X

“ The mere written characters, a, b, x, y, z, serve as well for representatives of Things in general, as any more complex and apparently more concrete conception. That we are conscious of them, however, in their character of things, and not of mere signs, is evident from the fact that our whole process of reasoning is carried on by predicating of them the properties of things. In resolving an algebraic equation, by what rules do we proceed? By applying at each step to a, b, and x, the proposition that equals added to equals make equals; that equals taken from equals leave equals ”

“ Hence the Trio is a Syllogism; the two Propositions, “No x are m” and “No y are m′”, are its Premisses; the Proposition “No x are y” is its Conclusion; the Terms m and m′ are its Eliminands; and the Terms x and y are its Retinends.
Hence we may write it thus:—
“No x are m;
No y are m′.
∴ No x are y”.
As a second specimen, let us take the Trio
“All cats understand French;
Some chickens are cats.
Some chickens understand French”.
These, put into normal form, are
“All cats are creatures understanding French;
Some chickens are cats.
Some chickens are creatures understanding French”. ”

“ X has generally the sound of ks. In all words beginning with x or ex, followed by a vowel, it has the sound of gz, as in Xavier, exil, which are pronounced gzavié, èg-zil. When final, it is joined to the following word, and sounds like z, as in beaux yeux, dix hommes, which are pronounced bô-zieû, dî-zom. X has the sound of ss in Auxerre, Auxonne, and Bruxelles. ”

“ With one symbol for an unknown, it will easily be understood what scope there is for adroit assumptions, for the required numbers, of expressions in the one unknown which are at once seen to satisfy some of the conditions, leaving only one or two to be satisfied by the particular value of x to be determined. ”

“ It might happen that one of the older pupils, specially clever at mathematics, but not very well disciplined, should start some point connected with the supposition that x is something different than unity. It would be the teacher's business to remind her: "At present we are dealing with the supposition that x is unity. ”

“ In consequence of the considerations I have here set forth at length, we can surely hardly avoid recognising the identity of this x, even on the lowest degrees of the scale, where it was but faintly perceptible; then higher up, where it extended its obscurity more and more; and finally on the highest degrees, where it cast a shadow upon all things till, at the very top, it reveals itself to our consciousness in our own phenomenal being, as the will. ”

“ Every variable is the sign of a formal concept.
For every variable presents a constant form, which all its values possess, and which can be conceived as a formal property of these values.
4.1272 OGD [→GER | →P/M]
So the variable name “x” is the proper sign of the pseudo-concept object.
Wherever the word “object” (“thing”, “entity”, etc.) is rightly used, it is expressed in logical symbolism by the variable name.
For example in the proposition “there are two objects which ...”, by “ (∃x,y) ...”. ”

“ Hence we give the name "Jones" to the whole set of particulars, but do not trouble to give separate names to the separate particulars that make up the set.
Passing on from proper names, we come next to general names, such as "man," "cat," "triangle." A word such as "man" means a whole class of such collections of particulars as have proper names. The several members of the class are assembled together in virtue of some similarity or common property. All men resemble each other in certain important respects; hence we want a word which shall be equally applicable to all of them. ”

“ There is no scientific evidence whatsoever of the existence of such a third entity, "X," but all our deductions have been by analogy, which proves nothing—that is, by speculation, dreaming, and unavoidably so—since in these conceptions we are close to the border line of the human mind where logical reasoning loses itself in the fog of contradiction. ”

“ The experiences of Miss X., in crystal-gazing are devoid of the interest imparted by exciting incident, and on that very account are the more valuable as illustrating our subject. She has friends of whose experiments she has carefully [Pg 210] observed the results, and she has some seventy cases or experiments of her own of which she has kept carefully prepared notes, always made directly or within an hour after each experiment. ”

“ Thus, ‘Every A is X, Every X is B,’ which yields ‘Every A is B,’ also yields ‘Some A is B.’ But writers on logic have always considered these syllogisms as useless, conceiving it better to draw from any premises their strongest conclusion. ”

“ The interest of life's equation arrives with the appearance of x, the unknown quantity. A settled, unchangeable, clearly foreseeable order of things does not suit our constitution. It tends to melancholy and a fatty heart. ”

“ I said.
“I'll tell you a story,” said the publisher—“speaking about youthful aspirations! I was talking to Mr. X—— last night, the author of ——. [Footnote: The manuscript names an extremely popular historical novel.] You wouldn't think X—— was the sort of man to be reforming the world, would you? ”

“ From the point of view of the hearer, a single instance of the use of a word consists of a certain series of sounds, each being approximately represented by a single letter in writing, though in practice a letter may represent several sounds, or several letters may represent one sound. The connection between the spoken word and the word as it reaches the hearer is causal. Let us confine ourselves to the spoken word, which is the more important for the analysis of what is called "thought." ”

“ In a word, so far as A and B contain l, m, n, and o, p, q, respectively, in addition to x, they are not explained by x. Each additional particularity makes its distinct appeal. A single explanation of a fact only explains it from a single point of view. ”

“ The above rules may be worked out not only with the symbols, as "All A is B," but also with any Judgments or Propositions, such as "All horses are animals;" "All men are mortal;" "Some men are artists;" etc. The principle involved is identical in each and every case. The "All A is B" symbology is merely adopted for simplicity, and for the purpose of rendering the logical process akin to that of mathematics. The letters play the same part that the numerals or figures do in arithmetic or the a, b, c; x, y, z, in algebra. Thinking in symbols tends toward clearness of thought and reasoning. ”

“ It is possible that some few exceedingly sober-minded mathematicians, who are impatient of any terminology in their favourite science but the academic, and who object to the elusive x and y appearing under any other names, will have wished that various problems had been presented in a less popular dress and introduced with a less flippant phraseology. ”

“ Thus, the expression
xm0 † ym′0 ¶ xy0
is a Formula, by which we can find the Conclusion to any Pair of Premisses whose subscript forms are
xm0 † ym′0
For example, suppose we had the Pair of Propositions
“No gluttons are healthy;
No unhealthy men are strong”.
proposed as Premisses. Taking “men” as our ‘Universe’, and making m = healthy; x = gluttons; y = strong ”

“ Before that there never was a letter X in the alphabet. It was then the king said, "We shall put the crossed arms in the alphabet; we shall have a new letter called X, so everybody will understand a funeral is beautiful if there are young singing playmates." ”

“ For example, suppose we have to find the Compartment assigned to ym. We say to ourselves “y has the West Half; and m has the Inner portion of that West Half.”
Again, suppose we have to find the Cell assigned to x′ym′. We say to ourselves “x′ has the South Half ”

“ Some tendencies, of course, appear to be more enduring than others, and thus may be thought to approximate more closely to laws of the timeless kind. But x, the unknown quantity, the something or other that is not physical, runs through them all, however much or little they may seem to endure. ”

“ So in the science of abstract number the fundamental assumptions, as that 1=1, x=x, etc., are contradicted by every fact of experience, for in the world as we know it, absolute equality is simply impossible to discover; and yet these fundamental conceptions are in their development most powerful instruments for the extension of man's command over his own experiences. ”

“ The argument à fortiori, may then be defined as a universally 26affirmative syllogism, in which both of the premises are shewn to be less than the whole truth, or greater. Thus, in ‘Every A is X, Every X is B, therefore Every A is B,’ we do not certainly imply that there are more Xs than As, or more Bs than Xs, so that we do not know that there are more Bs than As. ”

“ I am x in an indeterminate equation. ”

“ So that to make our generalization quite correct we must say, that while in certain sentences expressing strong feeling, the word which more especially implies that feeling may often with advantage be a many-syllabled or Latin one; in the immense majority of cases, each word serving but as a step to the idea embodied by the whole sentence, should, if possible, be a one-syllabled or Saxon one. ”

“ But, once more, grant that C is D, and it does not follow that A is B; so neither, if the conclusion of a syllogism be true, does it follow that the premises are. For example:—
Sociology is an exact science;
Mathematics is a branch of Sociology:
∴ Mathematics is an exact science. [Pg 160] Here the conclusion is true although the premises are absurd. Or again:—
Mathematics is an exact science;
Sociology is a branch of Mathematics:
∴ Sociology is an exact science. Here the major premise is true, but the minor is false, and the conclusion is false. ”

“ We have now seen how, from the physiological construction of our mental apparatus, the process of concept formation and the experience of concept connections are the basis of the whole of mental life. The laws of the mutual interaction of the most general or elementary concepts operated in the formation of the concepts, thing, group, co-ordination. Here were found the fundamentals of logic or the science of concepts. A special process of abstraction yielded the concept of number, and with it the corresponding field of mathematics, arithmetic, algebra, and the theory of numbers. ”

“ As we arrange all our words in a dictionary under twenty-four letters, so do they arrange all their words, or characters, under two hundred and fourteen radical signs; the simplest radicals being the first, and the more complex the last.”
“Does the Chinese resemble any of the European languages in words?” said I.
“I am scarcely competent to inform you,” said the old man; “but I believe not.”
“What does that character represent?” said I, pointing to one on the vase.
“A knife,” said the old man, “that character is one of the simplest radicals or keys.” ”

“ And just as in mathematics the symbols that are employed contain in an easily manipulated form enormous bodies of meaning—possibly, indeed, the entire meaning of a long calculation,—so in all other kinds of abstract ideation, the symbols which we employ—whether in gesture, speech, or writing—contain more or less condensed masses of signification. Or, to take another illustration, which, like the last example, I quote from Lewes, “It is the same with the development of commerce. Men begin by exchanging things. ”

“ In the common or Syllogistic Logic, a certain Formula of language is used in stating the reasoning, and is useful in enabling us more readily to apply the Criterion of Form to alleged demonstrations. This formula is the usual Syllogism; with its members, Major Premiss, Minor Premiss, and Conclusion. It may naturally be asked whether in Inductive Logic there is any such Formula? whether there is any standard form of words in which we may most properly express the inference of a general truth from particular facts? ”

“ We have examples of such ideas in symbols. A symbol, I take it, is always itself, existentially, a particular object. A word, an algebraic sign, is just as much a concrete existence as is a horse, a fire-engine, or a flyspeck. ”

“ It follows from the principle of Identity that, the matter of the propositions being the same, if A. is true I. is true, and that if E. is true O. is true; for A. and E. predicate something of All S or All men; and since I. and O. make [Pg 80] the same predication of Some S or Some men, the sense of these particular propositions has already been predicated in A. or E. If All S is P, Some S is P; if No S is P, Some S is not P; or, if All men are fond of laughing, Some men are; if No men are exempt from ridicule, Some men are not. ”

“ In Book I, Chapter II, I have adopted a new definition of ‘Classification’, which enables me to regard the whole Universe as a ‘Class,’ and thus to dispense with the very awkward phrase ‘a Set of Things.’
In the Chapter on ‘Propositions of Existence’ I have adopted a new ‘normal form,’ in which the Class, whose existence is affirmed or denied, is regarded as the Predicate, instead of the Subject, of the Proposition, thus evading a very subtle difficulty which besets the other form. ”

“ The treatment of the Latin derivatives in Part II. presents a new and important feature, to wit: the systematic analysis of the structure and organism of derivative words, together with the statement of their primary meaning in such form that the pupil inevitably perceives its relation with the root, and in fact makes its primary meaning by the very process of analyzing the word into its primitive and its modifying prefix or suffix. ”