The subject of modality, though a traditional department of logic, is one which has gradually come to occupy less and less of the attention of philosophers. This diminishing share of attention is due, I believe, to the fact that the division of judgments into necessary, assertorical, and problematic is, in the main, based upon error and confusion. I do not deny that it is possible to make valid distinctions among propositions, which will have some of the characteristics of the traditional modal distinctions; but the distinctions which are valid seem, so far as I can discover, to be none of them fundamental, and all of them better described in non-modal terms.

In order to recommend this view, I shall first consider the characteristics which ought to belong to a doctrine of necessity and possibility on the traditional theory; I shall then examine various suggested definitions of necessity and possibility, and shall try to show that they do not have the characteristics required.

Necessity and possibility, to begin with, must be primarily predicates of propositions. When we say (for example) "God is a necessary Being", we must be regarded as meaning "That God is is necessary". We must distinguish between a necessary proposition and a proposition which predicates necessity. A necessary proposition is the subject of a true proposition which predicates necessity. If somebody says to me "Well, you must be a fool", what is meant is "That you are a fool is necessary". What is said to be necessary is "You are a fool", not "You must be a fool"; though his latter may happen to be also necessary. There must have been some coonfusion on this point in Kant's mind when he contended that modality only affects the copula (Kritik der reinen Vernunft, Hartenstein p. 97). The copula in a proposition which predicates necessity is the same as in all other propositions: the predication of necessity is made by the proposition "p is necessary", not "p must be necessary". The confusion is rendered pardonable by the fact that a corresponding confusion is very common in regard to negation. It is supposed that "is not" is a copula, whereas all negations properly are of the form "p is false", where the copula is is, and the predicate is false.

We shall be in fairly general agreement with usage if we decide that a necessary proposition must be true, and that a possible proposition is one of which the contradictory is not necessary.¹ If a theory of modality is to be of much importance, we require that some true propositions should not be necessary, and that some false ones should be possible. We may define a contingent proposition as one which is true but not necessary, and an impossible proposition as one which is not possible. These definitions of possible, contingent, and impossible are intended as purely verbal definitions. They all presuppose that we know what is meant by a necessary proposition. They are thus merely preliminary, and serve to narrow the discussion to the meaning of the one word necessary.

Different definitions of necessity have been given by different authors, but as a rule these definitions have not been purely verbal. That is to say, the authors have believed that they had an idea of necessity, and that the definitions they gave were true, i. e. gave marks, other than necessity, which are common and peculiar to what is necessary If this had not been the case, different definitions would not have been, as in fact they have been, marks of philosophical disagreement For example, when one writer says that the good is pleasure, and another that it is virtue, they differ in opinion, because both attach the same meaning to the word good, though they differ as to the things that are good. The main question to be considered in regard to necessity is, therefore Is there any such predicate as necessity, as distinct from the various predicates which various definitions assert to be equivalent to it? If not, different definitions do not disagree philosophically, but only as regards the use of words. I do not myself believe that there is such a predicate as necessity, apart from definitions which are strictly verbal definitions, though I hardly see how my opinion is to be proved

The sort of view that I disagree with may be illustrated by a quotation Meinong, in discussing what he calls "objects of higher order", observes² "The colour which I now think as in this place I can also think as in another place, and I can equally think another colour in the same place. Not so as regards diversity: if once A and B are diverse, then they are so always, for they must be so, this word being understood in the sense of 'logical necessity', which here is grounded in the nature of A and B on the one hand and of diversity on the other " Meinong, as appears both here and elsewhere in his writings, regards necessity as a recognizable property of propositions, discoverable by inspection, and not standing in need of a definition. It appears to me that the propositions which he regards as necessary are really those not predicating existence at particular times, and that no more fundamental predicate is involved. But this predicate, as I shall try to show, derives its importance from theory of knowledge, and deserves no special place of honour in logic.

I shall now proceed to consider various definitions of necessity, with a view to discovering, if possible, what people really have in their minds when they affirm necessity.

The terms a priori and empirical seem to be more or less connected with the terms necessary and contingent, and to differ chiefly by the fact that they belong to theory of knowledge rather than to logic. It is rather our cognitions than the things we know that are spoken of as a priori or empirical; and the same proposition might be known both a priori and empirically. For example, historical revelation gives empirical evidence of propositions which natural religion endeavours to prove a priori. Moses by the burning bush knew the existence of God empirically; St. Anselm knew it a priori. The difference here depends upon the source of our knowledge. Empirical knowledge is knowledge derived, in whole or part, from perception; a priori knowledge is knowledge independent of perception. (We need not linger over these somewhat vague phrases, as they do not concern our main subject.) There is a certain feeling that what we know by perception might quite well have been otherwise, while what is known a priori is necessary; and it is through this feeling that the a priori and the empirical become connected with the necessary and the contingent.

Now if a given proposition may be known both a priori and empirically, the connection between the epistemological and the logical pair cannot be quite so simple as might be hoped. We must find a logical pair of terms roughly corresponding to "propositions known a priori" and "propositions known empirically", before we can hope to make the connection. This logical pair of terms may be found, approximately, in "propositions not predicating existence at particular times" and "propositions predicating existence at particular times". But among those regarded as predicating existence at particular times, we must include those which are concerned with actually occurring sequences, and those which (like the laws of motion) are concerned with all particular times, without being deducible from the nature of time. When such necessary extensions have been made, we obtain a class of propositions having, obviously, no specially notable logical characteristic, and important only on account of the way in which we come to know them. And the feeling of necessity which we have in regard to other propositions, but not in regard to those derived from perception, seems to me derived from two sources, one psychological, and the other a confusion. The psychological source is that, when a proposition is not concerned with a particular time, the knowledge of it, if attainable at all, is equally attainable at all times, so that there is not an enforced period of doubt while we wait to see what will happen. And where science has rendered prediction possible, as in astronomy, people feel that events are necessary: the motions of the heavenly bodies are often taken as the very type of necessity. Thus it would seem that the epistemological question of how we come to know a proposition is largely what determines our feeling of necessity or contingency. The other source, which is sheer confusion, is the ambiguity of sentences involving a tense without an assigned date. The sentence "it is raining" expresses a proposition which is true sometimes but not always (except in the Lake district). Hence it comes to be felt that the proposition expressed is sometimes true and sometimes false, and therefore may be true or may be false. But of course this is only because the same form of words expresses different propositions at different times, and each of these propositions is true or false independently of the date at which it is considered.

Necessary and contingent, therefore, in so far as these are connected with a priori and empirical, have a purely epistemological importance, and are not notions which logic need take account of.

The view of necessity which we have been hitherto considering is the one which connects it with independence of particular times. This is the view which caused necessary propositions to be spoken of as "eternal truths". But there is another view of necessity, according to which, stating the matter as broadly as possible, a proposition is necessary when it is demonstrable. In daily life, "it must be so" is commonly used to indicate an inference. "He must be in, for I saw his hat in the hall." According to the view of necessity we are considering, this remark asserts that the proposition "he is in" is necessary, because the proposition "I saw his hat in the hall" is true, and (what is understood but not expressed) the proposition "I saw his hat in the hall" implies the proposition "he is in".

In practice, people do not say "so-and-so must be true" unless they have inferred so-and-so by a process sufficiently difficult to be consciously felt as inference.

In answer to the question what day of the week it is, we should say "It is Tuesday: today's paper says so", but we should say "It must be Tuesday, because it is the iyth, and the month began on a Sunday." It is the feeling of having inferred that we express by "it must be so".

Mr. Bradley's theory of necessity is an attempt to express this state of things in logical terms. He says:³

The theory set forth in these two passages amounts to this: "A proposition q is said to be necessary if it is implied by a proposition p." Mr. Bradley makes it appear that it is the consequent, not the whole hypothetical, that he regards as necessary (though elsewhere he is ambiguous on this point), and that he does not hold it essential that either the antecedent or the consequent should be true. Hence whatever follows from anything is necessary.

The objection to this theory is that it makes every proposition necessary, true and false alike. For there is no proposition whatever which will not follow from some premiss, e.g. the premiss "all propositions are true, and this is a proposition". We cannot make the proviso that our premiss must be possible, since possibility is regarded by Mr. Bradley as "a form of hypothetical necessity" (p. 186), and it is admitted that "the necessary may be impossible or non-existent" (ib.). Hence there seems no escape from the conclusion that all propositions are necessary.

I do not mean to suggest that Mr. Bradley has not given a perfectly right account of the circumstances under which we say "it must be so".⁴ All that I mean to say is that, if we accept this account, necessity, in its only important sense, belongs to psychology, or at most to theory of knowledge, and not to logic in any degree. For to say "it must be so" is merely to say "I infer that it is so"; or, if we insist upon a logical meaning, "it must be so" may be taken to mean "it is demonstrable that it is so"; where "p is demonstrable" must be defined as meaning, not "it is possible to demonstrate p" (since possibility is to be defined by means of necessity), but "there is a true proposition which implies p". But this logical meaning is wholly destitute of importance, since it holds when and only when p is true.

From Mr. Bradley's language it is not always clear whether it is the consequent of a hypothetical, or the connection of antecedent and consequent, that he regards as necessary. Mr. Bosanquet (Logic, I, p. 391) adopts the view (if I have understood him aright) that it is the connection that is necessary, not the consequent per se. He speaks of "judgments of apodeictic character, i.e. hypothetical and disjunctive judgments". He does not make it clear whether he considers that when we assert a hypothetical or disjunctive judgment, we actually assert that the consequence or alternative concerned is necessary, or whether he considers that, when we truly assert a hypothetical or disjunctive judgment, the truth which we affirm is in fact a necessary truth. The former position could hardly be maintained; for it would make it impossible simply to assert a consequence or alternative, without asserting that it is necessary. Hence we may translate Mr. Bosanquet's position as implying that every true hypothetical or disjunctive proposition is necessary, and no other propositions are necessary.

Now hypothetical and disjunctive propositions are certainly an important class logically, and if we choose to give the name necessary to such propositions, certain conveniences may result. But it seems to me that by no means all such propositions would be commonly called necessary. "If it rains, I shall bring my umbrella"; "if I am in town tomorrow, I shall go to the play" - such hypotheticals may very well be true, and yet few people would call them necessary. It is needful that the connection should be of a certain kind in order that it should be felt to be necessary. What this kind is, I shall shortly try to define: it may be loosely described as the kind which makes the consequent logically deducible from the antecedent. Exactly the same remark applies to disjunctive propositions. It is true that "either Caesar was killed on the Ides of March, or he died of a surfeit of pickles", but this would not be commonly called a necessary proposition. On the other hand, "either Caesar was killed on the Ides of March, or he was not killed on the Ides of March" is a proposition of the sort commonly called necessary. For this reason, I cannot think that Mr. Bosanquet's account of necessity corresponds to what is commonly meant by the word, although I see no logical reason against using the word as he uses it.

Mr. G.E.Moore⁵ has advocated a theory of necessity which depends upon logical priority. According to this theory, a proposition is more or less necessary in proportion as there are more or fewer other propositions to which it is logically prior; and p is logically prior to q if q implies p but p does not imply q. This theory is not available with a doctrine of implication which holds that true propositions are implied by all propositions and false propositions imply all propositions. For then we find that, according to the definition, all true propositions are at one level of logical priority, and all false ones at another level; and all true propositions are logically prior to all false ones. This gives the same degree of necessity to all true propositions, and is therefore not suitable for discriminating among true propositions.

Before Kant, it would, I suppose, have been universally admitted that a necessary proposition is one whose truth can be deduced from the law of contradiction, and a possible proposition is one whose falsehood cannot be deduced from the law of contradiction. This view of necessity and possibility requires much modification before it can be harmonized with modern logic; but by the help of such modification it can, I believe, be still rendered more or less serviceable.

The old identification of the necessary with the analytic requires a twofold modification. In the first place, the meaning of analytic must be changed, if it is to apply to the sort of propositions that used to be called analytic. In the second place, some meaning other than "p is implied by q" must be found for "p is deducible from q", if propositions deducible from the law of contradiction, or from any other principle of logic, are to be not coextensive with all true propositions. Some such other meaning, we found, is also required for distinguishing between the sort of hypotheticals that we feel to be necessary and the sort that we feel to be not necessary. I shall begin by trying to decide upon the meaning of deducibility. A new and appropriate meaning of analytic, we shall find, results at once from the notion of deducibility.

The difficulty about deducibility is caused by the doctrine of implication, according to which "p implies q", or "ifp, then q", is equivalent to "p is not true or q is true" (the alternatives being not mutually exclusive). This view of implication is rendered unavoidable by various considerations, such, for example, as the following. Suppose p, q, r to be such that if p and q are true, then r is true. It follows that if p is true, then if q is true, r is true. (For example, if a person is male and married, he is a husband; hence if a person is male, then if he is married he is a husband.) Now if p and q are true, then r is true. Hence, by the above principle, if p is true, then if q is true, p is true; that is, if p is true, then q implies p; that is, a true proposition (p) is implied by every proposition (q). I shall not pursue the arguments in favour of this view of implication; I shall content myself by pointing out that it is accepted (though without a full realization of its consequences) by Shakspeare and Mr. Bradley in the following passage (Logic, p.121 ):

On the strength of these authorities, therefore, I shall henceforth assume that "p implies q" is equivalent to "p is not true or q is true".

It follows that, if we regard q as deducible from p whenever p implies q, all true propositions are deducible from the law of contradiction. But in the practice of inference, it is plain that something more than implication must be concerned. The reason that proofs are used at all is that we can sometimes perceive that q follows from p, when we should not otherwise know that q is true; while in other cases, "p implies q" is only to be inferred either from the falsehood of p or from the truth of q. In these other cases, the proposition "p implies q" serves no practical purpose; it is only when this proposition is used as a means of discovering the truth of q that it is useful. Given a true proposition p, there will be some propositions q such that the truth of "p implies q" is evident, and thence the truth of q is inferred; while in the case of other true propositions, their truth must be independently known before we can know that p implies them. What we require is a logical distinction between these two cases.

This distinction may be capable of statement in some simple form, but the only way of stating it that I have been able to discover is as follows. There are certain general propositions, which we may enumerate as the laws of deduction: such are "if not-p is false, then p is true", "if p implies not-q, then q implies not-p", "if p implies q and q implies r, then p implies r"; in all we need about ten such principles. These replace the old syllogism and its rules. We may then say that q is deducible from p if it can be shown by means of the above principles that p implies q.

This definition may be re-stated as follows. The laws of deduction tell us that two propositions having certain relations of form (e.g. that one is the negation of the negation of the other) are such that one of them implies the other. Thus q is deducible from p if p and q either have one of the relations contemplated by the laws of deduction, or are connected by any (finite) number of intermediaries each having one of these relations to its successor. This meaning of deducible is purely logical, and covers, I think, exactly the cases in which, in practice, we can deduce a proposition q from a proposition p without assuming either that p is false or that q is true.

It is to be observed that, although deducible from (as just defined) is a different notion from implied by, it cannot be made to replace the notion of implied by. For deducible from is defined by means of the laws of deduction, and these laws employ the notion of implication. Hence we cannot substitute deducible from for implied by in the laws of deduction, without incurring a vicious circle. The notion of implication thus remains fundamental, and the notion of deducibility is derivative from it.

Of the two definitions which we lately set out to seek, namely that of analytic and that of deducible from, the definition of deducible from is thus found. It remains to decide what meaning we are to give to the term analytic.

The traditional meaning of analytic combined two properties. The property from which the name was derived was that supposed to appear in analysis, namely that the subject of an analytic proposition, when anayzed, was to contain the predicate. With this property we need not concern ourselves. Another property, supposed to be equivalent to this one, though in reality not so, was the property of being deducible (in the sense just explained) from the law of contradiction, or, more generally, from what have been optimistically called "the laws of thought". Now formal logic, in the times when such views flourished, had no exact means of mowing what was deducible from what, having expended its slender store of exactitude and subtlety on the syllogism - a subject scarcely more useful or less amusing than heraldry. Consequently people supposed that they could deduce, from the laws of identity, contradiction, and excluded middle, many things which in fact require the help of other premisses in order to be proved. Very little indeed is deducible from the law of contradiction alone, and not very much is deducible from the laws of identity, contradiction and excluded middle. We want, therefore, a wider meaning of analytic than the traditional one, if its extension is to remain at all what it was formerly supposed to be. We may obtain such a meaning by adopting a suggestion due to M. Couturat. A certain large body of propositions, namely (approximately) those constituting formal logic and pure mathematics, all have some very important logical characteristics in common, and are all deducible from a small number of general logical premisses, among which are included the laws of deduction already spoken of. These general logical premisses fulfill the functions formerly supposed to be fulfilled by the so-called "laws of thought": they may be called the "laws of logic".⁶ From the laws of logic all the propositions of formal logic and pure mathematics will be deducible. We may, then, usefully define as analytic those propositions which are deducible from the laws of logic; and this definition is conformable in the spirit, though not in the letter, to the pre-Kantian usage. Certainly Kant, in urging that pure mathematics consists of synthetic propositions, was urging, among other things, that pure mathematics cannot be deduced from the laws of logic alone. In this we now know that he was wrong and Leibniz was right; to call pure mathematics analytic is therefore an appropriate way of marking dissent from Kant on this point.

If the proposition "p implies q" is analytic, we may say that q is an analytic consequence of p. To say that "p implies q" is analytic is equivalent, according to the definitions we have adopted, to saying that q is deducible from p. Thus q is an analytic consequence of p when and only when q is deducible from p. It is noteworthy that, in all actual valid deduction, whether or not the material is of a purely logical nature, the relation of premiss to conclusion, in virtue of which we make the deduction, is one of those contemplated by the laws of logic or deducible from them. Thus in all valid inferences, the conclusion is an analytic consequence of the premiss, in other words, the implication is analytic. Implications which are not analytic can only be practically discovered if it is independently known that the premiss is false or that the conclusion is true, or if the implication can be made analytic by adding to the premiss a proposition whose truth is independently known. But in such cases (except sometimes in the third case), implications have no practical utility.

It is now open to us, if we choose, to say that a necessary proposition is an analytic proposition, and a possible proposition is one of which the contradictory is not analytic. It was in order to be able to have some precise meaning in saying this that the long excursus into deducibility and the laws of logic was required. But the feeling of necessity does not answer to this definition; many propositions are felt to be necessary which are not analytic. Such are: "If a thing is good, it is not bad", "If a thing is yellow, it is not red", and so on. Bad does not mean the same as not-good, and therefore mere logic will never prove that good and bad are any more incompatible than round and blue. Hence, although the class of analytic propositions is an important class, it does not seem to be the same as the class of necessary propositions.

It is possible to regard a proposition as necessary when it is an instance of a type of propositions all of which are true. For example, "Socrates is either a man or not a man" may be called necessary on the ground that the statement remains true if we substitute anything else in place of Socrates. Similarly "If Socrates is a man, he is mortal" remains true if we put anything else in place of Socrates, and may therefore be called necessary. A correlative definition of possibility is: "A proposition in which (say) Socrates occurs is possible if there is something which can be substituted for Socrates so as to make the proposition true".⁷ The latter definition leads to more or less paradoxical consequences, for example, that Socrates might be a triangle, because there are triangles. These consequences, however, are only paradoxical because we know that Socrates is a man; and the proposition "Socrates is a man and a triangle" is impossible according to the definition. If we allow ourselves to take account of the actual properties of Socrates in estimating whether a proposition in which he occurs is possible or not, there is no ground for stopping short of all his properties; and then true propositions in which he occurs are possible, and false propositions in which he occurs are impossible. I think, however, that something like this theory covers a good many cases in which possibility is commonly affirmed. Suppose I take a cab, and its number has five figures; I shall feel that it might have had four figures. In this case, all that is meant seems to be: "This is a London cab, and some London cabs have numbers consisting of four figures." In such cases, the subject of the proposition is felt as a variable: it is not felt as fully determinate, but as an indefinite member of some class. To make our definitions of necessity and possibility precise, in this theory, it is natural to regard necessity and possibility as not attaching to propositions, but to propositional functions, that is, to propositions with an indeterminate subject. Thus we may define as follows:

"The propositional function 'x has the property φ' is necessary if it holds of everything; it is necessary throughout the class u if it holds of every member of u."

"The propositional function 'x has the property φ' is possible if it holds of something; it is possible within the class u if it holds of some member of u."

For example, "x is identical with x" is necessary; "x is mortal" is necessary throughout the class man; "x is an even prime" is possible, because 2 is an even prime; "x is a philosopher" is possible within the class man.

These definitions are, I think, substantially those of Mr. MacColl⁸, who, however, does not distinguish between propositions and propositional functions. There is, so far as I can see, no particular objection to these definitions, except that they do not make necessity and possibility a property of propositions. Some care is required in giving precision to the suggestion with which we started, that the definitions are to be applied to propositions by regarding the subject of the proposition as variable. For if the subject occurs two or more times in the proposition, the proposition will be an instance of three or more different types. E.g. "Socrates is identical with Socrates" is necessary (on this view) when regarded as an instance of "x is identical with x", but is only contingent when regarded as an instance of "Socrates is identical with x" or of "x is identical with Socrates". We can, however, circumvent this ambiguity by deciding that a proposition of which (say) Socrates is a constituent is to be called necessary with respect to Socrates if there is any type, consisting wholly of true propositions, of which the given proposition is the instance got by substituting Socrates for the variable. Thus in the case of "Socrates is identical with Socrates", the proposition is necessary with respect to Socrates because, of the three types obtainable by turning Socrates into a variable in the first place where he occurs, or the second, or both, the last is true for all values of the variable. The proposition is contingent with respect to identity, because the type "Socrates has the relation R to Socrates" is not true for all values of R, but only for some values.

There are many advantages in this theory, which embraces elements from our previous theories. Analytic propositions have the property that they are necessary with respect to all their constituents except such as are what I call logical constants. Thus e.g. "Socrates is identical with Socrates" is an analytic proposition, and identity (with respect to which it is not necessary) is a logical constant. Again, the view that a necessary proposition is one not concerned with particular parts of time receives a certain explanation, since propositions which are concerned with particular parts of time are in general not true of other parts of time, which is merely a way of stating that changes occur.

But there remain propositions which do not seem quite to fit this scheme. For example, we feel certain of the truth of all propositions of the type: "x either is not a moment of time, or is a moment of time subsequent to the death of Cromwell, or is a moment of time preceding the Restoration"; yet we should hesitate to call propositions of this type necessary. For we realize at once that the truth of all propositions of this type is a deduction from "the death of Cromwell preceded the Restoration", which must be a contingent proposition if any proposition is to be contingent. Yet perhaps this feeling could be turned into its opposite. For if anybody said "Such and such an event happened before the death of Cromwell but after the Restoration", we should reply "that is impossible, because Cromwell died before the Restoration". Thus the feeling of necessity on such points seems to be uncertain and vacillating.

The subject of probability is one which is naturally associated with modality: the probability of a proposition's being true may be supposed to be measure of its greater or less degree of possibility. Thus it would be necessary, in order to show that modal distinctions are never required, to produce a theory of probability in which no such distinctions are invoked. I am not prepared, in this paper, to advocate any view on such a thorny question as probability; and I confess that I do not know of any view which strikes me as tenable. But as against the view that probability is the measure of possibility, it seems a sufficient reply to point out that the answer to a question of probabilities is often, if not always, more or less arbitrary. Divide the circumference of a circle into two semi-circles, and choose a point at haphazard on the circumference. What is the chance that this point will be on the upper semi-circle? Obviously one half, one would suppose; yet all values from 0 to 1 can be proved by methods accepted as valid in other cases.

It is very likely that there are other possible definitions of necessity which are more satisfactory than those that I have discussed. But if any conclusion is warranted by the above arguments, it is this: That the feeling of necessity which we have is a complex and rather muddled feeling, compounded of such elements as the following:

(1).	The feeling that a proposition can be known without an appeal to perception;
(2).	The feeling that a proposition can be proved,
(3).	The feeling that a proposition can be deduced from the laws of logic;
(4).	The feeling that a proposition holds not only of its actual subject, but of all subjects more or less resembling its actual subject, or, as an extreme case, of all subjects absolutely.

Any one of these four may be used to found a theory of necessity. The first gives a theory whose importance is not logical, but epistemological; the second makes the necessary coextensive with the true. The third and fourth give important classes of propositions; but the third class (propositions deducible from the laws of logic) is better described as the class of analytic propositions, and the view underlying the fourth is more readily applicable to propositional functions than to propositions.

I conclude that, so far as appears, there is no one fundamental logical notion of necessity, nor consequently of possibility. If this conclusion is valid, the subject of modality ought to be banished from logic, since propositions are simply true or false, and there is no such comparative and superlative of truth as is implied by the notions of contingency and necessity.