W. E. Johnson, Logic: Part I (1921)
§ 1. Under the general problem of the nature of negation we may begin by considering the particular form of negation which has been called 'pure negation.' There appear to be several different meanings attached to the notion of pure negation: it may mean the simple attitude of rejection, as opposed to that of acceptance, towards a proposition taken as a unit and without further analysis. Such negation may be called pure, because the negative element does not enter within the content of the assertum, but expresses merely a certain mental attitude to the proposition itself. According to this definition of pure negation, the judgment which may be called purely negative has as its object precisely what I have called a secondary proposition in my previous discussion as to whether the statement 'p is false' is to be regarded as primary or as secondary. When then we enquire as to the importance or the relevance of pure negation, we may be raising the question whether a judgment expressed in this purely negative form really ever represents a genuine attitude of thought. No doubt there are not many cases in which this negative attitude towards an assertum taken as a unit could be illustrated; but we may at least insist that, when some assertum is proposed which can be clearly conceived in thought, and yet repels any attempt to accept it, then the attitude towards such an assertum to which our thinking process has led us is strictly to be called that of pure negation. For example, the proposition 'Matter exists' may appear to some philosophers to have in it a sufficiently clear content to enable them to reject it, without their having in mind any correspondingly clear substitute which they can accept. In this case their mental attitude towards the proposed assertum may be properly called one of mere negation; since the only positive element involved is the conceived content of the proposition rejected.
CHAPTER V
NEGATIONBut the term pure negation is more generally applied where a predicate is denied of some subject within the proposition. Under this head, the case where negation would seem to be quite pure may be illustrated by a proposition like -- 'Wisdom is not blue.' Such a proposition would have purpose only in a logical contex where we are pointing out that certain types of adjective cannot be predicated of certain types of substantive. A more common case which leads to a purely negative form of predication, is where, for instance, a distant object of perception, is considered as to whether it is blue or of some other colour, or as to whether it is a man or some other material body. Towards this proposed assertum -- that it is blue, or that it is a man -- our attitude may be that of mere denial, in the sense that we are perfectly clear what it is not, but we are not correspondingly clear as to what it is. We may admit that a judgment which in this sense is merely negative and without any positive content is rare, since when we deny of a flower that it is red, we are at least judging that it has some colour, and similarly when we deny of something in sight that it has the shape of a man we are at least judging that it has some shape, and this constitutes a positive element in our judgment. The above examples illustrate two applications of the notion of negation: first, in denying the proposition as a whole, and again in denying that an adjective of a certain type can be predicated of a certain type of substantive, where the positive element is evanescent; and secondly, in denying the more specific predicate proposed for a substantive while tacitly asserting some wider predicate under which it falls, where a positive element is properly to be recognised.
Some logicians, going one step further, have asserted that, in denying an object to be red, not only is the generic adjective colour a positive factor in the judgment, but that some specific colour other than red is tacitly affirmed: that is, they hold that we cannot deny unless we have some positive determinate ground for our denial. But this reason for asserting the universal presence of a positive factor in judgment must not be confused with the former; for it is one thing to say that the denying of any proposed adjective involves the affirming of some other adjective of the same generic kind, and another thing to say that it involves the affirming of a specific adjective. While admitting the first, I reject the view that in denying red we are affirming say green or blue as the case may be, on the ground that it involves a confusion between what is necessarily determined in fact with what may or may not be determinate in our knowledge of fact. There are countless cases of our denying a certain proposed adjective in which, while we know that some determinate adjective can be truly applied, yet we do not know which determinate adjective is to be substituted for that rejected. The most obvious illustration is in predications of place and time: thus we may say 'Mr Smith is not now in this room,' and, knowing that Mr Smith is alive, we know that in the necessities of nature he must be in some other determinate place. Thus we may in a rapid survey discover the absence of any object within a given place, independently of any knowledge -- by observation or otherwise -- of its presence in some other place; and this is sufficient to dispose of the contention that there must be positive ground for a negative judgment. In fact the strictly negative form of judgment is relevant for purposes of further development of thought, whether we are able to assert an opposed positive, or know only that some opposed positive could be affirmed if our knowledge were further extended. What is obviously true of time or place predications is also, though not always so obviously, true of qualitative predicates such as colour or tone: for instance, we may deny that a certain sound is that of a piano, because of our familiarity with that instrument, without being able to define the kind of musical instrument from which the sound proceeds, owing perhaps to our unfamiliarity with other instruments; although we may know, first, that it is a musical sound, and secondly on quite general grounds that it must come -- not from any instrument whatever -- but from some determinate kind of instrument.
§ 2. Having distinguished some of the different ways in which the phrase pure negation may be understood, we will briefly examine the dictum that pure negation has no significance. It may perhaps be at once said that this dictum is itself purely negative, and that therefore anyone who maintains its significance has committed himself to a contradiction. A more serious treatment of the contention shows that for the word 'significance' we should substitute 'having value' or 'importance' or 'relevance to a specific purpose.' The purpose, for instance, of the above negatively expressed dictum is to oppose some other philosophers who have attributed a false value or importance to the negative judgment. It will be seen that the whole question hinges on the meaning to be attached to the word 'significance.' A form of words maybe said to be absolutely non-significant when they fail to convey any precise content for thought-construction. This failure of a phrase to convey meaning may be due either to the substantial components themselves or to their mode of combination; thus it is a merely verbal expression that may be said to have or not have significance for thought in this absolute sense. But in attributing non-significance to a judgment apart from its verbal expression, the most probable meaning intended is that it does not represent any actual process in thought. But any of the examples taken above go to show that the purely negative judgment cannot be universally charged with non-significance in this sense.
§ 3. We have considered in turn, first the proposition as a whole unanalysed; secondly, the predication of an adjective of a given subject-term; and we now turn to the subject-term itself, apart from the adjective predicated, and raise the question whether any proposition can have significance in case there is no real thing corresponding to the subject-term, although there may be a word or phrase used professedly to denote such thing. Now I have regarded the substantive, which is ultimately the subject in all propositions, as a determinandum -- that is as something given to be determined in thought; if then here is nothing given to be so determined corresponding to the word or phrase by which we intend a certain substantive, then what becomes of the proposition? Consider for example the propositions: 'An integer between 3 and 4 is prime,' and again 'An integer between 3 and 4 is composite.' It must be said that neither of these propositions is true. Now since every integer is either prime or composite, it can be at once seen that any proposition predicating an adjective of the subject 'an integer between 3 and 4' must be false, even though the adjective is appropriate to integers as such. This statement needs only the qualification that we may correctly predicate of an integer between 3 and 4 that it is greater than 3 and less than 4; this, however, is not a genuine proposition but one that is implied in the meaning of the subject-term, and is thus merely verbal. We conclude then that of such a subject-term as 'an integer between 3 and 4' no adjective can be truly predicated in a real or genuine proposition.
We may therefore contrast two cases of a subject-term S: (1) where S is such that some adjective can be truly predicated of it in a genuine proposition, and (2) where S is such that no adjective can be truly predicated of it in a genuine proposition. These two cases may be briefly expressed -- 'S is' and 'S is not.' The significance of these two propositions is brought out in considering the process technically known as obversion. The fundamental problem of obversion I will symbolise as the problem of passing from 'S is-not P' to 'S is non-P.' Here, when we hyphen the negative with the copula, I understand it to mean that the proposition 'S is P' as a unit, is asserted to be false. But when we hyphen the negative with the predicate, we are affirming of the subject S the kind of predicate called negative; in other words 'S is non-P' is an affirmative proposition containing a negative predicate, while 'S is-not P' is a negative proposition in the sense that the attitude of negation applies to the proposition as a whole. Now this transformation from the negative proposition to the positive assertion of a negative predicate, has been assumed as almost trifling, and as only too obvious; but I would wish at once to raise the question as to the condition necessary for the validity of this process, called obversion, in its fundamental form.
As we have already stated, the incomplete proposition 'S is' really means, 'S denotes something of which some adjective may be predicated truly in a proposition not merely verbal.' Thus the scheme by which I express the condition under which obversion is valid, is to add to the explicit negative premiss 'S is-not P,' the additional premiss 'S is,' from which we may validly infer the affirmative conclusion 'S is non-P.' The incomplete form of proposition 'S is' means that S has some character which may be predicated of it, without defining what character can be positively asserted. The conclusion 'S is non-P' means that we predicate of S a character, determined so far as that it is an opponent of the proposed character P, but otherwise indeterminate. An illustration from history will show how this process may be applied. Thus the name William Tell is the name of a historical character about whose existence there appears to be doubt. In denying any proposition which predicates an adjective such as 'submissive' of the subject William Tell, we could not validly predicate of him the contrary adjective 'defiant,' unless we were able first to assert that Tell is, in the sense we have explained.
The problem of the ohversion of a singular proposition is the same as that of formulating accurately the contradictory of a singular proposition. Thus, in showing that, in order to pass from the denial (or contradictory) of 'S is P' to the affirmation 'S is non-P' we require the additional datum 'S is,' we have indicated that neither of the propositions 'S is P' and 'S is non-P' would be true, in the case that 'S is' were not true. In other words, the two propositions 'S is P' and 'S is non-P' are not properly contradictories. The contra-dictory of 'S is P' should be formulated in the alternative proposition 'Either 5 is-not or 5 is non-P'; as also the contradictory of 'S is non-P' in the alternative proposition 'Either S is-not or S is P.' Thus, in our historical illustration, neither of the two propositions 'A certain man named William Tell submitted to the Austrians' and 'A certain man named William Tell defied the Austrians' would be true, if it were the case that there was no such person as William Tell; and hence the proper contradictories of the two propositions must be respectively expressed in the alternative forms: 'Either there was no such person as Tell or he (Tell) defied the Austrians,' and 'Either there was no such person as Tell or he (Tell) submitted to the Austrians.'
§ 4. To illustrate the significance of this view wemust consider the different types of cases in which a proposition of the form -- 'S is' -- can be truly asserted. In every case, the term S must have sufficiently determinate meaning, to give rise to the alternative propositions 'S is' or 'S is not'; the question could not arise if S were treated as a mere symbol without significance. When this is agreed, it will be found that any apparent variations in the meaning of the word 'is,' will in reality be variations in the kinds of substantive category to which the name S is understood to apply. For instance, let us take the names of substantives under the category of number. We may say on the positive side that the number 3 is. This will mean that some true adjectives can be predicated of the number 3, beyond those which might be held as merely involved in the definition or connotation of the word 3; thus, if we should define 3 as meaning 2 + 1, the statement that the number 3 has the characteristic expressed by 2 +1 would be purely verbal. But the number 3, we say, is such that an indefinite number of other adjectives, not included in its definition, can be truly predicated, as for instance that 3 is prime or that 3 is a factor of 12. Contrast the name 3 with the phrase 'an integer between 4 and 5': in the sense in which we can significantly assert that 3 is, we may assert that an integer between 4 and 5 is not; in other words, no true character can be assigned to this proposed subject, except what is involved in our understanding of its meaning, namely that it belongs to the general category of integer, and that it is to be greater than 4 and less than 5. Generalising from this example, it will be seen that such a subject-term is defined first by reference to a general category (in the above case that of number) and next, by a proposed means of determining or selecting out of the members of that category, a particular example related in a defined way to other things.
With reference to the category to which the subject-term S by definition belongs, any difference of category is naturally associated with an apparent difference in the meaning of 'is.' In particular there is a range of subjects for which the word 'exists' would be naturally substituted for 'is.' Thus it may be agreed that what is manifested in space and time may be said to exist: hence we raise such questions as whether God exists, or whether the centaur Cheiron existed, or whether William Tell existed. The objects intended to be denoted by these subject-terms may be said to belong to the category of the existent whether the propositions asserting their existence are true or false. Thus we must maintain, in accordance with the nature of the definition of God, that 'God is an existent.' -- this being a merely verbal or analytic proposition; but the question of the truth of the synthetic or real proposition 'God exists' remains problematic. The same holds of Cheiron and William Tell. On the other hand what is denoted by such a subject-term as 3 or an integer between 4 and 5, would not be called an existent. Thus we maintain that there is no difference in the force of the word 'is' in its isolated usage; but that if any difference appears -- as when we substitute 'exists' for ' is' -- this is merely due to a difference in the category of the subject-term, which again presupposes a difference in the types of adjectives that are properly predicable of it.
When the proposition 'S is' is under consideration it must be understood that the term S is not an ordinary singular name but one of a peculiar nature that has not, I think, been recognised by logicians. Such a name will be designated by the prefix 'a certain.' Consider the following propositions: 'A certain man was both a philosopher and a historian,' 'a certain integer between 3 and 11 is prime,' 'a certain novel has no hero,' 'a certain flash of lightning was vivid.' The truth or falsity of these propositions could only be decided by the hearer if for the phrase 'a certain' is substituted 'some or other,' 'one or more,' so that for him the reference is indeterminate. Thus, of Hume and of Xenophon it is true that they were both historians and philosophers; between 3 and 11 there are two numbers -- 5 and 7 -- that are prime; and the other examples are equally ambiguous. We must suppose that the speaker has in mind a single determinate philosopher-historian, number, novel or flash, which has been identified by him, and to which therefore he may return in thought. From these examples we see that a term may be properly called uniquely singular for the asserter, although in fact there may be several objects answering to its explicit description. Thus from a proposition with the predesignation 'a certain' maybe inferred the corresponding proposition with the predesignation 'some or other,' though of course not conversely. This points to two modes in which what is technically called the particular proposition can be inferred: first, from premisses one of which is itself particular; and secondly, from a specific instance for which the predesignation 'a certain' stands.
Now it is the latter form of proposition which raises the problem of the significance of the proposition 'S is.' For the asserter, the contradictory of the proposition that 'A certain man was both an historian and a philosopher' would be that the person of whom he is thinking was not both an historian and a philosopher: or the contradictory of the proposition that a certain integer between 3 and 11 is prime, would be that that same integer is composite; whereas for the hearer, who can only understand the given propositions as particular, the contradictory in the first case would be: 'No man is both an historian and a philosopher' and in the second, 'No integer between 3 and 11 is prime.' In fact, in denying the proposition that a certain integer between 3 and 11 is prime, we must mentally specify the integer about which we are thinking, and assert that this integer is composite. The form of the statement thus reached is equivalent to that of the conclusion in the process of obversion, but it is not obtained here (as in obversion) by the medium of the purely negative premiss, but directly by mentally specifying the number under consideration.
§ 5. It remains to explain more precisely the nature of the denial of 'S is P' which combined with 'S is' yields the conclusion 'S is non-P.' The proposition which merely denies 'S is P' must be understood to involve a hypothetical element. Consider, for example, the statement 'Anyone who calls this afternoon is not to be admitted'; this proposition does not contain any categorical assumption that somebody will call, and may be otherwise expressed in an explicitly hypothetical form 'If anyone calls he is not to be admitted.' Combining this premiss with the further ascertainable fact that a certain person has called, the obvious conclusion, that this person is not to be admitted, follows. In general the symbols that we have used, namely that 'S is P' would be that the person of whom is false and that 'S is,' may be explained by making explicit the descriptive, adjectival, or connotative factor in the term symbolised by S, which factor we shall symbolise by M. The negative premiss then becomes: 'If anything is M it will not be P': the categorical premiss becomes: 'A certain thing is M.' In this formulation the symbol S does not appear. Now S stands for a certain thing which has not yet been identified and which is only presented in thought by the general description M; or briefly S is to mean 'a certain thing which is M.' In transforming the proposition 'Anything that is M will not be P' into the form 'S will not be P' we introduce the factor 'a certain thing.' And in transforming the proposition 'A certain given thing is M' into the form 'S is' we have transferred the whole of the adjectival component in the proposition from the predicate to the subject: or otherwise, the two propositions may be rendered 'Anything that may be given having the character M will not be P' and 'A certain thing having the character M is given.' Thus it is not strictly correct to use the same symbol S in our two propositions, since the only differentiating element of meaning in the term S in the negative premiss is the adjectival or descriptive component, whereas in the categorical premiss the substantival component enters along with the adjectival. It follows that the analysis given is not restricted to the negative form of our first premiss, since the same kind of syllogism would apply to an affirmative conclusion: the essential characteristic of the first premiss is its hypothetical character, as opposed to the other premiss which is categorical. Thus the affirmative case would be rendered 'Anything that may be given having the character M will be P,' 'A certain thing with the character M is given' therefore 'This thing having the character M will be P.'
§ 6. In the course of this final explanation it will be noted that for the formula 'S is' in which no determinate adjective is predicated, we might substitute 'S is given' or 'S is real.' Now the words 'given' and 'real' though of course grammatically adjectival, are not in the logical sense adjectival, for their meaning does not contain any indication of character or relation. It may be remarked in passing that the application of the term 'real' includes but goes beyond that of the word 'given.' The postulate that has to be assumed is that, however indeterminately we may have been able to characterise it, the real must have some determinate character. We thus return to our first exposition of the force of the incomplete predication 'S is ': namely that S, as being real, must have some determinate character although it may be that this character cannot be completely or exactly known by any finite intelligence.