Our Knowledge of the External World as a Field for Scientific Method in Philosophy Part 10

[50] _Mathematical Discourses concerning two new sciences relating to mechanics and local motion, in four dialogues._ By Galileo Galilei, Chief Philosopher and Mathematician to the Grand Duke of Tuscany. Done into English from the Italian, by Tho. Weston, late Master, and now published by John Weston, present Master, of the Academy at Greenwich.

See pp. 46 ff.

The way in which the problem is expounded in the above discussion is worthy of Galileo, but the solution suggested is not the right one. It is actually the case that the number of square (finite) numbers is the same as the number of (finite) numbers. The fact that, so long as we confine ourselves to numbers less than some given finite number, the proportion of squares tends towards zero as the given finite number increases, does not contradict the fact that the number of all finite squares is the same as the number of all finite numbers. This is only an instance of the fact, now familiar to mathematicians, that the _limit_ of a function as the variable _approaches_ a given point may not be the same as its _value_ when the variable actually _reaches_ the given point. But although the infinite numbers which Galileo discusses are equal, Cantor has shown that what Simplicius could not conceive is true, namely, that there are an infinite number of different infinite numbers, and that the conception of _greater_ and _less_ can be perfectly well applied to them. The whole of Simplicius's difficulty comes, as is evident, from his belief that, if _greater_ and _less_ can be applied, a part of an infinite collection must have fewer terms than the whole; and when this is denied, all contradictions disappear. As regards greater and less lengths of lines, which is the problem from which the above discussion starts, that involves a meaning of _greater_ and _less_ which is not arithmetical. The number of points is the same in a long line and in a short one, being in fact the same as the number of points in all s.p.a.ce. The _greater_ and _less_ of metrical geometry involves the new metrical conception of _congruence_, which cannot be developed out of arithmetical considerations alone. But this question has not the fundamental importance which belongs to the arithmetical theory of infinity.

(2) _Non-inductiveness._--The second property by which infinite numbers are distinguished from finite numbers is the property of non-inductiveness. This will be best explained by defining the positive property of inductiveness which characterises the finite numbers, and which is named after the method of proof known as "mathematical induction."

Let us first consider what is meant by calling a property "hereditary"

in a given series. Take such a property as being named Jones. If a man is named Jones, so is his son; we will therefore call the property of being called Jones hereditary with respect to the relation of father and son. If a man is called Jones, all his descendants in the direct male line are called Jones; this follows from the fact that the property is hereditary. Now, instead of the relation of father and son, consider the relation of a finite number to its immediate successor, that is, the relation which holds between 0 and 1, between 1 and 2, between 2 and 3, and so on. If a property of numbers is hereditary with respect to this relation, then if it belongs to (say) 100, it must belong also to all finite numbers greater than 100; for, being hereditary, it belongs to 101 because it belongs to 100, and it belongs to 102 because it belongs to 101, and so on--where the "and so on" will take us, sooner or later, to any finite number greater than 100. Thus, for example, the property of being greater than 99 is hereditary in the series of finite numbers; and generally, a property is hereditary in this series when, given any number that possesses the property, the next number must always also possess it.

It will be seen that a hereditary property, though it must belong to all the finite numbers greater than a given number possessing the property, need not belong to all the numbers less than this number. For example, the hereditary property of being greater than 99 belongs to 100 and all greater numbers, but not to any smaller number. Similarly, the hereditary property of being called Jones belongs to all the descendants (in the direct male line) of those who have this property, but not to all their ancestors, because we reach at last a first Jones, before whom the ancestors have no surname. It is obvious, however, that any hereditary property possessed by Adam must belong to all men; and similarly any hereditary property possessed by 0 must belong to all finite numbers. This is the principle of what is called "mathematical induction." It frequently happens, when we wish to prove that all finite numbers have some property, that we have first to prove that 0 has the property, and then that the property is hereditary, _i.e._ that, if it belongs to a given number, then it belongs to the next number. Owing to the fact that such proofs are called "inductive," I shall call the properties to which they are applicable "inductive" properties. Thus an inductive property of numbers is one which is hereditary and belongs to 0.

Taking any one of the natural numbers, say 29, it is easy to see that it must have all inductive properties. For since such properties belong to 0 and are hereditary, they belong to 1; therefore, since they are hereditary, they belong to 2, and so on; by twenty-nine repet.i.tions of such arguments we show that they belong to 29. We may _define_ the "inductive" numbers as _all those that possess all inductive properties_; they will be the same as what are called the "natural"

numbers, _i.e._ the ordinary finite whole numbers. To all such numbers, proofs by mathematical induction can be validly applied. They are those numbers, we may loosely say, which can be reached from 0 by successive additions of 1; in other words, they are all the numbers that can be reached by counting.

But beyond all these numbers, there are the infinite numbers, and infinite numbers do not have all inductive properties. Such numbers, therefore, may be called non-inductive. All those properties of numbers which are proved by an imaginary step-by-step process from one number to the next are liable to fail when we come to infinite numbers. The first of the infinite numbers has no immediate predecessor, because there is no greatest finite number; thus no succession of steps from one number to the next will ever reach from a finite number to an infinite one, and the step-by-step method of proof fails. This is another reason for the supposed self-contradictions of infinite numbers. Many of the most familiar properties of numbers, which custom had led people to regard as logically necessary, are in fact only demonstrable by the step-by-step method, and fail to be true of infinite numbers. But so soon as we realise the necessity of proving such properties by mathematical induction, and the strictly limited scope of this method of proof, the supposed contradictions are seen to contradict, not logic, but only our prejudices and mental habits.

The property of being increased by the addition of 1--_i.e._ the property of non-reflexiveness--may serve to ill.u.s.trate the limitations of mathematical induction. It is easy to prove that 0 is increased by the addition of 1, and that, if a given number is increased by the addition of 1, so is the next number, _i.e._ the number obtained by the addition of 1. It follows that each of the natural numbers is increased by the addition of 1. This follows generally from the general argument, and follows for each particular case by a sufficient number of applications of the argument. We first prove that 0 is not equal to 1; then, since the property of being increased by 1 is hereditary, it follows that 1 is not equal to 2; hence it follows that 2 is not equal to 3; if we wish to prove that 30,000 is not equal to 30,001, we can do so by repeating this reasoning 30,000 times. But we cannot prove in this way that _all_ numbers are increased by the addition of 1; we can only prove that this holds of the numbers attainable by successive additions of 1 starting from 0. The reflexive numbers, which lie beyond all those attainable in this way, are as a matter of fact not increased by the addition of 1.

The two properties of reflexiveness and non-inductiveness, which we have considered as characteristics of infinite numbers, have not so far been proved to be always found together. It is known that all reflexive numbers are non-inductive, but it is not known that all non-inductive numbers are reflexive. Fallacious proofs of this proposition have been published by many writers, including myself, but up to the present no valid proof has been discovered. The infinite numbers actually known, however, are all reflexive as well as non-inductive; thus, in mathematical practice, if not in theory, the two properties are always a.s.sociated. For our purposes, therefore, it will be convenient to ignore the bare possibility that there may be non-inductive non-reflexive numbers, since all known numbers are either inductive or reflexive.

When infinite numbers are first introduced to people, they are apt to refuse the name of numbers to them, because their behaviour is so different from that of finite numbers that it seems a wilful misuse of terms to call them numbers at all. In order to meet this feeling, we must now turn to the logical basis of arithmetic, and consider the logical definition of numbers.

The logical definition of numbers, though it seems an essential support to the theory of infinite numbers, was in fact discovered independently and by a different man. The theory of infinite numbers--that is to say, the arithmetical as opposed to the logical part of the theory--was discovered by Georg Cantor, and published by him in 1882-3.[51] The definition of number was discovered about the same time by a man whose great genius has not received the recognition it deserves--I mean Gottlob Frege of Jena. His first work, _Begriffsschrift_, published in 1879, contained the very important theory of hereditary properties in a series to which I alluded in connection with inductiveness. His definition of number is contained in his second work, published in 1884, and ent.i.tled _Die Grundlagen der Arithmetik, eine logisch-mathematische Untersuchung uber den Begriff der Zahl_.[52] It is with this book that the logical theory of arithmetic begins, and it will repay us to consider Frege's a.n.a.lysis in some detail.

[51] In his _Grundlagen einer allgemeinen Mannichfaltigkeitslehre_ and in articles in _Acta Mathematica_, vol. ii.

[52] The definition of number contained in this book, and elaborated in the _Grundgesetze der Arithmetik_ (vol. i., 1893; vol. ii., 1903), was rediscovered by me in ignorance of Frege's work. I wish to state as emphatically as possible--what seems still often ignored--that his discovery antedated mine by eighteen years.

Frege begins by noting the increased desire for logical strictness in mathematical demonstrations which distinguishes modern mathematicians from their predecessors, and points out that this must lead to a critical investigation of the definition of number. He proceeds to show the inadequacy of previous philosophical theories, especially of the "synthetic _a priori_" theory of Kant and the empirical theory of Mill.

This brings him to the question: What kind of object is it that number can properly be ascribed to? He points out that physical things may be regarded as one or many: for example, if a tree has a thousand leaves, they may be taken altogether as const.i.tuting its foliage, which would count as one, not as a thousand; and _one_ pair of boots is the same object as _two_ boots. It follows that physical things are not the subjects of which number is properly predicated; for when we have discovered the proper subjects, the number to be ascribed must be unambiguous. This leads to a discussion of the very prevalent view that number is really something psychological and subjective, a view which Frege emphatically rejects. "Number," he says, "is as little an object of psychology or an outcome of psychical processes as the North Sea....

The botanist wishes to state something which is just as much a fact when he gives the number of petals in a flower as when he gives its colour.

The one depends as little as the other upon our caprice. There is therefore a certain similarity between number and colour; but this does not consist in the fact that both are sensibly perceptible in external things, but in the fact that both are objective" (p. 34).

"I distinguish the objective," he continues, "from the palpable, the spatial, the actual. The earth's axis, the centre of ma.s.s of the solar system, are objective, but I should not call them actual, like the earth itself" (p. 35). He concludes that number is neither spatial and physical, nor subjective, but non-sensible and objective. This conclusion is important, since it applies to all the subject-matter of mathematics and logic. Most philosophers have thought that the physical and the mental between them exhausted the world of being. Some have argued that the objects of mathematics were obviously not subjective, and therefore must be physical and empirical; others have argued that they were obviously not physical, and therefore must be subjective and mental. Both sides were right in what they denied, and wrong in what they a.s.serted; Frege has the merit of accepting both denials, and finding a third a.s.sertion by recognising the world of logic, which is neither mental nor physical.

The fact is, as Frege points out, that no number, not even 1, is applicable to physical things, but only to general terms or descriptions, such as "man," "satellite of the earth," "satellite of Venus." The general term "man" is applicable to a certain number of objects: there are in the world so and so many men. The unity which philosophers rightly feel to be necessary for the a.s.sertion of a number is the unity of the general term, and it is the general term which is the proper subject of number. And this applies equally when there is one object or none which falls under the general term. "Satellite of the earth" is a term only applicable to one object, namely, the moon. But "one" is not a property of the moon itself, which may equally well be regarded as many molecules: it is a property of the general term "earth's satellite." Similarly, 0 is a property of the general term "satellite of Venus," because Venus has no satellite. Here at last we have an intelligible theory of the number 0. This was impossible if numbers applied to physical objects, because obviously no physical object could have the number 0. Thus, in seeking our definition of number we have arrived so far at the result that numbers are properties of general terms or general descriptions, not of physical things or of mental occurrences.

Instead of speaking of a general term, such as "man," as the subject of which a number can be a.s.serted, we may, without making any serious change, take the subject as the cla.s.s or collection of objects--_i.e._ "mankind" in the above instance--to which the general term in question is applicable. Two general terms, such as "man" and "featherless biped,"

which are applicable to the same collection of objects, will obviously have the same number of instances; thus the number depends upon the cla.s.s, not upon the selection of this or that general term to describe it, provided several general terms can be found to describe the same cla.s.s. But some general term is always necessary in order to describe a cla.s.s. Even when the terms are enumerated, as "this and that and the other," the collection is const.i.tuted by the general property of being either this, or that, or the other, and only so acquires the unity which enables us to speak of it as _one_ collection. And in the case of an infinite cla.s.s, enumeration is impossible, so that description by a general characteristic common and peculiar to the members of the cla.s.s is the only possible description. Here, as we see, the theory of number to which Frege was led by purely logical considerations becomes of use in showing how infinite cla.s.ses can be amenable to number in spite of being incapable of enumeration.

Frege next asks the question: When do two collections have the same number of terms? In ordinary life, we decide this question by counting; but counting, as we saw, is impossible in the case of infinite collections, and is not logically fundamental with finite collections.

We want, therefore, a different method of answering our question. An ill.u.s.tration may help to make the method clear. I do not know how many married men there are in England, but I do know that the number is the same as the number of married women. The reason I know this is that the relation of husband and wife relates one man to one woman and one woman to one man. A relation of this sort is called a one-one relation. The relation of father to son is called a one-many relation, because a man can have only one father but may have many sons; conversely, the relation of son to father is called a many-one relation. But the relation of husband to wife (in Christian countries) is called one-one, because a man cannot have more than one wife, or a woman more than one husband. Now, whenever there is a one-one relation between all the terms of one collection and all the terms of another severally, as in the case of English husbands and English wives, the number of terms in the one collection is the same as the number in the other; but when there is not such a relation, the number is different. This is the answer to the question: When do two collections have the same number of terms?

We can now at last answer the question: What is meant by the number of terms in a given collection? When there is a one-one relation between all the terms of one collection and all the terms of another severally, we shall say that the two collections are "similar." We have just seen that two similar collections have the same number of terms. This leads us to define the number of a given collection as the cla.s.s of all collections that are similar to it; that is to say, we set up the following formal definition:

"The number of terms in a given cla.s.s" is defined as meaning "the cla.s.s of all cla.s.ses that are similar to the given cla.s.s."

This definition, as Frege (expressing it in slightly different terms) showed, yields the usual arithmetical properties of numbers. It is applicable equally to finite and infinite numbers, and it does not require the admission of some new and mysterious set of metaphysical ent.i.ties. It shows that it is not physical objects, but cla.s.ses or the general terms by which they are defined, of which numbers can be a.s.serted; and it applies to 0 and 1 without any of the difficulties which other theories find in dealing with these two special cases.

The above definition is sure to produce, at first sight, a feeling of oddity, which is liable to cause a certain dissatisfaction. It defines the number 2, for instance, as the cla.s.s of all couples, and the number 3 as the cla.s.s of all triads. This does not _seem_ to be what we have hitherto been meaning when we spoke of 2 and 3, though it would be difficult to say _what_ we had been meaning. The answer to a feeling cannot be a logical argument, but nevertheless the answer in this case is not without importance. In the first place, it will be found that when an idea which has grown familiar as an una.n.a.lysed whole is first resolved accurately into its component parts--which is what we do when we define it--there is almost always a feeling of unfamiliarity produced by the a.n.a.lysis, which tends to cause a protest against the definition.

In the second place, it may be admitted that the definition, like all definitions, is to a certain extent arbitrary. In the case of the small finite numbers, such as 2 and 3, it would be possible to frame definitions more nearly in accordance with our una.n.a.lysed feeling of what we mean; but the method of such definitions would lack uniformity, and would be found to fail sooner or later--at latest when we reached infinite numbers.

In the third place, the real desideratum about such a definition as that of number is not that it should represent as nearly as possible the ideas of those who have not gone through the a.n.a.lysis required in order to reach a definition, but that it should give us objects having the requisite properties. Numbers, in fact, must satisfy the formulae of arithmetic; any indubitable set of objects fulfilling this requirement may be called numbers. So far, the simplest set known to fulfil this requirement is the set introduced by the above definition. In comparison with this merit, the question whether the objects to which the definition applies are like or unlike the vague ideas of numbers entertained by those who cannot give a definition, is one of very little importance. All the important requirements are fulfilled by the above definition, and the sense of oddity which is at first unavoidable will be found to wear off very quickly with the growth of familiarity.

There is, however, a certain logical doctrine which may be thought to form an objection to the above definition of numbers as cla.s.ses of cla.s.ses--I mean the doctrine that there are no such objects as cla.s.ses at all. It might be thought that this doctrine would make havoc of a theory which reduces numbers to cla.s.ses, and of the many other theories in which we have made use of cla.s.ses. This, however, would be a mistake: none of these theories are any the worse for the doctrine that cla.s.ses are fictions. What the doctrine is, and why it is not destructive, I will try briefly to explain.

On account of certain rather complicated difficulties, culminating in definite contradictions, I was led to the view that nothing that can be said significantly about things, _i.e._ particulars, can be said significantly (_i.e._ either truly or falsely) about cla.s.ses of things.

That is to say, if, in any sentence in which a thing is mentioned, you subst.i.tute a cla.s.s for the thing, you no longer have a sentence that has any meaning: the sentence is no longer either true or false, but a meaningless collection of words. Appearances to the contrary can be dispelled by a moment's reflection. For example, in the sentence, "Adam is fond of apples," you may subst.i.tute _mankind_, and say, "Mankind is fond of apples." But obviously you do not mean that there is one individual, called "mankind," which munches apples: you mean that the separate individuals who compose mankind are each severally fond of apples.

Now, if nothing that can be said significantly about a thing can be said significantly about a cla.s.s of things, it follows that cla.s.ses of things cannot have the same kind of reality as things have; for if they had, a cla.s.s could be subst.i.tuted for a thing in a proposition predicating the kind of reality which would be common to both. This view is really consonant to common sense. In the third or fourth century B.C. there lived a Chinese philosopher named Hui Tzu, who maintained that "a bay horse and a dun cow are three; because taken separately they are two, and taken together they are one: two and one make three."[53] The author from whom I quote says that Hui Tzu "was particularly fond of the quibbles which so delighted the sophists or unsound reasoners of ancient Greece," and this no doubt represents the judgment of common sense upon such arguments. Yet if collections of things were things, his contention would be irrefragable. It is only because the bay horse and the dun cow taken together are not a new thing that we can escape the conclusion that there are three things wherever there are two.

[53] Giles, _The Civilisation of China_ (Home University Library), p. 147.

When it is admitted that cla.s.ses are not things, the question arises: What do we mean by statements which are nominally about cla.s.ses? Take such a statement as, "The cla.s.s of people interested in mathematical logic is not very numerous." Obviously this reduces itself to, "Not very many people are interested in mathematical logic." For the sake of definiteness, let us subst.i.tute some particular number, say 3, for "very many." Then our statement is, "Not three people are interested in mathematical logic." This may be expressed in the form: "If _x_ is interested in mathematical logic, and also _y_ is interested, and also _z_ is interested, then _x_ is identical with _y_, or _x_ is identical with _z_, or _y_ is identical with _z_." Here there is no longer any reference at all to a "cla.s.s." In some such way, all statements nominally about a cla.s.s can be reduced to statements about what follows from the hypothesis of anything's having the defining property of the cla.s.s. All that is wanted, therefore, in order to render the _verbal_ use of cla.s.ses legitimate, is a uniform method of interpreting propositions in which such a use occurs, so as to obtain propositions in which there is no longer any such use. The definition of such a method is a technical matter, which Dr Whitehead and I have dealt with elsewhere, and which we need not enter into on this occasion.[54]

[54] Cf. _Principia Mathematica_, -- 20, and Introduction, chapter iii.

If the theory that cla.s.ses are merely symbolic is accepted, it follows that numbers are not actual ent.i.ties, but that propositions in which numbers verbally occur have not really any const.i.tuents corresponding to numbers, but only a certain logical form which is not a part of propositions having this form. This is in fact the case with all the apparent objects of logic and mathematics. Such words as _or_, _not_, _if_, _there is_, _ident.i.ty_, _greater_, _plus_, _nothing_, _everything_, _function_, and so on, are not names of definite objects, like "John" or "Jones," but are words which require a context in order to have meaning. All of them are _formal_, that is to say, their occurrence indicates a certain form of proposition, not a certain const.i.tuent. "Logical constants," in short, are not ent.i.ties; the words expressing them are not names, and cannot significantly be made into logical subjects except when it is the words themselves, as opposed to their meanings, that are being discussed.[55] This fact has a very important bearing on all logic and philosophy, since it shows how they differ from the special sciences. But the questions raised are so large and so difficult that it is impossible to pursue them further on this occasion.

[55] In the above remarks I am making use of unpublished work by my friend Ludwig Wittgenstein.

LECTURE VIII

ON THE NOTION OF CAUSE, WITH APPLICATIONS TO THE FREE-WILL PROBLEM

The nature of philosophic a.n.a.lysis, as ill.u.s.trated in our previous lectures, can now be stated in general terms. We start from a body of common knowledge, which const.i.tutes our data. On examination, the data are found to be complex, rather vague, and largely interdependent logically. By a.n.a.lysis we reduce them to propositions which are as nearly as possible simple and precise, and we arrange them in deductive chains, in which a certain number of initial propositions form a logical guarantee for all the rest. These initial propositions are _premisses_ for the body of knowledge in question. Premisses are thus quite different from data--they are simpler, more precise, and less infected with logical redundancy. If the work of a.n.a.lysis has been performed completely, they will be wholly free from logical redundancy, wholly precise, and as simple as is logically compatible with their leading to the given body of knowledge. The discovery of these premisses belongs to philosophy; but the work of deducing the body of common knowledge from them belongs to mathematics, if "mathematics" is interpreted in a somewhat liberal sense.

But besides the logical a.n.a.lysis of the common knowledge which forms our data, there is the consideration of its degree of certainty. When we have arrived at its premisses, we may find that some of them seem open to doubt, and we may find further that this doubt extends to those of our original data which depend upon these doubtful premisses. In our third lecture, for example, we saw that the part of physics which depends upon testimony, and thus upon the existence of other minds than our own, does not seem so certain as the part which depends exclusively upon our own sense-data and the laws of logic. Similarly, it used to be felt that the parts of geometry which depend upon the axiom of parallels have less certainty than the parts which are independent of this premiss. We may say, generally, that what commonly pa.s.ses as knowledge is not all equally certain, and that, when a.n.a.lysis into premisses has been effected, the degree of certainty of any consequence of the premisses will depend upon that of the most doubtful premiss employed in proving this consequence. Thus a.n.a.lysis into premisses serves not only a logical purpose, but also the purpose of facilitating an estimate as to the degree of certainty to be attached to this or that derivative belief. In view of the fallibility of all human beliefs, this service seems at least as important as the purely logical services rendered by philosophical a.n.a.lysis.

In the present lecture, I wish to apply the a.n.a.lytic method to the notion of "cause," and to ill.u.s.trate the discussion by applying it to the problem of free will. For this purpose I shall inquire: I., what is meant by a causal law; II., what is the evidence that causal laws have held hitherto; III., what is the evidence that they will continue to hold in the future; IV., how the causality which is used in science differs from that of common sense and traditional philosophy; V., what new light is thrown on the question of free will by our a.n.a.lysis of the notion of "cause."

I. By a "causal law" I mean any general proposition in virtue of which it is possible to infer the existence of one thing or event from the existence of another or of a number of others. If you hear thunder without having seen lightning, you infer that there nevertheless was a flash, because of the general proposition, "All thunder is preceded by lightning." When Robinson Crusoe sees a footprint, he infers a human being, and he might justify his inference by the general proposition, "All marks in the ground shaped like a human foot are subsequent to a human being's standing where the marks are." When we see the sun set, we expect that it will rise again the next day. When we hear a man speaking, we infer that he has certain thoughts. All these inferences are due to causal laws.

A causal law, we said, allows us to infer the existence of one _thing_ (or _event_) from the existence of one or more others. The word "thing"

here is to be understood as only applying to particulars, _i.e._ as excluding such logical objects as numbers or cla.s.ses or abstract properties and relations, and including sense-data, with whatever is logically of the same type as sense-data.[56] In so far as a causal law is directly verifiable, the thing inferred and the thing from which it is inferred must both be data, though they need not both be data at the same time. In fact, a causal law which is being used to extend our knowledge of existence must be applied to what, at the moment, is not a datum; it is in the possibility of such application that the practical utility of a causal law consists. The important point, for our present purpose, however, is that what is inferred is a "thing," a "particular,"

an object having the kind of reality that belongs to objects of sense, not an abstract object such as virtue or the square root of two.

[56] Thus we are not using "thing" here in the sense of a cla.s.s of correlated "aspects," as we did in Lecture III. Each "aspect" will count separately in stating causal laws.

But we cannot become acquainted with a particular except by its being actually given. Hence the particular inferred by a causal law must be only _described_ with more or less exactness; it cannot be _named_ until the inference is verified. Moreover, since the causal law is _general_, and capable of applying to many cases, the given particular from which we infer must allow the inference in virtue of some general characteristic, not in virtue of its being just the particular that it is. This is obvious in all our previous instances: we infer the unperceived lightning from the thunder, not in virtue of any peculiarity of the thunder, but in virtue of its resemblance to other claps of thunder. Thus a causal law must state that the existence of a thing of a certain sort (or of a number of things of a number of a.s.signed sorts) implies the existence of another thing having a relation to the first which remains invariable so long as the first is of the kind in question.

It is to be observed that what is constant in a causal law is not the object or objects given, nor yet the object inferred, both of which may vary within wide limits, but the _relation_ between what is given and what is inferred. The principle, "same cause, same effect," which is sometimes said to be the principle of causality, is much narrower in its scope than the principle which really occurs in science; indeed, if strictly interpreted, it has no scope at all, since the "same" cause never recurs exactly. We shall return to this point at a later stage of the discussion.

The particular which is inferred may be uniquely determined by the causal law, or may be only described in such general terms that many different particulars might satisfy the description. This depends upon whether the constant relation affirmed by the causal law is one which only one term can have to the data, or one which many terms may have. If many terms may have the relation in question, science will not be satisfied until it has found some more stringent law, which will enable us to determine the inferred things uniquely.