Chapter II: History (1)

Logic cannot dispense with the light afforded by its history so long as counter-solutions of the same fundamental problems continue to hold the field. A critical review of some of the chief types of logical theory, with a view to determine development, needs no further justification.

Logic arose, at least for the Western world, in the golden age of Greek speculation which culminated in Plato and Aristotle. There is an Indian logic, it is true, but its priority is more than disputable. In any case no influence upon Greek thought can be shown. The movement which ends in the logic of Aristotle is demonstrably self-contained. When we have shaken ourselves free of the prejudice that all stars are first seen in the East, Oriental attempts at analysis of the structure of thought may be treated as negligible.

It is with Aristotle that the bookish tradition begins to dominate the evolution of logic. The technical perfection of the analysis which he offers is, granted the circle of presuppositions within which it works, so decisive, that what precedes, even Plato's logic, is not unnaturally regarded as merely preliminary and subsidiary to it. What follows is inevitably, whether directly or indirectly, by sympathy or by antagonism, affected by the Aristotelian tradition.

A. GREEK LOGIC

i. _Before Aristotle_

The physical philosophers.

Logic needs as its presuppositions that thought should distinguish itself from things and from sense, that the problem of validity should be seen to be raised in the field of thought itself, and that analysis of the structure of thought should be recognized as the one way of solution. Thought is somewhat late in coming to self-consciousness. Implied in every contrast of principle and fact, of rule and application, involved as we see after the event, most decisively when we react correctly upon a world incorrectly perceived, thought is yet not reflected on in the common experience. Its so-called natural logic is only the potentiality of logic. The same thing is true of the first stage of Greek philosophy. In seeking for a single material principle underlying the multiplicity of phenomena, the first nature-philosophers, Thales and the rest, did indeed raise the problem of the one and the many, the endeavour to answer which must at last lead to logic. But it is only from a point of view won by later speculation that it can be said that they sought to determine the predicates of the single subject-reality, or to establish the permanent subject of varied and varying predicates.[1] The direction of their inquiry is persistently outward. They hope to explain the opposed appearance and reality wholly within the world of things, and irrespective of the thought that thinks things. Their universal is still a material one. The level of thought on which they move is still clearly pre-logical. It is an advance on this when Heraclitus[2] opposes to the eyes and ears which are bad witnesses "for such as understand not their language" a common something which we would do well to follow; or again when in the incommensurability of the diagonal and side of a square the Pythagoreans stumbled upon what was clearly neither thing nor image of sense, but yet was endowed with meaning, and henceforth were increasingly at home with symbol and formula. So far, however, it might well be that thought, contradistinguished from sense with its illusions, was itself infallible. A further step, then, was necessary, and it was taken at any rate by the Eleatics, when they opposed their thought to the thought of others, as the way of truth in contrast to the way of opinion. If Eleatic thought stands over against Pythagorean thought as what is valid or grounded against what is ungrounded or invalid, we are embarked upon dialectic, or the debate in which thought is countered by thought. Claims to a favourable verdict must now be substantiated in this field and in this field alone. It was Zeno, the controversialist of the Eleatic school, who was regarded in after times as the "discoverer" of dialectic.[3]

Zeno's amazing skill in argumentation and his paradoxical conclusions,
particular and general, inaugurate a new era. "The philosophical
mind," says waiter Pater,[4] "will perhaps never be quite in health,
quite sane or natural again." The give and take of thought had by a
swift transformation of values come by something more than its own.
Zeno's paradoxes, notably, for example, the puzzle of Achilles and the
Tortoise, are still capable of amusing the modern world. In his own
age they found him imitators. And there follows the sophistic
movement.

The Sophists.

The sophists have other claims to consideration than their service to
the development of logic. In the history of the origins of logic the
sophistic age is simply the age of the free play of thought in which
men were aware that in a sense anything can be debated and not yet
aware of the sense in which all things cannot be so. It is the age of
discussion used as a universal solvent, before it has been brought to
book by a deliberate unfolding of the principles of the structure of
thought determining and limiting the movement of thought itself. The
sophists furthered the transition from dialectic to logic in two ways.
In the first place they made it possible. Incessant questioning leads
to answers. Hair-splitting, even when mischievous in intent, leads to
distinctions of value. Paradoxical insistence on the accidents of
speech-forms and thought-forms leads in the end to perception of the
essentials. Secondly they made it necessary. The spirit of debate run
riot evokes a counter-spirit to order and control it. The result is a
self-limiting dialectic. This higher dialectic is a logic. It is no
accident that the first of the philosophical sophists, Gorgias, on the
one hand, is Eleatic in his affinities, and on the other raises in the
characteristic formula of his intellectual nihilism[5] issues which
are as much logical and epistemological as ontological. The meaning of
the copula and the relation of thoughts to the objects of which they
are the thoughts are as much involved as the nature of being. It is
equally no accident that the name of Protagoras is to be connected, in
Plato's view at least, with the rival school of Heracliteans. The
problems raised by the relativism of Protagoras are no less
fundamentally problems of the nature of knowledge and of the structure
of thought. The _Theaetetus_ indeed, in which Plato essays to deal
with them, is in the broad sense of the word logical, the first
distinctively logical treatise that has come down to us. Other
sophists, of course, with more practical interests, or of humbler
attainments, were content to move on a lower plane of philosophical
speculation. As presented to us, for example, in Plato's surely not
altogether hostile caricature in the _Euthydemus_, they mark the
intellectual preparation for, and the moral need for, the advance of
the next generation.

Socrates.

Among the pioneers of the sophistic age Socrates stands apart. He has
no other instrument than the dialectic of his compeers, and he is as
far off as the rest from a criticism of the instrument, but he uses it
differently and with a difference of aim. He construes the give and
take of the debate-game with extreme rigour. The rhetorical element
must be exorcised. The set harangue of teacher to pupil, in which
steps in argument are slurred and the semblance of co-inquiry is
rendered nugatory, must be eliminated. The interlocutors must in truth
render an account under the stimulus of organized heckling from their
equals or superiors in debating ability. And the aim is heuristic,
though often enough the search ends in no overt positive conclusion.
Something can be found and something is found. Common names are fitted
for use by the would-be users being first delivered from abortive
conceptions, and thereupon enabled to bring to the birth living and
organic notions.

Aristotle would assign to Socrates the elaboration of two logical
functions:--general definition and inductive method.[6] Rightly, if we
add that he gives no theory of either, and that his practical use of
the latter depends for its value on selection.[7] It is rather in
virtue of his general faith in the possibility of construction, which
he still does not undertake, and because of his consequent insistence
on the elucidation of general concepts, which in common with some of
his contemporaries, he may have thought of as endued with a certain
objectivity, that he induces the controversies of what are called the
Socratic schools as to the nature of predication. These result in the
formulation of a new dialectic or logic by Plato. Manifestly Socrates'
use of certain forms of argumentation, like their abuse by the
sophists, tended to evoke their logical analysis. The use and abuse,
confronted one with the other, could not but evoke it.

Antisthenes.

The one in the many, the formula which lies at the base of the
possibility of predication, is involved in the Socratic doctrine of
general concepts or ideas. The nihilism of Gorgias from the Eleatic
point of view of bare identity, and the speechlessness of Cratylus
from the Heraclitean ground of absolute difference, are alike
disowned. But the one in the many, the identity in difference, is so
far only postulated, not established. When the personality of Socrates
is removed, the difficulty as to the nature of the Socratic universal,
developed in the medium of the individual processes of individual
minds, carries disciples of diverse general sympathies, united only
through the practical inspiration of the master's life, towards the
identity-formula or the difference-formula of other teachers. The
paradox of predication, that it seems to deny identity, or to deny
difference, becomes a _pons asinorum_. Knowledge involves synthesis or
nexus. Yet from the points of view alike of an absolute pluralism, of
a flux, and of a formula of bare identity--and _a fortiori_ with any
blending of these principles sufficiently within the bounds of
plausibility to find an exponent--all knowledge, because all
predication of unity, in difference, must be held to be impossible.
Plato's problem was to find a way of escape from this impasse, and
among his Socratic contemporaries he seems to have singled out
Antisthenes[8] as most in need of refutation. Antisthenes, starting
with the doctrine of identity without difference, recognizes as the
only expression proper to anything its own peculiar sign, its name.
This extreme of nominalism for which predication is impossible is,
however, compromised by two concessions. A thing can be described as
like something else. And a compound can have a [Greek: logos] or
account given of it by the (literally) adequate enumeration of the
names of its simple elements or [Greek: prôta].[9] This analytical
[Greek: logos] he offers as his substitute for knowledge.[10] The
simple elements still remain, sensed and named but not known. The
expressions of them are simply the speech-signs for them. The account
of the compound simply sets itself taken piecemeal as equivalent to
itself taken as aggregate. The subject-predicate relation fails really
to arise. Euclides[11] found no difficulty in fixing Antisthenes' mode
of illustrating his simple elements by comparison, and therewith
perhaps the "induction" of Socrates, with the dilemma; so far as the
example is dissimilar, the comparison is invalid; so far as it is
similar, it is useless. It is better to say what the thing is. Between
Euclides and Antisthenes the Socratic induction and universal
definition were alike discredited from the point of view of the
Eleatic logic. It is with the other point of doctrine that Plato comes
to grips, that which allows of a certainty or knowledge consisting in
an analysis of a compound into simple elements themselves not known.
The syllable or combination is, he shows, not known by resolution of
it into letters or elements themselves not known. An aggregate
analysed into its mechanical parts is as much and as little known as
they. A whole which is more than its parts is from Antisthenes' point
of view inconceivable. Propositions analytical of a combination in the
sense alleged do not give knowledge. Yet knowledge is possible. The
development of a positive theory of predication has become quite
crucial.

Plato.

Plato's logic supplies a theory of universals in the doctrine of ideas. Upon this it bases a theory of predication, which, however, is compatible with more than one reading of the metaphysical import of the ideas. And it sets forth a dialectic with a twofold movement, towards differentiation and integration severally, which amounts to a formulation of inference. The more fully analysed movement, that which proceeds downward from less determinate to more determinate universals, is named Division. Its associations, accordingly, are to the modern ear almost inevitably those of a doctrine of classification only. Aristotle, however, treats it as a dialectical rival to syllogism, and it influenced Galilei and Bacon in their views of inference after the Renaissance. If we add to this logic of "idea," judgment and inference, a doctrine of categories in the modern sense of the word which makes the _Theaetetus_, in which it first occurs, a forerunner of Kant's _Critique of Pure Reason_, we have clearly a very significant contribution to logic even in technical regard. Its general philosophical setting may be said to enhance its value even as logic.

The "Idea."

(a) Of the idea we may say that whatever else it is, and apart from all puzzles as to ideas of relations such as smallness, of negative qualities such as injustice, or of human inventions such as beds, it is opposed to that of which it is the idea as its intelligible formula or law, the truth or validity--Herbart's word--of the phenomenon from the point of view of nexus or system. The thing of sense in its relative isolation is unstable. It is and is not. What gives stability is the insensible principle or principles which it holds, as it were, in solution. These are the ideas, and their mode of being is naturally quite other than that of the sensible phenomena which they order. The formula for an indefinite number of particular things in particular places at particular times, and all of them presentable in sensuous imagery of a given time and place, is not itself presentable in sensuous imagery side by side with the individual members of the group it orders. The law, e.g., of the equality of the radii of a circle cannot be exhibited to sense, even if equal radii may be so exhibited. It is the wealth of illustration with which Plato expresses his meaning, and the range of application which he gives the idea--to the class-concepts of natural groups objectively regarded, to categories, to aesthetic and ethical ideals, to the concrete aims of the craftsman as well as to scientific laws--that have obscured his doctrine, viz. that wherever there is law, there is an idea.

The one in the many.

(b) The paradox of the one in the many is none, if the idea may be regarded as supplying a principle of nexus or organization to an indefinite multiplicity of particulars. But if Antisthenes is to be answered, a further step must be taken. The principle of difference must be carried into the field of the ideas. Not only sense is a principle of difference. The ideas are many. The multiplicity in unity must be established within thought itself. Otherwise the objection stands: man is man and good is good, but to say that man is good is clearly to say the thing that is not. Plato replies with the doctrine of the interpenetration of ideas, obviously not of all with all, but of some with some, the formula of identity in difference within thought itself. Nor can the opponent fairly refuse to admit it, if he affirms the participation of the identical with being, and denies the participation of difference with being, or affirms it with not-being. The _Sophistes_ shows among other things that an identity-philosophy breaks down into a dualism of thought and expression, when it applies the predicate of unity to the real, just as the absolute pluralism on the other hand collapses into unity if it affirms or admits any form of relation whatsoever. Identity and difference are all-pervasive categories, and the speech-form and the corresponding thought-form involve both. For proposition and judgment involve subject and predicate and exhibit what a modern writer calls "identity of reference with diversity of characterization." Plato proceeds to explain by his principle of difference both privative and negative predicates, and also the possibility of false predication. It is obvious that without the principle of difference error is inexplicable. Even Plato, however, perhaps scarcely shows that with it, and nothing else but it, error is explained.

Division.

(c) Plato's Division, or the articulation of a relatively indeterminate and generic concept into species and sub-species with resultant determinate judgments, presumes of course the doctrine of the interpenetration of ideas laid down in the _Sophistes_ as the basis of predication, but its use precedes the positive development of that formula, though not, save very vaguely, the exhibition of it, negatively, in the antinomies of the one and the many in the _Parmenides_. It is its use, however, not the theory of it, that precedes. The latter is expounded in the _Politicus_ (260 sqq.) and _Philebus_ (16c sqq.). The ideal is progressively to determine a universe of discourse till true _infimae species_ are reached, when no further distinction in the determinate many is possible, though there is still the numerical difference of the indefinite plurality of particulars. The process is to take as far as possible the form of a continuous disjunction of contraries. We must bisect as far as may be, but the division is after all to be into limbs, not parts. The later examples of the _Politicus_ show that the permission of three or more co-ordinate species is not nugatory, and that the precept of dichotomy is merely in order to secure as little of a _saltus_ as possible; to avoid e.g. the division of the animal world into men and brutes. It is the middle range of the [Greek: mesa] of _Philebus_ 17a that appeals to Bacon, not only this but their mediating quality that appeals to Aristotle. The _media axiomata_ of the one and the _middle term_ of the other lie in the phrase. Plato's division is nevertheless neither syllogism nor _exclusiva_. It is not syllogism because it is based on the disjunctive, not on the hypothetical relation, and so extends horizontally where syllogism strikes vertically downward. Again it is not syllogism because it is necessarily and finally dialectical. It brings in the choice of an interlocutor at each stage, and so depends on a concession for what it should prove.[12] Nor is it Bacon's method of exclusions, which escapes the imputation of being dialectical, if not that of being unduly cumbrous, in virtue of the cogency of the negative instance. The Platonic division was, however, offered as the scientific method of the school. A fragment of the comic poet Epicrates gives a picture of it at work.[13] And the movement of disjunction as truly has a place in the scientific specification of a concept in all its differences as the linking of lower to higher in syllogism. The two are complementary, and the reinstatement of the disjunctive judgment to the more honourable rôle in inference has been made by so notable a modern logician as Lotze.

Combination.

(d) The correlative process of Combination is less elaborately sketched, but in a luminous passage in the _Politicus_ (§ 278), in explaining by means of an example the nature and use of examples, Plato represents it as the bringing of one and the same element seen in diverse settings to conscious realization, with the result that it is viewed as a single truth of which the terms compared are now accepted as the differences. The learner is to be led forward to the unknown by being made to hark back to more familiar groupings of the alphabet of nature which he is coming to recognize with some certainty. To lead on, [Greek: epagein], is to refer back, [Greek: anagein],[14] to what has been correctly divined of the same elements in clearer cases. Introduction to unfamiliar collocations follows upon this, and, only so, is it possible finally to gather scattered examples into a conspectus as instances of one idea or law. This is not only of importance in the history of the terminology of logic, but supplies a philosophy of induction.

Mental synthesis.

(e) Back of Plato's illustration and explanation of predication and dialectical inference there lies not only the question of their metaphysical grounding in the interconnexion of ideas, but that of their epistemological presuppositions. This is dealt with in the Theaetetus (184b sqq.). The manifold affections of sense are not simply aggregated in the individual, like the heroes in the Trojan horse. There must be convergence in a unitary principle, soul or consciousness, which is that which really functions in perception, the senses and their organs being merely its instruments. It is this unity of apperception which enables us to combine the data of more than one sense, to affirm reality, unreality, identity, difference, unity, plurality and so forth, as also the good, the beautiful and their contraries. Plato calls these pervasive factors in knowledge [Greek: koina], and describes them as developed by the soul in virtue of its own activity. They are objects of its reflection and made explicit in the few with pains and gradually.[15] That they are not, however, psychological or acquired categories, due to "the workmanship of the mind" as conceived by Locke, is obvious from their attribution to the structure of mind[16] and from their correlation with immanent principles of the objective order. Considered from the epistemological point of view, they are the implicit presuppositions of the construction or [Greek: syllogismos][17] in which knowledge consists. But as ideas,[18] though of a type quite apart,[19] they have also a constitutive application to reality. Accordingly, of the selected "kinds" by means of which the interpenetration of ideas is expounded in the _Sophistes_, only motion and rest, the ultimate "kinds" in the physical world, have no counterparts in the "categories" of the _Theaetetus_. In his doctrine as to [Greek: en to poioun] or [Greek: krinon], as generally in that of the activity of the [Greek: nous apathês], Aristotle in the _de Anima_[20] is in the main but echoing the teaching of Plato.[21]

ii. _Aristotle._

Plato's episodic use of logical distinctions[22] is frequent. His recourse to such logical analysis as would meet the requirements of the problem in hand[23] is not rare. In the "dialectical" dialogues the question of method and of the justification of its postulates attains at least a like prominence with the ostensible subject matter. There is even formal recognition of the fact that to advance in dialectic is a greater thing than to bring any special inquiry to a successful issue.[24] But to the end there is a lack of interest in, and therefore a relative immaturity of, technique as such. In the forcing atmosphere, however, of that age of controversy, seed such as that sown in the master's treatment of the uttered [Greek: logos][25] quickly germinated. Plato's successors in the Academy must have developed a system of grammatico-logical categories which Aristotle could make his own. Else much of his criticism of Platonic doctrine[26] does, indeed, miss fire. The gulf too, which the _Philebus_[27] apparently left unbridged between the sensuous apprehension of particulars and the knowledge of universals of even minimum generality led with Speusippus to a formula of knowledge in perception ([Greek: epistêmonikê aisthêsis]). These and like developments, which are to be divined from references in the Aristotelian writings, jejune, and, for the most part, of probable interpretation only, complete the material which Aristotle could utilize when he seceded from the Platonic school and embarked upon his own course of logical inquiry.

Syllogism.

This is embodied in the group of treatises later known as the _Organon_[28] and culminates in the theory of syllogism and of demonstrative knowledge in the _Analytics_. All else is finally subsidiary. In the well-known sentences with which the _Organon_ closes[29] Aristotle has been supposed to lay claim to the discovery of the principle of syllogism. He at least claims to have been the first to dissect the procedure of the debate-game, and the larger claim may be thought to follow. In the course of inquiry into the formal consequences from probable premises, the principle of mediation or linking was so laid bare that the advance to the analytic determination of the species and varieties of syllogism was natural. Once embarked upon such an analysis, where valid process from assured principles gave truth, Aristotle could find little difficulty in determining the formula of demonstrative knowledge or science. It must be grounded in principles of assured certainty and must demonstrate its conclusions with the use of such middle or linking terms only as it is possible to equate with the real ground or cause in the object of knowledge. Hence the account of axioms and of definitions, both of substances and of derivative attributes. Hence the importance of determining how first principles are established. It is, then, a fair working hypothesis as to the structure of the _Organon_ to place the _Topics_, which deal with dialectical reasoning, before the _Analytics_.[30] Of the remaining treatises nothing of fundamental import depends on their order. One, however, the _Categories_, may be regarded with an ancient commentator,[31] as preliminary to the dialectical inquiry in the _Topics_. The other, on thought as expressed in language ([Greek: Peri ermêneias]) is possibly spurious, though in any case a compilation of the Aristotelian school. If genuine, its naïve theory that thought copies things and other features of its contents would tend to place it among the earliest works of the philosopher.

The logical treatises.

Production in the form of a series of relatively self-contained treatises accounts for the absence of a name and general definition of their common field of inquiry. A more important lack which results is that of any clear intimation as to the relation in which Aristotle supposed it to stand to other disciplines. In his definite classification of the sciences,[32] into First Philosophy, Mathematics and Physics, it has no place. Its axioms, such as the law of contradiction, belong to first philosophy, but the doctrine as a whole falls neither under this head nor yet, though the thought has been entertained, under that of mathematics, since logic orders mathematical reasoning as well as all other. The speculative sciences, indeed, are classified according to their relation to form, pure, abstract or concrete, i.e. according to their objects. The logical inquiry seems to be conceived as dealing with the thought of which the objects are objects. It is to be regarded as a propaedeutic,[33] which, although it is in contact with reality in and through the metaphysical import of the axioms, or again in the fact that the categories, though primarily taken as forms of predication, must also be regarded as kinds of being, is not directly concerned with object-reality, but with the determination for the thinking subject of what constitutes the knowledge correlative to being. Logic, therefore, is not classed as one, still less as a branch of one, among the 'ologies, ontology not excepted.

The way in which logical doctrine is developed in the Aristotelian treatises fits in with this view. Doubtless what we have is in the main a reflex of the heuristic character of Aristotle's own work as pioneer. But it at least satisfies the requirement that the inquiry shall carry the plain man along with it. Actual modes of expression are shown to embody distinctions which average intelligence can easily recognize and will readily acknowledge, though they may tend by progressive rectification fundamentally to modify the assumption natural to the level of thought from which he begins. Thus we start[34] from the point of view of a world of separate persons and things, in which thought mirrors these concrete realities, taken as ultimate subjects of predicates. It is a world of communication of thought, where persons as thinkers need to utter in language truths objectively valid for the _mundus communis_. In these truths predicates are accepted or rejected by subjects, and therefore depend on the reflection of fact in [Greek: logoi] (propositions). These are combinatory of parts, attaching or detaching predicates, and so involving subject, predicate and copula.[35] At this stage we are as much concerned with speech-forms as the thought-forms of which they are conventional symbols, with Plato's analysis, for instance, into a noun and a verb, whose connotation of time is as yet a difficulty. The universal of this stage is the universal of fact, what is recognized as predicable of a plurality of subjects. The dialectical doctrine of judgment as the declaration of one member of a disjunction by contradiction, which is later so important, is struggling with one of its initial difficulties,[36] viz. the contingency of particular events future, the solution of which remains imperfect.[37]

The Categories.

The doctrine of the _Categories_ is still on the same level of thought,[38] though its grammatico-logical analysis is the more advanced one which had probably been developed by the Academy before Aristotle came to think of his friends there as "them" rather than "us." It is what in one direction gave the now familiar classification of parts of speech, in the other that of thought-categories underlying them. If we abstract from any actual combination of subject and predicate and proceed to determine the types of predicate asserted in simple propositions of fact, we have on the one hand a subject which is never object, a "first substance" or concrete thing, of which may be predicated in the first place "second substance" expressing that it is a member of a concrete class, and in the second place quantity, quality, correlation, action and the like. The list follows the forms of the Greek language so closely that a category emerges appropriated to the use of the perfect tense of the middle voice to express the relation of the subject to a garb that it dons. In all this the individual is the sole self-subsistent reality. Truth and error are about the individual and attach or detach predicates correctly and incorrectly. There is no committal to the metaphysics in the light of which the logical inquiry is at last to find its complete justification. The point of view is to be modified profoundly by what follows--by the doctrine of the class-concept behind the class, of the form or idea as the constitutive formula of a substance, or, again, by the requirement that an essential attribute must be grounded in the nature or essence of the substance of which it is predicated, and that such attributes alone are admissible predicates from the point of view of the strict ideal of science. But we are still on the ground of common opinion, and these doctrines are not yet laid down as fundamental to the development.

The Topics.

Dialectic then, though it may prove to be the ultimate method of establishing principles in philosophy,[39] starts from probable and conceded premises,[40] and deals with them only in the light of common principles such as may be reasonably appealed to or easily established against challenge. To the expert, in any study which involves contingent matter, i.e. an irreducible element of indetermination, e.g. to the physician, there is a specific form of this, but the reflection that this is so is something of an afterthought. We start with what is prima facie given, to return upon it from the ground of principles clarified by the sifting process of dialectic[41] and certified by [Greek: nous]. The _Topics_ deal with dialectic and constitute an anatomy of argumentation, or, according to what seems to be Aristotle's own metaphor, a survey of the tactical vantage-points ([Greek: topoi]) for the conflict of wits in which the prize is primarily victory, though it is a barren victory unless it is also knowledge. It is in this treatise that what have been called "the conceptual categories"[42] emerge, viz. the _predicables_, or heads of predication as it is analysed in relation to the provisional theory of definition that dialectic allows and requires. A predicate either is expressive of the essence or part of the essence of the subject, viz. that original group of mutually underivable attributes of which the absence of any one destroys its right to the class-name, or it is not. Either it is convertible with the subject or it is not. Here then judgment, though still viewed as combinatory, has the types which belong to coherent systems of implication discriminated from those that predicate coincidence or accident, i.e. any happening not even derivatively essential from the point of view of the grouping in which the subject has found a place. In the theory of dialectic any predicate may be suggested for a subject, and if not affirmed of it, must be denied of it, if not denied must be affirmed. The development of a theory of the ground on which subjects claim their predicates and disown alien predicates could not be long postponed. In practical dialectic the unlimited possibility was reduced to manageable proportions in virtue of the groundwork of received opinion upon which the operation proceeded. It is in the _Topics_, further, that we clearly have a first treatment of syllogism as formal implication, with the suggestion that advance must be made to a view of its use for material implication from true and necessary principles. It is in the _Topics_,[43] again, that we have hints at the devices of an inductive process, which, as dialectical, throw the burden of producing contradictory instances upon the other party to the discussion. In virtue of the common-stock of opinion among the interlocutors and their potentially controlling audience, this process was more valuable than appears on the face of things. Obviously tentative, and with limits and ultimate interpretation to be determined elsewhere, it failed to bear fruit till the Renaissance, and then by the irony of fate to the discrediting of Aristotle. In any case, however, definition, syllogism, induction all invited further determination, especially if they were to take their place in a doctrine of truth or knowledge. The problem of analytic, i.e. of the resolution of the various forms of inference into their equivalents in that grouping of terms or premises which was most obviously cogent, was a legacy of the _Topics_. The debate-game had sought for diversion and found truth, and truth raised the logical problem on a different plane.

Class concept.

The Prior Analytics.

At first the problem of formal analysis only. We proceed with the talk of instances and concern ourselves first with relations of inclusion and exclusion. The question is as to membership of a class, and the dominant formula is the _dictum de omni et nullo_. Until the view of the individual units with which we are so far familiar has undergone radical revision, the primary inquiry must be into the forms of a class-calculus. Individuals fall into groups in virtue of the possession of certain predicates. Does one group include, or exclude, or intersect another with which it is compared? We are clearly in the field of the diagrams of the text-books, and much of the phraseology is based upon an original graphic representation in extension. The middle term, though conceived as an intermediary or linking term, gets its name as intermediate in a homogeneous scheme of quantity, where it cannot be of narrower extension than the subject nor wider than the predicate of the conclusion.[44] It is also, as Aristotle adds,[45] middle in position in the syllogism that concludes to a universal affirmative.[45] Again, so long as we keep to the syllogism as complete in itself and without reference to its place in the great structure of knowledge, the nerve of proof cannot be conceived in other than a formal manner. In analytic we work with an ethos different from that of dialectic. We presume truth and not probability or concession, but a true conclusion can follow from false premises, and it is only in the attempt to derive the premises in turn from their grounds that we unmask the deception. The passage to the conception of system is still required. The _Prior Analytics_ then are concerned with a formal logic to be knit into a system of knowledge of the real only in virtue of a formula which is at this stage still to seek. The forms of syllogism, however, are tracked successfully through their figures, i.e. through the positions of the middle term that Aristotle recognizes as of actual employment, and all their moods, i.e. all differences of affirmative and negative, universal and particular within the figures, the cogent or legitimate forms are alone left standing, and the formal doctrine of syllogism is complete. Syllogism already defined[46] becomes through exhibition in its valid forms clear in its principle. It is a speech-and-thought-form ([Greek: logos]) in which certain matters being posited something other than the matters posited necessarily results because of them, and, though it still needs to receive a deeper meaning when presumed truth gives way to necessary truth of premises, the notion of the class to that of the class-concept, collective fact to universal law, its formal claim is manifest. "Certain matters being posited." Subject and predicate not already seen to be conjoined must be severally known to be in relation with that which joins them, so that more than one direct conjunction must be given. "Of necessity." If what are to be conjoined are severally in relation to a common third it does perforce relate or conjoin them. "Something other." The conjunction was by hypothesis not given, and is a new result by no means to be reached, apart from direct perception save by use of at least two given conjunctions. "Because of them," therefore. Yet so long as the class-view is prominent, there is a suggestion of a begging of the question. The class is either constituted by enumeration of its members, and, passing by the difficulty involved in the thought of "its" members, is an empirical universal of fact merely, or it is grounded in the class-concept. In the first case it is a formal scheme which helps knowledge and the theory of knowledge not at all. We need then to develop the alternative, and to pass from the external aspect of all-ness to the intrinsic ground of it in the universal [Greek: kath' auto kai ê auto], which, whatsoever the assistance it receives from induction in some sense of the word, in the course of its development for the individual mind, is secured against dependence on instances by the decisive fiat or guarantee of [Greek: nous], insight into the systematic nexus of things. The conception of linkage needs to be deepened by the realization of the middle term as the ground of nexus in a real order which is also rational.

Problem of inference.

Nous.

Aristotle's solution of the paradox of inference, viz. of the fact that in one sense to go beyond what is in the premises is fallacy, while in another sense not to go beyond them is futility, lies in his formula of implicit and explicit, potential and actual.[47] The real nexus underlying the thought-process is to be articulated in the light of the voucher by intelligence as to the truth of the principles of the various departments of knowledge which we call sciences, and at the ideal limit it is possible to transform syllogism into systematic presentation, so that, differently written down, it is definition. But for human thought sense, with its accidental setting in matter itself incognizable is always with us. The activity of [Greek: nous] is never so perfectly realized as to merge implication in intuition. Syllogism must indeed be objective, i.e. valid for any thinker, but it is also a process in the medium of individual thinking, whereby new truth is reached. A man may know that mules are sterile and that the beast before him is a mule, and yet believe her to be in foal "not viewing the several truths in connexion."[48] The doctrine, then, that the universal premise contains the conclusion not otherwise than potentially is with Aristotle cardinal. The datum of sense is only retained through the universal.[49] It is possible to take a universal view with some at least of the particular instances left uninvestigated.[50] Recognition that the class-concept is applicable may be independent of knowledge of much that it involves. Knowledge of the implications of it does not depend on observation of all members of the class. Syllogism as formula for the exhibition of truth attained, and construction or what not as the instrumental process by which we reach the truth, have with writers since Hegel and Herbart tended to fall apart. Aristotle's view is other. Both are syllogisms, though in different points of view. For this reason, if for no other, the conception of movement from the potential possession of knowledge to its actualization remains indispensable. Whether this is explanation or description, a problem or its solution, is of course another matter.

Posterior Analytics.

In the _Posterior Analytics_ the syllogism is brought into decisive connexion with the real by being set within a system in which its function is that of material implication from principles which are primary, immediate and necessary truths. Hitherto the assumption of the probable as true rather than as what will be conceded in debate[51] has been the main distinction of the standpoint of analytic from that of dialectic. But the true is true only in reference to a coherent system in which it is an immediate ascertainment of [Greek: nous], or to be deduced from a ground which is such. The ideal of science or demonstrative knowledge is to exhibit as flowing from the definitions and postulates of a science, from its special principles, by the help only of axioms or principles common to all knowledge, and these not as premises but as guiding rules, all the properties of the subject-matter, i.e. all the predicates that belong to it in its own nature. In the case of any subject-kind, its definition and its existence being avouched by [Greek: nous], "heavenly body" for example, the problem is, given the fact of a non-self-subsistent characteristic of it, such as the eclipse of the said body, to find a ground, a [Greek: meson] which expressed the [Greek: aition], in virtue of which the adjectival concept can be exhibited as belonging to the subject-concept [Greek: kath auto] in the strictly adequate sense of the phrase in which it means also [Greek: hê auto].[52] We are under the necessity then of revising the point of view of the syllogism of all-ness. We discard the conception of the universal as a predicate applicable to a plurality, or even to all, of the members of a group. To know merely [Greek: kata pantos] is not to know, save accidentally. The exhaustive judgment, if attainable, could not be known to be exhaustive. The universal is the ground of the empirical "all" and not conversely. A formula such as the equality of the interior angles of a triangle to two right angles is only scientifically known when it is not of isosceles or scalene triangle that it is known, nor even of all the several types of triangle collectively, but as a predicate of triangle recognized as the widest class-concept of which it is true, the first stage in the progressive differentiation of figure at which it can be asserted.[53]

Three points obviously need development, the nature of definition, its connexion with the syllogism in which the middle term is cause or ground, and the way in which we have assurance of our principles.

Definition.

The middle term.

Definition is either of the subject-kind or of the property that is grounded in it. Of the self-subsistent definition is [Greek: ousias tis gnôrismos][54] by exposition of genus and differentia.[55] It is indemonstrable. It presumes the reality of its subject in a postulate of existence. It belongs to the principles of demonstration. _Summa genera_ and groups below _infimae species_ are indefinable. The former are susceptible of elucidation by indication of what falls under them. The latter are only describable by their accidents. There can here be no true differentia. The artificiality of the limit to the articulation of species was one of the points to which the downfall of Aristotle's influence was largely due. Of a non-self-subsistent or attributive conception definition in its highest attainable form is a recasting of the syllogism, in which it was shown that the attribute was grounded in the substance or self-subsistent subject of which it is. Eclipse of the moon, e.g. is privation of light from the moon by the interposition of the earth between it and the sun. In the scientific syllogism the interposition of the earth is the middle term, the cause or "because" ([Greek: dioti]), the residue of the definition is conclusion. The difference then is in verbal expression, way of putting, inflexion.[56] If we pluck the fruit of the conclusion, severing its nexus with the stock from which it springs, we have an imperfect form of definition, while, if further we abandon all idea of making it adequate by exhibition of its ground, we have, with still the same form of words, a definition merely nominal or lexicographical. In the aporematic treatment of the relation of definition and syllogism identical as to one form and in one view, distinct as to another form and in another view, much of Aristotle's discussion consists. The rest is a consideration of scientific inquiry as converging in [Greek: mesou zêtêsis], the investigation of the link or "because" as ground in the nature of things. [Greek: To men gar aition to meson][57] real ground and thought link fall together. The advance from syllogism as formal implication is a notable one. It is not enough to have for middle term a _causa cognoscendi_ merely. We must have a _causa essendi_. The planets are near, and we know it by their not twinkling,[58] but science must conceive their nearness as the cause of their not twinkling and make the _prius_ in the real order the middle term of its syllogism. In this irreversible catena proceeding from ground to consequent, we have left far behind such things as the formal parity of genus and differentia considered as falling under the same predicable,[59] and hence justified in part Porphyry's divergence from the scheme of predicables. We need devices, indeed, to determine priority or superior claim to be "better known absolutely or in the order of nature," but on the whole the problem is fairly faced.[60]

Of science Aristotle takes for his examples sometimes celestial physics, more often geometry or arithmetic, sometimes a concrete science, e.g. botany.[61] In the field of pure form, free from the disconcerting surprises of sensible matter and so of absolute necessity, no difficulty arises as to the deducibility of the whole body of a science from its first principles. In the sphere of abstract form, mathematics, the like may be allowed, abstraction being treated as an elimination of matter from the [Greek: synolon] by one act. When we take into account relative matter, however, and traces of a conception of abstraction as admitting of degree,[62] the question is not free from difficulty. In the sphere of the concrete sciences where law obtains only [Greek: hôs epi to polu] this ideal of science can clearly find only a relative satisfaction with large reserves. In any case, however, the problem as to first principles remains fundamental.

Formal and scientific principles.

Induction and dialectic.

Knowledge and reality.

Conclusions as to induction.