89 I shall not attempt to give here any adequate philosophic analysis of the conception of objective necessity. It must suffice to say that we all have the conception of the operation of law, and that for our present purpose the validity of this conception is assumed.

90 Another solution, however, and one that may be made to yield a symmetrical scheme, is to utilise the conditional (as distinguished from the true hypothetical,87) proposition, and to differentiate it from the categorical, by interpreting it as modal,88 while the categorical remains merely assertoric.

Thus, we should have,—
   If anything is S it is P,—apodeictic;
   All S is P,—assertoric;
   If anything is S it may be P,—problematic.89

It is of course not pretended that the differentiation here proposed is adopted in the ordinary use of the propositional forms in question; we shall, for example, have presently to point out that in the customary usage of categoricals the universal affirmative has frequently an apodeictic force. We shall return to a discussion of the suggested scheme later on.

60. Modality in relation to Compound Judgments.—We may now consider the application of distinctions of modality to compound judgments, that is, to judgments which express a relation in which simple judgments stand one to another. It is one thing to say that as a matter of fact two judgments are not both true; it is another thing to say that two judgments are so related to one another that they cannot both be true. We may describe the one statement as assertoric, the other as apodeictic. An apodeictic judgment thus conceived expresses a relation of ground and consequence; an obligation, therefore, to affirm the truth of a certain proposition when the truth of a certain other proposition or combination of propositions is admitted. The obligation may sometimes depend upon the assistance of certain other propositions which are left unexpressed.90

91 In section 55 a threefold classification of compound judgments was given; the distinction now under consideration points, however, to a more fundamental twofold classification. From this point of view a scheme may be suggested in which conjunctives (P and Q) and so-called disjunctives (P or Q) would be regarded as assertoric, while hypotheticals (If P then Q) would be regarded as modal. The enquiry as to how far this is in accordance with the ordinary usage of the propositional forms in question must be deferred. It may, however, be desirable to point out at once that, if this scheme is adopted, certain ordinarily recognised logical relations are not valid. For the hypothetical If P then Q is ordinarily regarded as equivalent to the disjunctive Either not-P or Q, and this as equivalent to the denial of the conjunctive Both P and not-Q. If, however, the conjunctive (and, therefore, its denial) and also the disjunctive are merely assertoric, while the hypothetical is apodeictic, it is clear that this equivalence no longer holds good. The disjunctive can indeed still be inferred from the hypothetical, but not the hypothetical from the disjunctive. This result will be considered further at a later stage.

So far we have spoken only of the apodeictic form, If P then Q. The corresponding problematic form is, If P then possibly Q ; for example, If all S is P it is still possible that some P is not S. This denies the obligation to admit that all P is S when it has been admitted that all S is P. It is to be observed that in any treatment of modality, the apodeictic and the problematic involve one another, since the one form is always required to express the contradictory of the other.

61. The Quantity and the Quality of Propositions.—Propositions are commonly divided into universal and particular, according as the predication is made of the whole or of a part of the subject. This division of propositions is said to be according to their quantity.

Kant added a third subdivision, namely, singular ; and other logicians have added a fourth, namely, indefinite. Under the head of quantity there have also to be considered what are called plurative and numerically definite propositions; and the possibility of multiple quantification has to be recognised. The 92 question may also be raised whether there are not some propositions, e.g., hypothetical propositions, which do not admit of division according to quantity at all. The discussion of the various points here indicated may, however, conveniently be deferred until the traditional scheme of categorical propositions, which is based on the definitive division into universal and particular, has been briefly touched upon.

A categorical proposition consists of two terms (which are respectively the subject and the predicate), united by a copula, and usually preceded by a sign of quantity. It thus contains four elements, two of which—the subject and the predicate—constitute its matter, while the remaining two—the copula and the sign of quantity—constitute its form.91

The subject is that term about which affirmation or denial is made. The predicate is that term which is affirmed or denied of the subject.

When propositions are brought into one of the forms recognised in the traditional scheme the subject precedes the predicate. In ordinary discourse, however, this order is sometimes 93 inverted for the sake of literary effect, for example, in the proposition—Sweet are the uses of adversity.

The traditional scheme of propositions is obtained by a combination of the division (according to quantity) into universal and particular, and the division (according to quality) into affirmative and negative. This combination yields four fundamental forms of proposition as follows:—
 (1) the universal affirmative—All S is P (or Every S is P, or Any S is P, or All S’s are P’s)—usually denoted by the symbol A; 94
 (2) the particular affirmative—Some S is P (or Some S’s are P’s)—usually denoted by the symbol I;
 (3) the universal negative—No S is P (or No S’s are P’s)—usually denoted by the symbol E;
 (4) the particular negative—Some S is not P (or Not all S is P, or Some S’s are not P’s, or Not all S’s are P’s)—usually denoted by the symbol O.

It is better not to write the universal negative in the form All S is not P ;92 for this form is ambiguous and would usually be interpreted as being merely particular, the not being taken to qualify the all, so that we have All S is not P = Not-all S is P. Thus, “All that glitters is not gold” is intended for an O proposition, and is equivalent to “Some things that glitter are not gold.”

95 The traditional scheme of formulation is somewhat limited in its scope, and from more points of view than one it is open to criticism. It has, however, the merit of simplicity, and it has met with wide acceptation. For these reasons it is as a rule convenient to adopt it as a basis of discussion, though it is also not infrequently necessary to look beyond it.

63. The Distribution of Terms in a Proposition.—A term is said to be distributed when reference is made to all the individuals denoted by it; it is said to be undistributed when they are only referred to partially, that is, when information is given with regard to a portion of the class denoted by the term, but we are left in ignorance with regard to the remainder of the class. It follows immediately from this definition that the subject is distributed in a universal, and undistributed in a particular,93 proposition. It can further be shewn that the predicate is distributed in a negative, and undistributed in an affirmative proposition. Thus, if I say All S is P, I identify every member of the class S with some member of the class P, and I therefore imply that at any rate some P is S, but I make no implication with regard to the whole of P. It is left an open question whether there is or is not any P outside the class S. Similarly if I say Some S is P. But if I say No S is P, in excluding the whole of S from P, I am also excluding the whole of P from S, and therefore P as well as S is distributed. Again, if I say Some S is not P, although I make an assertion with regard to a part only of S, I exclude this part from the whole of P, and therefore the whole of P from it. In this case, then, the predicate is distributed, although the subject is not.94

96 Summing up our results, we find that
   A distributes its subject only,
   I  distributes neither its subject nor its predicate,
   E distributes both its subject and its predicate,
   O distributes its predicate only.

It follows that the subject must be given first in idea, since we cannot assert a predicate until we have something about which to assert it. Can it, however, be said that because the subject logically comes first in order of thought, it must necessarily do so in order of statement, the subject always preceding the copula, and the predicate always following it? In other words, can we consider the order of the terms in a proposition to suffice as a criterion? If the subject and the predicate are pure synonyms95 or if the proposition is practically reduced to an equation, as in the doctrine of the quantification of the predicate, it is difficult to see what other criterion can be taken; or it may rather be said that in these cases the distinction between subject and predicate loses all importance. The two are placed on an equality, and nothing is left by which to distinguish them except the order in which they are stated. This view is indicated by Professor Baynes in his Essay on the New Analytic of Logical Forms. In such a proposition, for example, as “Great is Diana of the Ephesians,” he would call “great” the subject, reading the proposition, “(Some) great is (all) Diana of the Ephesians.”

With reference to the traditional scheme of propositions, however, it cannot be said that the order of terms is always a 97 sufficient criterion. In the proposition just quoted, “Diana of the Ephesians” would generally be accepted as the subject. What further criterion then can be given? In the case of E and I propositions (propositions, as will be shewn, which can be simply converted) we must appeal to the context or to the question to which the proposition is an answer. If one term clearly conveys information regarding the other term, it is the predicate. It will be shewn also that it is more usual for the subject to be read in extension and the predicate in intension.96 If these considerations are not decisive, then the order of the terms must suffice. In the case of A and O propositions (propositions, as will be shewn, which cannot be simply converted) a further criterion may be added. From the rules relating to the distribution of terms in a proposition it follows that in affirmative propositions the distributed term (if either term is distributed) is the subject; whilst in negative propositions, if only one term is distributed, it is the predicate. It is doubtful if the inversion of terms ever occurs in the case of an O proposition; but in A propositions it is not infrequent. Applying the above considerations to such a proposition as “Workers of miracles were the Apostles,” it is clear that the latter term is distributed while the former is not; the latter term is, therefore, the subject. Since a singular term is equivalent to a distributed term, it follows further as a corollary that in an affirmative proposition if one and only one term is singular it is the subject. This decides such a case as “Great is Diana of the Ephesians.”

65. Universal Propositions.—In discussing the import of the universal proposition All S is P, attention must first be called to a certain ambiguity resulting from the fact that the word all may be used either distributively or collectively. In the proposition, All the angles of a triangle are less than two right angles, it is used distributively, the predicate applying to each and every angle of a triangle taken separately. In the proposition. All the angles of a triangle are equal to two right angles, it is used collectively, the predicate applying to all the 98 angles taken together, and not to each separately. This ambiguity attaches to the symbolic form All S is P, but not to the form All S’s are P’s. Ambiguity may also be avoided by using every instead of all, as the sign of quantity. In any case the ambiguity is not of a dangerous character, and it may be assumed that all is to be interpreted distributively, unless by the context or in some other way an indication is given to the contrary.

A more important distinction between propositions expressed in the form All S is P remains to be considered. For such propositions may be merely assertoric or they may be apodeictic, in the sense given to these terms in section 59.

It will be convenient here to commence with a threefold distinction.

(1) The proposition All S is P may, in the first place, make a predication of a limited number of particular objects which admit of being enumerated: e.g., All the books on that shelf are novels, All my sons are in the army, All the men in this year’s eleven were at public schools. A proposition of this kind may be called distinctively an enumerative universal. It is clear that such a proposition cannot claim to be apodeictic.

(2) The proposition All S is P may, in the second place, express what is usually described as an empirical law or uniformity: e.g., All lions are tawny, All scarlet flowers are without sweet scent, All violets are white or yellow or have a tinge of blue in them. Many propositions relating to the use of drugs, to the succession of certain kinds of weather to certain appearances of sky, and so on, fall into this class. A proposition of this kind expresses a uniformity which has been found to hold good within the range of our experience, but which we should hesitate to extend much beyond that range either in space or in time. The predication which it makes is not limited to a definite number of objects which can be enumerated, but at the same time it cannot be regarded as expressing a necessary relation between subject and predicate. Such a proposition is, therefore, assertoric, not apodeictic.

(3) The proposition All S is P may, in the third place, express a law in the strict sense, that is to say, a uniformity 99 that we believe to hold good universally and unconditionally: e.g., All equilateral triangles are equiangular, All bodies have weight, All arsenic is poisonous. A proposition of this kind is to be regarded as expressing a necessary relation between subject and predicate, and it is, therefore, apodeictic.

Propositions falling under the first two of the above categories may be described as empirically universal, and those falling under the third as unconditionally universal.97

If this distinction is regarded merely as a distinction between different ways in which judgments may be obtained (for example, by enumeration or empirical generalisation on the one hand, or by abstract reasoning or the aid of the principle of causality on the other hand), without any real difference of content, it becomes merely genetic and can hardly be retained as a 100 distinction between judgments considered in and by themselves. If we are so to retain it, it must be as a distinction between the merely assertoric and the apodeictic in the sense already explained. In order to be able to deal with it as a formal distinction, we must further be prepared to assign distinctive forms of expression to the two kinds of universal judgments respectively. Lotze appears to regard the forms All S is P and S is P as sufficiently serving this purpose. But this is hardly borne out by the current usage of these forms. All the S’s are P might serve for the enumerative universal and S as such is P for the unconditionally universal. These forms do not, however, fit into any generally recognised schedules; and our second class of universal would be left out. Another solution, which has been already indicated in section 59, would be to use the categorical form for the empirically universal judgment only, adopting the conditional form for the unconditionally universal judgment.

The most important outcome of the above discussion is that a proposition ordinarily expressed in the form All S is P may be either assertoric or apodeictic. It will be found that this distinction has an important bearing on several questions subsequently to be raised.

66. Particular Propositions.—In dealing with particular propositions it is necessary to assign a precise signification to the sign of quantity some.

In its ordinary use, the word some is always understood to be exclusive of none, but in its relation to all there is ambiguity. For it is sometimes interpreted as excluding all as well as none, while sometimes it is not regarded as carrying this further implication. The word may, therefore, be defined in two conflicting senses: first, as equivalent simply to one at least, that is, as the pure contradictory of none, and hence as covering every case (including all) which is inconsistent with none ; secondly, as any quantity intermediate between none and all and hence carrying with it the implication not all as well as not none. In ordinary speech the latter of these two meanings is probably the more usual.98 It has, however, been customary with 101 logicians in interpreting the traditional scheme to adopt the other meaning, so that Some S is P is not inconsistent with All S is P. Using the word in this sense, if we want to express Some, but not all, S is P, we must make use of two propositions—Some S is P, Some S is not P. The particular proposition as thus interpreted is indefinite, though with a certain limit; that is, it is indefinite in so far that it may apply to any number from a single one up to all, but on the other hand it is definite in so far as it excludes none. We shall henceforth interpret some in this indefinite sense unless an explicit indication is given to the contrary.

Mr Bosanquet regards the particular proposition as unscientific, on the ground that it always depends either upon imperfect description or upon incomplete enumeration.99 I may, for instance, know that all S’s of some particular description are P, but not caring or not troubling to define them I content myself with saying Some S is P, for example, Some truth is better kept to oneself.100 Contrasted with this, we have the particular proposition of incomplete enumeration where our ground for asserting it is simply the observation of individual instances in which the proposition is found to hold good.

It is true that the particular proposition is not in itself of much scientific importance; and its indefinite character naturally limits its practical utility. It seems, however, hardly correct to describe it as unscientific, since—as will subsequently be shewn in more detail—it may be regarded as possessing distinctive functions. Two such functions may be distinguished, though they are often implicated the one in the other. In the first place, the utility of the particular proposition often depends 102 rather on what it denies than on what it affirms, and the proposition that it denies is not indefinite. One of the principal functions of the particular affirmative is to deny the universal negative, and of the particular negative to deny the universal affirmative. In the second place, the distinctive purpose of the particular proposition may be to affirm existence; and this is probably as a rule the case with propositions which are described as resulting from incomplete description. If, for example, we say that “some engines can drag a train at a mile a minute for a long distance,” our object is primarily to affirm that there are such engines; and this would not be so clearly expressed in the universal proposition of which the particular is said to be the incomplete and imperfect expression.

Singular propositions may be regarded as forming a sub-class of universals, since in every singular proposition the affirmation or denial is of the whole of the subject.101 More definitely, the singular proposition may be said to fall into line, as a rule, with the enumerative universal proposition.

Hamilton distinguishes between universal and singular propositions, the predication being in the former case of a whole undivided, and in the latter case of a unit indivisible. The 103 distinction here indicated is sometimes useful; but it can with advantage be expressed somewhat differently. A singular proposition may generally without risk of confusion be denoted by one of the symbols A or E; and in syllogistic inferences a singular may ordinarily be treated as equivalent to a universal proposition. The use of independent symbols for singular propositions (affirmative and negative) would introduce considerable additional complexity into the treatment of the syllogism; and for this reason it seems desirable as a rule to include singulars under universals. Universal propositions may, however, be divided into general and singular, and there will then be terms whereby to call attention to the distinction whenever it may be necessary or useful to do so.

68. Plurative Propositions and Numerically Definite Propositions.—Other signs of quantity besides all and some are sometimes recognised by logicians. Thus, propositions of the forms Most S’s are P’s, Few S’s are P’s, are called plurative propositions. Most may be interpreted as equivalent to at least one more than half. Few has a negative force; and Few S’s are P’s may be regarded as equivalent to Most S’s are not 104 P’s.102 Formal logicians (excepting De Morgan and Hamilton) have not as a rule recognised these additional signs of quantity; and it is true that in many logical combinations they cannot be regarded as yielding more than particular propositions, Most S’s are P’s being reduced to Some S’s are P’s, and Few S’s are P’s to Some S’s are not P’s. Sometimes, however, we are able to make use of the extra knowledge given us; e.g., from Most M’s are P’s, Most M’s are S’s, we can infer Some S’s are P’s, although from Some M’s are P’s, Some M’s are S’s, we can infer nothing.

Numerically definite propositions are those in which a predication is made of some definite proportion of a class; e.g., Two-thirds of S are P. A certain ambiguity may lurk in numerically definite propositions; e.g., in the above proposition is it meant that exactly two-thirds of S neither more nor less are P, so that we are also given implicitly one-third of S are not P, or is it merely meant that at least two-thirds of S but perhaps more are P? In ordinary discourse we should no doubt mean sometimes the one and sometimes the other. If we are to fix our interpretation, it will probably be best to adopt the first alternative, on the ground that if figures are introduced at all we should aim at being quite determinate.103 Some such words 105 as at least can then be used when it is not professed to state more than the minimum proportion of S’s that are P’s.