Studies and Exercises in Formal Logic

173 Comparing the above with the five ordinary Eulerian diagrams (which may be designated α, β &c. as in section 126), it will be seen that (i) corresponds to α; (ii) to β; (iii) to γ; (iv) and (v) represent the two cases now yielded by δ; (vi) and (vii) the two yielded by ε.

Our seven diagrams might also be arrived at as follows:—Every part of the universe must be either S or Sʹ, and also P or Pʹ ; and hence the mutually exclusive combinations SP, SPʹ, SʹP, SʹPʹ must between them exhaust the universe. The case in which these combinations are all to be found is represented by diagram (iv); if one but one only is absent we obviously have four cases which are represented respectively by (ii), (iii), (v), and (vi); if only two are to be found it will be seen that we are limited to the cases represented by (i) and (vii) or we should not fulfil the condition that neither S nor Sʹ, P nor Pʹ, is to be altogether non-existent; for the same reason the universe cannot contain less than two of the four combinations. We thus have the seven cases represented by the diagrams, and these are shewn to exhaust the possibilities.

174 The four traditional propositions are related to the new scheme as follows:—
 A  limits us to (i) or (ii);
 I   to (i), (ii), (iii), (iv), or (v);
 E  to (vi) or (vii);
 O  to (iii), (iv), (v), (vi), or (vii).

It will be observed that the new scheme is in itself more symmetrical than Euler’s, and also that it succeeds better in bringing out the symmetry of the fourfold schedule of propositions.176 A and E give two alternatives each, I and O give five each; whereas with Euler’s scheme E gives only one alternative, A two, O three, I four, and it might, therefore, seem as if E afforded more definite and unambiguous information than A, and O than I, which is not really the case. Further, the problem of expressing each diagram propositionally yields a more symmetrical result than the corresponding problem in the case of Euler’s diagrams.

This sevenfold scheme of class relations should be compared with the sevenfold scheme of relations between propositions given in section 84.

131. Lambert’s diagram and the class-relations between S, not-S, P, not-P.—The following is a compact diagrammatic representation of the seven possible class-relations between S, not-S, P, not-P, based upon Lambert’s scheme. 175

176 In each case the full extent of a line represents the entire universe of discourse; any portion of a line that is dotted may be either S or Sʹ (or P or Pʹ, as the case may be).

135. Fourfold Implication of Propositions in Connotation and Denotation.—In dealing with the question whether propositions assert a relation between objects or between attributes or between objects and attributes, logicians have been apt to commit the fallacy of exclusiveness, selecting some one of the given alternatives, and treating the others as necessarily excluded thereby. It follows, however, from the double aspect of names—in extension and intension—that the different relations really involve one another, so that all of them are implied in any categorical proposition whose subject and predicate are both general names.177 If any one of the relations is selected as constituting the meaning of the proposition, the other relations are at any rate involved as implications.

The problem will be made more definite if we confine ourselves to a consideration of connotation and denotation in the strict sense, as distinguished from comprehension and exemplification, our terms being supposed to be defined intensively.178 Both subject and predicate will then have a denotation determined by their connotation, and hence our 178 proposition may be considered from four different points of view, which are not indeed really independent of one another, but which serve to bring different aspects of the proposition into prominence. (1) The subject may be read in denotation and the predicate in connotation; (2) both terms may be read in denotation; (3) both terms may be read in connotation; (4) the subject may be read in connotation and the predicate in denotation.

As an example, we may take the proposition, All men are mortal.179 According to our point of view, this proposition may be read in any of the following ways:
 (1) The objects denoted by man possess the attributes connoted by mortal ;
 (2) The objects denoted by man are included within the class of objects denoted by mortal ;
 (3) The attributes connoted by man are accompanied by the attributes connoted by mortal ;
 (4) The attributes connoted by man indicate the presence of an object belonging to the class denoted by mortal.

It should be specially noticed that a different relation between subject and predicate is brought out in each of these four modes of analysing the proposition, the relations being respectively (i) possession, (ii) inclusion, (iii) concomitance, (iv) indication.

It may very reasonably be argued that a certain one of the above ways of regarding the proposition is (a) psychologically the most prominent in the mind in predication; or (b) the most fundamental in the sense of making explicit that relation which ultimately determines the other relations; or (c) the most convenient for a given purpose, e.g., for dealing with the problems of formal logic. We need not, however, select the same mode of interpretation in each case. There would, for example, be nothing inconsistent in holding that to read the 179 subject in denotation and the predicate in connotation is most correct from the psychological standpoint; to read both terms in connotation the most ultimate, inasmuch as connotation determines denotation and not vice versâ, and to read both terms in denotation the most serviceable for purposes of logical manipulation. To say, however, that a certain one of the four readings alone can be regarded as constituting the import of the proposition to the exclusion of the others cannot but be erroneous. They are in truth so much implicated in one another, that the difficulty may rather be to justify a treatment which distinguishes between them.180

(1) Subject in denotation, predicate in connotation.
 If we read the subject of a proposition in denotation and the predicate in connotation, we have what is sometimes called the predicative mode of interpreting the proposition. This way of regarding propositions most nearly corresponds in the great majority of cases with the course of ordinary thought;181 that is to say, we naturally contemplate the subject as a class of objects of which a certain attribute or complex of attributes is predicated. Such propositions as All men are mortal, Some violets are white, All diamonds are combustible, may be taken as examples. Dr Venn puts the point very clearly with reference to the last of these three propositions: “If I say that ‘all diamonds are combustible,’ I am joining together two connotative terms, each of which, therefore, implies an attribute and denotes a class; but is there not a broad distinction in respect of the prominence with which the notion of a class is presented to the mind in the two cases? As regards the diamond, we think at once, or think very speedily, of a class of things, the distinctive attributes of the subject being mainly used to carry the mind on to the contemplation of the objects referred to by them. But as regards the combustibility, the attribute itself is the prominent thing … Combustible things, other than the diamond itself, come scarcely, if at all, under 180 contemplation. The assertion in itself does not cause us to raise a thought whether there be other combustible things than these in existence” (Empirical Logic, p. 219).

Two points may be noticed as serving to confirm the view that generally speaking the predicative mode of interpreting propositions is psychologically the most prominent:
 (a) The most striking difference between a substantive and an attributive (i.e., an adjective or a participle) from the logical point of view is that in the former the denotation is usually more prominent than the connotation, even though it may be ultimately determined by the connotation, whilst in the latter the connotation is the more prominent, even though the name must be regarded as the name of a class of objects if it is entitled to be called a name in the strict logical sense at all. Corresponding to this we find that the subject of a proposition is almost always a substantive, whereas the predicate is more often an attributive.
 (b) It is always the denotation of a term that we quantify, never the connotation. Whether we talk of all men or of some men, the complex of attributes connoted by man is taken in its totality; the distinction of quantity relates entirely to the denotation of the term. Corresponding to this, we find that we naturally regard the quantity of a proposition as pertaining to its subject, and not to its predicate. It will be shewn in the following chapter that the doctrine of the quantification of the predicate has at any rate no psychological justification.

There are, however, numerous exceptions to the statement that the subject of a proposition is naturally read in denotation and the predicate in connotation; for example, in the classificatory sciences. The following propositions may be taken as instances: All palms are endogens, All daisies are compositae, None but solid bodies are crystals, Hindoos are Aryans, Tartars are Turanians. In such cases as these most of us would naturally think of a certain class of objects as included in or excluded from another class rather than as possessing or not possessing certain definite attributes; in other words, as Dr Venn puts it, “the class-reference of the predicate is no less definite than that of the subject” (Empirical Logic, p. 220). 181 In the case of such a proposition as No plants with opposite leaves are orchids, the position is even reversed, that is to say, it is the subject rather than the predicate that we should more naturally read in connotation. We may pass on then to other ways of regarding the categorical proposition.

The class mode of interpreting categorical propositions is nevertheless treated by some writers as being positively 182 erroneous. But the arguments used in support of this view will not bear examination.

(ii) It is asked what we mean by a class, by the class of birds, for example, when we say All owls are birds. “It is nothing existing in space; the birds of the world are nowhere collected together so that we can go and pick out the owls from amongst them. The classification is a mental abstraction of our own, founded upon the possession of certain definite attributes. The class is not definite and fixed, and we do not find out whether any individual belongs to it by going over a list of its members, but by enquiring whether it possesses the necessary attributes.”182 In so far as this argument applies against reading the predicate in denotation, it applies equally against reading the subject in denotation. It is in effect the argument used by Mill (Logic, i. 5, § 3) in order to lead up to his position that the ultimate interpretation of the categorical proposition requires us to read both subject and predicate in connotation, since denotation is determined by connotation. But if this be granted, it does not prove the class reading of the 183 proposition erroneous; it only proves that in the class reading, the analysis of the import of the proposition has not been carried as far as it admits of being carried.

(iii) It is argued that when we regard a proposition as expressing the inclusion of one class within another, even then the predicate is only apparently read in denotation. “On this view, we do not really assert P but ‘inclusion in P,’ and this is therefore the true predicate. For example, in the proposition ‘All owls are birds,’ the real predicate is, on this view, not ‘birds’ but ‘included in the class birds.’ But this inclusion is an attribute of the subject, and the real predicate, therefore, asserts an attribute. It is meaningless to say ‘Every owl is the class birds,’ and it is false to say ‘The class owls is the class birds.’”183 This argument simply begs the question in favour of the predicative mode of interpretation. It overlooks the fact that the precise kind of relation brought out in the analysis of a proposition will vary according to the way in which we read the subject and the predicate. An analogous argument might also be used against the predicative reading itself. Take the proposition, “All men are mortal.” It is absurd to say that “Every man is the attribute mortality,” or that “The class men is the attribute mortality.”

(iv) It is said that a class interpretation of both S and P would lead properly to a fivefold, not a fourfold, scheme of propositions, since there are just five relations possible between any two classes, as is shewn by the Eulerian diagrams. This contention has force, however, only upon the assumption that we must have quite determinate knowledge of the class relation between S and P before being able to make any statement on the subject; and this assumption is neither justifiable in itself nor necessarily involved in the interpretation in question. It may be added that if a similar view were taken on the adoption of the predicative mode of interpretation, we should have a threefold, not a fourfold scheme. For then the quantity of our subject at any rate would have to be perfectly determinate, and with S and P for subject and predicate, the three possible statements would be—All S is P, Some S is P and 184 some is not, No S is P. The point here raised will presently be considered further in connexion with the quantification of the predicate.

(3) Subject in connotation, predicate in connotation.
 If we read both the subject and the predicate of a proposition in connotation, we have what may be called the connotative mode of interpreting the proposition. In the proposition All S is P, the relation expressed between the attributes connoted by S and those connoted by P is one of concomitance—“the attributes which constitute the connotation of S are always found accompanied by those which constitute the connotation of P.”184 Similarly, in the case of Some S is P,—“the attributes 185 which constitute the connotation of S are sometimes found accompanied by those which constitute the connotation of P”; No S is P,—“the attributes which constitute the connotation of S are never found along with those which constitute the connotation of P”; Some S is not P,—“the attributes which constitute the connotation of S are sometimes found unaccompanied by those which constitute the connotation of P.”

It will be noticed that in the connotative reading we have always to take the attributes which constitute the connotation collectively. In other words, by the attributes constituting the connotation of a term we mean those attributes regarded as a whole. Thus, No S is P does not imply that none of the attributes connoted by S are ever accompanied by any of those connoted by P. This is apparent if we take such a proposition as No oxygen is hydrogen. It follows that when the subject is read in connotation the quantity of the proposition must appear as a separate element, being expressed by the word “always” or “sometimes,” and must not be interpreted as meaning “all” or “some” of the attributes included in the connotation of the subject.

It is argued by those who deny the possibility of the connotative mode of interpreting propositions, that this is not really reading the subject in connotation at all; always and sometimes are said to reduce us to denotation at once. In reply to this, it must of course be allowed that real propositions affirm no relation between attributes independently of the objects to which they belong. The connotative reading implies the denotative, and we must not exaggerate the nature of the distinction between them. Still the connotative reading presents the import of the proposition in a new aspect, and there is at any rate a prima facie difference between regarding one class as included within another, and regarding one attribute as always accompanied by another, even though a little 186 consideration may shew that the two things mutually involve one another.185

(4) Subject in connotation, predicate in denotation.
 Taking the proposition All S is P, and reading the subject in connotation and the predicate in denotation, we have, “The attributes connoted by S are an indication of the presence of an individual belonging to the class P.” This mode of interpretation is always a possible one, but it must be granted that only rarely does the import of a proposition naturally present itself to our minds in this form. There are, however, exceptional cases in which this reading is not unnatural. The proposition No plants with opposite leaves are orchids has already been given as an example. Another example is afforded by the proposition All that glitters is not gold. Taking the subject in connotation and the predicate in denotation we have, The attribute of glitter does not always indicate the presence of a gold object ; and it will be found that this reading of the proverb serves to bring out its meaning really better than any of the three other readings which we have been discussing.

It is worth while noticing here by way of anticipation that on any view of the existential interpretation of propositions, as discussed in chapter 8, we shall still have a fourfold reading of categorical propositions in connotation and denotation. The universal negative and the particular affirmative may be taken as examples, on the supposition that the former is interpreted as existentially negative and the latter as existentially affirmative. The universal negative yields the following: (1) There is no individual belonging to the class S and possessing the attributes connoted by P ; (2) There is no individual common to the two classes S and P ; (3) The attributes 187 connoted by S and P respectively are never found conjoined; (4) There is no individual possessing the attributes connoted by S and belonging to the class P. Similarly the particular affirmative yields: (1) There are individuals belonging to the class S and possessing the attributes connoted by P ; (2) There are individuals common to the two classes S and P ; (3) The attributes connoted by S and P respectively are sometimes found conjoined; (4) There are individuals possessing the attributes connoted by S and belonging to the class P. We see, therefore, that the question discussed in this section is independent of that which will be raised in chapter 8; and that for this reason, if for no other, no solution of the general problem raised in the present chapter can afford a complete solution of the problem of the import of categorical propositions.

136. The Reading of Propositions in Comprehension.—If, in taking the intensional standpoint, we consider comprehension instead of connotation, our problem is to determine what relation is implied in any proposition between the comprehension of the subject and the comprehension of the predicate. This question being asked with reference to the universal affirmative proposition All S is P, the solution clearly is that the comprehension of S includes the comprehension of P. The interpretation in comprehension is thus precisely the reverse of that in denotation (the denotation of S is included in the denotation of P); and we might be led to think that, taking the different propositional forms, we should have a scheme in comprehension, analogous throughout to that in denotation. But this is not the case, for the simple reason that in our ordinary statements we do not distributively quantify comprehension in the way in which we do denotation; in other words, comprehension is always taken in its totality. Thus, reading an I proposition in denotation we have—the classes S and P partly coincide ; and corresponding to this we should have—the comprehensions of S and P partly coincide. But this is clearly not what we express by Some S is P ; for the partial coincidence of the comprehensions of S and P is quite compatible with No S is P, that is to say, the classes S and P may be mutually exclusive, and yet some attributes may be common to the whole of S and 188 also to the whole of P ; for example, No Pembroke undergraduates are also Trinity undergraduates. Again, given an E proposition, we have in denotation—the classes S and P have no part in common ; but for the reason just given, it does not follow that the comprehension of S and the comprehension of P have nothing in common.

It is, however, important to recognise that in thus borrowing a symbol from mathematics we do not retain its meaning unaltered, and that a so-called logical equation is, therefore, something very different from a mathematical equation. In mathematics the symbol of equality generally means numerical or quantitative equivalence. But clearly we do not mean to express mere numerical equality when we write equilateral triangles = equiangular triangles. Whatever this so-called equation signifies, it is certainly something more than that there are precisely as many triangles with three equal sides as there are triangles with three equal angles. It is further clear that we do not intend to express mere similarity. Our meaning is that the denotations of the terms which are equated are absolutely identical; in other words, that the class of objects denoted by the term equilateral triangle is absolutely identical with the class of objects denoted by the term equiangular triangle.186 It may, however, be objected that, if this 190 is what our equation comes to, then inasmuch as a statement of mere identity is empty and meaningless, it strictly speaking leaves us with nothing at all; it contains no assertion and can represent no judgment. The answer to this objection is that whilst we have identity in a certain respect, it is erroneous to say that we have mere identity. We have identity of denotation combined with diversity of connotation, and, therefore, with diversity of determination (meaning thereby diversity in the ways in which the application of the two terms identified is determined).187 The meaning of this will be made clearer by the aid of one or two illustrations. Taking, then, as examples the logical equations already given, we may analyse their meaning as follows. If out of all triangles we select those which possess the property of having three equal sides, and if again out of all triangles we select those which possess the property of having three equal angles, we shall find that in either case we are left with precisely the same set of triangles. Thus, each side of our equation denotes precisely the same class of objects, but the class is determined or arrived at in two different ways. Similarly, if we select all plants that are exogenous and again all plants that are dicotyledonous, our results are precisely the same although our mode of arriving at them has been different. Once more, if we simply take the class of objects which possess the attribute of humanity, and again the class which possess both this attribute and also the attribute of mortality, the objects selected will be the same; none will be excluded by our second method of selection although an additional attribute is taken into account.

Since the identity primarily signified by a logical equation is an identity in respect of denotation, any equational mode of reading propositions must be regarded as a modification of the 191 “class” mode. What has been said above, however, will make it clear that here as elsewhere denotation is considered not to the exclusion of connotation but as dependent upon it; and we again see how denotative and connotative readings of propositions are really involved in one another, although one side or the other may be made the more prominent according to the point of view which is taken.

138. Types of Logical Equations188—Jevons (Principles of Science, chapter 3) recognises three types of logical equations, which he calls respectively simple identities, partial identities, and limited identities.