W. E. Johnson, Logic: Part I (1921)§ 1. Logic is most comprehensively and least controversially defined as the analysis and criticism of thought. This definition involves the least possible departure from the common understanding of the term and is not intended to restrict or extend its scope in any unusual way. The scope of logic has tended to expand in two directions -- backwards into the domain of metaphysics, and forwards into that of science. These tendencies show that no rigid distinction need be drawn on the one side between logic and metaphysics, nor on the other between logic and science. The limits imposed by any writer are justified so far as his exposition exhibits unity; it is, in fact, much more important to remove confusions and errors within the subjects discussed under the head of logic, than to assign precise limits to its scope. It is, I hold, of less importance to determine the line of demarcation between logic and philosophy than that between logic and science; so that my treatment of logic might be called philosophical in comparison with that of those who implicitly or explicitly separate their criticism and analysis from what in their view should be relegated to epistemology and ontology.
INTRODUCTION This account of the scope of logic does not differ in any essential respects from that given, for example, in Mill's long introductory chapter. The special feature of Mill's logic is the great prominence given to the theory of induction, in contrast to most of his predecessors and contemporaries, including Whately. Whately does not omit reference to induction any more than Mill omits syllogism: where they differ is that Whately asserts that in order to be valid any inductive inference must be formulated syllogistically, and that therefore the principle for induction is dependent on the principle of syllogism. Mill opposes this view; but as regards the scope of logic there is no disagreement between them: they differ simply on the question of the relations of deduction to induction.
If any writer deliberately or on principle dismisses from logic the theory of inductive inference, it must be on one of three grounds: either
- that no inductive inference is valid; or
- that different criteria of validity apply to different sciences; or
- that the problem of the validity of induction constitutes a topic to be included in some study other than that named logic.
§ 2. As regards (a), this is the view which seems to be held by Venn in his Empirical Logic where, in the chapter on the subjective foundations of induction, he acknowledges that as a matter of fact human beings do make directly inductive inferences, even with a feeling of conviction, but that no warrant for such conviction can be found. Another aspect of his view of induction is expounded in the chapter on the objective foundations of induction, in which he classifies the different kinds of uniformity -- such as sequence, co-existence, permanence, rhythm -- which are used as major premisses, expressive of actual fact, by means of which specific uniformities under each general head are established as valid. When then he is asked what reasonable ground there is for accepting these major premisses as true, he maintains in effect that they have to be assumed, in order to give security to the conclusions inductively inferred. In using the word assumption, there seems to be some ambiguity, namely whether it is to be understood to mean 'assumed to be true although known to be false' or 'assumed to be true although unprovable.' I take Venn to mean the latter, and that the attitude towards this assumption is merely one of felt certainty -- felt, indeed, by all human beings, but having no root in our rational nature, and only exhibiting a common psychological disposition or character. This view, that there is no inductive principle that is self-evidently or demonstrably true, seems to be held by many other logicians, though none of them, I think, put it as explicitly as Venn. So while he and others include induction in their logical exposition, they neglect what I take to be the essential justification for its inclusion, namely as affording a systematic criticism of the question of its validity.
As regards (b), many excellent text-books have been written in these days treating of the principles and methods peculiar to different sciences; it is not denied by their authors that this treatment is logical; but, if not explicitly stated, yet it seems to be suggested that in comparing the logic of one science with that of another the sole result is to exhibit differences, and that no one set of principles applies to all the different sciences. If this were the fact there would be some excuse for excluding the treatment of induction from the scope of logic, on the ground that the discussion of each of the separate principles should be relegated to its own department of science. But if, as I hold in agreement with most other logicians, there must be a community of principle discoverable in all sciences, then the discussion of this must be included in logic.
As regards (c) the question raised seems to be: 'Given the topic induction, what name shall be given to the science that includes it in its treatment?' rather than the converse question 'Given the name logic, shall it be defined so as to include, or so as to exclude, induction?' If we put the question in the first form, the answer is of course purely arbitrary; we might give it the name Epagogics. But if the question is put in the second form, the answer is not in the same sense arbitrary, assuming that there is general unanimity as regards the usage of the name logic to denote a science whose central or essential function is to criticise thought as valid or invalid. That induction should be included in logic thus defined follows from the undeniable fact that we do infer inductively, and that some persons in reference to some problems do infer invalidly. Even if this were not the fact, it is certainly of scientific importance to render explicit what everyone implicitly recognises in their inferences -- as much for the case of induction as for that of syllogism or other formal types of inference. It has even been hinted that nobody makes mistakes in formal inference; and yet -- in despite of this, if true -- no one questions the value of systematising the principles under which people may unconsciously reason; and what holds of formal inference would certainly hold a fortiori of the processes of inductive inference which present many more serious opportunities for fallacy.
As regards the term 'thought' which enters into my definition, its application is intended to include perceptual judgments which are commonly contrasted with rather than subsumed under thought, for the reason that thought is conceived as purely abstract while perception contains an element of concreteness. But properly speaking even in perceptual judgment there is an element of abstraction; and on the other hand no thought involves mere abstraction. It follows, therefore, that the processes of thinking and of perceptual judgment have an essential identity of character which justifies their treatment in a single systematic whole. It is the distinction between sense-experience and perceptual judgment and not that between perceptual judgment and thought, that must be emphasised. The essential feature of perceptual judgment in contrast to mere sense-experience is that it involves activity, and that this activity is controlled by the purpose of attaining truth; further it is the presence of this purpose which distinguishes thought from other forms of mental activity. Thought may therefore be defined as mental activity controlled by a single purpose, the attainment of truth.
§ 3. Now it is true, as often urged, that thought is motived not solely by the purpose of attaining truth, but rather by the intention of realising a particular end in some specific form and under certain specific circumstances. But I have to maintain that any other or further purpose which may prompt us to undertake the activity of thinking is irrelevant to the nature of thought as such, this other purpose serving only to determine the direction of activity. When such activity is actually in operation its course is wholly independent of the prompting motive and guided by the single purpose of attaining truth. For instance, our desire for food may prompt us to search for it; but this resolve, once taken, leads to a thinking process the purpose of which is to come to some conclusion as to where food is likely to be found, and the sole aim of this process is to discover what is true on the matter in hand. This being so, the logical treatment of thought must be disencumbered from all reference to any ulterior purpose.
Whether truth is ever pursued without any ulterior purpose is a psychological question which may fairly be asked; and if introspection is to be trusted must certainly be answered in the affirmative, although the enquiry whether true knowledge has intrinsic value or not belongs to ethics. That the attainment of truth for its own sake constitutes a genuine motive force is further confirmed by recognising the fact that people do actually attach value to true knowledge, as is incontestably proved by their willingness to defy the prospect of social disapprobation, persecution, and even martyrdom incurred by the utterance and promulgation of what they hold to be true. At the same time, it must be pointed out that the aim of the thinking process is not the attainment of truth in general, but always of truth in regard to some determinate question under consideration. This is closely analogous to the psychological fact that what we desire is never pleasure in general, but always -- if the doctrine of psychological hedonism is to be accepted -- some specific experience which is represented as pleasurable.
Any thinking process is normally initiated by a question and terminated by an answer; what distinguishes one thinking process from another is the difference of the question proposed. The bond of unity amongst the phases of a single process does not necessarily entail unbroken temporal continuity, but only identity of the question proposed. Indeed any thought process may be temporarily interrupted before the proposed question has been answered. It must be left as a topic for psychology to investigate the causes of such suspension, and how far the advance made serves as a starting point for further advances. Logic, on the other hand, is concerned with the nature of the advance as an advance and criticises the process from the point of view of validity or invalidity.
§ 4. The above definition of logic as the analysis and criticism of thought should be compared with that of the Scholastics, who laid emphasis on the point that logic is concerned with the art of thinking, where art is nearly equivalent to the modern term technique, and has an understood reference to activity with an end in view. The study of the art of thinking as thus understood is of use in instructing us how to proceed when thinking out any problem: for instance, it lays down rules of classification and division for the clearing up of obscurities and inconsistencies in thought; rules for the recall and selection of knowledge appropriate to any given problem; etc. Descartes' Discourse on Method is a classical illustration of this species of science. Modern examples of excellent treatises on these lines are to be found in Alfred Sidgwick, and other neo-pragmatists. It is a science of the highest value, and need only be separated from logic on the ground of the difference of purpose; inasmuch as its direct purpose is the attainment of valid thought, whereas logic is the study of the conditions of valid thought, and as such it does not exclude the study of the art.
§ 5. Alongside of the use of the term 'art' to mean technique, there is a more modern usage where it implies reference to aesthetic feelings and judgments. Nowadays discussions as to whether an objective standard for these feelings and judgments should be recognised are very prominent. The nature of the feelings and judgments that enter into aesthetic appreciation belongs to psychology; but if we agree that there is a discoverable objective standard, then the treatment of the subject of aesthetics is to be distinguished from the psychological treatment, precisely as the treatment of thought in logic is distinguished from that in psychology.
Aesthetics, in this sense, raises very similar problems to those presented in Ethics; and it is frequently said that as normative Logic, Aesthetics and Ethics are related in the same way to the three psychological factors, thought, feeling and volition respectively. Each of the normative studies may be said to be based on a standard of value, the precise determination of which it is their function to formulate; in each, imperatives are laid down which are acknowledged by the individual, not on any external authority, but as self-imposed; and, in each, the ultimate appeal is to the individual's intuitive judgment. There is, however, a closer resemblance between Ethics and Aesthetics in their relations to volitions and feelings respectively, than between either of them and Logic; inasmuch as there are apparently fundamental differences of opinion as to the ultimate ethical and aesthetical standards, that give to the studies of Ethics and Aesthetics a controversial character absent from Logic about whose standards there is no genuine disagreement. As regards the relation of Ethics to Logic, the question sometimes arises as to which subject is supreme. The answer to this question depends entirely upon the nature, of the supremacy intended: the imperatives for thought become imperatives for conduct only on condition that true judgments have intrinsic value and false judgments intrinsic disvalue; and thus, from the point of view of conduct, Logic is subordinate to Ethics. On the other hand, ethical enquiry -- like any other scientific investigation -- has to avoid violating logical principles, so that from the point of view of true thought Logic is supreme over Ethics.
§ 6. Our discussion so far has led us to consider the relations of Logic to Philosophy in general, Psychology, Aesthetics and Ethics. Another subject to which it is closely allied and from which it is yet distinct is Grammar, the alliance being prima facie accounted for by the common concern of the two studies with language. The connection between thought and language presents a problem for the science of Psychology; but, so far as thinking or the communication of thought involves the use of words, the provinces of Logic and Grammar coincide; that is to say universal Grammar, which excludes what pertains to different languages and includes only what is common to all languages, should be subsumed under Logic. For the modes in which words are combined -- which constitute the subject matter for Grammar -- cannot be expounded or understood except as reflecting the modes in which thoughts are combined; and this combination is effected by means of such logical operations as negation, conjunction, disjunction, alternation, implication and so on, represented by the words not, and, not both, or, if, etc. To justify the subordination of Grammar to Logic we have only to realise that the analysis of the sentence in Grammar corresponds to the analysis of thought in Logic, and that grammatical criticism is confined to securing that the sentence precisely represents the thought, any further criticism of the proposition coming exclusively within the province of Logic. It may be pointed out in this connection as specially significant both for the linguist and for the logician, that languages differ in the degree of their capacity to exhibit through their structure intimacy between words and thoughts.
§ 7. Amongst all the sciences over which logic must rule, there is one that occupies a unique place. The constituents of thought which are in the most narrow sense logical are those which give form to the construct, connecting alien elements by modes which give specific significance to the whole. The first, group of these is expressed by ties, conjunctional words, prepositional words, and modes of verbal inflection. But as the form of thought is further elaborated there enter new kinds of terms, namely specific adjectives which have a constant meaning definable in terms of pure thought, or else are to be admitted and understood as indefinables. The most generic form of such adjectives directly expresses the result of such mental acts of comparison as like, unlike, different from, agreeing with. Owing to the purely logical nature of these relations, universal formulae in which they are introduced can be constructed by mere abstract thought. The preliminary condition for this construction is the separating of what is given to constitute a plurality, and thus to introduce a formal factor which can only be verbally expressed by the separations and juxtapositions of the substantial words. The very general relation that separation effects is that most indeterminate relation otherness. When the complementary notions of separateness and togetherness are joined to constitute a unity, there enters the idea of number, and we are in the domain of mathematics.
The extraordinary capacity for development that marks mathematics is due to the precision with which the relations of comparison are capable of being amplified. Through the substitutions that are thus rendered possible, the range of application of mathematical formulae is extended beyond the bounds which would otherwise delimit logic. Any material that might be presented to thought upon which the same precise operations of comparison could be performed, would lead to the same forms as mathematics. For example ideas, not only of difference, but of determinable degrees of difference, bring the material into relations of intrinsic order, and out of these relations emanate relations between relations, so that theoretically the science develops into a highly complicated system. The point then, where we may venture to say that logic actually passes into mathematics is where the specific indefinable adjectives above referred to give new material for further logical combinations.
Here it is of great importance to point to the relative nature of the distinction between form and matter. Logic begins with a sharp contrast between matter, as what is given as merely shapeless, and form, as that which thought imposes. But as we advance to mathematics, we impose a new element of form in introducing the relation otherness and its developments; and this being operated on by thought takes the place of new matter: in short, what is introduced as matter is form in the making. All this could be summed up by saying that for elemental logic, mathematical notions would constitute matter; whereas when the step into mathematics is once taken these same elements are just those in accordance with which thought advances in constructing more and more complicated forms. This view of the relation of logic to mathematics will be worked out in Part II of the present work under 'Demonstration,' where the procedure of building up mathematical science is shown to involve the very same principles as are used in the logical structure.
All the sciences, including mathematics, over which logic has supreme control, have been properly described asjapplied logic. But mathematics is applied logic in a certain very unique sense, for mathematics is nothing but an extension of logical formulae introducing none but purely logical factors; while every other science borrows its material from experiential sources, and can only use logical principles when or after such material is supplied. Within mathematics we have again the same kind of distinction, namely that between pure and applied mathematics, as it has been called. In pure mathematics, the mathematician can give free play to his imagination in constructing forms that are restricted only by principles of logical consistency, and he develops the implications that are derivable from what may be indifferently regarded either as definitions of his fictitious constructs or as hypothetically entertained first axioms. In order that these axioms and the theorems therefrom derived may be considered as true, recourse must be had to the real world, and if applicable, the axioms come to be assertorically entertained as premisses, and the derived propositions as the developed conclusions. This application of mathematics to reality constitutes applied mathematics. Taking geometry as our first example, while there is no limit to constructing conceived spaces other than Euclidian, their application to reality demands the enquiry whether our space is or is not Euclidian. This is answered by an appeal to our immediate intuitions directed to our spatial experiences, and it is this appeal that is outside the range of pure mathematics. Again the merely logical conception of betweenness, which develops into that of serial orders of lower or higher forms of complexity, is in the first instance a product of pure logical constructiveness, and would yield implications from which a system of implicates could be developed. But such a hypothetically conceived body of propositions would have no basis in the real but for the applicability of the defined conceptions to what is given in non-mathematical intuition. This applicability holds not only in the domain of spatial order, but also in that of the qualitative relations of difference which impose serial order amongst sense impressions.
Regarded in the light of its control over all sciences logic has been called by the name 'Methodology'; that is to say while the forms of logic implicitly control the conclusions of science, logic itself includes the study which renders explicit the ways according to which its authority is exercised. The department of logic known as methodology constitutes the third part of the present work, which is entitled 'The logical foundations of Science.'
Another illustration of applied mathematics is to quantity. Quantity is not a mere direct development from number, since a new conception, namely that of equality of units, enters as a distinctive factor which is not purely logical. It is true that equality for merely formal developments could be defined as a certain relation having the formal properties of symmetry and transitiveness, and if to this conception is added the fundamental operation plus (+), definable as a certain relation having the formal properties commutative and associative, the whole system of quantitative science could be developed without recourse to any but pure mathematical principles. But even in this range of thought quantities of different types would need recognition. For example, given the notion of length as the first spatial quantity, a new quantity is derived by multiplying length by length, which is called area; here 'multiplied' need not be more specifically defined than a certain relation having the formal properties commutative and associative. Again where a quantum of space is divided by a quantum of time, we have velocity, and in this way a totally new type of quantity is constructed and we pass from geometry to kinematics. Another quantity called mass is such that when multiplied by velocity there is engendered the new quantity called momentum, and when multiplied by velocity squared, energy: and in the introduction of these new species of quantity we pass from kinematics to dynamics. This is the terminus on these lines of applied mathematics; and dynamics may be defined as the science that uses the three independently definable species of quantity time, space and mass. In every extension, then, of mathematics no new idea or mode of thought need accompany the work of the calculus. It is only when the formulae have to be applied to reality, and thus to be entertained categorically, that a process of thought other than merely mathematical enters in, and intuition is directed to what is given in some form of experience. The ideas which enter into the mathematical sciences thus constructed have a form which renders them amenable to purely logical processes of indefinite degrees of complexity; this distinguishes them from the non-mathematical or 'natural' sciences that introduce ideas dependent simply upon brute matter, unamenable to logical analysis, logic entering only in the application to these ideas of classification, and the principles of inductive inference.
§ 8. Having considered logic in its relation to the different sciences, we may now pass to a discussion of its more philosophical aspects. Logicians have been classified as nominalists, conceptualists, and realists or materialists, according as they think it worth while to discuss words, thoughts or things. Names that are apt to be understood as synonyms for these have been applied to different philosophical opinions; and this fact is indicative of the change which has occurred in the course of the history of philosophy, where the ground has been shifted from ontology to psychology, and later from psychology to logic. To take realism first. It is the name given to the Platonic view which formed the basis of Aristotle's controversy with Plato. Plato in discussing the relation between the universal and the individual, attributes real existence in the truest or most ultimate sense to the universal, holding that the particular individual has reality only so far as it partakes of the nature of the universal, towards which it strives as the end (εντελεκη) of its existence. Aristotle, opposing this view, holds that the universal exists not from the particular but in it.
A new psychological significance came to be attached to the term Realism, when the question of reality was raised not about the thing, but about the possible idea of the thing, these two concepts being taken to be equivalent. The so-called nominalist school of philosophers maintained the psychological view that we had no idea corresponding to a general name, along with the ontological view according to which the particular individual or concrete alone existed, and no existence could be attributed to the universal; generality, for them, attached only to names in use, and had no objective application. On the psychological point at issue the opponents of this view have been known as conceptualists, and in maintaining their opposition were led to make a psychological distinction of great importance between images and ideas. In common with the nominalists, they held that images are necessarily concrete, particular or individual, but they maintained that we can also frame ideas which can properly be called abstract or general. Both schools assumed that images were equivalent to or at least resembled perceptions, and further that the latter were obviously concrete and particular. Berkeley represents the nominalist school, and his subtle difference from Locke -- who definitely held that we can frame_general ideas, though with difficulty -- comes out clearly when he disputes the possibility of a general idea of a triangle (instanced by Locke) which shall be neither equilateral nor isosceles nor scalene, and from which we can in thought abstract the shape from variations of colour. In my view Locke and Berkeley were both wrong, even where they agreed; inasmuch as neither images nor perceptions reflect the concreteness and particularity of the individual thing, which should be described as determinate, in contrast to the indeterminateness of the mental processes. In fact there has been a confusion in the description of our thoughts, images and percepts, between the distinction of the universal from the particular, and that of the indeterminate from the determinate. The modern term 'generic,' which has been applied to images, should be extended also to percepts, on the ground that they share with images the character of indeterminateness -- a character which must be rigidly distinguished from general or universal as properly applied to ideas or concepts.
Nominalism has yet another meaning when applied as a special logical theory; in this sense it denotes the theory according to which the proposition is an indication of the names that have been arbitrarily chosen to denote things or classes of things, and predicates merely what follows from the consistent use of these names. Propositions are thus used as mere formulae and repeated in thought when necessary, without demanding any consideration of their meaning; so that the only ultimate foundations or premisses of knowledge are definitions, no other propositions of the nature of axioms being required. This view still clings to some modern philosophical expositions of arithmetic and pure logic, and is rather subtly akin to the view that the first premisses for science are nothing but postulates or hypotheses which, if consistently held, lead to the discovery of truth.
As regards Conceptualism, it is doubtful whether, as applied to the work of such writers as Hamilton and Sigwart, it can be properly regarded as a distinctive logical theory. For the prominent use of the word concept and its associate judgment points not necessarily to any difference of logical theory between those who use these words, and those who prefer the words 'term' or 'name' and 'proposition,' but merely to the common recognition that thought has form as well as verbal expression. If, however, the conceptualist proceeds to limit the scope of logic to the consideration of the forms of thought alone, then he must maintain that the truth of a judgment is tested by the form that connects the content as conceived; and conceptualism becomes equivalent to formalism. The criterion for the formalist is indeed mere consistency or coherence in fact; that for the conceptualist proper, clearness or distinctness in thought. The latter is expressed negatively by Herbert Spencer: what is clearly not conceivable is false; positively by Descartes: what is clearly conceivable is true. It follows immediately from this view that truth concerns only conceived content; so that the direct objects of thought are not things, but our ideas about things, and judgment contains no reference to things but only to adjectives. On this understanding, the conceptualist's view is that we can only deal with things as conceived, and that it is the mode under which we conceive them that determines the adjectives themselves and their relations as constituting the content of the judgment. In this way they are led to deny all relations as subsisting between things -- a denial which is simply equivalent to denying the one supreme relation otherness; for otherness may be said to be the one determinable relation to which all specific relations stand as determinates. Hence it is enough for this school of philosophy to deny the single relation otherness, and in this denial to adopt the position of monism. The view, if carried out rigidly, goes beyond that of Spinoza, who asserted that thought was other than extension, and even that the one Substance had an infinity of other attributes, though not conceivable by us. It is an odd fact that Lotze, in particular, explicitly rejects relations only, as expressive of the nature of Reality; but in consistency he ought to have included in his rejection ordinary adjectives. From this point of view, the only kind of singular categorical judgment concerns Reality as a whole and not any one of its several separable parts: it predicates character of the indivisible one, not of this or that unit in the one. Individual units, in fact, are conceived as the result of the imposition of thought to which nothing in the one corresponds. Thus the Monist's first principle is to deny the Pluralisms fundamental assumption that the Real, as given to thought, is given as many and as such involves existential otherness.
The conceptualist's account of the character of the singular judgment leads to a similar account of that of the particular and the universal judgment. The view is consistently borne out by his interpretation of particulars as possible conjunctions, i.e. of adjectives that we can conjoin in conception; and of universals as necessary conjunctions, i.e. of adjectives that we must conjoin in conception. Symbolically: 'Some things that are p are q' is to mean 'p and q can be conjoined and can be disjoined'; 'Everything or nothing that is p is q' is to mean 'p and q must be conjoined or must be disjoined.' What is true in this view is that the operations not, and, not-both, if, or, are supplied by thought: and that nothing in the merely objective world manifests the mere absence of a character, or the mere indeterminateness of the alternative operation, or dependence as expressed by implication. These relations are not manifested to thought, but analytically or synthetically discovered or rather imposed by thought. The view is most strikingly expressed by Mr Bradley in his dictum: only if what is possible is necessitated will it be actualised; and again, only if what is necessary is possible will it be actualised.
From conceptualism we pass back again to realism in its new sense as applying to logic, and in this application it is usually denoted by the term rnaterialism or empiricism. We are thus led back again to Venn, and less explicitly to Mill, who contrasts the formalism or conceptualism of Hamilton with his own logical standpoint. Taking empiricism to mean that all knowledge is obtained by experience alone (as Mill only seems to have held) the doctrine amounts to maintaining that all inference is ultimately of the nature of pure induction. But taking it to mean that no knowledge gained by experience can be validly universalised (as Venn seems to hold) then the doctrine amounts to maintaining that no inference of the nature of pure induction is valid, and that hence only deduction is guaranteed by logic. In default of any explication of which of these two views is meant by empiricism or materialism, we can only conclude that the term stands for that department of logic that is concerned with an analysis of the process of induction. But here we must note that the distinction in character between induction and deduction is not properly expressed by the antithesis of matter and form; since the relations amongst premisses and conclusion which constitute the form of an inference hold for the validity of induction as for that of deduction; and conversely, reference to the matter of the propositions is required equally for the truth of a deductive inference as for that of an inductive inference. This obvious fact has been forgotten, owing to the great prominence given by inductive logicians to the treatment of the preliminary processes of observation, search, arrangement, comparison of material data, and the formation of formulae that shall hold for the facts collected, and the aid required by experimentation. In consequence, stress is laid on the securing of correctly described premisses in the case of induction; whereas in the case of deduction stress is laid only on securing validity for the form of inference.
§ 9. In conclusion I propose to enumerate the most important features in the treatment of logical theory to be developed in the course of this work:
(a) The epistemic aspect of thought is included within the province of logic, and contrasted with the constitutive aspect; the former is a recognition that knowledge depends upon the variable conditions and capacities for its acquisition; the latter refers to the content of knowledge which has in itself a logically analysable form. Such fallacies as petitio principii really require reference to the epistemic aspect of thought, while fallacies of the strictly formal type refer exclusively to the constitutive aspect. Again the whole theory of modality which develops into probability is essentially epistemic, indicating as it does the relation of the content of the proposition to the thinker. Thus a distinction is clearly drawn between the proposition and the attitude of assertion or judgment; and while, on this view, the proposition is identifiable when in variable relations to different thinkers, the necessity is emphasised of conceiving the proposition in terms of assertion, the act of assertion being thus taken as the complete fact to be analysed and criticised. It is this intimate connection between the assertion and the proposition which gives meaning to the identification of the adjectives true and false with the imperatives 'to be accepted' and 'to be rejected.'
(b) The proposition itself, which is customarily resolved into subject and predicate, is more precisely analysed by showing that the substantive alone can function as subject, and the adjective as predicate, and that these stand to one another in the relation of characterisation: the substantive being that which is characterised, the predicate that which characterises. Since an appropriate adjective can be predicated of a subject belonging to any category, including adjective, relation and proposition, the subject as thus functioning becomes a quasi-substantive. The substantive proper seems to coincide with the category 'existent,' while if any category other than substantive stands as subject its logical nature is not thereby altered, but rather the adjectives proper to it fall under correspondingly special sub-categories determined by the category to which the subject belongs.
(c) Adjectives are fundamentally distinguishable into determinables and determinates, the relation between which is primarily a matter of degree, a determinable being the extreme of indeterminateness under which adjectives of different degrees of determinateness are subsumed. The relation of a determinate to its determinable resembles that of an individual to a class, but differs in some important respects. For instance, taking any given determinate, there is only one determinable to which it can belong. Moreover any one determinable is a literal summum genus not subsumable under any higher genus; and the absolute determinate is a literal infima species under which no other determinate is subsumable.
(d) Relations are treated as a specific kind of adjective, and are called transitive adjectives in distinction from ordinary adjectives which are intransitive. The adjectival nature of relations is apt to be obscured by the inclusion under relative terms of what are merely substantives defined by relational characterisation. All that holds universally of adjectives, including the relation of determinates to their determinable, holds of relations as such.
(e) Under the head of induction, fundamentally different types are distinguished. First: the very elementary process of intuitive induction, which lies at the basis of the distinction between form and matter, and by which all the formal principles of logic are established. Next: summary induction, more usually called perfect induction, which establishes conclusions of limited universality by means of mere enumeration. Such a summary universal stands as premiss for an unlimited universal conclusion, obtained by what is called inductio per simplicem enumerationem. What is specially important in my treatment is the function of summary induction in the specifically geometrical form of inference. Thirdly: demonstrative induction, which employs no other principles than those which have been recognised in deduction. This species of induction is directly employed in inferring from a single experimental instance an unlimited universal; and it is this species of induction which gives the true form to the methods formulated by Mill and Bacon. Lastly we distinguish induction proper, which is conceived as essentially problematic, and as thus re-introducing the epistemic aspect in the form of probability.
(f) The specific notion of cause as applying to events is distinguished from the generic notion of mere determination according to a universal formula. As specific, cause relates exclusively to states or conditions temporally alterable and also referable to place; and, in this application of the notion of determination, the effect and cause are homogeneous. Not only is the character of the effect regulated by that of the cause, but the date and place of the latter is determined by the date and place of the former. The universal positional relation, as it may be called, of cause to effect is that of contiguity, which is to be conceived in the form of the coincidence of the temporal or spatial boundary of that which constitutes the cause with that which constitutes the effect. This absolute contiguity disallows any gap between the cause process and the effect process; so that contiguity is strictly defined as equivalent to continuity. This further implies that when, as is always permissible, we conceive a phase of the causal process as temporally or spatially separated from a phase of the effect process, we must also conceive of that which goes on in the interval bridging cause and effect to be part of one continuous process. This is possible because time and space are themselves continuous. Thus change and movement are connectionally continuous, in the special sense that the character manifested at one instant of time or at one point of space differs from that manifested at another instant of time or at another point of space, in a degree the smallness of which depends upon that of the temporal or spatial interval. Again superimposed upon the continuity of this process, there is a discontinuity of the second order, ultimately due to the discontinuous occupation of space by different kinds of matter.
(g) The notions of cause and substance reciprocally imply one another, the latter being that which continues to exist and in which alterable states or conditions inhere. These alterable states constitute what may be called the occurrent or, in accordance with scholastic usage, the occasional causal factor. The occurrent is distinguished from and essentially connected with the continuant or the material factor in causation. The occurrent and continuant factors are thus united in our complete conception of substance, neither being conceivable apart from the other. This analysis gives meaning to the conception of the properties of the continuant, as potential causes which are actualised in accordance with unchanging rules by the relatively incidental occurrences that come into being either from within or from without the continuant. In the former case the process is immanent, cause and effect being manifestations of the changeless nature of the continuant, and the temporal relation between cause and effect is here that of succession. In the latter case, the causality is transeunt, the patient being that whose state is determined, the agent being that whose alterable relation to the agent is determinative. In transeunt causality the temporal relation of cause to effect is literal simultaneity, and the critical instant at which the cause operates is that in which there is also literal geometrical contact of cause agent with effect patient. There are two fundamentally distinct types of transeunt causality. In the one case no change of state in the agent accompanies the change of state in the patient, and we have action without any direct reaction; in the other case change of state in the one directly entails change of state in the other of such a nature that the latter may be formulated as a function of the former, and here action always involves an assignable reaction. The latter case holds invariably of inter-physical causality, and again of inter-psychical causality within the sphere of a single individual's experience. But in physico-psychical causality, as also in psycho-pjrysical causality, action never directly determines reaction, owing to the absolute disparateness between the physical and psychical in regard to the characters of the states which are predicable of the one and of the other. It is here where my treatment of logical questions transgresses into the domain of ontology; but it must be admitted that all logicians who treat these subjects inevitably transgress in the same manner.
(h) The position assumed by probability in logical discussion has always been dubious. On the one side the topic has been assumed to be the exclusive property of the mathematician, or rather more precisely, the arithmetician. On this view the quantity called probability is a mere abstract fraction, and the rules of probability are merely those of arithmetic. The fraction is, in short, the ratio of two numbers, the number holding for a species to that holding for its proximate genus, this ratio being necessarily a proper fraction, the limits of which are zero and unity. If this view were correct, there would be no separate topic to be called probability. A precisely reverse account of probability is that it is a measure of a certain psychological attitude of thought to which the most obvious names that could be given are belief or doubt, taken as subject to different degrees. On either of these two extreme views probability would have no particular connection with logic. The psychological account would be separated from logic, inasmuch as it would concern solely the causal explanation of different degrees of belief, and would thus give rise to no principle of rational criticism. The mere arithmetical account of probability ought in the first instance to be corrected by the recognition that the topic has its mental side. This correction requires that probability should not be expressed by a merely abstract fraction, but rather as a fraction of a certain mental quantity which may be called certainty. The psychological conditions of the variable degrees in which doubt may approximate to certainty are as such outside the province of logic; but when these various degrees are such as reason would dictate, we may speak of reasonable doubt as an assignable fraction of certitude, thus bringing the subject into the sphere of logic. Further the quantity or degree called probability attaches exclusively to the proposition; not however to the proposition as such, but to the proposition regarded as based upon rationally certified knowledge acquired by any supposed thinker. The degree of probability is therefore referential to such knowledge, but is wholly independent of the individual thinker, being dependent solely on his rational nature, and the knowledge which he has rationally acquired.
The whole development of this aspect of the subject is to be called formal probability, and constitutes the one subject of the fourth Part of this work. The treatment of probability there developed must be distinguished from that of informal probability, that is required in discussing the foundations of science as treated in my third Part; for there, while the logic of inductive inference is made to depend upon the principles of probability and not upon any big fact about nature, yet probability is only introduced on broad and indeterminately quantitative lines. This treatment leads to an attempted enumeration of broadly formulated criteria for the evaluation of the degrees of probability to be attached to the generalisations of inductive inference. These criteria are merely expressions of what is popularly felt, and their rational justification can only be represented as depending upon postulates: that is, speculations that are neither intuitively self-evident nor experientially verifiable, but merely demanded by reason in order to supply an incentive to the endeavour to systematise the world of reality and thus give to practical action an adequate prompting motive.