Our Knowledge of Universals

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

For any person's knowledge at a given time, universals, like particulars, may be divided into three groups. These are universals known by acquaintance, universals known only by description, and universals not known in either way.

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

Let us first consider knowledge of universals through acquaintance. To begin with, we are obviously acquainted with universals such as white, red, black, sweet, sour, loud, and hard. These are qualities found in sense-data. When we see a white patch, we are first acquainted with that particular patch. But by seeing many white patches, we easily learn to identify the whiteness they all share, apart from the patches themselves. In learning to do so, we become acquainted with whiteness. A similar process makes us acquainted with any other universal of the same kind. Universals of this kind may be called 'sensible qualities'. Understanding sensible qualities requires less effort of abstraction than understanding any other universals. Sensible qualities also seem less distant from particular things than other universals do.

We come next to relations. The easiest relations to apprehend are those which 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 am writing; thus the whole page is included in one sense-datum. But I perceive that some parts of the page are to the left of other parts, and some parts are above other parts. The process of abstraction in this case seems to proceed somewhat as follows: I see successively a number of sense-data in which one part is to the left of another; I perceive, as in the case of different white patches, that all these sense-data have something in common, and by abstraction I find that what they have in common is a certain relation between their parts, namely the relation which I call 'being to the left of'. In this way I become acquainted with the universal relation.

Next we turn to relations. The easiest relations to grasp are those between different parts of a single complex sense-datum. For example, I can see the whole page I am writing on at a glance. So the whole page is included in one sense-datum. But I see that some parts of the page are to the left of other parts. Some parts are above other parts. In this case, the process of abstraction seems to work something like this: I see, one after another, several sense-data in which one part is to the left of another. As with different white patches, I recognize that all these sense-data have something in common. Through abstraction, I find that what they share is a particular relation between their parts. This is the relation I call 'being to the left of'. In this way, I become acquainted with the 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, I can retain the whole chime before my mind, and I can perceive that the earlier bells came before the later ones. Also in memory I perceive that what I am remembering came before the present time. From either of these sources I can abstract the universal relation of before and after, just as I abstracted the universal relation 'being to the left of'. Thus time-relations, like space-relations, are among those with which we are acquainted.

In the same way, I become aware of the relation of before and after in time. Suppose I hear a chime of bells. When the last bell sounds, I can hold the whole chime in my mind. I can recognize that the earlier bells sounded before the later ones. In memory, too, I recognize that what I remember happened before the present moment. From either experience, I can draw out the universal relation of before and after. This is just as I drew out the universal relation 'being to the left of'. So we are directly acquainted with relations in time, just as we are with relations in space.

Another relation with which we become acquainted in much the same way is resemblance. If I see simultaneously two shades of green, I can see that they 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 than either has to the red. In this way I become acquainted with the universal resemblance or similarity.

Resemblance is another relation we come to know in much the same way. If I see two shades of green at the same time, I can see that they resemble each other. If I also see a shade of red at the same time, I can compare the three shades. I can see that the two greens resemble each other more than either resembles the red. In this way, I come to know the universal resemblance or similarity.

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

We may be directly aware of relations between universals, just as we may be aware of relations between particulars. We have just seen that we can perceive a greater resemblance between two shades of green than between red and green. Here, 'greater than' is a relation between two relations. Knowing such relations requires more abstract thinking than perceiving the qualities of sense-data. Even so, our knowledge of these relations seems just as direct and (at least in some cases) just as certain. So we have direct knowledge of universals as well as of sense-data.

Returning now to the problem of a priori knowledge, which we left unsolved when we began the consideration of universals, we find ourselves in a position to deal with it in a much more satisfactory manner than was possible 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 proposition states 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 the relations of universals. This proposition is of great importance, and goes a long way towards solving our previous difficulties concerning a priori knowledge.

We now return to the problem of a priori knowledge, left unsolved when we began discussing universals. We can now address this problem much more satisfactorily than before. Let us return to the statement 'two and two are four'. In light of what we have said, this statement fairly obviously expresses a relation between the universals 'two' and 'four'. This relation suggests a claim that we will now try to establish: All a priori knowledge deals exclusively with the relations of universals. This claim is very important. It goes a long way toward resolving our earlier difficulties with a priori knowledge.

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

Only one case might at first seem to contradict our claim. An a priori proposition may state that all members of one class of particular things belong to another class. Or, equivalently, it may state that all particular things with one property also have another. In this case, we might seem to be discussing the particular things with that property rather than the property itself. The proposition 'two and two are four' is an example of this case. We can express it as 'any two and any other two are four'. Or we can say 'any collection formed of two twos is a collection of four'. If we can show that statements like these really concern only universals, we may regard our claim as proved.

One way of discovering what a proposition deals with is to ask ourselves what words we must understand—in other words, what objects we must be acquainted with—in order to see what the proposition means. As soon as we see what the proposition means, even if we do not yet know whether it is true or false, it is evident that we must have acquaintance with whatever is really dealt with by the proposition. By applying this test, it appears that many propositions which might seem to be concerned with particulars are really concerned only with universals. In the special case of 'two and two are four', even when we interpret it as meaning 'any collection formed of two twos is a collection of four', it is plain that we can understand the proposition, i.e. we can see what it is that it asserts, 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: if it were necessary, obviously we could never understand the proposition, since the couples are infinitely numerous and therefore cannot all be known to us. Thus although our general statement implies statements about particular couples, as soon as we know that there are such particular couples, yet it does not itself assert or imply that there are such particular couples, and thus fails to make any statement whatever about any actual particular couple. The statement made is about 'couple', the universal, and not about this or that couple.

One way to discover what a proposition is about is to ask what words we must understand to grasp its meaning. In other words, what objects must we be acquainted with? Once we understand the proposition, we must clearly be acquainted with whatever it is really about. We must have this acquaintance even if we do not yet know whether the proposition is true or false. Applying this test shows that many propositions that seem to concern particulars really concern only universals. Take 'two and two are four'. We can interpret this as meaning 'any collection formed of two twos is a collection of four'. Even then, we can understand what the proposition asserts once we know what 'collection', 'two', and 'four' mean. We do not need to know all the pairs in the world. If such knowledge were necessary, we could obviously never understand the proposition. There are infinitely many pairs, so we cannot know them all. Thus, our general statement implies statements about particular pairs, as soon as we know that there are such particular pairs. Yet the general statement does not itself assert or imply that such particular pairs exist. So it says nothing at all about any actual particular pair. The statement is about 'pair', the universal, not about this or that pair.

Thus the statement 'two and two are four' deals exclusively with universals, and therefore may be known by anybody who is acquainted with the universals concerned and can perceive the relation between them which the statement asserts. It must be taken as a fact, discovered by reflecting upon our knowledge, that we have the power of sometimes perceiving such relations between universals, and therefore of sometimes knowing general a priori propositions such as those of arithmetic and logic. The thing that seemed mysterious, when we formerly considered such knowledge, was that it seemed to anticipate and control experience. This, however, we can now see to have been an error. No fact concerning anything capable of being experienced can be known independently of experience. We know a priori that two things and two other things together make four things, but we do not know a priori that if Brown and Jones are two, and Robinson and Smith are two, then Brown and Jones and Robinson and Smith are four. The reason is that this proposition cannot be understood at all unless we know that there are such people as Brown and Jones and Robinson and Smith, and this we can only know by experience. Hence, although our general proposition is a priori, all its applications to actual particulars involve experience and therefore contain an empirical element. In this way what seemed mysterious in our a priori knowledge is seen to have been based upon an error.

Thus the statement 'two and two are four' deals only with universals. Anyone familiar with those universals may therefore know the statement to be true, if they can see the relationship it asserts. Reflection on our knowledge reveals a fact we must accept: we can sometimes perceive such relationships between universals. We can therefore sometimes know general a priori propositions, such as those of arithmetic and logic. When we considered this knowledge earlier, it seemed mysterious because it appeared to anticipate and control experience. But we can now see that this appearance was misleading. No fact about anything we can experience can be known independently of experience. We know a priori that two things and two other things together make four things. But we do not know a priori that if Brown and Jones are two, and Robinson and Smith are two, then Brown and Jones and Robinson and Smith are four. The reason is that we cannot understand this proposition at all unless we know that these people exist. And we can know that Brown, Jones, Robinson, and Smith exist only through experience. So, although our general proposition is a priori, all its applications to particular things involve experience. Every such application therefore contains an empirical element. We can thus see that the apparent mystery in our a priori knowledge rested on an error.

It will serve to make the point clearer if we contrast our genuine a priori judgement with an empirical generalization, such as 'all men are mortals'. Here as before, we can understand what the proposition means as soon as we understand the universals involved, namely man and mortal. It is obviously unnecessary to have an individual acquaintance with the whole human race in order to understand what our proposition means. Thus the difference between an a priori general proposition and an empirical generalization does not come in the meaning of the proposition; it comes in the nature of the evidence for it. In the empirical case, the evidence consists in the particular instances. We believe that all men are mortal because we know that there are innumerable instances of men dying, and no instances of their living beyond a certain age. We do not believe it because we see a connexion between the universal man and the universal mortal. It is true that if physiology can prove, assuming the general laws that govern living bodies, that no living organism can last for ever, that gives a connexion between man and mortality which would enable us to assert our proposition without appealing to the special evidence of men dying. But that only means that our generalization has been subsumed under a wider generalization, for which the evidence is still of the same kind, though more extensive. The progress of science is constantly producing such subsumptions, and therefore giving a constantly wider inductive basis for scientific generalizations. But although this gives a greater degree of certainty, it does not give a different kind: the ultimate ground remains inductive, i.e. derived from instances, and not an a priori connexion of universals such as we have in logic and arithmetic.

To make the point clearer, we can contrast our genuine a priori judgment with an empirical generalization, such as 'all men are mortals'. Here, as before, we can understand what the statement means once we understand the universals involved, namely man and mortal. We obviously do not need to know every individual in the human race to understand what our statement means. So the difference between an a priori general statement and an empirical generalization is not in the statement’s meaning. The difference lies in the nature of the evidence supporting it. In the empirical case, the evidence consists of individual examples. We believe that all men are mortal because we know of countless examples of men dying. We also know of no examples of men living beyond a certain age. We do not believe the statement because we see a connection between the universal man and the universal mortal. Physiology might prove that no living organism can last forever, assuming the general laws governing living bodies. If so, that proof does establish a connection between man and mortality. This connection would let us make our statement without relying on the specific evidence of men dying. But this only means that our generalization now falls under a broader generalization. The evidence for the broader generalization is still of the same kind, though more extensive. Scientific progress constantly brings generalizations under broader ones, giving scientific generalizations an ever-wider basis in evidence from individual examples. But although this broader basis gives a greater degree of certainty, it does not give a different kind. The ultimate basis remains inductive, that is, drawn from individual examples. It is not an a priori connection between universals, such as we find in logic and arithmetic.

Two opposite points are to be observed concerning a priori general propositions. The first is that, if many particular instances are known, our general proposition may be arrived at in the first instance by induction, and the connexion of universals may be only subsequently perceived. For example, it is known that if we draw perpendiculars to the sides of a triangle from the opposite angles, all three perpendiculars meet in a point. It would be quite possible to be first led to this proposition by actually drawing perpendiculars in many cases, and finding that they always met in a point; this experience might lead us to look for the general proof and find it. Such cases are common in the experience of every mathematician.

Two contrasting points should be noted about a priori general propositions. The first is that, if many particular examples are known, we may initially reach a general proposition through induction. Only later may we recognize the connection between universals. For example, we know that perpendicular lines drawn to a triangle’s sides from the opposite corners all meet at one point. We could quite possibly first discover this proposition by drawing perpendiculars in many triangles and finding they always meet at one point. This experience might lead us to look for a general proof and find it. Such cases are common in every mathematician’s experience.

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

The other point is more interesting and more important philosophically. We may sometimes know a general statement is true without knowing a single example of it. Consider the following case: We know that any two numbers can be multiplied together to give a third number, called their product. We know that all pairs of integers whose product is less than 100 have actually been multiplied together. The results have been recorded in the multiplication table. But we also know that there are infinitely many integers. And human beings have only ever thought of, or will ever think of, a finite number of pairs of integers. So there are pairs of integers that no human being has ever thought of or will ever think of. All these pairs have products greater than 100. So we reach this statement: 'All products of two integers that no human being has ever thought of or will ever think of are greater than 100.' This general statement is undeniably true, yet we can never give an example, because of the very nature of the case. Any two numbers we might think of are excluded by the statement's own terms.

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

People often deny that we can know general propositions without being able to give a single instance. They fail to see that knowing such propositions requires only knowledge of the relations between universals. It does not require knowledge of any instances of those universals. Yet knowledge of such general propositions is essential to much of what is generally accepted as known. For example, we saw in the early chapters that knowledge of physical objects, unlike knowledge of sense-data, comes only through inference. Physical objects are not things we know directly. So we can never know any proposition of the form 'this is a physical object', where 'this' is something immediately known. It follows that all our knowledge of physical objects is knowledge for which we cannot give an actual instance. We can give instances of the associated sense-data, but we cannot give instances of the physical objects themselves. So all our knowledge of physical objects depends on the possibility of general knowledge without any instance we can give. The same applies to our knowledge of other people's minds. It also applies to any other class of things of which we know no instance directly.

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

We can now review the sources of our knowledge as they have emerged during our analysis. First, we must distinguish knowledge of things from knowledge of truths. Each has two kinds: immediate and derived. Our immediate knowledge of things, which we called acquaintance, has two forms. These depend on whether the things known are particulars or universals. Among particulars, we are acquainted with sense-data and (probably) with ourselves. Among universals, there seems to be no principle for deciding which universals we can know by acquaintance. But clearly, we can know sensible qualities, relations of space and time, similarity, and certain abstract logical universals this way. Our derived knowledge of things, which we call knowledge by description, always involves both acquaintance with something and knowledge of truths. Our immediate knowledge of truths may be called intuitive knowledge. The truths known this way may be called self-evident truths. These include truths that simply state what our senses present to us, and certain abstract logical and arithmetical principles. They also include some ethical propositions, though with less certainty. Our derived knowledge of truths consists of everything we can deduce from self-evident truths using self-evident principles of deduction.

If the above account is correct, all our knowledge of truths depends upon our intuitive knowledge. It therefore becomes important to consider the nature and scope of intuitive knowledge, in much the same way as, at an earlier stage, we considered the nature and scope of knowledge by acquaintance. But knowledge of truths raises a further problem, which does not arise in regard to knowledge of things, namely the problem of error. Some of our beliefs turn out to be erroneous, and therefore it becomes necessary to consider how, if at all, we can distinguish knowledge from error. This problem does not arise with regard to knowledge by acquaintance, for, whatever may be the object of acquaintance, even in dreams and hallucinations, there is no error involved so long as we do not go beyond the immediate object: error can only arise when we regard the immediate object, i.e. the sense-datum, as the mark of some physical object. Thus the problems connected with knowledge of truths are more difficult than those connected with knowledge of things. As the first of the problems connected with knowledge of truths, let us examine the nature and scope of our intuitive judgements.

If the account above is correct, all our knowledge of truths depends on our intuitive knowledge. So we need to consider the nature and scope of intuitive knowledge. Earlier, we considered the nature and scope of knowledge by acquaintance in much the same way. But knowledge of truths raises another problem, one that does not arise with knowledge of things: the problem of error. Some of our beliefs turn out to be wrong. So we need to consider how, if at all, we can distinguish knowledge from error. This problem does not arise with knowledge by acquaintance, whatever the object of acquaintance may be. Even in dreams and hallucinations, no error is involved as long as we stay with the immediate object. Error can arise only when we treat the immediate object, or sense-datum, as a sign of some physical object. Thus, the problems surrounding knowledge of truths are more difficult than those surrounding knowledge of things. As the first problem concerning knowledge of truths, let us examine the nature and scope of our intuitive judgments.