iii. The Problem of Logic

(a) Contingency and Necessity

(b) What is Logical Necessity?

(c) Logical Positivism and Necessity

(d) Criticism of Logical Positivism

(a) Contingency and Necessity — In both the preceding examples the mind was confronted with certain "brute facts," in the one case, unreachable fruit, and in the other, recalcitrant "data" of astronomical observation. It also saw certain connections between these facts and others. Reasoning is always "about" something given, something other than the actual operation of reasoning. It works on "data" which, so far as this particular act of reasoning is concerned, are simply accepted, not proved. And though some- times its data may themselves be partly products of past reasonings, those past reasonings themselves must have operated on merely given and unprovable facts. In the last analysis reason deals with data that are simply "given," and are not susceptible of proof. All the immediate data of sense-experience (and therefore the whole superstructure of theory that natural science builds thereon) are of this unprovable kind. We can see no logical necessity in the events of the external world. They just happen. In technical language, they are "contingent," not "necessary." It is true, of course, that they happen in a more or less systematic manner, and that we expect them to continue doing so, and that, on the assumption that they will continue to happen as before, we can construct very complicated formulae by means of which we can predict how in detail they will "probably" happen. But we can see no necessity that they should do so. Stones might all leap from the ground to-morrow. Heated water might freeze. Pigs might sprout wings. If these things happened we should not, if we were wise, simply adopt the attitude of the man who said of the ostrich, "There's no such bird." We should laboriously begin to collect the data for a whole new natural science.

Contingent facts, then, are simply given, and must be accepted, after due scrutiny to determine precisely what is given, and what its actual relation is to other given facts. Logical necessity itself is also in a sense simply given, and must be accepted after due scrutiny; but what is given in the case of logical necessity is of a different order, and it is given in a different manner. What we grasp when we seize upon a logical connection is always some fact of the type, "If A is true, then B must be true also." Thus "if the law of gravity is true, then this stone, if I let it go, will fall." Or again, " If a definable law of 'anti-gravity' were true, then so-and-so,would happen." Or again, "Given certain fundamental arithmetical postulates and axioms, then 7 x 4² = 7 x 2 x 8." Or "Given the postulates and axioms of Euclidean geometry, then the internal angles of a plane triangle are equal to two right angles."

Neither of these last two truths is self-evident to average human intelligence, but in each case the premise can be shown to involve the conclusion by means of a process of reasoning. The steps of this process consist of intuitive advances from one "self-evident" truth to another. This principle of implication by linked self-evidences, or logical necessity, is essential to all reasoning.

There is a perennial dispute between the champions of intellect and the champions of intuition. Let us never forget that intellect itself is intuitive through and through. Not only does it work upon data which must be intuitively apprehended; its actual operations also are intuitive. Each apprehension of self-evidence is a flash of logical intuition. Let us now consider in more detail the nature of self-evident logical necessity.

(b) What is Logical Necessity? — Logic is generally regarded as the science of true thinking. Is logical necessity, the essential "therefore" of our thinking, simply a necessity in our thinking itself; or is it a necessity in the things about which we think?

There are serious difficulties in the theory that when we experience logical necessity or self-evidence we are simply observing a necessary connection between objective facts. If this is the case, how is it that people sometimes disagree about self-evidence? Is it possible to have an illusory logical necessity? If so, how are we to distinguish between true and false logical necessity? All we can do is to check each fresh bit of seeming necessity by reference to other parts of the system of rational thought. Does it fit into the system or not? If not, we must look very carefully at it again to see if it still seems a logical necessity; and at the system, to see if there is any way by which it can accommodate the awkward intuition. And if the intuition does still seem self-evident, and the system still recalcitrant to it, we must choose between .the isolated intuition and the system. And since the total system of our thought is overwhelmingly better established, we shall provisionally reject the isolated experience.

One thing we need not do. We need not roundly deny the validity of logical thinking in those spheres in which it does prove effective.

Nor need we suppose that because we cannot prove the validity of the principle of logical thinking, therefore it is unsound. In a hot bath I feel warm. I cannot prove that warmth is happening. It just happens. Proof is not needed. Similarly with logical thinking. In principle its validity needs no proof. We cannot logically use reason itself to prove reason's, validity; nor to disprove it. For the principle itself would have to be used to construct the very argument that seeks to defend or destroy it. Only in particular instances, when our logical intuitions seem to conflict with one another, need we doubt and seek proof; and then, of the particular instance, not of reason in general.

Logic certainly is the science of true thinking. It does in some sense study a necessity in our thinking. But this is not the whole story. When we use reason upon the external world (as the chimpanzee and Einstein did) it very often proves effective. Whether the world is systematic through and through or not, it certainly contains a good deal of system. Things do with great accuracy behave in a regular and logical manner. It does seem as though there were, for instance, some kind of necessary connection between the falling of stones and the mass of the earth and the movements of planets and stars. In fact, it seems, at least on the level of common sense, reasonable to hold that logical necessity does actually in some sense hold good not merely of thoughts but of things. We have no obvious reason to deny it, and some reason to believe it, since action based on the belief is often successful.

Further, it seems at least plausible that the mental disposition toward logical thought should have been evoked in us through the impact of a world whose structure was itself logical; a world in which a thing cannot-both be and not be, and in which two and two must make four, not five.

(c) Logical Positivism and Necessity — The tentative account of logical necessity which I gave in the preceding section was in principle Realist. We must now consider the radical criticism brought against this kind of theory by the Logical Positivists. According to them the mysterious thing, "logical necessity," is not a characteristic of the objective world but is simply the consequence of the definitions which we ourselves formulate to describe the various subjects of our thinking. Thus 2 + 2 = 4 just because we have so defined the symbol 2 + 2 and the symbol 4 that they have identical meaning.

The Logical Positivist begins by distinguishing in an orthodox manner between two kinds of statements, or propositions, namely, those which are statements of fact, and are not logically necessary, and those which are purely logical, and necessary. The former are generally called empirical propositions, the latter "a priori"propositions. Examples of "empirical" propositions are: "Water flows down hill," and "Your behaviour annoys me." Examples of a priori propositions are: "Twice two is four," and "A man's father is his son's grandfather." Empirical propositions are statements of observed fact. They are not expressions of a necessary connection between the subject and predicate. A priori propositions, on the other hand, though they may be indirectly based on observation of fact, are statements of logical implication. In them the predicate merely analyses out certain logical implications of the definition of the subject. They are therefore said to be "analytic" propositions; whereas empirical propositions are said to be "synthetic," because in them the predicate, so to speak, puts things together, adds new facts to the subject.

Many propositions are seemingly empirical, but really a priori. From their ambiguity arises a danger that the necessity in them, which is in fact merely a logical consequence of a definition, may seem to be a necessity somehow belonging to the objective world. The proposition "All men are mortal" may be interpreted either synthetically or analytically. If the proposition means simply to state the result of prolonged observation of the fate of members of the human race, it is empirical and synthetic. There is no necessary in it. But if we mean by the word " man " a mortal animal of a special kind, then it is analytic and necessary. Its predicate is contained in its subject's definition. The empirical sense of the proposition records an actual addition to knowledge; the analytic sense adds nothing to knowledge, but merely draws attention to one factor included in the definition of "man." In fact, like all a priori propositions, this proposition (taken in it’s a priori sense) is tautological. If the two senses of the proposition "All men are mortal" are confused, we may be tempted to think that mortality is necessarily involved not merely by the definition of man but by the rest of human nature; which is not true. It is merely an observed fact about men.

If the Logical Positivist is right, a famous problem raised by Kant turns out to be unreal. It seemed to Kant that mathematical propositions, such as 5+ 7 = 12, were at once necessary and synthetic. They seemed synthetic because apparently they really do add to knowledge, because 12 is not simply identical with 5 +7. On the other hand they were obviously necessary, since the subject logically involved the predicate.

The Logical Positivist claims to undermine this problem by asserting that "5 + 7" and "12" are simply two ways of saying one and the same thing, or two names for the same thing. If this view is correct, then the whole of mathematics, which we regard (according to our temperament) as a majestic edifice either of pure thought or of objective necessity, consists merely of ways of saying more clearly what was already obscurely said in the basic propositions on which the great science is based.

Not only so, but all deductive reasoning, we are told, is of the same type. This is not to say that it is worthless. If our minds were incomparably more lucid than they are, mathematics (in this view) would indeed be worthless, because we should see at a glance all that the basic definitions contained. We should therefore take no further interest in the subject. But since we are merely human, and our insight is limited, laborious calculation is needed for the discovery of the full content of the basic definitions. Similarly with all other kinds of deductive reasoning. When we have by repeated observation and experiment established a scientific law (say the Law of Gravity), we can deduce from it the sequence of future events. The law itself is simply a summary of past observation, a definition of a principle which is observed to have held good in the world of fact up to the present date, and is expected to hold good in the future. We say in effect, "If the law is true, then so-and-so will happen." The mysterious necessity in this reasoning lies, we are told, simply in the fact that the particular expected event is of the kind already included within the definition of the law.

Thus logical necessity is reduced strictly to a purely linguistic phenomenon, namely to tautology, to the fact that different symbols may have identical meanings.

(d) Criticism of Logical Positivism — The foregoing account of logical necessity is to my mind very impressive; but with the diffidence proper to the mere amateur in logical analysis, I suspect that it falls seriously short of the whole truth of the matter. The definitions which we call scientific laws are, of course, formulae which are true of sense-experience. The law of gravity is not merely a definition implying certain verbally diverse but logically identical consequences.; it is a description of the way in which certain kinds of events have been observed to happen in the experienced world. This description is repeatedly corroborated by succeeding experiences. So far as it is verified, it plainly does in some sense describe a real factor in the experienced world. It is not merely a form of words. Regarded linguistically, logical necessity may appear as sheer tautology; but regarded in its application to the experienced world it appears, not merely as the fact that the same meaning may be expressed in different words, but as the fact that in the actual world the same identical principle may be in different manners. There is, of course, no observable necessity that the principle must continue to hold good; but, so long as the general character of the world appears not to have changed, particular events may reasonably be expected to occur in certain predictable manners. The tautology, so to speak, is not merely a tautology of language but an identity of fact occurring in diverse kinds of situations.

Such statements would be heartily condemned by the Logical Positivists. Clearly, all turns on the word "principle." Logical Positivists deny the existence of such vague entities. But in the present connection the word "principle" does not mean a mysterious and occult "something" behind the experienced world. It means simply an identity of character inherent in a number of diverse events, a "universal character" in many "particular instances." We shall presently enquire whether there is any justification for the belief in such entities, which Logical Positivists reject. Meanwhile, let us assume that there is an identity in all cases of gravitation, as there is for common sense an identity in all cases of warmth, or animality, or justice. An all-powerful intelligence might see at a glance this identity in all gravitational events, as we see the identity in all cases of "warmth"; but human intelligence can only by toilful observation and calculation gradually discover this gravitational identity.

Let us now revert to the Logical Positivists' account of mathematics. It is necessary first to form a clear idea of number. For our purpose it is enough to say that number is the distinctive character of groups. In saying this we assume that there can indeed be identity of character in a number of particular things or events; that, for instance, in all couples there is a certain identity, and again in triplets, and so on. This assumption we shall consider critically in the next section. Meanwhile, let us make use of it. "One," then, is the character common to all single things, whether stones or days or desires, or what not; "two" is the character common to all couples; "three," to all triplets; and so on. Let us note also that "one" is a character of every group as a whole, since it is a single group, however many units it contains. Thus we can have a single couple, a single triplet, a single century. The same applies to "couple." We may have a couple of couples, or of triplets, or centuries. And so on with all the numbers.

In the first instance some of these characters must be observed in actual concrete groups of things. The basic operations of mathematics, adding and subtracting, must also, in the first instance, be carried out with actual objects, and observed. But once we have discovered that by adding a single thing to a single thing we get a couple, we have opened the way to the whole of mathematics, and could in theory construct all the mathematical systems without further experiment. In fact, mathematical reasoning is "necessary," not "contingent."

The Logical Positivists say that mathematical necessity consists in sheer tautology. Are they right? For simplicity, we will consider the proposition 2 + 2 = 4. In a sense 2 + 2 does mean the same thing as four. But in a sense it does not. Strictly what it means is that if you take a couple and then another couple you will have a quadruplet. The operation of "adding two to two" is not the same as the result, "four." The operation of taking two pills twice a day is not the same as the operation of taking four at a time. Now clearly the symbol 2 + 2 means not merely a number but an operation performed with numbers. The result of the operation 2 + 2 is the number 4. Numerically the symbols 2 + 2, and 4, and 1 + 3, and 2 x 2, and so on, have identical significance, but "operationally" they have not.

Apparently, then, the attempt to explain away the seemingly objective necessity of mathematics by reducing it to tautology of symbols has failed. Tautology there is, but there is something else. And the problem lies not in the tautology but in the something else. Mathematical necessity consists in the fact that certain operations with certain numbers produce certain numerical results. Of course, this happens because, in spite of the difference of operation, there is a numerical identity in both sides of the equation. But this identity is not, in the final analysis, a linguistic identity; it is an identity of actual number, an identity of character in any group of things with which the respective operations are performed.

Clearly this vague talk about "character" forces on us a discussion of the whole problem of "universal characters" and their "particular instances."

iv. Universals and Particulars

(a) The Distinction between them

(b) Types of Theory

(c) Impossibility of Denying Particularity

(d) Impossibility of Denying Universality

(e) "Distributive Unity" of Universals in Particulars

(a) The Distinction between them — We have seen that thinking involves noticing the identities and differences in the characters of things. To say "This rose and that flag are both red" is to do more than be aware of a red flag and a red rose without recognising that they are both red, that they both have a certain identical character called "red." The rose and the flag are two things, or events, consisting in each case of a number of characters; and one character is identical in them both. Does this kind of statement describe the matter truly? What is the relation between a particular instance of a character and the universal character of which it is an instance?

Clearly, the characters that constitute any particular thing or event are in a sense more than the particular example of them. Each of the thing's characters seems to have some sort of being beyond the thing in which it occurs, since it occurs in other things.

The red of this rose is indistinguishable (let us suppose) from the red of that flag. An identical something occurs in two situations, a something in virtue of which we relate the two situations, and contrast both with that green grass.

Does this distinction between a. universal character and its particular instances rest on a distinction in the nature of reality, or is it merely a consequence of the nature of our thinking, or our speaking?

(b) Types of Theory — Let us glance at some of the answers that have been made to this question.

In the first place there are theories which accept the reality both of universals and particulars, and attempt to state the relation between them.

In one view, originated by Plato, universals have a special kind of being of their own, quite apart from their instances. They "subsist" in a peculiar sphere out of relation with time and space. They are the perfect types or patterns or forms to which particular things approximate, or in which they participate, and without. which they could have no features. In this view the pure universal character, "redness," subsists independently of all red things.

The Greek word which Plato used to signify a universal was the original of our word "idea"; but to Plato it meant an objective "form," not a mental state. On the other hand, the Platonic "idea" or "form" did mean something more than mere character. The "form" of a thing was the ideal or perfect pattern toward which it somehow strove. A man, for instance, participated to some extent in the form of manhood; and also in some sense he strove toward the perfection of this form. The supreme form was the form of the Good. From this all other forms were derived. God himself was subordinate to the form of the Good.

This introduction of perfection and striving toward perfection is irrelevant to the problem of universal characters. It overlooks the fact that the idea of perfection is derived from human need. Thus the ideal form of the circle, to which actual circles merely approximate, is "perfect" simply in relation to our need for good wheels or for a useful geometrical concept. The child's rough drawing of a circle does not itself aspire to approximate to the ideal of circularity. In fact, circularity is simply an abstraction from our experience of actual round objects. This (as I shall argue later) does not mean that it is a mere figment of our own minds; but it does mean that we know the character 'of circularity only in its particular instances, and that we have no evidence of a distinct realm of purely logical entities.

Medieval philosophy, which was largely derived from Plato's pupil and critic Aristotle, inclined to conceive the Platonic forms as actually mental ideas in the mind of God. The Good was good because God willed it. And all created things were embodiments of the ideas that God conceived.

In both these types of theory universals are regarded as more fundamental than the concrete things which exemplify them. To use the old phrase, both theories put the universals before the particular thing (universalia ante rem). In this view, even if there were no red things, and never had been any, redness would still "subsist."

In another kind of theory universals have no being at all save in their instances. Redness is simply a character which is observed in all red things (or events). Manhood is in men. Apart from actual men, manhood has no being at all. This theory, that universals are simply in the thing (universalia in re), has somehow to bridge the gulf between the identity of the universal and the separateness of its instances.

All these theories accept the reality (in some sense) both of universals and particulars. Even the medieval view that universals were ideas in God's mind allowed that at least they were objective to the minds of men.

But from this position it is easy to pass to the theory that universals are creatures of our own minds, and that they have no objective being. We are said to form in our minds "concepts" or "general ideas" about things. This is the theory of universals after the thing (universalia post rem) or conceptualism.

Two other kinds of view have to be recorded. First there is that which denies the being of universals entirely, preserving only particulars. In this view there are not even such things as general ideas in the mind. What actually happens is that we use one and the same name for similar things or similar characters in things. Redness is just a name, and a name is just a noise or a mark on paper. This is the theory of "nominalism."

Some philosophers have gone to the other extreme and denied the reality of particulars, preserving universals alone. In this view, a concrete thing, such as a particular stone, or a tree, or Oliver Cromwell, is simply a very complex system of universal characters, of all the characters that go to make up this individual thing. Not only is it claimed that Oliver Cromwell is a system of universals occurring in a particular historical context; but also that even his historical relationships are universals. For instance, the date of his birth is a universal character belonging to all events that happened before one set of events and after another. Further, we are told that he himself is constituted by his relations to other things, by their effects on him and his reactions on them. He is what the environment makes of him, and what he does to the environment. Even the shape of his body is the shape as it affects other things, and his own and other people's minds. No particular thing, it is said, is fully real. It involves a context. And the simpler and less self-complete the thing is, the less "real," the less concrete and more abstract it is. Thus an electron is less "real" than an amoeba, and this than a man; and a man is less "real" than a community of men. The only completely " real " thing is the Whole that comprises all things. For this alone (according to the theory) is a self-complete system of universal characters.

Thus we arrive at the Idealist's theory of the "concrete universal." A distinction is made between a relatively more abstract universal, such as "redness," and a relatively more concrete universal, such as " this red patch," which is redness combined with certain other universal characters and universal relations. A particular man is a very much more complex (and therefore concrete) universal. The British nation is more concrete still. The only fully concrete universal is the universe itself.

Such are the main theories of universals and particulars. I shall examine the two extreme theories, and endeavour to show that both universals and particulars must somehow be retained. I shall then summarise a theory which seems to me to give a credible account of their relation.

(c) Impossibility of Denying Particularity — If there is no such thing as particularity, two exactly similar systems of character must be in fact one and the same system. There is no meaning in saying that there are two of them. This is the theory of "the identity of indiscernibles." The theory assumes that even the relationships in which a thing stands are themselves universal characters, and are moreover intrinsic to the thing itself; in fact, that the thing is constituted by its relationships. Oliver Cromwell is the sum of his intercourse with the world. As we have seen, there is a sense in which even Cromwell's bodily shape is constituted by its relations to other volumes. If this theory of relations is granted, clearly there logically cannot be two identical Cromwells in the same universe, since they could not have the same relations with the rest of the universe. Nor could there be two identical stones.

But there is a serious difficulty in the theory that things are constituted by their relations to other things. If all things are thus constituted, all things exist, as it were, "by taking in each other's washing," and nothing in fact exists at all. We may admit that our knowledge of a thing is wholly constituted by our knowledge of its relations to other things, but that the thing itself is thus constituted we must not allow. Or rather, since in one sense a finite thing certainly does seem to be constituted by its relations, we must expect to find some other sense in which it is not. If things are constituted by their relations, it is equally true that relations are constituted by things. They are essentially relations of things. Or, since the word "thing" is in bad odour in philosophy, let us substitute the word "event."

But to return to the problem of universals, we must try to see the whole matter from a fresh angle. Is not the theory that denies particulars merely playing with words? What we actually experience is particular examples of universal characters. This rose, that flag, and that nose, we say, are all "red." Redness is the character in respect of which these particulars are identical. Of redness unparticularised we know nothing, save by our power of abstracting, of attending to the identity of red things while ignoring their differences. It is true that the only way in which redness can be particularised is by "entering into" particular situations or relationships with other characters; but "entering into" is metaphorical. Of redness apart from particular situations we know nothing. It is of the very nature of redness to be particularised. Particularity is as essential to it as universality.

Observing one red thing after another, and attending to their identity of character (and their difference from green things) we abstract the universality of their redness, and "hypostatise" it, or treat it as a self-complete thing. We think of redness as something other than this red and that red. In fact, we set up the universal and the particular as two independent "things." And having done this we find it impossible to relate them. In despair we have to abolish either one or the other. Thus we may actually come to think of the universality of red as in some manner the whole reality of red. But this is unnatural. and artificial, and arose through the initial mistake of detaching the universality of red from its particularity in its instances.

(d) Impossibility of Denying Universality — If, on the other hand, we retain particulars and deny that universals have any being at all, we land ourselves in another set of difficulties. According to this theory " red" is just a name, and a name is a kind of behaviour which we adopt in relation to a certain class of situations; and "green" is another name, for other situations. Each class of situations is made up of many particular members, and there is nothing whatever which can be called the "universal."

This view lays itself open to a simple criticism which, so far as I can see, vast ingenuity has entirely failed to answer. In virtue of what distinguishing mark do we assign all red things to one class and one name, and all green things to another? If there is nothing in respect of which all dogs, in spite of their differences, are identical, and distinguishable from all cats, how do we know which animals to call "dogs" and which "cats"? Similarly, if there is nothing in respect of which all couples are identical, and, again, all triplets, and so on, how do we know what to put into the class of couples, what into the class of triplets, and so on?

Further, what is a name? It is a noise or mark or action of a special kind. The noise "dog" is only a name in virtue of the fact that all instances of this noise have an identity of character, are in fact instances of one and the same name, and distinct from other names.

We saw that the denial of particulars arose from the hypostatisation of the universality in characters. We now see that the denial of universality arises from the hypostatisation of particularity in characters, the assertion that a thing or event is nothing whatever but its particularity.

Similar objections can be brought against the theory that a universal is a "concept," a mental thing, made of the stuff of the mind itself. If that is all it is, how do we know which concepts to apply to which things? The things must have similarity and difference, and so must the concepts.

(e) "Distributive Unity" of Universals in Particulars — In some sense, then, we must retain both universals and particulars. But we must avoid cutting them adrift from one another, and regarding each as an independent thing. Then what are they, and how are they related?

Let us begin by insisting that there is nothing what- ever in a particular thing (or event) save characters, and that these characters have universality, that they can occur identically in more than one instance. On the other hand, let us insist that these universal characters have no being save in their instances. The problem, then, is to state the relation between these two abstractions, namely, the particularity and the universality of a concrete character.

The following remarks are based on the theory of Professor G. F. Stout, according to which the being of the universal simply is the "distributive unity" of a character in many particular instances of the character. Thus the universal character "redness" is not a disembodied abstraction or ideal form, inhabiting a timeless realm of pure being, and mysteriously conferring itself upon its instances. But neither is "redness" a mere figment of the mind, or a mere noise. On the other hand, a particular existent "red," say in this rose, is not a completely isolated thing without objective relations (of similarity and difference) with other particulars. But neither is it describable wholly in terms of its universality. "Redness" just is the identity of character in all red things. It is not above them or before them or between them. It consists, let us say, of "the respect in which all red things are identical." It is the distributive unity of all red things.

It may be objected that this theory does not really solve the difficulty, and that we must still ask how this identity of a single something (red) in many instances can be. But the question is improper. The difficulty arises through stating the problem wrongly at the outset, through cutting universals and particulars apart and hypostatising both of them. All that a theory can be expected to do is to describe the facts of experience faithfully; and this, we may claim, our theory succeeds in doing.

Postscript on Truth

Some readers may feel the lack of a discussion of this subject. I therefore append a note, as a sketch- map of the territory.

When we say that a statement is true, we generally mean that, in some sense, it corresponds with some fact other than the statement itself. Idealists, however, maintain that the truth of a statement or idea is constituted by its coherence with the total system of knowledge. Pragmatists hold that any object (such as a word, a sign-post, a flag) may be used as an idea, may assume the office of "idea," in so far as it serves as a symbol or guide for our activities. The truth of the "idea" is simply its successful functioning in that capacity. Logical Positivists insist that a statement is true only if it can be verified in sense-perception. In this respect they and the Realists agree with common sense, though with many qualifications.

It is important to distinguish between the meaning of truth, which, I submit, involves "correspondence," and the test of truth, which is very often "coherence" with the established system of human experience.

(Left Keyword <-) Previous: Chapter 4 The External World and i Back Next: Chapter 6 The Scope and Limitations of Reason (Right Keyword:->)