56 Thus, Mansel calls attention to “a class of propositions which are not, in the strict sense of the word, analytical, viz., those in which the predicate is a single term synonymous with the subject” (Mansel’s Aldrich, p. 170).
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.
32. Nature of the Analysis involved in Analytic Propositions.—Confusion is not unfrequently introduced into discussions relating to analytic propositions by a want of agreement as to the nature of the analysis involved. If identified, as above, with a division of the verbal proposition, an analytic proposition gives an analysis, partial or complete, of the connotation of the subject-term. Some writers, however, appear to have in view an analysis of the subjective intension of the subject-term. There is of course nothing absolutely incorrect in this interpretation, if consistently adhered to, but it makes the distinction between analytic and synthetic propositions logically valueless and for all practical purposes nugatory. “Both intension and extension,” says Mr Bradley, “are relative to our knowledge. And the perception of this truth is fatal to a well-known Kantian distinction. A judgment is not fixed as ‘synthetic’ or ‘analytic’: its character varies with the knowledge 54 possessed by various persons and at different times. If the meaning of a word were confined to that attribute or group of attributes with which it set out, we could distinguish those judgments which assert within the whole one part of its contents from those which add an element from outside; and the distinction thus made would remain valid for ever. But in actual practice the meaning itself is enlarged by synthesis. What is added to-day is implied to-morrow. We may even say that a synthetic judgment, so soon as it is made, is at once analytic.”62
62 Principles of Logic, p. 172. Professor Veitch expresses himself somewhat similarly. “Logically all judgments are analytic, for judgment is an assertion by the person judging of what he knows of the subject spoken of. To the person addressed, real or imaginary, the judgment may contain a predicate new—a new knowledge. But the person making the judgment speaks analytically, and analytically only; for he sets forth a part of what he knows belongs to the subject spoken of. In fact, it is impossible anyone can judge otherwise. We must judge by our real or supposed knowledge of the thing already in the mind” (Institutes of Logic, p. 237).
If by intension is meant subjective intension, and by an analytic judgment one which analyses the intension of the subject, the above statements are unimpeachable. It is indeed so obviously true that in this sense synthetic judgments are only analytic judgments in the making, that to dwell upon the distinction itself at any length would be only waste of time. It is, however, misleading to identify subjective intension with meaning ;63 and this is especially the case in the present connexion, since it may be maintained with a certain degree of plausibility that some synthetic judgments are only analytic judgments in the making, even when by an analytic judgment is meant one which analyses the connotation of the subject. For undoubtedly the connotation of names is not in practice unalterably fixed. As our knowledge progresses, many of our 55 definitions are modified, and hence a form of words which is synthetic at one period may become analytic at another.
63 Compare the following criticism of Mill’s distinction between real and verbal propositions: “If every proposition is merely verbal which asserts something of a thing under a name that already presupposes what is about to be asserted, then every statement by a scientific man is for him merely verbal” (T. H. Green, Works, ii. p. 233). This criticism seems to lose its force if we bear in mind the distinction between connotation and subjective intension.
But, in the first place, it is very far indeed from being a universal rule that newly-discovered properties of a class are taken ultimately into the connotation or intensive definition of the class-name. Dr Bain (Logic, Deduction, pp. 69 to 73) seems to imply the contrary; but his doctrine on this point is not defensible on the ground either of logical expediency or of actual practice. As to logical expediency, it is a generally recognised principle of definition that we ought to aim at including in a definition the minimum number of properties necessary for identification rather than the maximum which it is possible to include.64 And as to what actually occurs, it is easy to find cases where we are able to say with confidence that certain common properties of a class never will as a matter of fact be included in the definition of the class-name; for example, equiangularity will never be included in the definition of equilateral triangle, or having cloven hoofs in the definition of ruminant animal.
64 If we include in the definition of a class-name all the common properties of the class, how are we to make any universal statement of fact about the class at all? Given that the property P belongs to the whole of the class S, then by hypothesis P becomes part of the meaning of S, and the proposition All S is P merely makes this verbal statement, and is no assertion of any matter of fact at all. We are, therefore, involved in a kind of vicious circle.
In the second place, even when freshly discovered properties of things come ultimately to be included in the connotation of their names, the process is at any rate gradual, and it would, therefore, be incorrect to say—in the sense in which we are now using the terms—that a synthetic judgment becomes in the very process of its formation analytic. On the other hand, it may reasonably be assumed that in any given discussion the meaning of our terms is fixed, and the distinction between analytic and synthetic propositions then becomes highly significant and important. It may be added that when a name changes its meaning, any proposition in which it occurs does not strictly speaking remain the same proposition as before. We ought 56 rather to say that the same form of words now expresses a different proposition.65
65 This point is brought out by Mr Monck in the admirable discussion of the above question contained in his Introduction to Logic, pp. 130 to 134.
EXERCISES.
33. State which of the following propositions you consider real, and which verbal, giving your reasons in each case:
| (i) | All proper names are singular; |
| (ii) | A syllogism contains three and only three terms; |
| (iii) | Men are vertebrates; |
| (iv) | All is not gold that glitters; |
| (v) | The dodo is an extinct bird; |
| (vi) | Logic is the science of reasoning; |
| (vii) | Two and two are four; |
| (viii) | All equilateral triangles are equiangular; |
| (ix) | Between any two points one, and only one, straight line can be drawn; |
| (x) | Any two sides of a triangle are together greater than the third side. |
[C.]
34. Enquire whether the following propositions are real or verbal: (a) Homer wrote the Iliad, (b) Milton wrote Paradise Lost. [C.]
35. How would you characterise a proposition which is formally inferred from the conjunction of a verbal proposition with a real material proposition? Explain your view by the aid of an illustration. [J.]
36. If all x is y, and some x is z, and p is the name of those z’s which are x ; is it a verbal proposition to say that all p is y? [V.]
37. Is it possible to make any term whatever the subject (a) of a verbal proposition, (b) of a real proposition? [J.]
CHAPTER IV.
NEGATIVE NAMES AND RELATIVE NAMES.
38. Positive and Negative Names.—A pair of names of the forms A and not-A are commonly described as positive and negative respectively. The true import of the negative name not-A, including the question whether it really has any signification at all, has, however, given rise to much discussion.
Strictly speaking neither affirmation nor negation has any meaning except in reference to judgments or propositions. A concept or a term cannot be itself either affirmed or denied. If I affirm, it must be a judgment or a proposition that I affirm; if I deny, it must be a judgment or a proposition that I deny.
Starting from this position, Sigwart is led to the conclusion that, “taken literally, the formula not-A, where A denotes any idea, has no meaning whatever” (Logic, I. p. 134). Apart from the fact that the mere absence of an idea is not itself an idea, not-A cannot be interpreted to mean the absence of A in thought; for, on the contrary, it implies the presence of A in thought. We cannot, for instance, think of not-white except by thinking of white. Nor again can we interpret not-A as denoting whatever does not necessarily accompany A in thought. For, if so, A and not-A would not as a rule be exclusive or incompatible. For example, square, solid, do not necessarily accompany white in thought; but there is no opposition between these ideas and the idea of white. In order to interpret not-A as a real negation we must, says Sigwart, tacitly introduce a judgment or rather a series of judgments, 58 meaning by not-A “whatever is not A,” that is, everything whatsoever of which A must be denied. “I must review in thought all possible things in order to deny A of them, and these would be the positive objects denoted by not-A. But even if there were any use in this, it would be an impossible task” (p. 135).
Whilst agreeing with much that Sigwart says in this connexion, I cannot altogether accept his conclusion. We shall return to the question from the more controversial point of view in the following section. In the meantime we may indicate the result to which Sigwart’s general argument really seems to lead us.
We must agree that not-A cannot be regarded as representing any independent concept; that is to say, we cannot form any idea of not-A that negates the notion A. It is, therefore, true that, taken literally (that is, as representing an idea which is the pure negation of the idea A), the formula not-A is unintelligible. Regarding not-A, however, as equivalent to whatever is not A, we may say that its justification and explanation is to be found primarily by reference to the extension of the name. The thinking of anything as A involves its being distinguished from that which is not A. Thus on the extensive side every concept divides the universe with reference to which it is thought (whatever that may be) into two mutually exclusive subdivisions, namely, a portion of which A can be predicated and a portion of which A cannot be predicated. These we designate A and not-A respectively. While it may be said that A and not-A involve intensively only one concept, they are extensively mutually exclusive.
Confining ourselves to connotative names, we may express the distinction between positive and negative names somewhat differently by saying that a positive name implies the presence in the things called by the name of a certain specified attribute or set of attributes, while a negative name implies the absence of one or other of certain specified attributes. A negative name, therefore, has its denotation determined indirectly. The class denoted by the positive name is determined positively, and then the negative name denotes what is left.
59 39. Indefinite Character of Negative Names.—Infinite and indefinite are designations that have been applied to negative names when interpreted in such a way as not to involve restriction to a limited universe of discourse. For without such restriction (explicit or implicit) a negative name, for example, not-white, must be understood to denote the whole infinite or indefinite class of things of which white cannot truly be affirmed, including such entities as virtue, a dream, time, a soliloquy, New Guinea, the Seven Ages of Man.
Many logicians hold that no significant term can be really infinite or indefinite in this way.66 They say that if a term like not-white is to have any meaning at all, it must be understood as denoting, not all things whatsoever except white things, but only things that are black, red, green, yellow, etc., that is, all coloured things except such as are white. In other words, the universe of discourse which any pair of contradictory terms A and not-A between them exhaust is considered to be necessarily limited to the proximate genus of which A is a species; as, for example, in the case of white and not-white, the universe of colour.
66 This is at the root of Sigwart’s final difficulty with regard to negative names, as indicated in the preceding section. Later on he points out that in division we are justified in including negative characteristics of the form not-A in a concept, although we cannot regard not-A itself as an independent concept. Thus we may divide the concept organic being into feeling and not-feeling, a specific difference being here constituted by the absence of a characteristic which is compatible with the remaining characteristics, but is not necessarily connected with them (Logic, I. p. 278). Compare also Lotze, Logic, § 40.
It is doubtless the case that we seldom or never make use of negative names except with reference to some proximate genus. For instance, in speaking of non-voters we are probably referring to the inhabitants of some town or locality whom we subdivide into those who have votes and those who have not. In a similar way we ordinarily deny red only of things that are coloured, squareness only of things that have some figure, etc., so that there is an implicit limitation of sphere. It may be granted further that a proposition containing a negative name interpreted as infinite can have little or no practical value. But it does not follow that some limitation 60 of sphere is necessary in order that a negative term may have meaning. The argument is used that it is an utterly impossible feat to hold together in any one idea a chaotic mass of the most different things. But the answer to this argument is that we do not profess to hold together the things denoted by a negative name by reference to any positive elements which they may have in common: they are held together simply by the fact that they all lack some one or other of certain determinate elements. In other words, the argument only shews that a negative name has no positive concept corresponding to it.67 It may be added that if this argument had force, it would apply also to the subdivision of a genus with reference to the presence or absence of a certain quality. If we divide coloured objects into red and not-red, we may say equally that we cannot hold together coloured objects other than red by any positive element that they have in common: the fact that they are all coloured is obviously insufficient for the purpose.
67 For a good statement of the counter-argument, compare Mrs Ladd Franklin in Mind, January, 1892, pp. 130, 1.
A somewhat different argument is implied by Sigwart when he says, “If A = mortal, where will justice, virtue, law, order, distance find a place? They are neither mortal beings, nor yet not-mortal beings, for they are not beings at all.” The answer seems clear. They are not-(mortal beings), and therefore not-A. As a rule, it is needless to exclude explicitly from a species what does not even belong to some higher genus. But the fact of the exclusion remains.
Granting then that in practice we rarely, if ever, employ a negative name except with reference to some proximate genus, we nevertheless hold that not-A is perfectly intelligible whatever the universe of discourse may be and however wide it may be. For it denotes in that universe whatever is not denoted by the corresponding positive name. Moreover in formal processes we should be unnecessarily hampered if not allowed to pass unreservedly from X is not A to X is not-A.68
68 Writers who take the view which we are here criticising must in consistency deny the universal validity of the process of immediate inference called obversion. Thus Lotze, rightly on his own view, will not allow us to pass from spirit is not matter to spirit is not-matter ; in fact he rejects altogether the form of judgment S is not-P (Logic, § 40). Some writers, who follow Lotze on the general question here raised, appear to go a good deal further than he does, not merely disallowing such a proposition as virtue is not-blue but also such a proposition as virtue is not blue, on the ground that if we say “virtue is not blue,” there is no real predication, since the notion of colour is absolutely foreign to an unextended and abstract concept such as “virtue.” Lotze, however, expressly draws a distinction between the two forms S is non-Q and S is not Q, and tells us that “everything which it is wished to secure by the affirmative predicate non-Q is secured by the intelligible negation of Q” (Logic, § 72; cf. § 40). On the more extreme view it is wrong to say that Virtue is either blue or it is not blue ; but Lotze himself does not thus deny the universality of the law of excluded middle.
61 From this point of view attention may be called to the difference in ordinary use between such forms as unholy, immoral, discourteous and such forms as non-holy, non-moral, non-courteous. The latter may be used with reference to any universe of discourse, however extensive. But not so the former; in their case there is undoubtedly a restriction to some universe of discourse that is more or less limited in its range. We can, for example, speak of a table as non-moral, although we cannot speak of it as immoral. A want of recognition of this distinction may be partly responsible for the denial that any terms can properly be described as infinite or indefinite.69
69 It should be added that in the ordinary use of language the negative prefix does not always make a term negative as here defined. Thus, as Mill points out, “the word unpleasant, notwithstanding its negative form, does not connote the mere absence of pleasantness, but a less degree of what is signified by the word painful, which, it is hardly necessary to say, is positive.” On the other hand, some names positive in form may, with reference to a limited universe of discourse, be negative in force; e.g., alien, foreign. Another example is the term Turanian, as employed in the science of language. This term has been used to denote groups lying outside the Aryan and Semitic groups, but not distinguished by any positive characteristics which they possess in common.
40. Contradictory Terms.—A positive name and the corresponding negative are spoken of as contradictory. We may define contradictory terms as a pair of terms so related that between them they exhaust the entire universe to which reference is made, whilst in that universe there is no individual of which both can be affirmed at the same time. It is desirable to repeat here that contradiction can exist primarily between 62 judgments or propositions only, so that as applied to terms or ideas the notion of contradiction must be interpreted with reference to predication. A and not-A are spoken of as contradictory because they cannot without contradiction be predicated together of the same subject. Thus it is in their exclusive character that they are termed contradictory; as between them exhausting the universe of discourse they might rather be called complementary.70
70 Dr Venn (Empirical Logic, p. 191) distinguishes between formal contradictories and material contradictories, according as the relation in which the pair of terms stand to one another is or is not apparent from their mere form. Thus A and not-A are formal contradictories; so are human and non-human. Material contradictories, on the other hand, are not constructed “for the express purpose of indicating their mutual relation.” No formal contradiction, for example, is apparent between British and Foreign, or between British and Alien ; and yet “within their range of appropriate application—which in the latter case includes persons only, and in the former case is extended to produce of most kinds—these two pairs of terms fulfil tolerably well the conditions of mutual exclusion and collective exhaustion.”
41. Contrary Terms.—Two terms are usually spoken of as contrary71 to one another when they denote things which can be regarded as standing at opposite ends of some definite scale in the universe to which reference is made; e.g., first and last, black and white, wise and foolish, pleasant and painful.72 Contraries differ from contradictories in that they admit of a mean, and therefore do not between them exhaust the entire universe of discourse. It follows that, although two contraries cannot both be true of the same thing at the same time, they may both be false. Thus, a colour may be neither black nor 63 white, but blue; a feeling may be neither pleasant nor painful, but indifferent.
71 De Morgan uses the terms contrary and contradictory as equivalent, his definition of them corresponding to that given in the preceding section.
72 It has been already pointed out that the negative prefix does not always make a term really negative in force. Thus pleasant and unpleasant are not contradictories, for they admit of a mean; when we say that anything is unpleasant, we intend something more than the mere denial that it is pleasant. It should be added that a pair of terms of this kind may also fail to be contraries as above defined, since while admitting of a mean they may at the same time not denote extremes. Unpleasant, for example, denotes only that which is mildly painful: unless intended ironically, it would be a misuse of terms to speak of the tortures of the Inquisition as merely unpleasant. Compare Carveth Read, Logic, p. 49.
It will be observed that not every term has a contrary as above defined, for the thing denoted by a term may not be capable of being regarded as representing the extreme in any definite scale. Thus blue can hardly be said to have a contrary in the universe of colour, or indifferent in the universe of feeling.
By some writers, the term contrary is used in a wider sense than the above, contrariety being identified with simple incompatibility (a mean between the two incompatibles being possible); thus, blue and yellow equally with black, would in this sense be called contraries of white.73 Other writers use the term repugnant to express the mere relation of incompatibility; thus red, blue, yellow are in this sense repugnant to one another.74
73 There is much to be said in favour of this wider use of the term contrary. Compare the discussion of contrary propositions in section 81.
74 So long as we are confined to simple terms the relations of contrariety and repugnancy cannot be expressed formally or in mere symbols. But it is otherwise when we pass on to the consideration of complex terms. Thus, while XY and not-X or not-Y are formal contradictories, XY and X not-Y may be said to be formal repugnants, XY and not-X not-Y formal contraries (in the narrower of the two senses indicated above).
42. Relative Names.—A name is said to be relative, when, over and above the object that it denotes, it implies in its signification another object, to which in explaining its meaning reference must be made. The name of this other object is called the correlative of the first. Non-relative names are sometimes called absolute.
Jevons considers that in certain respects all names are relative. “The fact is that everything must really have relations to something else, the water to the elements of which it is composed, the gas to the coal from which it is manufactured, the tree to the soil in which it is rooted “ (Elementary Lessons in Logic, p. 26). Again, by the law of relativity, consciousness is possible only in circumstances of change. We cannot think of any object except as distinguished from something else. Every term, therefore, implies its negative as an object 64 of thought. Take the term man. It is an ambiguous term, and in many of its meanings is clearly relative,—for example, as opposed to master, to officer, to wife. If in any sense it is absolute it is when opposed to not-man; but even in this case it may be said to be relative to not-man. To avoid this difficulty, Jevons remarks, “Logicians have been content to consider as relative terms those only which imply some peculiar and striking kind of relation arising from position in time or space, from connexion of cause and effect, &c.; and it is in this special sense, therefore, that the student must use the distinction.”
A more satisfactory solution of the difficulty may be found by calling attention to the distinction already drawn between the point of view of connotation (which has to do with the signification of names) and the subjective and objective points of view respectively. From the subjective point of view all notions are relative by the law of relativity above referred to. Again, from the objective point of view all things, at any rate in the phenomenal world, are relative in the sense that they could not exist without the existence of something else; e.g., man without oxygen, or a tree without soil. But when we say that a name is relative, we do not mean that what it denotes cannot exist or be thought about without something else also existing or being thought about; we mean that its signification cannot be explained without reference to something else which is called by a correlative name, e.g., husband, parent. It cannot be said that in this sense all names are relative.
The fact or facts constituting the ground of both correlative names is called the fundamentum relationis. For example, in the case of partner, the fact of partnership; in the case of husband and wife, the facts which constitute the marriage tie; in the case of ruler and subject, the control which the former exercises over the latter.
Sometimes the relation which each correlative bears to the other is the same; for example, in the case of partner, where the correlative name is the same name over again. Sometimes it is not the same; for example, father and son, slave-owner and slave. 65
The consideration of relative names is not of importance except in connexion with the logic of relatives, to which further reference will be made subsequently.
EXERCISES.
43. Give one example of each of the following,—(i) a collective general name, (ii) a singular abstract name, (iii) a connotative singular name, (iv) a connotative abstract name. Add reasons justifying your example in each case. [K.]
44. Discuss the logical characteristics of the following names:—beauty, fault, Mrs Grundy, immortal, nobility, slave, sovereign, the Times, truth, ungenerous. [K.]
[In discussing the character of any name it is necessary first of all to determine whether it is univocal, that is, used in one definite sense only, or equivocal (or ambiguous), that is, used in more senses than one. In the latter case, its logical characteristics may vary according to the sense in which it is used.]
45. It has been maintained that the doctrine of terms is extra-logical. Justify or controvert this position. [J.]
PART II.
PROPOSITIONS.
CHAPTER I.
IMPORT OF JUDGMENTS AND PROPOSITIONS.
46. Judgments and Propositions.—In passing to the next division of our subject we are confronted, first of all, with a question which is partly, but not entirely, a question of phraseology. Shall we speak of the judgment or of the proposition? The usage of logicians differs widely. Some treat almost exclusively of judgments; others almost exclusively of propositions. It will be found that for the most part the former are those who tend to emphasise the psychological or the metaphysical aspects of logic, while the latter are those who are more inclined to develop the symbolic or the material aspects.
To a certain extent it is a matter of little importance which of the alternatives is ostensively adopted. Those who deal with judgments from the logical standpoint must when pressed admit that they can deal with them only as expressed in language, and all their illustrations necessarily consist of judgments expressed in language. But a judgment expressed in language is precisely what is meant by a proposition. Hence in treating of judgments it is impossible not to treat also of propositions. 67
On the other hand, so far as we treat of propositions in logic, we treat of them not as grammatical sentences, but as assertions, as verbal expressions of judgments. The logical proposition is the proposition as understood; and a proposition as understood is a judgment. Hence in treating of propositions in logic we necessarily treat also of judgments.
In a large degree, then, the problem does resolve itself into a merely verbal question. At the same time, reasons and counter-reasons may be adduced in favour of the one alternative and in favour of the other.
On the one side, it is said that the use of the term proposition tends to confuse the sentence as a grammatical combination of words with the proposition as apprehended and intellectually affirmed; and it is urged that in treating of propositions the logician tends to become a mere grammarian.
On the other side, it is submitted that the logician is primarily concerned, not with the process of judgment, the discussion of which belongs to the sphere of psychology, but with judgment as a product, and moreover that he is concerned with this product only in so far as it assumes a fixed and definite form, which it cannot do until it receives verbal expression; and it is urged that if we concentrate our attention on judgments without explicit regard to their expression in language, our treatment tends to become too psychological.
It has been said above that logically we can deal with judgments only as expressed in propositions; and no doubt all judgments can with more or less effort be so expressed. But as a matter of fact we constantly judge in a vague sort of way without the precision that is necessary even in loose modes of expression, and we find that to give expression to our judgments may sometimes require very considerable effort. It must be remembered that logic has in view an ideal. Its object is to determine the conditions to which valid judgments must conform, and it is concerned with the characteristics of actual judgments only in subordination to this end. From this point of view it is specially important that we should deal with judgments in the only form in which it is possible for them to attain precision; and this consideration appears to be conclusive in favour of our 68 treating explicitly of propositions in some part at any rate of a logical course.
No doubt in dealing with propositions we have to raise certain questions that relate to the usage of language. Unfortunately the same propositional form may be understood as expressing very different judgments. It is therefore requisite that in any scientific treatment of logic we should discuss the interpretation of the propositional forms that we recognise. This problem is akin to the problem of definition which has to be faced sooner or later in every science; and, as is also true of a definition, the solution in any particular case is largely of the nature of a convention. But this does not detract from its importance as conducing to clearness of thought.
The question of the interpretation of propositional forms is as a matter of fact one that cannot be altogether avoided on any treatment of logic; and it is of importance to recognise explicitly that in discussing this question we are not dealing with judgments pure and simple. Words are like mathematical symbols, and the meaning of a given form of words is not something inherent either in the words themselves or in the thoughts that they may represent, but is dependent on a convention established by those who employ the words. In the force of a given judgment, however, there can be nothing that is dependent on convention. This distinction is not always remembered by those who confine their attention mainly to judgments, and they are consequently sometimes led to express themselves with an appearance of dogmatism on questions that do not really admit of dogmatic treatment.
But while in certain aspects of logical enquiry it is requisite to deal explicitly with propositions, it must never be forgotten that as logicians we are concerned with propositions only as the expressions of judgments; and there are numerous occasions when we have to go behind propositional forms and ask what are the fundamental characteristics of the judgments that they express.
47. The Abstract Character of Logic.—Reference has been made in the preceding section to the necessity for logical purposes of making our judgments precise. For only if they 69 are precise is it possible to determine with accuracy what are their logical implications considered either individually or in conjunction with one another. It has also been pointed out that we can make our judgments precise only by expressing them in propositional forms, the interpretation of which has been agreed upon.
But this is not without its disadvantages. Sometimes the full force of an actual judgment hardly admits of being expressed in words, and even the force of a proposition as understood may not be found exclusively in the words of which it composed, but may depend partly on the context in which it is placed. Hence the isolated proposition must frequently be regarded as in a sense an abstraction, leaving behind it some portion of the actual judgment for which it stands.
This is indeed much less true of the propositions of science than of those of everyday life; and the more fully a statement is independent of context the more fully may it be regarded as fulfilling its purpose from the scientific standpoint. Still the abstract character of logic must be frankly recognised. “Just as thought is abstract in its dealings with reality, so logic is abstract in its dealings with ordinary thought.”75