W. E. Johnson, Logic: Part I (1921)
§ 1. The division of propositions into simple and compound is to be distinguished from another division to which we shall now turn, namely that into primary and secondary. A secondary proposition is one which predicates some characteristic of a primary proposition. While it is unnecessary to give a separate definition of a primary proposition, a tertiary proposition may be defined as one which predicates a certain characteristic of a secondary proposition, just as a secondary proposition predicates some characteristic of a primary proposition. Theoretically this succession of propositions of higher and higher order could be carried on indefinitely. But it should be observed that any adjective that can be predicated of a primary proposition can be significantly predicated of a proposition as such, i.e. equally of a primary, a secondary, and a tertiary, etc. proposition ; and that, in consequence, although propositions may be ranged into higher and higher orders, adjectives predicable of propositions are of only one order, and will be called "pre-propositional." Taking p to stand for any proposition we may construct such secondary propositions as: p is true, p is false, p is certainly true, p is experientially certified, p has been maintained by Berkeley. Here we are predicating various adjectives (the precise meaning of which will be considered later) of any given proposition p; and we define each of these propositions -- of which the subject-term is a proposition and the predicate-term an appropriate adjective -- as secondary. One example may be given which has historic interest. Take (A) the proposition 'two straight lines cannot enclose a space' -- to illustrate a primary proposition; again take (B) the proposition 'A is established by experience' as a secondary proposition; and thirdly take (C) 'B is held by Mill' as a tertiary proposition; namely -- 'It is held by Mill that the theorem that two straight lines cannot enclose a space is established solely by experience.' It is at once obvious that, all these three propositions, the primary, the secondary, the tertiary, which include the same matter (viz. that expressed in the primary) might be attacked or defended on totally distinct grounds. We may defend the primary proposition: 'two straight lines cannot enclose a space,' by showing perhaps that it is involved in the definition of 'straight'; again we might attack or defend the secondary proposition: 'this geometrical theorem is established by experience,' by considering the general nature of experience, and the possibilities of proving generalisations; and lastly, if we are to examine the tertiary proposition, namely 'Mill held the experiential view on the subject of this geometrical axiom,' we have only to read Mill's book and try, if possible, to understand what was the precise view that he wished to maintain. CHAPTER IV
SECONDARY PROPOSITIONS AND MODALITY§ 2. In connection with a larger and wider treatment of secondary propositions in general, it will be useful here to introduce the subject of Modality. We shall throughout speak of modal adjectives, instead of modal propositions; it being understood that these adjectives fall under the general head of what we have called pre-propositional adjectives. We propose provisionally to include under modais the adjectives 'true' and 'false.' But a question of some interest arises as to whether the two very elementary cases 'p is true' and 'p is false' where p is a proposition are legitimate illustrations of secondary propositions. It may be held that the proposition 'p is true' is in general reducible to the simple proposition p; so that, if this were so, 'p is true' would only have the semblance of a secondary proposition, and would be equivalent for all ordinary purposes to the primary proposition p. It appears to me futile to enter into much controversy on this point, because it will be universally agreed that anyone who asserts the proposition p is implicitly committing himself to the assertion that p is true. And again the consideration of the proposition p is indistinguishable from the consideration of the proposition p as being true; or the attitude of doubt in regard to the proposition p simply means the attitude of doubt as regards p being true. These illustrations, in my view, show that we may say strictly that the adjective true is redundant as applied to the proposition p; which illustrates the principle, which I have put forward, that a proposition by itself is, in a certain sense, incomplete and requires to be supplemented by reference to the assertive attitude. Thus the assertion of p is equivalent to the assertion that p is true; though of course the assertum p is not the same as the assertion that p is true. The adjective true has thus an obvious analogy to the multiplier one in arithmetic: a number is unaltered when multiplied by unity, and therefore in multiplication the factor one may be dropped; and in the same way the introduction of the adjective true may be dropped without altering the value or significance of the proposition taken as asserted or considered.
More interest attaches to the apparently secondary proposition 'p is false.' It certainly appears that p-false is indistinguishable from not-p, and the majority of logicians rather assume that not-p is on a level with p, and may be at once co-ordinated with p as a primary proposition. Now it appears to me that, while p-true is practically indistinguishable from the primary proposition p, on the other hand p-false is essentially a secondary proposition, and can only be co-ordinated with primary propositions after a certain change of attitude has been adopted. This problem will come up again in the general treatment of negation and obversion.
§ 3. We may now turn to what have been always known as modal adjectives such as necessary, contingent, possible, etc. The discussion of modality is complicated rather unfortunately owing to certain merely formal confusions which have not been explicitly recognised. Hence, before plunging into the really difficult philosophic problems, these formal confusions must be cleared away. The simplest of these occurs in the controversy between those who hold that contradictories belong to the same sphere of modality, and those who hold that they belong to opposite spheres of modality. This controversy is resolved by explicitly realising the distinction between a primary and a secondary proposition. Thus taking, for purposes of illustration, the antithesis between necessary and contingent, we may consider the primary proposition 'It is raining now' and its contradictory 'It is not raining now'; if one of these primary propositions is contingent, so also is the other. But the contradictory of the secondary proposition affirming contingency of the primary -- i.e. 'that it is raining now is contingent' -- is the secondary proposition which affirms necessity of the primary -- i.e. 'that it is raining now is necessary.' Thus, in doubting or contradicting a secondary proposition, we use the opposite or contrary modal predicate; but in denying the primary proposition we should attach the same modal adjective to the proposition and to its contradictory. There can really be no difference of opinion on this subject; the opposition of modality is expressed in the secondary propositions that contradict one another; the agreement in modality holds of the primary propositions that contradict one another. Summarising: if a given primary proposition is necessarily true, its contradictory, which is also a primary proposition, is necessarily false; and if a given primary proposition is contingently true, its contradictory, which is also a primary proposition, is contingently false. Thus in both cases the contradictory primary propositions belong to the same sphere of modality. But the contradictory of a secondary proposition affirming necessity or contingency of a primary, will be the secondary proposition which affirms contingency or necessity of the primary. Thus the contradictories of the secondary propositions assert opposite modals.
It is necessary to enter into the more philosophical aspect of modality, if only in a preliminary and introductory way, because, apart from the confusion between a secondary and a primary proposition, there is, it would appear, considerable confusion in regard to the terminology adopted by different logicians or philosophers in their treatment of modals. To do this we feel bound to reconsider entirely the terminology. Since Kant it has been customary to make a three-fold division, using the terms apodictic, assertoric, and problematic; and this trichotomous division at once leads to some unfortunate confusions. The precise significance of assertoric in particular is peculiarly ambiguous: thus the proposition '2 and 3 make 5' as it stands, would appear to be merely assertoric; so that assertoric would include apodictic as one of its species1. Let us then begin our investigation without any bias derived from the traditional terminology.
§ 4. The first antithesis that immediately impresses us in this connection is that between a certified and an uncertified proposition. A proposition which is uncertified appears to be what Kant and others have sometimes meant by a problematic proposition; hence we begin by replacing the term 'problematic' by the term 'uncertified.' The contradictory of uncertified is certified, so that all propositions may be divided into the two exclusive classes of certified and uncertified. It is of course obvious that these terms are what is called relative; that is to say, at one stage in the acquisition of knowledge a given proposition may be uncertified, while at a later or higher stage, or with increased opportunity of observation, etc., it may become certified. The distinction therefore is of course not permanent or absolute, but temporal and relative to individuals and their means of acquiring knowledge. It might be held that such distinctions should be excluded from Logic; but this, in our opinion, is unsound, in as much as reference to the mental powers and the individual opportunities of acquiring knowledge turns out in many discussions to be a most essential topic for logical treatment. The whole doctrine of probability hinges upon our realising the changeable or relative opportunities and means, which differ, from one'situation to another, in the extent of attainable knowledge. The further discussion then of uncertified propositions will later introduce the logical topic of probability. Returning to certified propositions, a distinction is required according as the given proposition is certified as true or certified as false; and thus we have a triple division: uncertified, certified as true, and certified as false. But for most purposes this latter distinction is unnecessary, because for the given proposition that has been certified as false we might substitute the contradictory proposition that has been certified as true. It would be enough therefore to use the two divisions uncertified and certified, understanding by certified 'certified as true.'
§ 5. The above division leads to a fundamentally important subdivision under the term 'certified'; for we must recognise, in epistemology or general philosophy, that there are essentially different principles or modes, by which the truth of a proposition may be certified; and a rough two-fold classification will conveniently introduce this subject: thus we may contrast a proposition whose truth is certified by pure thought or reason with a proposition which is certified on the ground of actual experience. Briefly we shall call these two classes 'formally certified' and 'experientially certified.' The range of these two modes of certification will be a matter of dispute: some philosophers hold that all the principles and formulae of logic, and all those of arithmetic and mathematics, are to be regarded as certified by pure thought or reason. This gives perhaps the widest range for the propositions that may be said to be formally certified. But even amongst these, we may have to distinguish those which have been formally certified, from amongst the entire range which may be regarded as formally certifiable. Others would hold that many mathematical principles, such as those of geometry, can only be certified by an appeal to sense-perception -- a form of experience; and thus the limits to be ascribed to the range of formal certification would open up serious controversy. Again, on the other hand, the range of propositions immediately certifiable by experience raises serious problems. Some may hold that the only truths guaranteed by mere experience are the characterisations of actual sense-impressions experienced by the thinker at the moment in which he asserts the proposition; many would extend this to judgments on the individual's past experiences revived in memory; but the most universally understood range of experientially certified propositions is still wider: it would include sense-perceptions, and observations of physical phenomena, and even judgments on mental phenomena, -- these supplying the required data for science in general. We will not then profess to draw the line precisely between propositions that are to be regarded as formally certifiable and those that are to be regarded as experientially certifiable; but there is one explanation of the relation between these two classes which will probably be admitted by all; namely, that propositions which are admittedly based on experience, will also involve processes of thought or reasoning, and that therefore no propositions of any importance are based upon experience alone; since an element of thought or reason enters into the certification of all such propositions. This leads to a simple, more precise definition of the antithesis -- formal and experiential: while we define a formally certifiable proposition as one which can be certified by thought or reason alone, we do not define experiential propositions as those which can be certified by experience alone, but rather as those which can only be certified with the aid of experience. In this way we imply that experience alone would be inadequate2.
A certain relation between the two antithetical modals, formal and experiential, will be found to apply over and over again to other antitheses in the characteristics of propositions. It may be illustrated by reference to the syllogism. Thus a certain syllogism may contain one formal premiss and one experiential premiss; and the conclusion deducible from these two premisses must be called experiential, because it has been certified by at least one experiential premiss. To put it otherwise, if all the premissesof an inference were formal, the conclusion would be formal; but if only one premiss is experiential (even though the others may be formal), the conclusion must be experiential. This particular characteristic of the syllogism is not arbitrary, but follows from the common understanding of what is meant by 'experientially certified,' namely something which could not be certified without experience, -- not something which could be certified by experience alone.
§ 6. One of the chief sources of confusion is the use of the term 'necessary' in various different senses as an adjective predicable of propositions. It has sometimes been said that all propositions should be conceived as necessary; in the sense that the asserter of a proposition represents to himself an objective ground or reference to which he submits and which restrains the free exercise of his will in the act of judgment. This contention is indisputable, and may be regarded as one of the many ways in which the nature of judgment or assertion as such may be philosophically expounded. But obviously necessity as so conceived cannot serve as a predicate for distinguishing between propositions of different kinds. We pass, therefore, to the next and more usual meaning of the term necessary which will perhaps best be indicated by a quotation from Kant: 'Mathematical propositions are always judgments a priori and not empirical, because they carry with them the conception of necessity, which cannot be given by experience.' Here necessary is opposed to empirical; and the antithesis that Kant has in view coincides approximately with that between the formally certified and the experientially certified (as I have preferred to express it). But still another meaning has been attached to the term necessary, viz., that accordi to which the necessary is opposed to the contingent. If, however, the term contingent is interpreted as equivalent to (what I have called) experientially certified, then we might agree that necessary should be interpreted as equivalent to formally certified; and in this case we should not have found a third meaning to the term. The question therefore arises whether a use can be found for the antithesis 'necessary' and 'contingent,' within the sphere of the experientially certified. Now it has been maintained as a fundamental philosophical postulate that 'All that happens is necessitated'; and this may be taken as equivalent to saying that 'Nothing that happens is contingent." It should here be pointed out that this contention is to be clearly distinguished from the view that 'All judgments or propositions are necessary.' For the necessity ascribed to judgments is conceived as a compulsion exercised by the objective or real upon the thinker; whereas the necessitation attributed to events is conceived (more or less metaphorically) as, a compulsion exercised by nature as a unity upon natural phenomena as a plurality. The former necessity is so to speak objectivo-subjective; the latter objectivo-objective. But an elementary criticism must be directed against the use made of the postulate 'All that happens is necessitated' to deduce that there is no proper scope for the term contingent. For we inevitably conceive of that which happens as being necessitated by something else that happens in accordance with (what is popularly called) a law of nature. In other words, the laws of nature taken alone do not necessitate any event whatever; we should have rather to say that a law of nature necessitates that the happening of some one thing should necessitate the happening of a certain other thing. Hence, I should propose that nomic (from νομος a law) should be substituted for necessary as contrasted with contingent. Thus a nomic proposition is one that expresses a pure law of nature; and a contingent proposition is one that expresses a concrete event. In this way we have eliminated the ambiguous term necessary, and have substituted formally certified when, the term is opposed to experientially certified; and nomic when opposed to contingent. Finally the term possible must be coupled with the word necessary in its three usages. For 'possible' has three obviously distinct meanings: (i) what is not known to be false; i.e. what does not contradict the necessary in the first sense, applicable to all assertions; (2) what does not conflict with any formally certified proposition, i.e. with any proposition necessary in the second sense; (3) what does not conflict with any law of nature, i.e. with any proposition necessary in the third sense. The word 'possible' in these three senses may be distinguished respectively as 'the epistemicallv possible,' 'the formally possible' and 'the nomically possible.'
§ 7. It will now be apparent that the antithesis between nomic and contingent is of a totally different nature from that between certified and uncertified, or between the different modes of certification. The latter has been called subjective, the former objective; but the terms epistermc and constitutive are preferable: for the characteristics 'nomic' and 'contingent' apply within the content of the proposition, and are therefore properly to be regarded as constitutive; whereas the characteristics 'certified' and 'uncertified' apply to the relation of the proposition to the thinker, and should therefore be called epistemic. Taking for example, the proposition 'Nature is uniform': if this is held to be necessary in the sense that our reason alone establishes its truth, then the attribution of necessity is in this case of the same kind as what we have called formally certified and is thus epistemic. But the necessity involved in the laws of nature is generally attributed to Nature itself, and not merely to our grounds for asserting such uniformity: and is thus constitutive. Thus, if we say, as a specific example of the necessity attributed to Nature's processes, that 'bodies attract one another in obedience to the necessities of nature,' this statement is quite independent of any view we may hold as to the reasonable grounds for asserting the fact of universal gravitation. In short, referring back to the distinction between the fact and the proposition, such modals as certified and uncertified are adjectives directly characterising the proposition, whereas modals of the other kind, typified by nomic and contingent, directly characterise the fact.
§ 8. It remains now to introduce a certain familiar distinction amongst propositions not included in the understood meaning of modal, viz. that between real and verbal. These terms were used by Mill, and are generally understood as equivalent to Kant's terms 'synthetic' and 'analytic.' Mill's point of view is very different from Kant's, for Mill is thinking of the nature of language, of the definition of words, etc., while Kant is thinking of ideas and the various constructive acts of thought. Mill's usage is more easy to expound than Kant's, and gives rise to less serious conflict of view. A verbal proposition is one which can be affirmed from a mere knowledge of the meanings of words and their modes of combination; a real proposition, on the other hand, requires for its acceptance, not only a knowledge of the meanings of words, but also a knowledge of matters of fact. We may therefore note the same relation between verbal and real as between formal and experiential3: namely, that two premisses, both of which are verbal, can only yield a verbal conclusion; and that a single real premiss, even though joined with any number of verbal premisses, will impose upon the conclusion its own character as real.
The definition so far given of verbal propositions seems fairly clear; it is therefore surprising that it should have proved a stumbling-block to some logicians. The people who have raised difficulty on this point are those who have preferred the Kantian terms 'analytic' and 'synthetic' in place of Mill's terms 'verbal' and 'real': ('analytic' Kant illustrates by the proposition 'Material bodies are extended,' 'synthetic' by the proposition 'Material bodies attract one another'). The controversy has arisen through a tacit confusion between 'verbal or analytic' and 'familiar' on the one hand, and between 'real or synthetic' and 'unfamiliar' on the other hand, due to the kind of examples chosen to illustrate each type of proposition. This confusion is apparent in the well-known dictum of Bradley -- 'that synthetic judgments are analytic in the making' -- where it is clear that by a 'synthetic judgment' he means the newly-constructed proposition, and by 'in the making,' the process of rendering the proposition familiar. But, it needs only a little reflection to show that familiarity with a matter of fact does not render the proposition which expresses such fact verbal or analytic; nor does unfamiliarity with the meanings of words render a proposition which explains such meaning real or synthetic: a proposition about the meanings of words is verbal, and a proposition about matters of fact is real, whether the hearer is unfamiliar or familiar with the words or with the facts4. Thus the proposition '7 and 5 make 12' is familiar enough, but whether or not it is verbal (or analytic) has absolutely nothing to do with its familiarity; on the other hand, a technical definition given by a scientist will probably be quite unfamiliar, but if the scientist puts it forward as an expression of his intention to use a word with a certain significance, the proposition which states his intention is verbal, although it is ipso facto unfamiliar.
Perhaps a better way of indicating the nature of a verbal proposition, is to say that it is not quite what is ordinarily meant by a proposition; that is, as verbal, it cannot strictly be said to be either true or false, because it does not declare a fact, but rather expresses an intention, a command, or a request. The technical scientist puts forward his definitions in this spirit, when he asks readers to allow him to use a term with a certain signification which is explained by his definition. Thus a verbal proposition is neither true nor false, because it is properly expressed, not in the indicative, but in the imperative or other similar mood. But if by a verbal proposition is meant one that assigns the meaning of a word as conventionally used in any wider or narrower context, then, inasmuch as the proposition asserts the fact that such or such is the convention, it must be either true or false.
§ 9. At the beginning of this chapter we defined a secondary proposition as one that predicates one or other of the adjectives significantly predicated of a proposition as such. We proceeded to consider in turn different kinds of adjectives that are thus predicable: this has led to a discussion of modal adjectives, and has included in particular a consideration of the adjectives true and false, and finally of the predicates 'verbal' or 'analytic' and 'real' or 'synthetic'
Notes 1 This confusion is, of course, due merely to the failure to distinguish between a primary proposition as such and a secondary. It is totally independent of any question as to what the adjectives assertoric and apodictic mean respectively.
2 Even this distinction requires amendment; for it may be maintained that just as experience alone can certify nothing, so thought alone can certify nothing. Thus formal certification would coincide with what requires only experience in general (to use Kantian terminology) whilst experiential certification would involve in addition special or particular experience.
3 See above, last paragraph of § 5.
4 An important explanation of all this should be given. What Bradley means by "an analytic judgment" -- not "a verbal proposition" -- is a judgment that could be discovered by introspective analysis, so that his pronouncement is an obvious truism. But it is strange that he does not perceive that this is not in the least the same as what Kant meant. Kant's distinction is epistemological, Bradley's merely psychological.