The distinction between real and verbal propositions as above given assumes that the use of terms is fixed by their connotation and that this connotation is determinate.[57] Whether any given proposition is as a matter of fact verbal or real will depend on the meaning attached to the terms which it contains; and it is clear that logic cannot lay down any rule for determining under which category any given proposition should be placed.[58] Still, while we cannot with certainty distinguish a real proposition by its form, it may be observed that the attachment of a sign of quantity, such as all, every, some, &c., to the subject of a proposition may in general be regarded as an indication that in the view of the person laying down the 52 proposition a fact is being stated and not merely a term explained. Verbal propositions, on the other hand, are usually unquantified or indesignate (see section [69]). For example, in order to give a partially correct idea of the meaning of such a name as square, we should not say “all squares are four-sided figures,” or “every square is a four-sided figure,” but “a square is a four-sided figure.”[59]
[57] We can, however, adapt the distinction to the case in which the use of terms is fixed by extensive definition. We may say that whilst a proposition (expressed affirmatively and with a copula of inclusion) is intensively verbal when the connotation of the predicate is a part or the whole of the connotation of the subject, it is extensively verbal when the subject taken in extension is a part or the whole of the extensive definition of the predicate. Thus, if the use of the term metal is fixed by an extensive definition, that is to say, by the enumeration of certain typical metals, of which we may suppose iron to be one, then it is a verbal proposition to say that iron is a metal. If, however, tin is not included amongst the typical metals, then it is a real proposition to say that tin is a metal.
[58] It does not follow from this that the distinction between verbal and real propositions is of no logical importance. Although the logician cannot quâ logician determine in doubtful cases to which category a given proposition belongs, he can point out what are the conditions upon which this depends, and he can shew that in any discussion or argument no progress is possible until it is clearly understood by all who are taking part whether the propositions laid down are to be interpreted as being real or merely verbal. To refer to an analogous case, it will not be said that the distinction between truth and falsity is of no logical importance because the logician cannot quâ logician determine whether a given proposition is true or false.
[59] It should be added that we may formally distinguish a full definition from a real proposition by connecting the subject and the predicate by the word “means” instead of the word “is.”
(3) There are propositions usually classed as verbal which ought rather to be placed in a class by themselves, namely, those which are valid whatever may be the meaning of the terms involved; e.g., All A is A, No A is not-A, All Z is either B or not-B, If all A is B then no not-B is A, If all A is B and all B is C then all A is C. These may be called formal propositions, since their validity is determined by their bare form.[60]
[60] Propositions which are in appearance purely tautologous have sometimes an epigrammatic force and are used for rhetorical purposes, e.g., A man’s a man (for a’ that). In such cases, however, there is usually an implication which gives the proposition the character of a real proposition; thus, in the above instance the true force of the proposition is that Every man is as such entitled to respect. “In the proposition, Children are children, the subject-term means only the age characteristic of childhood; the predicate-term the other characteristics which are connected with it. By the proposition, War is war, we mean to say that when once a state of warfare has arisen, we need not be surprised that all the consequences usually connected with it appear also. Thus the predicate adds new determinations to the meaning in which the subject was first taken” (Sigwart, Logic, I. p. 86).
Formal propositions are the only propositions whose validity is examined and guaranteed by logic itself irrespective of other sources of knowledge, and many of the results reached in formal logic may be summed up in such propositions; for any formally valid reasoning can be expressed by a formal hypothetical proposition as in the last two of the examples given above.
A formal proposition as here defined must not be confused with a proposition expressed in symbols. A formal proposition need not indeed be expressed in symbols at all. Thus, the proposition An animal is an animal is a formal proposition; 53 All S is P is not. Strictly speaking, a symbolic expression, such as All S is P, is to be regarded as a propositional form, rather than as a proposition per se. For it cannot be described as in itself either true or false. What we are largely concerned with in logic are relations between propositional forms; because these involve corresponding relations between all propositions falling into the forms in question.
We have then three classes of propositions—formal, verbal, and real—the validity or invalidity of which is determined respectively by their bare form, by the mere meaning or application of the terms involved, by questions of fact concerning the things denoted by these terms.[61]
[61] Real propositions are divided into true and false according as they do or do not accurately correspond with facts. By verbal and formal propositions we usually mean propositions which from the point of view taken are valid. A proposition which from either of these points of view is invalid is spoken of as a contradiction in terms. Properly speaking we ought to distinguish between a verbal contradiction in terms and a formal contradiction in terms, the contradiction depending in the first case upon the force of the terms employed and in the second case upon the mere form of the proposition; e.g., Some men are not animals, A is not-A. Any purely formal fallacy may be said to resolve itself into a formal contradiction in terms. It should be added that a mere term, if it is complex, may involve a contradiction in terms; e.g., Roman Catholic (if the separate terms are interpreted literally), A not-A.