Equivalence

“ The second principle I shall take notice of is that of comparison, or the variation of our judgments concerning objects, according to the proportion they bear to those with which we compare them. We judge more of objects by comparison than by their intrinsic worth and value; and regard every thing as mean, when set in opposition to what is superior of the same kind. ”

“ It remains to note that such phrases as 'Being and Nothing are the same,' or 'The unity of Being and Nothing'—like all other such unities, that of subject and object, and others—give rise to reasonable objection. ”

“ Why rob one to feed the other?
“If you adopt, as absolute, a standard of evenness which is so only relatively, your results will not be absolutely even. If you adopt, as absolute, a criterion of right which is so only relatively, your results will not be absolutely right. ”

“ If our principle, put in practice, would wrong your section for the benefit of ours, or for any other object, then our principle, and we with it, are sectional, and are justly opposed and denounced as such. ”

“ There are many philosophers, who refuse to assign any standard of equality, but assert, that ‘tis sufficient to present two objects, that are equal, in order to give us a just notion of this proportion. ”

“ Clearly in the affirmation of equality, no attention is paid to the form in which the quantities are expressed, but to the quantities themselves alone; otherwise we should be unable to affirm not only identity, but also equality ”

“ Let us see how, in these special classes of cases, we shall succeed in verifying the natural presupposition of common sense, which regards error as possible only when our object is not wholly present to mind, and which assumes that a judgment can have an object that is yet only partially present to mind. In choosing the classes of cases, we shall first follow common sense as to their definition. We shall take just the assumptions of daily life, and shall show that they lead us into difficulty. We are not for the first bound to explain why these assumptions are made. ”

“ Even what in mathematics is called equality, also means identity. Thus in this class of judgments, besides what we have observed of general in them all, to wit: the identity saved by expressing the predicate in the concrete, the very relation of equality denotes identity. ”

“ Each category, or determination, is a general principle by which we seek to make experience a consistent whole and render it a concrete universal; and each has its own way of relating the two root factors, viz., unity and difference. ”

“ All the relations which can naturally be represented as equality in any respect, or as possession of a common property, are transitive and symmetrical—this applies, for example, to such relations as being of the same height or weight or colour. ”

“ And from hence it may be understood, that the proportions of excess and defect are quantity, being capable of more and less; but the proportion of equality is not quantity, because not capable neither of more, nor of less. ”

“ Let us suppose that the generalisation to be justified is one which asserts that the existence of a kind of object, which we will call A, is generally preceded, accompanied, or followed by the existence of a kind of object, which we call B. A, for instance, might be the hearing of a certain word by one person, and B the thought of that which we call the meaning of the word, in another person; and the generalisation to be justified might be that when one person hears a word, not spoken by himself, someone else has generally thought of the meaning of that word. ”

“ By another law, flowing from the same primary principle, the parts of a whole diverge from one another in proportion to their diversity, and group themselves together in proportion to their resemblances. ”

“ Tell a child that things which are equal to the same thing are equal to one another, and the child, lacking a sufficiently abstract notion of equality, and lacking, too, the needful practice in comparing relations, will fail to grasp the axiom. Similarly, a rustic, never having thought much about forces and their results, cannot form a definite conception answering to the axiom that action and reaction are equal and opposite. ”

“ The inquiry is, Are the angles at the base of an isosceles triangle equal or unequal? The first thing to be considered is, what inductions we have, from which we can infer equality or inequality. For inferring equality we have the following formulæ: Things which being applied to each other coincide, are equals. Things which are equal to the same thing are equals. A whole and the sum of its parts are equals. The sums of equal things are equals. The differences of equal things are equals. There are no other original formulæ to prove equality. ”

“ The general principle of right arrangement in sentences, which we have traced in its application to the leading divisions of them, equally determines the proper order of their minor divisions. In every sentence of any complexity the complement to the subject contains several clauses, and that to the predicate several others ”

“ In this view, therefore, the material principle of any individual man, for example, is numerically and individually distinct from that of any other individual, of itself and independently of its relation either to the formative principle or to quantity. ”

“ As has been already pointed out, this is all that art can do. To test a generalization, by showing that it either follows from, or conflicts with, some stronger induction, some generalization resting on a broader foundation of experience, is the beginning and end of the logic of induction. ”

“ The distinction between positive, categorical and assertory judgments, is either a pure invention of fancy, or is left undetermined. On the right theory, the different judgments follow necessarily from one another, and present the continuous specification of the notion; for the judgment itself is nothing but the notion specified. ”

“ As we judge by comparing- concepts with other concepts or with impressions, one judgment by a faculty is associated with other judgments by the same faculty, and as one property is concomitant with all the others, one property becomes a symbol of all the others, so that there is association by comparison of concepts and association by symbolism. ”

“ As to equality or any exact proportion, we can only guess at it from a single consideration; except in very short numbers, or very limited portions of extension; which are comprehended in an instant, and where we perceive an impossibility of falling into any considerable error. In all other cases we must settle the proportions with some liberty, or proceed in a more artificial manner. ”

“ What is the difference between those that are similar and those that are equal? Can one equal be distinguished from the other? So those who are equal are not unlike. If then those who are unlike are not equals, what can those who are like be but equals? ”

“ We are then forced to look beyond them to some common cause or common origin to explain why this singular relation of equality exists, rather than any one of the infinite number of possible relations of inequality. ”

“ In all our knowledge of general principles, what actually happens is that first of all we realise some particular application of the principle, and then we realise that the particularity is irrelevant, and that there is a generality which may equally truly be affirmed. ”

“ Before, then, we began to see, and hear, and use our other senses, we must have had a knowledge of equality itself—what it is, if we were to refer to it those equal things that come under the senses, and observe that all such things aim at resembling that, but fall far short of it. ”

“ For we have seen that the Uniformity of Nature is not really a principle, but a merely nominal generalisation, since it cannot be definitely stated; and, therefore, the principles of Contradiction, Mediate Equality, and Causation remain incapable of subsumption ”

“ Thus, then, there originates the notion of equality. The things which among ourselves are called equal—whether lines, angles, weights, temperatures, sounds or colours—are things which produce in us sensations which cannot be distinguished from each other. It is true that we now apply the word equal chiefly to the separate traits or relations which objects exhibit, and not to those combinations of them constituting our conceptions of the objects; but this limitation of the idea has evidently arisen by analysis. ”

“ Hence the word (as Civilization, for example) which professes to be the designation of the unknown common property, conveys scarcely to any two minds the same idea. No two persons agree in the things they predicate of it; and when it is itself predicated of any thing, no other person knows, nor does the speaker himself know with precision, what he means to assert. ”

“ Now when we consider that by one of the general axioms, any equality, when ascertained, is proof of as many other equalities as there are other things equal to either of the two equals; and that by another of those axioms, any ascertained equality is proof of the equality of as many pairs of magnitudes as can be formed by the numerous operations [pg 436] which resolve themselves into the addition of the equals to themselves or to other equals; we cease to wonder that in proportion as a science is conversant about equality, it should afford a more copious supply of marks of marks ”

“ In its abstract terms a Judgment is expressible in the proposition: 'The individual is the universal.' These are the terms under which the subject and the predicate first confront each other, when the functions of the notion are taken in their immediate character or first abstraction. [Propositions such as, 'The particular is the universal,' and 'The individual is the particular,' belong to the further specialisation of the judgment.] ”

“ The absolute, as unconditioned, identical, and one, cannot be known, under conditions, by difference and plurality. It cannot, therefore be known, if the subject of knowledge be distinguished from the object of knowledge; in the knowledge of the absolute, existence and knowledge must be identical; the absolute can only be known, if adequately known, by the absolute itself. ”

“ Equality being predicated of things which give us indistinguishable impressions, and no accurate comparison of impressions being possible unless they occur in immediate succession, it results that exactness of equality is ascertainable in proportion to the closeness of the compared things. ”

“ in principle and in practice, in a right track and in a wrong one, the rarest of all human qualities is consistency. ”

“ As all modifiers are joined by slanting lines, to the words they modify, you learn from this diagram that who is wise is a modifier of man.
+Oral Analysis+.—This is a complex sentence, because it consists of an independent clause and a dependent clause. A man will be honored is the independent clause; who is wise is the dependent clause. Man is the subject of the independent clause; will be honored is the predicate. The word A and the clause, who is wise, are modifiers of the subject. A points out man, and who is wise tells the kind of man. A man who is wise is the modified subject ”

“ The second class, on the contrary,—that of objective relations,—expresses a proportion which is not merely constituted between things by our thinking in an arbitrary way, but which is really extant for the things themselves in such manner that they are reciprocally affected in this same proportion. ”

“ In limited, close historical argument—in the power to infer a third proposition from a second, a second from a first—the power to expand a fact, either proved or assumed as a trifling postulate, into a series of facts, with undeniable cogency—I think we must go far to find his equal. ”

“ These are some of those Difficulties which arise from Discourses which our finite Understanding makes about Infinites, by ascribing to them Attributes which we give to Things finite and terminate, which I think most improper, because those Attributes of Majority, Minority, and Equality, agree not with Infinities, of which we can’t say that one is greater than, less than, or equal to another. ”

“ Resemblance and its opposite, except in the case in which they assume the names of Equality and Inequality, are seldom regarded as subjects of science; they are supposed to be perceived by simple apprehension; by merely applying our senses or directing our attention to the two objects at once, or in immediate succession. And this simultaneous, or virtually simultaneous, application of our faculties to the two things which are to be compared, does necessarily constitute the ultimate appeal, wherever such application is practicable. ”

“ For, being apprehensive that from the demand of equality of character in some point or other only it would follow that, as all things have certain characteristics in common, anything might originate from anything, you have declared that the equality of character necessary for the relation of cause and effect is constituted by the persistence, in the effect, of those characteristic points which differentiate the cause from other things. ”

“ The sums of equals are equal, The differences of equals are equal. These two are sufficient; for the corresponding propositions respecting unequals may be proved from these, by a reductio ad absurdum.
These axioms, and likewise the so-called definitions, are, as already shown, results of induction; true of all objects whatever, and, as it may seem, exactly true, without the hypothetical assumption of unqualified truth where an approximation to it is all that exists. ”