laws of cor-relation of the mental and the physical, in virtue of which, given the stateof all the matter in the world, and therefore of all the brains and livingorganisms, the state of all the minds in the world could be inferred, whileconversely the state of all the matter in the world could be inferred if thestate of all the minds were given. It is obvious that there is some degree ofcorrelation between brain and mind, and it is impossible to say how com-plete it may be. This, however, is not the point which I wish to elicit. WhatI wish to urge is that, even if we admit the most extreme claims of de-terminism and of correlation of mind and brain, still the consequencesinimical to what is worth preserving in free will do not follow. The beliefthat they follow results, I think, entirely from the assimilation of causes tovolitions, and from the notion that causes compel their effects in some senseanalogous to that in which a human authority can compel a man to dowhat he would rather not do. This assimilation, as soon as the true natureof scientific causal laws is realised, is seen to be a sheer mistake. But thisbrings us to the second of the two questions which we raised in regard tofree will, namely, whether, assuming determinism, our actions can be inany proper sense regarded as compelled by outside forces.

(z) Are human actions subject to an erternal compulsion? Wehave,in deliberation, a subjective sense of freedom, which is sometimes allegedagainst the view that volitions have causes. This sense of freedom, however,is only a sense that we can choose which we please of a number of alter-natives: it does not show us that there is no causal connection betweenwhat we please to choose and our previous history. The supposed incon-sistency of these two springs from the habit of conceiving causes as anal-ogous to volitions-a habit which often survives unconsciously in thosewho intend to conceive causes in a more scientific manner. If a cause isanalogous to a volition, outside causes will be analogous to an alien will, andacts predictable from outside causes will be subject to compulsion. But thisview of cause is one to which science lends no countenance. Causes. we haveseen, do not compel their effecs, any more than effects compel their causes.There is a mutual relation, so that either can be inferred from the other.When the geologist infers the past state of the eanh from its present state,we should not say that the present state compels the past state to have beenwhat it was; yet it renders it necessary as a consequence of the data, in theonly sense in which effects are rendered nec€ssary by their causes. The

difference which we f eel, in this respect, between causes and effects is amere confusion due to the fact that we remember past events but do nothappen to have memory of the future.

The apparent indeterminateness of the future, upon which some

4c6 CAUSALTTY, DETERMTNISM, AND PROBABTLITY

advocates of free will rely, is merely a result of our ignorance. It is plainthat no desirable kind of free will can be dependent simply upon ourignorance; for if that were the case, animals would be more free thanmen, and savages than civilised people. Free will in any valuable sense mustbe compatible with the fullest knowledge. Now, quite apart from anyassumption as to causality, it is obvious that complete knowledge woulCembrace the future as well as the past. Our knowledge of the past is notwholly based upon causal inferences, but is partly derived from memory.It is a mere accident that we have no memory of the future. We might-as in the pretended visions of seers-see future events immediately, in theway in which we see past events. They certainly will be what they willbe, and are in this sense just as determined as the past. If we saw futureevents in the same immediate way in which we see past events, what kindof free will would still be possible? Such a kind would be wholly inde-pendent of determinism: it could not be contrary to even the most entirelyuniversal reign of causality. And such a kind must contain whatever isworth having in free will, since it is impossible to believe that mere ig-norance can be the essential conditon of any good thing. Let us thereforeimagine a set of beings who know the whole future with absolute certainty,and let us ask ourselves whether they could have anything that we shouldcall free will.

Such beings as we are imagining would not have to wait for the eventin order to know what decision they were going to adopt on some futureoccasion. They would know now what their volitions were going to be.But would they have any reason to regxet this knowledge? Surely not,unless the foreseen volitions were in themselves regrettable. And it is lesslikely that the foreseen volitions would be regrettable if the steps whichwould lead to them were also foreseen. It is difficult not to suppose thatwhat is foreseen is fated, and must happen however much it r?ray be dreaded.But human actions are the outcome of desire, and no foreseeing can betrue unless it takes account of desire. A foreseen volition will have to beone which does not become odious through being foreseen. The beingswe are imagining would easily come to know the causal connections ofvolitions, and therefore their volitions would be better calculated to satisfytheir desires than ours are. Since volitions are the outcome of desires, aprevision of volitions contrary to desires could not be a true one. It mustbe remembered that the supposed prevision would not create the frgureany more than memory creates the past. We do not think we were neces-sarily not free in the past, merely because we can now remember our pastvolitions. Similarly, we might be free in the future, even if we could nowsee what our future volitions were going to be. Freedom, in short, in anyvaluable sense, demands only that our volitions shall be, as they are, the re-sult of our own desires, not of an outside force compelling us to will whatwe would rather not will. Everything else is confusion of thought, due to

oN THE NOTION OF CAUSE +o7