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.
[101] It is argued by Father Clarke that singulars ought to be included under particulars, on the ground that when a predicate is asserted of one member only of a class, it is asserted of a portion only of the class. “Now if I say, This Hottentot is a great rascal, my assertion has reference to a smaller portion of the Hottentot nation than the proposition Some Hottentots are great rascals. The same is the case even if the subject be a proper name. London is a large city must necessarily be a more restricted proposition than Some cities are large cities ; and if the latter should be reckoned under particulars, much more the former” (Logic, p. 274). This view fails to recognise that what is really characteristic of the particular proposition is not its restricted character—since the particular is not inconsistent with the universal—but its indefinite character.
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.
There is also a certain class of propositions which, while singular, inasmuch as they relate but to a single individual, possess also the indefinite character which belongs to the particular proposition: for example, A certain man had two sons ; A great statesman was present ; An English officer was killed. Having two such propositions in the same discourse we cannot, apart from the context, be sure that the same individual is referred to in both cases. Carrying the distinction indicated in the preceding paragraph a little further, we have a fourfold division of propositions:—general definite, “All S is P”; general indefinite, “Some S is P”; singular definite, “This S is P”; singular indefinite, “A certain S is P.” This classification admits of our working with the ordinary twofold distinction into universal and particular—or, as it is here expressed, definite and indefinite—wherever this is adequate, as in the traditional doctrine of the syllogism; while at the same time it introduces a further distinction which may in certain connexions be of importance.
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.
[102] With perhaps the further implication “although some S’s are P’s”; thus, Few S’s are P’s is given by Kant as an example of the exponible proposition (that is, a proposition which, though not compound in form, can nevertheless be resolved into a conjunction of two or more simpler propositions, which are independent of one another), on the ground that it contains both an affirmation and a negation, though one of them in a concealed way. It should be added that a few has not the same signification as few, but must be regarded as affirmative, and generally, as simply equivalent to some ; e.g., A few S’s are P’s = Some S’s are P’s. Sometimes, however, it means a small number, and in this case the proposition is perhaps best regarded as singular, the subject being collective. Thus, “a few peasants successfully defended the citadel” may be rendered “a small band of peasants successfully defended the citadel,” rather than “some peasants successfully defended the citadel,” since the stress is intended to be laid at least as much on the paucity of their numbers as on the fact that they were peasants. Whilst the proposition interpreted in this way is singular, not general, it is singular indefinite, not singular definite; for what small band is alluded to is left indeterminate.
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.
[103] De Morgan remarks that “a perfectly definite particular, as to quantity, would express how many X’s are in existence, how many Y’s, and how many of the X’s are or are not Y’s; as in 70 of the 100 X’s are among the 200 Y’s” (Formal Logic, p. 58). He contrasts the definite particular with the indefinite particular which is of the form Some X’s are Y’s. It will be noticed that De Morgan’s definite particular, as here defined, is still more explicit than the numerically definite proposition, as defined in the text.
69. Indefinite Propositions.—According to quantity, propositions have by some logicians been divided into (1) Universal, (2) Particular, (3) Singular, (4) Indefinite. Singular propositions have already been discussed.
By an indefinite proposition is meant one “in which the quantity is not explicitly declared by one of the designatory terms all, every, some, many, &c.”; e.g., S is P, Cretans are liars. We may perhaps say with Hamilton, that indesignate would be a better term to employ. At any rate the so-called indefinite proposition is not the expression of a distinct form of judgment. It is a form of proposition which is the imperfect expression of a judgment. For reasons already stated, the particular has more claim to be regarded as an indefinite judgment.