Chapter VII. On Our Knowledge of General Principles

by Bertrand Russell

　　We saw in the preceding chapter that the principle of induction, whilenecessary to the validity of all arguments based on experience, is itselfnot capable of being proved by experience, and yet is unhesitatinglybelieved by every one, at least in all its concrete applications. In thesecharacteristics the principle of induction does not stand alone. There area number of other principles which cannot be proved or disproved byexperience, but are used in arguments which start from what isexperienced.

　　Some of these principles have even greater evidence than the principle ofinduction, and the knowledge of them has the same degree of certainty asthe knowledge of the existence of sense-data. They constitute the means ofdrawing inferences from what is given in sensation; and if what we inferis to be true, it is just as necessary that our principles of inferenceshould be true as it is that our data should be true. The principles ofinference are apt to be overlooked because of their very obviousness—theassumption involved is assented to without our realizing that it is anassumption. But it is very important to realize the use of principles ofinference, if a correct theory of knowledge is to be obtained; for ourknowledge of them raises interesting and difficult questions.

　　In all our knowledge of general principles, what actually happens is thatfirst of all we realize some particular application of the principle, andthen we realize that the particularity is irrelevant, and that there is agenerality which may equally truly be affirmed. This is of course familiarin such matters as teaching arithmetic: 'two and two are four' is firstlearnt in the case of some particular pair of couples, and then in someother particular case, and so on, until at last it becomes possible to seethat it is true of any pair of couples. The same thing happens withlogical principles. Suppose two men are discussing what day of the monthit is. One of them says, 'At least you will admit that if yesterdaywas the 15th to-day must be the 16th.' 'Yes', says the other, 'I admitthat.' 'And you know', the first continues, 'that yesterday was the 15th,because you dined with Jones, and your diary will tell you that was on the15th.' 'Yes', says the second; 'therefore to-day is the 16th.'

　　Now such an argument is not hard to follow; and if it is granted that itspremisses are true in fact, no one will deny that the conclusion must alsobe true. But it depends for its truth upon an instance of a generallogical principle. The logical principle is as follows: 'Suppose it knownthat if this is true, then that is true. Suppose it also known thatthis is true, then it follows that that is true.' When it is thecase that if this is true, that is true, we shall say that this 'implies'that, and that that 'follows from' this. Thus our principle states that ifthis implies that, and this is true, then that is true. In other words,'anything implied by a true proposition is true', or 'whatever followsfrom a true proposition is true'.

　　This principle is really involved—at least, concrete instances of itare involved—in all demonstrations. Whenever one thing which webelieve is used to prove something else, which we consequently believe,this principle is relevant. If any one asks: 'Why should I accept theresults of valid arguments based on true premisses?' we can only answer byappealing to our principle. In fact, the truth of the principle isimpossible to doubt, and its obviousness is so great that at first sightit seems almost trivial. Such principles, however, are not trivial to thephilosopher, for they show that we may have indubitable knowledge which isin no way derived from objects of sense.

　　The above principle is merely one of a certain number of self-evidentlogical principles. Some at least of these principles must be grantedbefore any argument or proof becomes possible. When some of them have beengranted, others can be proved, though these others, so long as they aresimple, are just as obvious as the principles taken for granted. For novery good reason, three of these principles have been singled out bytradition under the name of 'Laws of Thought'.

　　They are as follows:

　　(1) The law of identity: 'Whatever is, is.'

　　(2) The law of contradiction: 'Nothing can both be and not be.'

　　(3) The law of excluded middle: 'Everything must either be or notbe.'

　　These three laws are samples of self-evident logical principles, but arenot really more fundamental or more self-evident than various othersimilar principles: for instance, the one we considered just now, whichstates that what follows from a true premiss is true. The name 'laws ofthought' is also misleading, for what is important is not the fact that wethink in accordance with these laws, but the fact that things behave inaccordance with them; in other words, the fact that when we think inaccordance with them we think truly. But this is a large question,to which we must return at a later stage.

　　In addition to the logical principles which enable us to prove from agiven premiss that something is certainly true, there are otherlogical principles which enable us to prove, from a given premiss, thatthere is a greater or less probability that something is true. An exampleof such principles—perhaps the most important example is theinductive principle, which we considered in the preceding chapter.

　　One of the great historic controversies in philosophy is the controversybetween the two schools called respectively 'empiricists' and'rationalists'. The empiricists—who are best represented by theBritish philosophers, Locke, Berkeley, and Hume—maintained that allour knowledge is derived from experience; the rationalists—who arerepresented by the Continental philosophers of the seventeenth century,especially Descartes and Leibniz—maintained that, in addition towhat we know by experience, there are certain 'innate ideas' and 'innateprinciples', which we know independently of experience. It has now becomepossible to decide with some confidence as to the truth or falsehood ofthese opposing schools. It must be admitted, for the reasons alreadystated, that logical principles are known to us, and cannot be themselvesproved by experience, since all proof presupposes them. In this,therefore, which was the most important point of the controversy, therationalists were in the right.

　　On the other hand, even that part of our knowledge which is logicallyindependent of experience (in the sense that experience cannot prove it)is yet elicited and caused by experience. It is on occasion of particularexperiences that we become aware of the general laws which theirconnexions exemplify. It would certainly be absurd to suppose that thereare innate principles in the sense that babies are born with a knowledgeof everything which men know and which cannot be deduced from what isexperienced. For this reason, the word 'innate' would not now be employedto describe our knowledge of logical principles. The phrase 'a priori'is less objectionable, and is more usual in modern writers. Thus, whileadmitting that all knowledge is elicited and caused by experience, weshall nevertheless hold that some knowledge is a priori, in thesense that the experience which makes us think of it does not suffice toprove it, but merely so directs our attention that we see its truthwithout requiring any proof from experience.

　　There is another point of great importance, in which the empiricists werein the right as against the rationalists. Nothing can be known to existexcept by the help of experience. That is to say, if we wish to prove thatsomething of which we have no direct experience exists, we must have amongour premisses the existence of one or more things of which we have directexperience. Our belief that the Emperor of China exists, for example,rests upon testimony, and testimony consists, in the last analysis, ofsense-data seen or heard in reading or being spoken to. Rationalistsbelieved that, from general consideration as to what must be, they coulddeduce the existence of this or that in the actual world. In this beliefthey seem to have been mistaken. All the knowledge that we can acquire apriori concerning existence seems to be hypothetical: it tells us thatif one thing exists, another must exist, or, more generally, that if oneproposition is true, another must be true. This is exemplified by theprinciples we have already dealt with, such as 'if this is true,and this implies that, then that is true', or 'if this and thathave been repeatedly found connected, they will probably be connected inthe next instance in which one of them is found'. Thus the scope and powerof a priori principles is strictly limited. All knowledge thatsomething exists must be in part dependent on experience. When anything isknown immediately, its existence is known by experience alone; whenanything is proved to exist, without being known immediately, bothexperience and a priori principles must be required in the proof.Knowledge is called empirical when it rests wholly or partly uponexperience. Thus all knowledge which asserts existence is empirical, andthe only a priori knowledge concerning existence is hypothetical,giving connexions among things that exist or may exist, but not givingactual existence.

　　A priori knowledge is not all of the logical kind we have beenhitherto considering. Perhaps the most important example of non-logical apriori knowledge is knowledge as to ethical value. I am not speakingof judgements as to what is useful or as to what is virtuous, for suchjudgements do require empirical premisses; I am speaking of judgements asto the intrinsic desirability of things. If something is useful, it mustbe useful because it secures some end; the end must, if we have gone farenough, be valuable on its own account, and not merely because it isuseful for some further end. Thus all judgements as to what is usefuldepend upon judgements as to what has value on its own account.

　　We judge, for example, that happiness is more desirable than misery,knowledge than ignorance, goodwill than hatred, and so on. Such judgementsmust, in part at least, be immediate and a priori. Like ourprevious a priori judgements, they may be elicited by experience,and indeed they must be; for it seems not possible to judge whetheranything is intrinsically valuable unless we have experienced something ofthe same kind. But it is fairly obvious that they cannot be proved byexperience; for the fact that a thing exists or does not exist cannotprove either that it is good that it should exist or that it is bad. Thepursuit of this subject belongs to ethics, where the impossibility ofdeducing what ought to be from what is has to be established. In thepresent connexion, it is only important to realize that knowledge as towhat is intrinsically of value is a priori in the same sense inwhich logic is a priori, namely in the sense that the truth of suchknowledge can be neither proved nor disproved by experience.

　　All pure mathematics is a priori, like logic. This was strenuouslydenied by the empirical philosophers, who maintained that experience wasas much the source of our knowledge of arithmetic as of our knowledge ofgeography. They maintained that by the repeated experience of seeing twothings and two other things, and finding that altogether they made fourthings, we were led by induction to the conclusion that two things and twoother things would always make four things altogether. If, however,this were the source of our knowledge that two and two are four, we shouldproceed differently, in persuading ourselves of its truth, from the way inwhich we do actually proceed. In fact, a certain number of instances areneeded to make us think of two abstractly, rather than of two coins or twobooks or two people, or two of any other specified kind. But as soon as weare able to divest our thoughts of irrelevant particularity, we becomeable to see the general principle that two and two are four; any oneinstance is seen to be typical, and the examination of otherinstances becomes unnecessary.(1)

　　(1) Cf. A. N. Whitehead, Introduction to Mathematics (HomeUniversity Library).

　　The same thing is exemplified in geometry. If we want to prove someproperty of all triangles, we draw some one triangle and reasonabout it; but we can avoid making use of any property which it does notshare with all other triangles, and thus, from our particular case, weobtain a general result. We do not, in fact, feel our certainty that twoand two are four increased by fresh instances, because, as soon as we haveseen the truth of this proposition, our certainty becomes so great as tobe incapable of growing greater. Moreover, we feel some quality ofnecessity about the proposition 'two and two are four', which is absentfrom even the best attested empirical generalizations. Suchgeneralizations always remain mere facts: we feel that there might be aworld in which they were false, though in the actual world they happen tobe true. In any possible world, on the contrary, we feel that two and twowould be four: this is not a mere fact, but a necessity to whicheverything actual and possible must conform.

　　The case may be made clearer by considering a genuinely-empiricalgeneralization, such as 'All men are mortal.' It is plain that we believethis proposition, in the first place, because there is no known instanceof men living beyond a certain age, and in the second place because thereseem to be physiological grounds for thinking that an organism such as aman's body must sooner or later wear out. Neglecting the second ground,and considering merely our experience of men's mortality, it is plain thatwe should not be content with one quite clearly understood instance of aman dying, whereas, in the case of 'two and two are four', one instancedoes suffice, when carefully considered, to persuade us that the same musthappen in any other instance. Also we can be forced to admit, onreflection, that there may be some doubt, however slight, as to whether allmen are mortal. This may be made plain by the attempt to imagine twodifferent worlds, in one of which there are men who are not mortal, whilein the other two and two make five. When Swift invites us to consider therace of Struldbugs who never die, we are able to acquiesce in imagination.But a world where two and two make five seems quite on a different level.We feel that such a world, if there were one, would upset the whole fabricof our knowledge and reduce us to utter doubt.

　　The fact is that, in simple mathematical judgements such as 'two and twoare four', and also in many judgements of logic, we can know the generalproposition without inferring it from instances, although some instance isusually necessary to make clear to us what the general proposition means.This is why there is real utility in the process of deduction,which goes from the general to the general, or from the general to theparticular, as well as in the process of induction, which goes fromthe particular to the particular, or from the particular to the general.It is an old debate among philosophers whether deduction ever gives newknowledge. We can now see that in certain cases, at least, it does do so.If we already know that two and two always make four, and we know thatBrown and Jones are two, and so are Robinson and Smith, we can deduce thatBrown and Jones and Robinson and Smith are four. This is new knowledge,not contained in our premisses, because the general proposition, 'two andtwo are four', never told us there were such people as Brown and Jones andRobinson and Smith, and the particular premisses do not tell us that therewere four of them, whereas the particular proposition deduced does tell usboth these things.

　　But the newness of the knowledge is much less certain if we take the stockinstance of deduction that is always given in books on logic, namely, 'Allmen are mortal; Socrates is a man, therefore Socrates is mortal.' In thiscase, what we really know beyond reasonable doubt is that certain men, A,B, C, were mortal, since, in fact, they have died. If Socrates is one ofthese men, it is foolish to go the roundabout way through 'all men aremortal' to arrive at the conclusion that probably Socrates ismortal. If Socrates is not one of the men on whom our induction is based,we shall still do better to argue straight from our A, B, C, to Socrates,than to go round by the general proposition, 'all men are mortal'. For theprobability that Socrates is mortal is greater, on our data, than theprobability that all men are mortal. (This is obvious, because if all menare mortal, so is Socrates; but if Socrates is mortal, it does not followthat all men are mortal.) Hence we shall reach the conclusion thatSocrates is mortal with a greater approach to certainty if we make ourargument purely inductive than if we go by way of 'all men are mortal' andthen use deduction.

　　This illustrates the difference between general propositions known apriori such as 'two and two are four', and empirical generalizationssuch as 'all men are mortal'. In regard to the former, deduction is theright mode of argument, whereas in regard to the latter, induction isalways theoretically preferable, and warrants a greater confidence in thetruth of our conclusion, because all empirical generalizations are moreuncertain than the instances of them.

　　We have now seen that there are propositions known a priori, andthat among them are the propositions of logic and pure mathematics, aswell as the fundamental propositions of ethics. The question which mustnext occupy us is this: How is it possible that there should be suchknowledge? And more particularly, how can there be knowledge of generalpropositions in cases where we have not examined all the instances, andindeed never can examine them all, because their number is infinite? Thesequestions, which were first brought prominently forward by the Germanphilosopher Kant (1724-1804), are very difficult, and historically veryimportant.