Domain

“ In general, the customs of the various domains are all different from one another, each having its own peculiarities. To divulge the secrets of one's own domain is a sure indication of an intent to curry favour. ”

“ If is a one-many relation, the range of possible arguments to the function is the converse domain of , and the range of values is the domain. Thus the range of possible arguments to the function "the father of " is all who have fathers, i.e. the converse domain of the relation father, while the range of possible values for the function is all fathers, i.e. the domain of the relation.
Many of the most important notions in the logic of relations are descriptive functions, for example: converse, domain, converse domain, field. Other examples will occur as we proceed. ”

“ Still we can distinguish broadly between knowledge which has no direct, immediate bearing on our acts, and knowledge that has.10 Hence the possibility of distinguishing between two great domains of philosophical knowledge—Theoretical or Speculative Philosophy, and Practical Philosophy. There are, in fact, two great domains into which the data of all human experience may be divided; and for each distinct domain submitted to philosophical investigation there will be a distinct department of philosophy. A first domain is the order realized in the universe independently of man ”

“ Why should not the Elements of Speech have been brought by Nature herself into some sort of parallelism with the Elements of Thought which it is the special province of Speech to represent? Why, again, should not the Prime Elements of every new domain of Being be merely a Repetition in new form of the Prime Elements of the Universe, as a whole, and of those especially of Language, its representative domain?—Language being the literal word, as Universal Law is the Logos or the Word par excellence, and Divine. ”

“ Hence the idea of a general order of nature, everywhere the same, hovering over life and over matter alike. Hence our habit of designating by the same word and representing in the same way the existence of laws in the domain of inert matter and that of genera in the domain of life. ”

“ When we have devised idealized constructions, therefore, we should not therewith be content but should scrutinize them, examine and study them for their implications; for thereby we may discover the path and the guide-posts to a new domain, a new ideal, following which we shall, in time, come to a point in our search for the real where the fluxional is at a minimum ”

“ Everything is bi-lateral in the domain of thought. Ideas are binary. ”

“ To apply the name of Science to domains in which real knowledge is not attainable, is, in some sense, an abuse of terms. To denominate Positive Sciences, domains which are not strictly Scientific, and in which positive certainty, in reference to Principles and ulterior Facts, cannot be attained, is still more incongruous. ”

“ That relationship between systems of ideas in which the dissimilarity of every two homologous concepts as well as the agreement in logical relations of every two homologous pairs of concepts, is clearly brought to light, is called an analogy. It is an effective means of mastering heterogeneous fields of facts in unitary comprehension. The path is plainly shown in which a universal physical phenomenology embracing all domains, will be developed. ”

“ In the technical language of Logic, the underlying principle of any system is called its Ground, the detail in which the principle finds systematic expression is called its Consequence. Ground and Consequence are thus one and the same systematic whole, only considered from two different points of view. The Ground is the pervading common nature of the system, thought of as an identity pervading and determining the character of its detail ”

“ In so far as they are genuine systems at all, they are not mere “parts” of a whole, but each is the expression, in a concrete conscious life, of the nature of a larger whole from a special “point of view.” The whole, if not equally in every part, is yet as a whole present in every part, and precisely for that reason the category of part and whole is inadequate to express their relation. Somewhat after this fashion we must conceive the structure of any individual whole of lesser individuals. ”

“ The Ground is the unity of identity and difference, the truth of what difference and identity have turned out to be,—the reflection-into-self, which is equally a reflection-into-an-other, and vice versâ. It is essence put explicitly as a totality.
The maxim of the Ground runs thus: Everything has its Sufficient Ground: that is, the true essentiality of any thing is not the predication of it as identical with itself, or as different (various) , or merely positive, or merely negative, but as having its Being in an other, which, being its self-same, is its essence. ”

“ It observes that when you declare any object to be real, you are in possession of an idea, however exact, or however inexact, however transient and relative, or however universal and eternal, — an idea to which you attribute an essentially teleological significance; since you assert that this idea is true, is valid, or in other words, is adapted to its ideal end. Our present conception regards this adaptation of the idea to its own end as the primary topic of any ontological assertion, and as the object which any one who asserts Being first of all inevitably means. ”

“ Our detailed theory of the relation between the unity of the world and its multiplicity must do equal justice to both, and it must be consistent with our previous recognition of the experiential nature of the real.
We may perhaps work out our theory in detail as follows. The world for knowledge must, we have seen, be an orderly whole or system. To be a system at all, it must be the development or expression in detail of a single principle. ”

“ The clear possession of the Idea of Space is the first requisite for all geometrical reasoning; and this clearness of idea may be tested by examining whether the axioms offer themselves to the mind as evident.
5. The necessity of ideas added to sensations, in order to produce knowledge, has often been overlooked or denied in modern times. The ground of necessary truth which ideas supply being thus lost, it was conceived that there still remained a ground of necessity in definitions;—that we might have necessary truths, by asserting especially what the definition implicitly involved in general. ”

“ More particularly do we illustrate the ideality of Space as a cognition precedent to experience. It is because general laws constantly operative regulate the transmutations which constitute the individual's Presentment that it is possible for him to abstract from and generalise the data of sense; and it is because the subjective process of Ideation, by which we mean our representative mental activity in its widest sense, consists also in transmutations under the same general laws of the same portion of the energetic organism, that it is possible to frame general ideas. ”

“ Taking a survey of the system to which we belong; the only one accessible to us; from thence we are the better enabled to judge of the nature and end of the other systems of the universe. For although there is almost an infinite variety in the parts of the creation which we have opportunities of examining, yet there is a general analogy running through and connecting all the parts into one scheme, one design, one whole! ”

“ Our idea of a point, I apprehend to be simply our idea of the minimum visibile, the smallest portion of surface which we can see. A line, as defined by geometers, is wholly inconceivable. We can reason about a line as if it had no breadth; because we have a power, which is the foundation of all the control we can exercise over the operations of our minds; the power, when a perception is present to our senses, or a conception to our intellects, of attending to a part only of that perception or conception, instead of the whole. ”

“ Albeit we see the boundary point of aught Is that which to our senses seems its least, Whereby thou mayst conjecture, that, because The things thou canst not mark have boundary points, They surely have their minimums. Then, too, Since these philosophers ascribe to things Soft primal germs, which we behold to be Of birth and body mortal, thus, throughout, The sum of things must be returned to naught, And, born from naught, abundance thrive anew— Thou seest how far each doctrine stands from truth. ”

“ Inclusiveness, exclusiveness, and intersection, are three different aspects of the purely formal relation of extension, and are therefore equally free of all spatial and temporal content.
12. Every entity is defined by a number of attributes, by the totality of which it is distinguished from all entities not identical with it. These attributes can be divided into two classes: those attributes which are common to all parts of a given entity, and by virtue of which (with respect to which) this entity has extension ”

“ Every idea is either an image or a symbol—it has therefore objective397 reference, a material aspect. Every object is a synthesis of feelings—it has therefore subjective reference, a material aspect. Thus while all the evidence points to the identity of Object and Subject, there is ample evidence for the logical necessity of their ideal separation as Aspects. This I have explained as a case of the general principle which determines all distinctions—namely, the diversity in the modes of production of the products, which—subjectively—is diversity in the modes of apprehending them. ”

“ A in part = non-A, non-A in part = A. Wherever two opposites are alike in one characteristic, we consider the ground as a ground of relation, and wherever two similar things are opposite in one characteristic, we consider the ground as a ground of distinction.—With these three principles we have now exhausted the measure of that which is unconditioned and absolutely certain. ”

“ If the concrete object or individual is carefully selected it will be a type, that is, it illustrates a whole class of similar objects. Such a typical concrete object really combines the particular and the general. It has all the advantage of object-teaching, the powerful attraction of real things, but its comparison with other objects will also show that it illustrates a general law or principle of wide-reaching scientific importance. ”

“ Likewise every arrangement and formation that is not perfect in its order we designate as accidental, and that which is orderly, regular, perfect in its relations and every part of which is in its proper place and is the essential requisite of the other constituent parts, this we call a composition formed through will and knowledge. There is no doubt that these infinite beings and the association of these diverse elements arranged in countless forms must have proceeded from a Reality that could in no wise be bereft of will or understanding. ”

“ Now is there any object which we know which, considered as a whole consisting of parts, is exactly like another whole, the two having all their parts similarly arranged, so as to form in themselves two identical systems, and yet the one incapable of being made to coincide with the other, even in thought? ”

“ Thus our world, however many and various its objects, possesses what we may call Ontological Unity, in so far as all its types of Being, concrete and abstract, appear as various aspects of one type of Being. Nor can you sunder any single idea of an isolated real object from the network established by ideas of reality in general. The whole of this world stands or falls together. ”

“ But the same internal relations found in the objective globe will be preserved in the subjective, and the transcript of the globe that is held in thought will be truth in exact proportion as it corresponds to the material globe that rolls out of the mind. That an objective globe, which is a thing, may become a subjective globe, which is a thought and not a thing, implies that there is something in common between thoughts and things; that is, the mind, by its constitution, is capable of apprehending and taking into itself the constitution and relations of things. ”

“ If the symbols of our system stand for entities for which the axioms hold, then all the propositions of the system will be true of those entities. Or, in other words: If entities can be found which satisfy the axioms of the system, then the system will be science of these entities. ”

“ The first principle of structure, Unity, has to do with the substance of a sentence or a composition. It is the law which requires that every composition shall be informed with a general intention, shall centre around one fundamental idea; that every paragraph and every sentence shall be dominated by one essential thought or purpose. It is the principle which produces the difference between a well-ordered whole and an unorganized collection of scraps ”

“ Such a relation as we have asserted between the individual whole of Reality and the elements or terms within the whole is necessarily unique, and cannot be adequately illustrated from any less perfect type of systematic unity recognised by everyday or by scientific thought. ”

“ Nothing so starts and prolongs the shock felt by the thinker as those mysterious exfoliations of abstraction into realities in the double region, the one positive, the other infinite, of human thought. A region double, and nevertheless one; the infinite is a precision. The profound word number is at the base of man's thought. ”

“ First, we have shown that the essentials of properties are their substrates severally; unity is the substrate of number, extension is the substrate of space, speed is the substrate of motion, consciousness is the substrate of judgment.
Second, we have shown that the quantitative properties are the substrates of the categoric properties; number is the substrate of kind, space is the substrate of form, motion is the substrate of force, time is the substrate of causation, and judgment is the substrate of conception. ”

“ From the existence of a thing having the properties, or from the existence of a name meaning them?
Take, for instance, any of the definitions laid down as premises in Euclid's Elements; the definition, let us say, of a circle. This, being analysed, consists of two propositions; the one an assumption with respect to a matter of fact, the other a genuine definition. "A figure may exist, having all the points in the line which bounds it equally distant from a single point within it:" "Any figure possessing this property is called a circle." ”