Notation

“ The aim of the paper is not to add one more to the numerous systems of notation and symbolic method that have been already worked out more or less independently, but rather to bring out some underlying principles and assumptions which belong equally to the ordinary Formal Logic, to Symbolic Logic, and to the so-called Logic of Relatives. ”

“ When functions are mentioned or implied in the notation, the reader will always understand single-valued functions, unless the contrary is distinctly intimated, or the case is one in which the distinction is obviously immaterial. ”

“ The algebraic notation, considered as a philosophical language, is perfect in its adaptation to the subjects for which it is commonly employed, namely those of which the investigations have already been reduced to the ascertainment of a relation between numbers. ”

“ We are therefore free to teach the elements of the Higher Logic, to those unacquainted with the notation of Algebra, by the same methods as were used of old in schools of Free-Masonry and of Prophecy. ”

“ When only small sections are analyzed singly, it is possible to understand something, at least, of the intricate pattern of forces which are interwoven in the rhythms of ordinary language. When one undertakes to analyze and express the combined rhythms—musical, logical, emotional—of connected sentences and paragraphs, one finds no system of notation adequate ”

“ Note also
the bibliographies given by Alden and Patterson.
I have emphasized in this chapter the desirability of compromise in some hotly contested disputes over terminology and methods of metrical notation. Perhaps I have gone farther in this direction than some teachers will wish to go. But all classroom discussion should be accompanied by oral reading of verse, by the teacher and if possible by pupils, and the moment oral interpretations begin, it will be evident that "a satisfied ear" is more important than an exact agreement upon methods of notation. ”

“ The miraculous powers of modern calculation are due to three inventions: the Arabic Notation, Decimal Fractions, and Logarithms. ”

“ Other systems of notation have been invented from time to time in the course of the last two or three centuries, but in most cases they have died with their inventors, and in no case has any such system been accepted with anything even approaching unanimity. ”

“ Any one who understands algebraic notation, reads at a glance in an equation results reached arithmetically only with great labour and pains. ”

“ Communication was much slower a thousand years ago than now and ideas about new methods of doing things did not spread rapidly, consequently it is entirely possible that various men or groups of men in various places worked out a system of notation differing somewhat in details of origin and development but alike in final result. ”

“ It needs scarcely be observed, that general propositions of which no one can tell exactly what they assert, can not possibly have been brought to the test of a correct induction. Whether a name is to be used as an instrument of thinking, or as a means of communicating the result of thought, it is imperative to determine exactly the attribute or attributes which it is to express; to give it, in short, a fixed and ascertained connotation. ”

“ What algebra is to arithmetic, the notation we now allude to is to mechanism. The various parts of the machinery under consideration being once expressed upon paper by proper symbols, the enquirer dismisses altogether from his thoughts the mechanism itself, and attends only to the symbols; the management of which is so extremely simple and obvious, that the most unpractised person, having once acquired an acquaintance with the signs, cannot fail to comprehend their use. ”

“ But the author believes that this history constitutes a mirror of past and present conditions in mathematics which can be made to bear on the notational problems now confronting mathematics. The successes and failures of the past will contribute to a more speedy solution of the notational problems of the present time. ”

“ When these common attributes have been ascertained and specified, the name which belongs in common to the resembling objects acquires a distinct instead of a vague connotation; and by possessing this distinct connotation, becomes susceptible of definition.
In giving a distinct connotation to the general name, the philosopher will endeavor to fix upon such attributes as, while they are common to all the things usually denoted by the name, are also of greatest importance in themselves ”

“ Although the first kind of imperfect definition (which defines a connotative term by a part only of what it connotes, but a part sufficient to mark out correctly the boundaries of its denotation) , has been considered by the ancients, and by logicians in general, as a complete definition; it has always been deemed necessary that the attributes employed should really form part of the connotation ”

“ Names with indeterminate connotation are not to be confounded with names which have more than one connotation, that is to say, ambiguous words. ”

“ It is the province of the part of Grammar called Accidence to tell us what are the Inflexions of a word. It is the province of the part called Syntax to tell us how and when to use them.
The different inflexions that a Noun can take are called its Declensions, and to give the inflexions of a Noun is to Decline it. The different inflexions that a Verb can take are called its Conjugations, and to give the inflexions of a Verb is to Conjugate it.
§ 13. Inflexion of Nouns.—Nouns belong to different Genders; and their inflexions denote differences of Number and Case. ”

“ Then, resolving that each word ought to be classed according to the meaning which its etymology fixes upon it, he refers the number of classes to nature, thus: "If, then, each [word] has a meaning, and is capable of raising an idea in the mind, that idea must have its prototype in nature. It must either denote an exertion, and is therefore a verb; or a quality, and is, in that case, an adjective; or it must express an assemblage of qualities, such as is observed to belong to some individual object, and is, on this supposition, the name of such object, or a noun. ”

“ 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? ”

“ In former lectures we have confined our remarks to things and a description of their characters and relations, so that every entity of which we can conceive a thought, or concerning which we can form an expression, has been defined and described in the use of nouns and adjectives. Every thing in creation, of which we think, material or immaterial, real or imaginary, and to which we give a name, to represent the idea of it, comes under the class of words called nouns. ”

“ The sciences of grammar and of logic are little more than attempts methodically to classify all such acquired ideas and to trace certain laws of relationship among them. The forms of relation between them, becoming themselves in turn noticed by the mind, are treated as conceptions of a higher and more abstract order, as when we speak of a syllogistic relation' between propositions, or of four quantities making a 'proportion,' or of the 'inconsistency' of two conceptions, or the 'implication' of one in the other. ”

“ Moreover, such rules, or definitions, as, "A noun is the name of anything"; "A fraction is a number which expresses one or more equal parts of a whole," are general truths, because they express in the form of a statement the general qualities which have been read into the ideas, noun and fraction. When the mind, from a study of particulars, thus either forms a class notion as noun, triangle, hepatica, etc., or draws a general conclusion as, "Air has weight," "Any two sides of a triangle are together greater than the third side," it is said to gain general knowledge. ”

“ It will be hereafter seen that when any important consequences seem to follow, as in mathematics, from an essential proposition, or, in other words, from a proposition involved in the meaning of a name, what they really flow from is the tacit assumption of the real existence of the objects so named. ”

“ It may be objected to our definition of an abstract name, that not only the names which we have called abstract, but adjectives, which we have placed in the concrete class, are names of attributes; that white, for example, is as much the name of the colour, as whiteness is. But (as before remarked) a word ought to be considered as the name of that which we intend to be understood by it when we put it to its principal use, that is, when we employ it in predication. ”

“ The sound ta or sa, for this or there, is as involuntary, as natural, as independent an expression as any [pg 268] of the predicative roots, and although some of these demonstrative, or pronominal, or local roots, for all these names have been applied to them, may be traced back to a predicative source, we must admit a small class of independent radicals, not predicative in the usual sense of the word, but simply pointing, simply expressive of existence under certain more or less definite, local or temporal prescriptions. ”

“ An abstract name is the name of an attribute, or combination of attributes. The corresponding concrete is a name given to things, because of, and in order to express, their possessing that attribute, or that combination of attributes. When, therefore, we predicate of any thing a concrete name, the attribute is what we in reality predicate of it. But it has now been shown that in all propositions of which the predicate is a concrete name, what is really predicated is one of five things: Existence, Co-existence, Causation, Sequence, or Resemblance. ”

“ We can therefore easily understand why the mathematician and the symbolic logician are driven to discard the word and to build up their thought with the help of symbols which have, each of them, a rigidly unitary value. ”

“ For some individual objects we require, and consequently have, separate distinguishing names; there is a name for every person, and for every remarkable place. Other objects, of which we have not occasion to speak so frequently, we do not designate by a name of their own; but when the necessity arises for naming them, we do so by putting together several words, each of which, by itself, might be and is used for an indefinite number of other objects ”

“ It is sometimes very difficult to tell the case of a noun. I shall, therefore, take up this subject again, when I come to give you an explanation of the participle and preposition.
Besides the three cases already explained, nouns are sometimes in the nominative case independent, sometimes in the nominative case absolute, sometimes in apposition in the same case, and sometimes in the nominative or objective case after the neuter to be, or after an active-intransitive or passive verb. These cases are illustrated in Lecture X. and in the 21 and 22 rules of Syntax. ”

“ The subject of denoting is of very great importance, not only in logic and mathematics, but also in the theory of knowledge. For example, we know that the center of mass of the solar system at a definite instant is some definite point, and we can affirm a number of propositions about it ”

“ The chalk was still there, but it no longer served the purpose of a distinctive mark.
When we impose a proper name, we perform an operation in some degree analogous to what the robber intended in chalking the house. We put a mark, not indeed upon the object itself, but, so to speak, upon the idea of the object. A proper name is but an unmeaning mark which we connect in our minds with the idea of the object, in order that whenever the mark meets our eyes or occurs to our thoughts, we may think of that individual object. ”

“ The degrees in which qualities may exist in nature, are infinitely various; but the only degrees with which the grammarian is concerned, are those which our variation of the adjective or adverb enables us to express--including, as of course we must, the state or sense of the primitive word, as one. The reasoning which would make the positive degree to be no degree, would also make the nominative case, or the casus rectus of the Latins, to be no case. ”

“ Unless I give you the noun, you do not understand. Thus every object is represented by a word which is its name; and this name is a noun. To understand whether a word is a noun or not, you simply ask: Is it a thing? Would it answer if I spoke to it? or Could I carry it to the teacher? For instance, bread: yes, bread is an object; table: yes, it is an object; conductor: yes, the conductor would answer, if I were to speak to him.
Let us look through our cards now. I take several cards from different boxes and shuffle them. ”

“ I shall hereafter have occasion to show how a whole phrase may be used to name an idea, and as such stand as the agent or object of a verb.
Some nouns are specifically used to designate certain objects, and distinguish them from the class to which they usually belong. In this way they assume a distinctive character, and are usually denominated proper nouns. They apply to persons, places and things; as, John Smith, Boston, Hylax. Boy is applied in common to all young males of the human species, and as such is a common noun or name. ”

“ It is scarcely necessary to give instances of a fact so familiar as the double meaning of a word. In reality, as has been already observed, an æquivocal or ambiguous word is not one name, but two names, accidentally coinciding in sound. File meaning a steel instrument, and file meaning a line of soldiers, have no more title to be considered one word, because written alike, than grease and Greece have, because they are pronounced alike. ”