Chapter X. On Our Knowledge of Universals

by Bertrand Russell

　　In regard to one man's knowledge at a given time, universals, likeparticulars, may be divided into those known by acquaintance, those knownonly by description, and those not known either by acquaintance or bydescription.

　　Let us consider first the knowledge of universals by acquaintance. It isobvious, to begin with, that we are acquainted with such universals aswhite, red, black, sweet, sour, loud, hard, etc., i.e. with qualitieswhich are exemplified in sense-data. When we see a white patch, we areacquainted, in the first instance, with the particular patch; but byseeing many white patches, we easily learn to abstract the whiteness whichthey all have in common, and in learning to do this we are learning to beacquainted with whiteness. A similar process will make us acquainted withany other universal of the same sort. Universals of this sort may becalled 'sensible qualities'. They can be apprehended with less effort ofabstraction than any others, and they seem less removed from particularsthan other universals are.

　　We come next to relations. The easiest relations to apprehend are thosewhich hold between the different parts of a single complex sense-datum.For example, I can see at a glance the whole of the page on which I amwriting; thus the whole page is included in one sense-datum. But Iperceive that some parts of the page are to the left of other parts, andsome parts are above other parts. The process of abstraction in this caseseems to proceed somewhat as follows: I see successively a number ofsense-data in which one part is to the left of another; I perceive, as inthe case of different white patches, that all these sense-data havesomething in common, and by abstraction I find that what they have incommon is a certain relation between their parts, namely the relationwhich I call 'being to the left of'. In this way I become acquainted withthe universal relation.

　　In like manner I become aware of the relation of before and after in time.Suppose I hear a chime of bells: when the last bell of the chime sounds, Ican retain the whole chime before my mind, and I can perceive that theearlier bells came before the later ones. Also in memory I perceive thatwhat I am remembering came before the present time. From either of thesesources I can abstract the universal relation of before and after, just asI abstracted the universal relation 'being to the left of'. Thustime-relations, like space-relations, are among those with which we areacquainted.

　　Another relation with which we become acquainted in much the same way isresemblance. If I see simultaneously two shades of green, I can see thatthey resemble each other; if I also see a shade of red: at the same time,I can see that the two greens have more resemblance to each other thaneither has to the red. In this way I become acquainted with the universalresemblance or similarity.

　　Between universals, as between particulars, there are relations of whichwe may be immediately aware. We have just seen that we can perceive thatthe resemblance between two shades of green is greater than theresemblance between a shade of red and a shade of green. Here we aredealing with a relation, namely 'greater than', between two relations. Ourknowledge of such relations, though it requires more power of abstractionthan is required for perceiving the qualities of sense-data, appears to beequally immediate, and (at least in some cases) equally indubitable. Thusthere is immediate knowledge concerning universals as well as concerningsense-data.

　　Returning now to the problem of a priori knowledge, which we leftunsolved when we began the consideration of universals, we find ourselvesin a position to deal with it in a much more satisfactory manner than waspossible before. Let us revert to the proposition 'two and two are four'.It is fairly obvious, in view of what has been said, that this propositionstates a relation between the universal 'two' and the universal 'four'.This suggests a proposition which we shall now endeavour to establish:namely, All a priori knowledge deals exclusively with therelations of universals. This proposition is of great importance, andgoes a long way towards solving our previous difficulties concerning apriori knowledge.

　　The only case in which it might seem, at first sight, as if ourproposition were untrue, is the case in which an a prioriproposition states that all of one class of particulars belong tosome other class, or (what comes to the same thing) that allparticulars having some one property also have some other. In this case itmight seem as though we were dealing with the particulars that have theproperty rather than with the property. The proposition 'two and two arefour' is really a case in point, for this may be stated in the form 'anytwo and any other two are four', or 'any collection formed of two twos isa collection of four'. If we can show that such statements as this reallydeal only with universals, our proposition may be regarded as proved.

　　One way of discovering what a proposition deals with is to ask ourselveswhat words we must understand—in other words, what objects we mustbe acquainted with—in order to see what the proposition means. Assoon as we see what the proposition means, even if we do not yet knowwhether it is true or false, it is evident that we must have acquaintancewith whatever is really dealt with by the proposition. By applying thistest, it appears that many propositions which might seem to be concernedwith particulars are really concerned only with universals. In the specialcase of 'two and two are four', even when we interpret it as meaning 'anycollection formed of two twos is a collection of four', it is plain thatwe can understand the proposition, i.e. we can see what it is that itasserts, as soon as we know what is meant by 'collection' and 'two' and'four'. It is quite unnecessary to know all the couples in the world: ifit were necessary, obviously we could never understand the proposition,since the couples are infinitely numerous and therefore cannot all beknown to us. Thus although our general statement implies statementsabout particular couples, as soon as we know that there are suchparticular couples, yet it does not itself assert or imply that thereare such particular couples, and thus fails to make any statement whateverabout any actual particular couple. The statement made is about 'couple',the universal, and not about this or that couple.

　　Thus the statement 'two and two are four' deals exclusively withuniversals, and therefore may be known by anybody who is acquainted withthe universals concerned and can perceive the relation between them whichthe statement asserts. It must be taken as a fact, discovered byreflecting upon our knowledge, that we have the power of sometimesperceiving such relations between universals, and therefore of sometimesknowing general a priori propositions such as those of arithmeticand logic. The thing that seemed mysterious, when we formerly consideredsuch knowledge, was that it seemed to anticipate and control experience.This, however, we can now see to have been an error. No factconcerning anything capable of being experienced can be knownindependently of experience. We know a priori that two things andtwo other things together make four things, but we do not know apriori that if Brown and Jones are two, and Robinson and Smith aretwo, then Brown and Jones and Robinson and Smith are four. The reason isthat this proposition cannot be understood at all unless we know thatthere are such people as Brown and Jones and Robinson and Smith, and thiswe can only know by experience. Hence, although our general proposition isa priori, all its applications to actual particulars involveexperience and therefore contain an empirical element. In this way whatseemed mysterious in our a priori knowledge is seen to have beenbased upon an error.

　　It will serve to make the point clearer if we contrast our genuine apriori judgement with an empirical generalization, such as 'all menare mortals'. Here as before, we can understand what theproposition means as soon as we understand the universals involved, namelyman and mortal. It is obviously unnecessary to have anindividual acquaintance with the whole human race in order to understandwhat our proposition means. Thus the difference between an a priorigeneral proposition and an empirical generalization does not come in themeaning of the proposition; it comes in the nature of the evidencefor it. In the empirical case, the evidence consists in the particularinstances. We believe that all men are mortal because we know that thereare innumerable instances of men dying, and no instances of their livingbeyond a certain age. We do not believe it because we see a connexionbetween the universal man and the universal mortal. It istrue that if physiology can prove, assuming the general laws that governliving bodies, that no living organism can last for ever, that gives aconnexion between man and mortality which would enable us toassert our proposition without appealing to the special evidence of mendying. But that only means that our generalization has been subsumed undera wider generalization, for which the evidence is still of the same kind,though more extensive. The progress of science is constantly producingsuch subsumptions, and therefore giving a constantly wider inductive basisfor scientific generalizations. But although this gives a greater degreeof certainty, it does not give a different kind: the ultimateground remains inductive, i.e. derived from instances, and not an apriori connexion of universals such as we have in logic andarithmetic.

　　Two opposite points are to be observed concerning a priori generalpropositions. The first is that, if many particular instances are known,our general proposition may be arrived at in the first instance byinduction, and the connexion of universals may be only subsequentlyperceived. For example, it is known that if we draw perpendiculars to thesides of a triangle from the opposite angles, all three perpendicularsmeet in a point. It would be quite possible to be first led to thisproposition by actually drawing perpendiculars in many cases, and findingthat they always met in a point; this experience might lead us to look forthe general proof and find it. Such cases are common in the experience ofevery mathematician.

　　The other point is more interesting, and of more philosophical importance.It is, that we may sometimes know a general proposition in cases where wedo not know a single instance of it. Take such a case as the following: Weknow that any two numbers can be multiplied together, and will give athird called their product. We know that all pairs of integers theproduct of which is less than 100 have been actually multiplied together,and the value of the product recorded in the multiplication table. But wealso know that the number of integers is infinite, and that only a finitenumber of pairs of integers ever have been or ever will be thought of byhuman beings. Hence it follows that there are pairs of integers whichnever have been and never will be thought of by human beings, and that allof them deal with integers the product of which is over 100. Hence wearrive at the proposition: 'All products of two integers, which never havebeen and never will be thought of by any human being, are over 100.' Hereis a general proposition of which the truth is undeniable, and yet, fromthe very nature of the case, we can never give an instance; because anytwo numbers we may think of are excluded by the terms of the proposition.

　　This possibility, of knowledge of general propositions of which noinstance can be given, is often denied, because it is not perceived thatthe knowledge of such propositions only requires a knowledge of therelations of universals, and does not require any knowledge of instancesof the universals in question. Yet the knowledge of such generalpropositions is quite vital to a great deal of what is generally admittedto be known. For example, we saw, in our early chapters, that knowledge ofphysical objects, as opposed to sense-data, is only obtained by aninference, and that they are not things with which we are acquainted.Hence we can never know any proposition of the form 'this is a physicalobject', where 'this' is something immediately known. It follows that allour knowledge concerning physical objects is such that no actual instancecan be given. We can give instances of the associated sense-data, but wecannot give instances of the actual physical objects. Hence our knowledgeas to physical objects depends throughout upon this possibility of generalknowledge where no instance can be given. And the same applies to ourknowledge of other people's minds, or of any other class of things ofwhich no instance is known to us by acquaintance.

　　We may now take a survey of the sources of our knowledge, as they haveappeared in the course of our analysis. We have first to distinguishknowledge of things and knowledge of truths. In each there are two kinds,one immediate and one derivative. Our immediate knowledge of things, whichwe called acquaintance, consists of two sorts, according as thethings known are particulars or universals. Among particulars, we haveacquaintance with sense-data and (probably) with ourselves. Amonguniversals, there seems to be no principle by which we can decide whichcan be known by acquaintance, but it is clear that among those that can beso known are sensible qualities, relations of space and time, similarity,and certain abstract logical universals. Our derivative knowledge ofthings, which we call knowledge by description, always involvesboth acquaintance with something and knowledge of truths. Our immediateknowledge of truths may be called intuitive knowledge, andthe truths so known may be called self-evident truths. Among suchtruths are included those which merely state what is given in sense, andalso certain abstract logical and arithmetical principles, and (thoughwith less certainty) some ethical propositions. Our derivativeknowledge of truths consists of everything that we can deduce fromself-evident truths by the use of self-evident principles of deduction.

　　If the above account is correct, all our knowledge of truths depends uponour intuitive knowledge. It therefore becomes important to consider thenature and scope of intuitive knowledge, in much the same way as, at anearlier stage, we considered the nature and scope of knowledge byacquaintance. But knowledge of truths raises a further problem, which doesnot arise in regard to knowledge of things, namely the problem of error.Some of our beliefs turn out to be erroneous, and therefore it becomesnecessary to consider how, if at all, we can distinguish knowledge fromerror. This problem does not arise with regard to knowledge byacquaintance, for, whatever may be the object of acquaintance, even indreams and hallucinations, there is no error involved so long as we do notgo beyond the immediate object: error can only arise when we regard theimmediate object, i.e. the sense-datum, as the mark of some physicalobject. Thus the problems connected with knowledge of truths are moredifficult than those connected with knowledge of things. As the first ofthe problems connected with knowledge of truths, let us examine the natureand scope of our intuitive judgements.