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NEW ELEMENTS (Kaina Stoicheia)
MS 517 (1904); EP2, 300-324; /page numbers/ here are from Essential Peirce 2.
I deem it useful to say a few words about this piece. Some years ago I wrote a book entitled "New Elements of Mathematics." It was such a book as a man with considerable natural aptitude for logic and mathematics, who had devoted the best of his time for forty years to the study of the former and all that has been written about it, and had not neglected the latter, was able to write by devoting a year exclusively to it. If the author had been a German, he would have shared the loose ideas of logic that naturally are associated with subjectivism, and consequently could not have written the same book; but had he written it, it would have been in print long ago. As it was, he carried it to three publishers, one of whom had asked him to prepare the book. All of them were very modest men. They did not pretend to know much except about the elements of mathematics. One of them was at that time the publisher of a treatise on geometry which professed to show how to inscribe /301/ a regular polygon of any number of sides in a circle by the aid of rule and compasses only. Both the others published treatises on geometry of similar startling pretensions. None of them approved of my book, because it put perspective before metrical geometry, and topical geometry before either. This was the fault of the book; namely, that a publisher who was so well versed in the elements of mathematics was not convinced by it that this arrangement was logical, even though he took the book home with him and glanced at it during the evening. A writer on the logic of mathematics in America must meet American requirements.
Personally, I regret that manuscript has been lost; for it was the record of much close thought. I can never reproduce it, because it was written in the strictest mathematical style, and with advancing years I have lost the power of writing about logic in mathematical style, although in my youth it was natural to me. In losing the power of writing this style, I have equally lost my admiration of it. I beg permission to offer a criticism of the mathematical style in logic.
I say in "logic" because it is only with a view of presenting the logic of the subject that mathematicians employ the Euclidean style. When they are simply intent on the solution of difficult problems they forget all about it. I call it Euclidean because the first book of Euclid's Elements is the earliest and the most perfect model of that style. Euclid follows it, in some measure, in his other writings; but it is only in the first book of the Elements that it is polished with endless labor and thought. It is easy to see that this style took its origin in the esthetic taste of the Greeks. Everything they did, in literature and in art, shows the predominance of their horror of the too "much." Perhaps this horror was due to the irrepressible activity of Greek minds, and their consequent impatience with useless considerations, together with the expensiveness, at that age in energy of every kind, of the mechanical processes of writing and reading. They took it for granted that the reader would actively think; and the writer's sentences were to serve merely as so many blazes to enable him to follow the track of that writer's thought. The modern book, which I only mention as a foil to the other, in order to be approved, must be approved by a densely stupid and unspeakably indolent young lady as she skims its pages while looking out of the window to be admired. In order to put an idea into such a shape that it cannot fail to be apprehended by her, the first requisite is that it shall fill a certain number of lines, and the second is that not the smallest step shall be left to her own intellectual activity.
The dominating idea of Euclid in writing his first book was plainly that the first elements of geometry can only be comprehended by understanding the logical structure of the doctrine. Yet, in his horror of the too much, he never says a single word about logic from beginning to end. He begins with a couple of dozen "definitions," which are followed by five "postulates," and these by several axioms, or "notions of common sense"; yet he never tells us what a "definition," a "postulate," or a "common notion" is supposed to be; and his meanings in all three cases have been seriously misapprehended. The /302/ forty-eight propositions of the book are set forth and arranged in a manner which betrays a profound understanding of their logical relations. Herein is the principal value of the work; today, its only value. Yet this knowledge is so concealed that it requires the same knowledge to detect it.
This profound work is put into the hands of boys who are not Greek, but overfed and logy. They meet with difficulties which they carry to a teacher who is far more incompetent than they, since he knows nothing of the logical structure which is the cryptic subject of the book, and long familiarity has rendered him incapable of perceiving the difficulties which his scholars can, at least, perceive. The old pedagogical method was to thrash the boys till they did understand; and that was tolerably efficient as an antidote to their overdoses of beef. Since that method has been abandoned it has been necessary to abandon the pedagogical use of the book.
It is extremely difficult to treat fully and clearly of the logic of mathematics in the Euclidean style, since this strictly requires that not a word should be said about logic. As an exact logician, however, I approve of addressing an actively intelligent reader in the ancient method by means of first, definitions; second, postulates; third, axioms; fourth, corollaries; fifth, diagrams; sixth, letters; seventh, theorems; eighth, scholiums. The distinction between a general proposition (which, if a postulate, is often erroneously called an axiom) and an indefinite proposition (to which, if indemonstrable, the word "Postulate" is restricted in German) may also be maintained.
A definition is the logical analysis of a predicate in general terms. It has two branches, the one asserting that the definitum is applicable to whatever there may be to which the definition is applicable, the other (which ordinarily has several clauses), that the definition is applicable to whatever there may be to which the definitum is applicable. A definition does not assert that anything exists.
A postulate is an initial hypothesis in general terms. It may be arbitrarily assumed provided that (the definitions being accepted) it does not conflict with any principle of substantive possibility or with any already adopted postulate. By a principle of substantive possibility, I mean, for example, that it would not be admissible to postulate that there was no relation whatever between two points, or to lay down the proposition that nothing whatever shall be true without exception. For though what this means involves no contradiction it is in contradiction with the fact that it is itself asserted.
An axiom is a self-evident truth, the statement of which is superfluous to the conclusiveness of the reasoning, and which only serves to show a principle involved in the reasoning. It is generally a truth of observation; such as the assertion that something is true.
A corollary, as I shall use the word, is an inference drawn in general terms without the use of any construction.
[CSP footnote: At present, corollary is not a scientific term. The Latin word, meaning a gratuity, was applied to obvious deductions added by commentators to Euclid's propositions. Those his proofs compelled them to grant; and they added admissions of the corollaries without requiring proof. I propose to use the word in a definite sense as a term of logic.] /303/
A diagram is an icon or schematic image embodying the meaning of a general predicate; and from the observation of this icon we are supposed to construct a new general predicate.
A letter is an arbitrary definite designation specially adopted in order to identify a single object of any kind.
A theorem, as I shall use the word, is an inference obtained by constructing a diagram according to a general precept, and after modifying it as ingenuity may dictate, observing in it certain relations, and showing that they must subsist in every case, retranslating the proposition into general terms.
[footnote: This may exclude some propositions called theorems. But I do not think that mathematicians will object to that, in view of my making a sharp distinction between a corollary and a theorem, and thus furnishing the logic of mathematics with two exact and convenient technical terms in place of vague, unscientific words.]
A theorem regularly begins with, first, the general enunciation. There follows, second, a precept for a diagram, in which letters are employed. Then comes, third, the ecthesis, which states [that] what it will be sufficient to show must, in every case, be true concerning the diagram. The fourth article is the subsidiary construction, by which the diagram is modified in some manner already shown to be possible. The fifth article is the demonstration, which traces out the reasons why a certain relation must always subsist between the parts of the diagram. Finally, and sixthly, it is pointed out, by some such expression as Euclid's hoper edei deiksai, or by the usual Q.E.D., or otherwise, that this was all that it was required to show.
A scholium is a comment upon the logical structure of the doctrine. This preface is a scholium.
1. I now proceed to explain the difference between a theoretical and a practical proposition, together with the two important parallel distinctions between definite and vague, and individual and general, noting, at the same time, some other distinctions connected with these. A sign is connected with the "Truth," i.e. the entire Universe of being, or, as some say, the Absolute, in three distinct ways. In the first place, a sign is not a real thing. It is of such a nature as to exist in replicas. Look down a printed page, and every the you see is the same word, every e the same letter. A real thing does not so exist in replica. The being of a sign is merely being represented. Now really being and being represented are very different. Giving to the word sign the full scope that reasonably belongs to it for logical purposes, a whole book is a sign; and a translation of it is a replica of the same sign. A whole literature is a sign. The sentence "Roxana was the queen of Alexander" is a sign of Roxana and of Alexander, and though there is a grammatical emphasis on the former, logically the name "Alexander" is as much a subject as is the name "Roxana"; and the real persons Roxana and Alexander are real objects of the sign. Every sign that /304/ is sufficiently complete refers to sundry real objects. All these objects, even if we are talking of Hamlet's madness, are parts of one and the same Universe of being, the "Truth." But so far as the "Truth" is merely the object of a sign, it is merely the Aristotelian Matter of it that is so. In addition however to denoting objects, every sign sufficiently complete signifies characters, or qualities. We have a direct knowledge of real objects in every experiential reaction, whether of Perception or of Exertion (the one theoretical, the other practical). These are directly hic et nunc. But we extend the category, and speak of numberless real objects with which we are not in direct reaction. We have also direct knowledge of qualities in feeling, peripheral and visceral. But we extend this category to numberless characters of which we have no immediate consciousness. All these characters are elements of the "Truth." Every sign signifies the "Truth." But it is only the Aristotelian Form of the universe that it signifies. The logician is not concerned with any metaphysical theory; still less, if possible, is the mathematician. But it is highly convenient to express ourselves in terms of a metaphysical theory; and we no more bind ourselves to an acceptance of it than we do when we use substantives such as "humanity," "variety," etc. and speak of them as if they were substances, in the metaphysical sense. But, in the third place, every sign is intended to determine a sign of the same object with the same signification or meaning. Any sign, B, which a sign, A, is fitted so to determine, without violation of its, A's, purpose, that is, in accordance with the "Truth," even though it, B, denotes but a part of the objects of the sign, A, and signifies but a part of its, A's, characters, I call an interpretant of A. What we call a "fact" is something having the structure of a proposition, but supposed to be an element of the very universe itself. The purpose of every sign is to express "fact," and by being joined with other signs, to approach as nearly as possible to determining an interpretant which would be the perfect Truth, the absolute Truth, and as such (at least, we may use this language) would be the very Universe. Aristotle gropes for a conception of perfection, or entelechy, which he never succeeds in making clear. We may adopt the word to mean the very fact, that is, the ideal sign which should be quite perfect, and so identical,—in such identity as a sign may have,—with the very matter denoted united with the very form signified by it. The entelechy of the Universe of being, then, the Universe qua fact, will be that Universe in its aspect as a sign, the "Truth" of being. The "Truth," the fact that is not abstracted but complete, is the ultimate interpretant of every sign.
2. Of the two great tasks of humanity, Theory and Practice, the former sets out from a sign of a real object with which it is acquainted, passing from this, as its matter, to successive interpretants embodying more and more fully its form, wishing ultimately to reach a direct perception of the entelechy; while the latter, setting out from a sign signifying a character of which it has an idea, passes from this, as its form, to successive interpretants realizing more and more precisely its matter, hoping ultimately to be able to make a direct effort, producing the entelechy. But of these two movements, logic very properly /305/ prefers to take that of Theory as the primary one. It speaks of an antecedent as that which, being known, something else, the consequent, may also be known. In our vernacular, the latter is inaccurately called a consequence, a word that the precise terminology of logic reserves for the proposition expressing the relation of any consequent to its antecedent, or for the fact which this proposition expresses. The conception of the relation of antecedent and consequent amounts, therefore, to a confusion of thought between the reference of a sign to its meaning, the character which it attributes to its object, and its appeal to an interpretant. But it is the former of these which is the more essential. The knowledge that the sun has always risen about once in each 24 hours (sidereal time) is a sign whose object is the sun, and (rightly understood) a part of its signification is the rising of the sun tomorrow morning. The relation of an antecedent to its consequent, in its confusion of the signification with the interpretant, is nothing but a special case of what occurs in all action of one thing upon another, modified so as to be merely an affair of being represented instead of really being. It is the representative action of the sign upon its object. For whenever one thing acts upon another it determines in that other a quality that would not otherwise have been there. In the vernacular we often call an effect a "consequence," because that which really is may correctly be represented; but we should refuse to call a mere logical consequent an "effect," because that which is merely represented, however legitimately, cannot be said really to be. If we speak of an argumentation as "producing a great effect," it is not the interpretant itself, by any means, to which we refer, but only the particular replica of it which is made in the minds of those addressed.
If a sign, B, only signifies characters that are elements (or the whole) of the meaning of another sign, A, then B is said to be a predicate (or essential part) of A. If a sign, A, only denotes real objects that are a part or the whole of the objects denoted by another sign, B, then A is said to be a subject (or substantial part) of B. The totality of the predicates of a sign, and also the totality of the characters it signifies, are indifferently each called its logical depth. This is the oldest and most convenient term. Synonyms are the comprehension of the Port-Royalists, the content (Inhalt) of the Germans, the force of DeMorgan, the connotation of J.S. Mill. (The last is objectionable.) The totality of the subjects, and also, indifferently, the totality of the real objects of a sign is called the logical breadth. This is the oldest and most convenient term. Synonyms are the extension of the Port-Royalists (ill-called extent by some modern French logicians), the sphere (Umfang) of translators from the German, the scope of DeMorgan, the denotation of J.S. Mill.
Besides the logical depth and breadth, I have proposed (in 1867) the terms information and area to denote the total of fact (true or false) that in a given state of knowledge a sign embodies.
3. Other distinctions depend upon those that we have drawn. I have spoken of real relations as reactions. It may be asked how far I mean to say that all real /306/ relations are reactions. It is seldom that one falls upon so fascinating a subject for a train of thought as the analysis of that problem in all its ramifications, mathematical, physical, biological, sociological, psychological, logical, and so round to the mathematical again. The answer cannot be satisfactorily given in a few words; but it lies hidden beneath the obvious truth that any exact necessity is expressible by a general equation; and nothing can be added to one side of a general equation without an equal addition to the other. Logical necessity is the necessity that a sign should be true to a real object; and therefore there is logical reaction in every real dyadic relation. If A is in a real relation to B, B stands in a logically contrary relation to A, that is, in a relation at once converse to and inconsistent with the direct relation. For here we speak not of a vague sign of the relation but of the relation between two individuals, A and B. This very relation is one in which A alone stands to any individual, and it to B only. There are, however, degenerate dyadic relations,—degenerate in the sense in which two coplanar lines form a degenerate conic,—where this is not true. Namely, they are individual relations of identity, such as the relation of A to A. All mere resemblances and relations of reason are of this sort.
Of signs there are two different degenerate forms. But though I give them this disparaging name, they are of the greatest utility, and serve purposes that genuine signs could not. The more degenerate of the two forms (as I look upon it) is the icon. This is defined as a sign of which the character that fits it to become a sign of the sort that it is, is simply inherent in it as a quality of it. For example, a geometrical figure drawn on paper may be an icon of a triangle or other geometrical form. If one meets a man whose language one does not know and resorts to imitative sounds and gestures, these approach the character of an icon. The reason they are not pure icons is that the purpose of them is emphasized. A pure icon is independent of any purpose. It serves as a sign solely and simply by exhibiting the quality it serves to signify. The relation to its object is a degenerate relation. It asserts nothing. If it conveys information, it is only in the sense in which the object that it is used to represent may be said to convey information. An icon can only be a fragment of a completer sign.
The other form of degenerate sign is to be termed an index. It is defined as a sign which is fit to serve as such by virtue of being in a real reaction with its object. For example, a weather-cock is such a sign. It is fit to be taken as an index of the wind for the reason that it is physically connected with the wind. A weather-cock conveys information; but this it does because in facing the very quarter from which the wind blows, it resembles the wind in this respect, and thus has an icon connected with it. In this respect it is not a pure index. A pure index simply forces attention to the object with which it reacts and puts the interpreter into mediate reaction with that object, but conveys no information. As an example, take an exclamation "Oh!" The letters attached to a geometrical figure are another case. Absolutely unexceptionable examples of degenerate forms must not be expected. All that is possible is to give examples which tend sufficiently towards those forms to suggest /307/ what is meant. It is remarkable that while neither a pure icon nor a pure index can assert anything, an index which forces something to be an icon, as a weather-cock does, or which forces us to regard it as an icon, as the legend under a portrait does, does make an assertion, and forms a proposition. This suggests the true definition of a proposition, which is a question in much dispute at this moment. A proposition is a sign which separately, or independently, indicates its object. No index, however, can be an argumentation. It may be what many writers call an argument; that is, a basis of argumentation; but an argument in the sense of a sign which separately shows what interpretant it is intended to determine it cannot be.
It will be observed that the icon is very perfect in respect to signification, bringing its interpreter face to face with the very character signified. For this reason, it is the mathematical sign par excellence. But in denotation it is wanting. It gives no assurance that any such object as it represents really exists. The index on the other hand does this most perfectly, actually bringing to the interpreter the experience of the very object denoted. But it is quite wanting in signification unless it involves an iconic part.
We now come to the genuine sign, for which I propose the technical designation symbol, following a use of that word not infrequent among logicians including Aristotle. A symbol is defined as a sign which is fit to serve as such simply because it will be so interpreted.
|is a sign fit to be used as such because
|it possesses the quality
|it is in real reaction
with the object denoted.
|it determines the
Language and all abstracted thinking, such as belongs to minds who think in words, is of the symbolic nature. Many words, though strictly symbols, are so far iconic that they are apt to determine iconic interpretants, or as we say, to call up lively images. Such, for example, are those that have a fancied resemblance to sounds associated with their objects; that are onomatopoetic, as they say. There are words, which although symbols, act very much like indices. Such are personal, demonstrative, and relative pronouns, for which A, B, C, etc. are often substituted. A Proper Name, also, which denotes a single individual well known to exist by the utterer and interpreter, differs from an index only in that it is a conventional sign. Other words refer indirectly to indices. Such is "yard" which refers to a certain bar in Westminster, and has no meaning unless the interpreter is, directly or indirectly, in physical reaction with that bar. Symbols are particularly remote from the Truth itself. They are abstracted. They neither exhibit the very characters signified as icons do, nor assure us of the reality of their objects, as indices do. Many proverbial sayings /308/ express a sense of this weakness; as "Words prove nothing," and the like. Nevertheless, they have a great power of which the degenerate signs are quite destitute. They alone express laws. Nor are they limited to this theoretical use. They serve to bring about reasonableness and law. The words justice and truth, amid a world that habitually neglects these things and utterly derides the words, are nevertheless among the very greatest powers the world contains. They create defenders and animate them with their strength. This is not rhetoric or metaphor: it is a great and solid fact of which it behooves a logician to take account.
A symbol is the only kind of sign which can be an argumentation.
[footnote: I commonly call this an argument; for nothing is more false historically than to say that this word has not at all times been used in this sense. Still, the longer word is a little more definite.]
4. I have already defined an argument as a sign which separately monstrates what its intended interpretant is, and a proposition as a sign which separately indicates [what] its object is, and we have seen that the icon alone cannot be a proposition while the symbol alone can be an argument. That a sign cannot be an argument without being a proposition is shown by attempting to form such an argument. "Tully, c'est-à-dire a Roman," evidently asserts that Tully is a Roman. Why this is so is plain. The interpretant is a sign which denotes that which the sign of which it is interpretant denotes. But, being a symbol, or genuine sign, it has a signification and therefore it represents the object of the principal sign as possessing the characters that it, the interpretant, signifies.
It will be observed that an argument is a symbol which separately monstrates (in any way) its purposed interpretant. Owing to a symbol being essentially a sign only by virtue of its being interpretable as such, the idea of a purpose is not entirely separable from it. The symbol, by the very definition of it, has an interpretant in view. Its very meaning is intended. Indeed, a purpose is precisely the interpretant of a symbol. But the conclusion of an argument is a specially monstrated interpretant, singled out from among the possible interpretants. It is, therefore, of its nature single, although not necessarily simple.
If we erase from an argument every monstration of its special purpose, it becomes a proposition; usually a copulate proposition, composed of several members whose mode of conjunction is of the kind expressed by "and," which the grammarians call a "copulative conjunction." If from a propositional symbol we erase one or more of the parts which separately denote its objects, the remainder is what is called a rhema; but I shall take the liberty of calling it a term. Thus, from the proposition "Every man is mortal," we erase "Every man," which is shown to be denotative of an object by the circumstance that if it be replaced by an indexical symbol, such as "That" or "Socrates," the symbol is reconverted into a proposition, we get the rhema or term "_____ is mortal." Most logicians will say that this is not a term. The term, they will say, is "mortal," while I have left the copula "is" standing with it. Now while it is true that one of Aristotle's memoirs dissects a proposition into subject, predicate, and verb, yet as long as Greek was the language which
logicians had in view, no importance was attached to the substantive verb, "is," because the Greek permits it to be omitted. It was not until the time of Abelard, when Greek was forgotten, and logicians had Latin in mind, that the copula was recognized as a constituent part of the logical proposition.
I do not, for my part, regard the usages of language as forming a satisfactory basis for logical doctrine. Logic, for me, is the study of the essential conditions to which signs must conform in order to function as such. How the constitution of the human mind may compel men to think is not the question; and the appeal to language appears to me to be no better than an unsatisfactory method of ascertaining psychological facts that are of no relevancy to logic. But if such appeal is to be made (and logicians generally do make it; in particular their doctrine of the copula appears to rest solely upon this), it would seem that they ought to survey human languages generally and not confine themselves to the small and extremely peculiar group of Aryan speech. Without pretending, myself, to an extensive acquaintance with languages, I am confident that the majority of non-Aryan languages do not ordinarily employ any substantive verb equivalent to "is." Some place a demonstrative or relative pronoun; as if one should say: "_____ is a man that is translated" for "A man is translated." Others have a word, syllable, or letter, to show that an assertion is intended. I have been led to believe that in very few languages outside the Aryan group is the common noun a well-developed and independent part of speech. Even in the Shemitic languages, which are remarkably similar to the Aryan, common nouns are treated as verbal forms and are quite separated from proper names. The ordinary view of a term, however, supposes it to be a common noun in the fullest sense of the term. It is rather odd that of all the languages which I have examined in a search for some support of this ordinary view, so outlandish a speech as the Basque is the only one I have found that seems to be constructed thoroughly in the manner in which the logicians teach us that every rational being must think.
[footnote: While I am on the subject of languages I may take occasion to remark with reference to my treatment of the direct and indirect "objects" of a verb as so many subjects of the proposition, that about nine out of every ten languages regularly emphasize one of the subjects, and make it the principal one, by putting it in a special nominative case, or by some equivalent device. The ordinary logicians seem to think that this, too, is a necessity of thought, although one of the living Aryan languages of Europe habitually puts that subject in the genitive which the Latin puts in the nominative. This practice was very likely borrowed from a language similar to the Basque spoken by some progenitors of the Gaels. Some languages employ what is, in effect, an ablative for this purpose. It no doubt is a rhetorical enrichment of a language to have a form "B is loved by A" in addition to "A loves B." The language will be still richer if it has a third form in which A and B are treated as equally the subjects of what is said. But logically, the three are identical.]
What is the difference between "_____ is a man" and "man"? The logicians hold that the essence of the latter lies in a definition describing its characters; which doctrine virtually makes "man" equivalent to "what is a man." It thus differs from "_____ is a man" by the addition of the badly named "indefinite pronoun," what. The rhema "_____ is a man" is a fragmentary sign. But "man" is never used alone, and would have no meaning by /310/ itself. It is sometimes written upon an object to show the nature of that object; but in such case, the appearance of the object is an index of that object; and the two taken together form a proposition. In respect to being fragmentary, therefore, the two signs are alike. It may be said that "Socrates wise" does not make a sentence in the language at present used in logic, although in Greek it would. But it is important not to forget that no more do "Socrates" and "is wise" make a proposition unless there is something to indicate that they are to be taken as signs of the same object. On the whole, it appears to me that the only difference between my rhema and the "term" of other logicians is that the latter contains no explicit recognition of its own fragmentary nature. But this is as much as to say that logically their meaning is the same; and it is for that reason that I venture to use the old, familiar word "term" to denote the rhema.
It may be asked what is the nature of the sign which joins "Socrates" to "_____ is wise," so as to make the proposition "Socrates is wise." I reply that it is an index. But, it may be objected, an index has for its object a thing hic et nunc, while a sign is not such a thing. This is true, if under "thing" we include singular events, which are the only things that are strictly hic et nunc. But it is not the two signs "Socrates" and "wise" that are connected, but the replicas of them used in the sentence. We do not say that "_____ is wise," as a general sign, is connected specially with Socrates, but only that it is so as here used. The two replicas of the words "Socrates" and "wise" are hic et nunc, and their junction is a part of their occurrence hic et nunc. They form a pair of reacting things which the index of connection denotes in their present reaction, and not in a general way; although it is possible to generalize the mode of this reaction like any other. There will be no objection to a generalization which shall call the mark of junction a copula, provided it be recognized that, in itself, it is not general, but is an index. No other kind of sign would answer the purpose; no general verb "is" can express it. For something would have to bring the general sense of that general verb down to the case in hand. An index alone can do this. BEGIN HERE But how is this index to signify the connection? In the only way in which any index can ever signify anything; by involving an icon. The sign itself is a connection. I shall be asked how this applies to Latin, where the parts of the sentence are arranged solely with a view to rhetorical effect. I reply that, nevertheless, it is obvious that in Latin, as in every language, it is the juxtaposition which connects words. Otherwise they might be left in their places in the dictionary. Inflexion does a little; but the main work of construction, the whole work of connexion, is performed by putting the words together. In Latin much is left to the good sense of the interpreter. That is to say, the common stock of knowledge of utterer and interpreter, called to mind by the words, is a part of the sign. That is more or less the case in all conversation, oral and scriptal. It is, thus, clear that the vital spark of every proposition, the peculiar propositional element of the proposition, is an indexical proposition; an index involving an icon. The rhema, say "_____ loves _____," /311/ has blanks which suggest filling; and a concrete actual connection of a subject with each blank monstrates the connection of ideas.
It is the Proposition which forms the main subject of this whole scholium; for the distinctions of vague and distinct, general and individual are propositional distinctions. I have endeavored to restrain myself from long discussions of terminology. But here we reach a point where a very common terminology overlaps an erroneous conception. Namely those logicians who follow the lead of Germans, instead of treating of propositions, speak of “judgments” (Urtheile). They regard a proposition as merely an expression in speech or writing of a judgment. More than one error is involved in this practice. In the first place, a judgment, as they very correctly teach, is a subject of psychology. Since psychologists, now-a-days, not only renounce all pretension to knowledge of the soul, but also take pains to avoid talking of the mind, the latter is at present not a scientific term, at all; and therefore I am not prepared to say that logic does not, as such, treat of the mind. I should like to take mind in such a sense that this could be affirmed; but in any sense in which psychology,—the scientific psychology now recognized,—treats of mind, logic, I maintain, has no concern with it. Without stopping here to discuss this large question, I will say that psychology is a science which makes special observations; and its whole business is to make the phenomena so observed (along with familiar facts allied to those things), definite and comprehensible. Logic is a science little removed from pure mathematics. It cannot be said to make any positive phenomena known, although it takes account and rests upon phenomena of daily and hourly experience, which it so analyzes as to bring out recondite truths about them. One might think that a pure mathematician might assume these things as an initial hypothesis and deduce logic from these; but this turns out, upon trial, not to be the case. The logician has to be recurring to reexamination of the phenomena all along the course of his investigations. But logic is all but as far remote from psychology as is pure mathematics. Logic is the study of the essential nature of signs. A sign is something that exists in replicas. Whether the sign "it is raining" or "all pairs of particles of matter have component accelerations toward one another inversely proportional to the square of the distance" happens to have a replica in writing, in oral speech, or in silent thought, is a distinction of the very minutest interest to logic, which is a study, not of replicas, but of signs. But this is not the only, nor the most serious error involved in making logic treat of "judgments" in place of propositions. It involves confounding two things which must be distinguished if a real comprehension of logic is to be attained. A proposition, as I have just intimated, is not to be understood as the lingual expression of a judgment. It is, on the contrary, that sign of which the judgment is one replica and the lingual expression another. But a judgment is distinctly more than the mere mental replica of a proposition. It not merely expresses the proposition, but it goes further and accepts it. I grant that the normal use of a proposition is to affirm it; and its chief logical properties relate to /312/ what would result in reference to its affirmation. It is, therefore, convenient in logic to express propositions in most cases in the indicative mood. But the proposition in the sentence, "Socrates est sapiens," strictly expressed, is "Socratem sapientum esse." The defence of this position is that in this way we distinguish between a proposition and the assertion of it; and without such distinction it is impossible to get a distinct notion of the nature of the proposition. One and the same proposition may be affirmed, denied, judged, doubted, inwardly inquired into, put as a question, wished, asked for, effectively commanded, taught, or merely expressed, and does not thereby become a different proposition. What is the nature of these operations? The only one that need detain us is affirmation, including judgment, or affirmation to oneself. As an aid in dissecting the constitution of affirmation I shall employ a certain logical magnifying-glass that I have often found efficient in such business. Imagine, then, that I write a proposition on a piece of paper, perhaps a number of times, simply as a calligraphic exercise. It is not likely to prove a dangerous amusement. But suppose I afterwards carry the paper before a notary public and make affidavit to its contents. That may prove to be a horse of another color. The reason is that this affidavit may be used to determine an assent to the proposition it contains in the minds of judge and jury;—an effect that the paper would not have had if I had not sworn to it. For certain penalties here and hereafter are attached to swearing to a false proposition; and consequently the fact that I have sworn to it will be taken as a negative index that it is not false. This assent in judge and jury's minds may effect in the minds of sheriff and posse a determination to an act of force to the detriment of some innocent man's liberty or property. Now certain ideas of justice and good order are so powerful that the ultimate result may be very bad for me. This is the way that affirmation looks under the microscope; for the only difference between swearing to a proposition and an ordinary affirmation of it, such as logic contemplates, is that in the latter case the penalties are less and even less certain than those of the law. The reason there are any penalties is, as before, that the affirmation may determine a judgment to the same effect in the mind of the interpreter to his cost. It cannot be that the sole cause of his believing it is that there are such penalties, since two events cannot cause one another, unless they are simultaneous. There must have been, and we well know that there is, a sort of hypnotic disposition to believe what one is told with an air [of] command. It is Grimes's credenciveness, which is the essence of hypnotism. This disposition produced belief; belief produced the penalties; and the knowledge of these strengthens the disposition to believe.
I have discussed the nature of belief in the Popular Science Monthly for November 1877. On the whole, we may set down the following definitions:
A belief in a proposition is a controlled and contented habit of acting in ways that will be productive of desired results only if the proposition is true.
An affirmation is an act of an utterer of a proposition to an interpreter, and consists, in the first place, in the deliberate exercise, in uttering the /313/ proposition, of a force tending to determine a belief in it in the mind of the interpreter. Perhaps that is a sufficient definition of it; but it involves also a voluntary self-subjection to penalties in the event of the interpreter's mind (and still more the general mind of society) subsequently becoming decidedly determined to the belief at once in the falsity of the proposition and in the additional proposition that the utterer believed the proposition to be false at that time he uttered it.
A judgment is a mental act deliberately exercising a force tending to determine in the mind of the agent a belief in the proposition: to which should perhaps be added that the agent must be aware of his being liable to inconvenience in the event of the proposition's proving false in any practical aspect.
In order fully to understand the distinction between a proposition and an argument, it will be found important to class these acts, affirmation, etc. and ascertain their precise nature. The question is a purely logical one; but it happens that a false metaphysics is generally current, especially among men who are influenced by physics but yet are not physicists enough fully to comprehend physics, which metaphysics would disincline those who believe in it from readily accepting the purely logical statement of the nature of affirmation. I shall therefore be forced to touch upon metaphysics. Yet I refuse to enter here upon a metaphysical discussion; I shall merely hint at what ground it is necessary to take in opposition to a common doctrine of that kind. Affirmation is of the nature of a symbol. It will be thought that this cannot be the case since an affirmation, as the above analysis shows, produces real effects, physical effects. No sign, however, is a real thing. It has no real being, but only being represented. I might more easily persuade readers to think that affirmation was an index, since an index is, perhaps, a real thing. Its replica, at any rate, is in real reaction with its object, and it forces a reference to that object upon the mind. But a symbol, a word, certainly exists only in replica, contrary to the nature of a real thing; and indeed the symbol only becomes a sign because its interpreter happens to be prepared to represent it as such. Hence, I must and do admit that a symbol cannot exert any real force. Still, I maintain that every sufficiently complete symbol governs things, and that symbols alone do this. I mean that though it is not a force, it is a law. Now those who regard the false metaphysics of which I speak as the only clear opinion on its subject are in the habit of calling laws "uniformities," meaning that what we call laws are, in fact, nothing but common characters of classes of events. It is true that they hold that they are symbols, as I shall endeavor to show that they are; but this is to their minds equivalent to saying that they are common characters of events; for they entertain a very different conception of the nature of a symbol from mine. I begin, then, by showing that a law is not a mere common character of events. Suppose that a man throwing a pair of dice, which were all that honest dice are supposed to be, were to throw sixes a hundred times running. Every mathematician will admit that that would be no ground for expecting the next throw to turn up sixes. It is true that in any actual case in which we should see sixes thrown a hundred times /314/ running we should very rightly be confident that the next throw would turn up sixes likewise. But why should we do so? Can anybody sincerely deny that it would be because we should think the throwing of a hundred successive sixes was an almost infallible indication of there being some real connection between those throws, so that the series not merely a uniformity in the common character of turning up sixes, but something more, a result of a real circumstance about the dice connecting the throws? This example illustrates the logical principle that mere community of character between the members of a collection is no argument, however slender, tending to show that the same character belongs to another object not a member of that collection and not (as far as we have any reason to think) having any real connection with it, unless perchance it be in having the character in question. For the usual supposition that we make about honest dice is that there will be no real connection (or none of the least significance) between their different throws. I know that writer has copied writer in the feeble analysis of chance as consisting in our ignorance. But the calculus of probabilities is pure nonsense unless it affords assurance in the long run. Now what assurance could there be concerning a long run of throws of a pair of dice, if, instead of knowing they were honest dice, we merely did not know whether they were or not, or if, instead of knowing that there would be no important connection between the throws, we merely did not know that there would be? That certain objects A, B, C, etc. are known to have a certain character is not the slightest reason for supposing that another object, [Greek capital xi], quite unconnected with the others so far as we know, has that character. Nor has this self evident proposition ever been denied. A "law," however, is taken very rightly by everybody to be a reason for predicting that an event will have a certain character although the events known to have that character have no other real connection with it than the law. This shows that the law is not a mere uniformity but involves a real connection. It is true that those metaphysicians say that if A, B, C, etc. are known to have two common characters and [Greek capital xi] is known to have one of these, this is a reason for believing that it has the other. But this is quite untenable. Merely having a common character does not constitute a real connection; and those very writers virtually acknowledge this, in reducing law to uniformity, that is, to the possession of a common character, as a way of denying that "law" implies any real connection. What is a law, then? It is a formula to which real events truly conform. By "conform," I mean that, taking the formula as a general principle, if experience shows that the formula applies to a given event, then the result will be confirmed by experience. But that such a general formula is a symbol, and more particularly, an asserted symbolical proposition, is evident. Whether or not this symbol is a reality, even if not recognized by you or me or any generations of men, and whether, if so, it implies an Utterer, are metaphysical questions into which I will not now enter. One distinguished writer seems to hold that, although events conform to the formula, or rather, although it conforms to the Truth of facts, yet it /315/ does not influence the facts. This comes perilously near to being pure verbiage; for, seeing that nobody pretends that the formula exerts a compulsive force on the events, what definite meaning can attach to this emphatic denial of the law's "influencing" the facts? The law had such mode of being as it ever has before all the facts had come into existence, for it might already be experientially known; and then the law existing, when the facts happen there is agreement between them and the law. What is it, then, that this writer has in mind? If it were not for the extraordinary misconception of the word "cause" by Mill, I should say that the idea of metaphysical sequence implied in that word, in "influence," and in other similar words was perfectly clear. Mill's singularity is that he speaks of the cause of a singular event. Everybody else speaks of the cause of a "fact," which is an element of the event. But, with Mill, it is the event in its entirety which is caused. The consequence is that Mill is obliged to define the cause as the totality of all the circumstances attending the event. This is, strictly speaking, the Universe of being in its totality. But any event, just as it exists, in its entirety, is nothing else but the same Universe of being in its totality. It strictly follows, therefore, from Mill's use of the words, that the only causatum is the entire Universe of being and that its only cause is itself. He thus deprives the word of all utility. As everybody else but Mill and his school more or less clearly understands the word, it is a highly useful one. That which is caused, the causatum, is, not the entire event, but such abstracted element of an event as is expressible in a proposition, or what we call a "fact." The cause is another "fact." Namely, it is, in the first place, a fact which could, within the range of possibility, have its being without the being of the causatum; but, secondly, it could not be a real fact while a certain third complementary fact, expressed or understood, was realized, without the being of the causatum; and thirdly, although the actually realized causatum might perhaps be realized by other causes or by accident, yet the existence of the entire possible causatum could not be realized without the cause in question. It may be added that a part of a cause, if a part in that respect in which the cause is a cause, is also called a cause. In other respects, too, the scope of the word will be somewhat widened in the sequel. If the cause so defined is a part of the causatum, in the sense that the causatum could not logically be without the cause, it is called an internal cause; otherwise, it is called an external cause. If the cause is of the nature of an individual thing or fact, and the other factor requisite to the necessitation of the causatum is a general principle, I would call the cause a minor, or individuating, or perhaps a physical cause. If, on the other hand, it is the general principle which is regarded as the cause and the individual fact to which it is applied is taken as the understood factor, I would call the cause a major, or defining, or perhaps a psychical cause. The individuating internal cause is called the material cause. Thus the integrant parts of a subject or fact form its matter, or material cause. The individuating external cause is called the efficient, or efficient cause; and the causatum is called the effect. The defining internal cause is called the formal /316/ cause, or form. All those facts which constitute the definition of a subject or fact make up its form. The defining external cause is called the final cause, or end. It is hoped that these statements will be found to hit a little more squarely than did those of Aristotle and the scholastics the same bull's eye at which they aimed. From scholasticism and the medieval universities, these conceptions passed in vaguer form into the common mind and vernacular of Western Europe, and especially so in England. Consequently, by the aid of these definitions I think I can make out what it is that the writer mentioned has in mind in saying that it is not the law which influences, or is the final cause of, the facts, but the facts that make up the cause of the law. He means that the general fact which the law of gravitation expresses is composed of the special facts that this stone at such a time fell to the ground as soon as it was free to do so and its upward velocity was exhausted, that each other stone did the same, that each planet at each moment was describing an ellipse having the centre of mass of the solar system at a focus, etc. etc.; so that the individual facts are the material cause of the general fact expressed by the law; while the propositions expressing those facts are the efficient cause of the law itself. This is a possible meaning in harmony with the writer's sect of thought; and I believe it is his intended meaning. But this is easily seen not to be true. For the formula relates to all possible events of a given description; which is the same as to say that it relates to all possible events. Now no collection of actual individual events or other objects of any general description can amount to all possible events or objects of that description; for it is possible that an addition should be made to that collection. The individuals do not constitute the matter of a general; those who with Kant, or long before him, said that they do were wanting in the keen edge of thought requisite for such discussions. On the contrary, the truth of the formula, its really being a sign of the indicated object, is the defining cause of the agreement of the individual facts with it. Namely, this truth fulfills the first condition, which is that it might logically be although there were no such agreement. For it might be true, that is, contains no falsity, that whatever stone there might be on earth would have a real downward component [of] acceleration even although no stone actually existed on earth. It fulfills the second condition, that as soon as the other factor (in this case the actual existence of each stone on earth) was present, the result of the formula, the real downward component of acceleration would exist. Finally, it fulfills the third condition, that while all existing stones might be accelerated downwards by other causes or by an accidental concurrence of circumstances, yet the downward acceleration of every possible stone would involve the truth of the formula.
It thus appears that the truth of the formula, that is, the law, is, in the strictest sense, the defining cause of the real individual facts. But the formula, if a symbol at all, is a symbol of that object which it indicates as its object. Its truth, therefore, consists in the formula being a symbol. Thus a symbol may be the cause of real individual events and things. It is easy to see that nothing /317/ but a symbol can be such a cause, since a cause is by its definition the premiss of an argument; and a symbol alone can be an argument. Every sufficiently complete symbol is a final cause of, and "influences," real events, in precisely the same sense in which my desire to have the window open, that is, the symbol in my mind of the agreeability of it, influences the physical facts of my rising from my chair, going to the window, and opening it. Who but a Millian or a lunatic will deny that that desire influences the opening of the window? Yet the sense in which it does so is none other than that in which every sufficiently complete and true symbol influences real facts.
A symbol is defined as a sign which becomes such by virtue of the fact that it is interpreted as such. The signification of a complex symbol is determined by certain rules of syntax which are part of its meaning. A simple symbol is interpreted to signify what it does from some accidental circumstance or series of circumstances, which the history of any word illustrates. For example, in the latter half of the fifteenth century, a certain model of vehicle came into use in the town of Kots (pronounced, kotch) in Hungary. It was copied in other towns, doubtless with some modifications, and was called a kotsi szeker, or Kots cart. Copied in still other towns, and always more or less modified, it came to be called, for short, a cotch. It thus came about that the [word] coach was used, first, for a magnificent vehicle to be drawn by horses for carrying persons in state and in such comfort as that state required; then, for a large and pretentious vehicle to be drawn by four or more horses for conveying passengers from one town to another; and finally, to any large vehicle for conveying passengers at a fare by the seat from one town to another. In all ordinary cases, it is, and must be, an accidental circumstance which causes a symbol to signify just the characters it does; for were there any necessary, or nearly necessary, reason for it, it would be this which would render the sign a sign, and not the mere fact that so it would be interpreted, as the definition of a symbol requires. It will be well here to interpose a remark as to the identity of a symbol. A sign has its being in its adaptation to fulfill a function. A symbol is adapted to fulfill the function of a sign simply by the fact that it does fulfill it; that is, that it is so understood. It is, therefore, what it is understood to be. Hence, if two symbols are used, without regard to any differences between them, they are replicas of the same symbol. If the difference is looked upon as merely grammatical (as with he and him), or as merely rhetorical (as with money and spondesime), or as otherwise insignificant, then logically they are replicas of one symbol. Hardly any symbol directly signifies the characters it signifies; for whatever it signifies it signifies by its power of determining another sign signifying the same character. If I write of the "sound of sawing," the reader will probably do little more than glance sufficiently at the words to assure himself that he could imagine the sound I referred to if he chose to do so. If, however, what [I] proceed to say about that sound instigates him to do more, a sort of auditory composite will arise in his imagination of different occasions when he has been near a saw; and this will serve as /318/ an icon of the signification of the phrase "sound of a saw." If I had used, instead of that phrase, the word "buzz," although this would have been less precise, yet, owing to the sound of the word being itself a sort of buzz, it would have more directly called up an iconic interpretation. Thus some symbols are far superior to others in point of directness of signification. This is true not only of outward symbols but also of general ideas. When a person remembers something, as for example in trying in a shop to select a ribbon whose color shall match that of an article left at home, he knows that his idea is a memory and not an imagination by a certain feeling of having had the idea before, which he will not be very unlikely to find has been somewhat deceptive. It is a sort of sense of similarity between the present and the past. Even if he had the two colors before his eyes, he could only know them to be similar by a peculiar feeling of similarity; because as two sensations they are different. But in the case supposed, it is not the mere general feeling of similarity which is required but that peculiar variety of it which arises when a present idea is pronounced to be similar to one not now in the mind at all, but formerly in the mind. It is clear that this feeling functions as a symbol. To call it an icon of a past idea would be preposterous. For instead of the present idea being serviceable as a substitute for the past idea by virtue of being similar to it, it is on the contrary only known to be similar to it by means of this feeling that it is so. Neither will it answer to call it an index. For it is the essence of an index to be in real connection with its object; so that it cannot be mendacious in so far as it is indexical; while this feeling is not infrequently deceptive; sometimes, in everybody's life, absolutely baseless. It is true that it is on the whole veracious; and veracity,—necessitated truth,—can belong to no sign except insofar as it involves an index. But a symbol, if sufficiently complete, always involves an index, just as an index sufficiently complete always involves an icon. There is an infallible criterion for distinguishing between an index and a symbol. Namely, although an index, like any other sign, only functions as a sign when it is interpreted, yet though it never happen to be interpreted, it remains equally fitted to be the very sign that [it?] would be if interpreted. A symbol, on the other hand, that should not be interpreted, would either not be a sign at all, or would only be a sign in an utterly different way. An inscription that nobody ever had interpreted or ever would interpret would be but a fanciful scrawl, an index that some being had been there, but not at all conveying or apt to convey its meaning. Now imagine the feeling which tells us that a present idea has been experienced before not to be interpreted as having that meaning, and what would it be? It would be like any other feeling. No study of it could ever discover that it had any connection with an idea past and gone. Even if this should be discovered, and if it should further be discovered that that connection was of such a nature as to afford assurance that the present idea with which the feeling is connected is similar to that former idea, still this would be by an additional discovery, not involved in the sign itself; a discovery, too, of the nature of a symbol, since it would be /319/ the discovery of a general law. The only way in which an index can be a proposition is by involving an icon. But what icon does this feeling represent? Does it exhibit anything similar to similarity? To suppose that the feeling in question conveys its meaning by presenting [in] a new idea a vague duplicate of the idea first present, gratuitous as this hypothesis would be, would not suffice to prove the feeling to be an index, since a symbol would be requisite to inform us that the first idea and the newly presented idea were similar; and even then there would be the element of preteritness to be conveyed, which no icon and consequently no index could signify. It is quite certain therefore that in this feeling we have a definite instance of a symbol which, in a certain sense, necessarily signifies what it does. We have already seen that it can only be by an accident, and not by inherent necessity, that a symbol signifies what it does. The two results are reconciled by the consideration that the accident in this case is that we are so constituted that that feeling shall be so interpreted by us. A little psychological examination will vindicate the first assertion. For although it is not a very rare experience to have a strong feeling of having been in a present situation before, when in fact one never was in such a situation, yet everybody, unless he be a psychologist, is invariably deeply impressed by such an experience (or at least by his first experience of this sort), will not forget it for long years, and is importuned by the notion that he is here confronted with a phenomenon profoundly mysterious, if not supernatural. The psychologist may waive the matter aside, in his debonair satisfaction with the state of his science; but he seems to me to overlook the most instructive part of the phenomenon. That is, that though internal feelings generally are testimonies proverbially requiring to be received with reserve and caution, yet when this particular one plays us false, people feel as if the bottom had fallen out of the universe of being. Why should they take the matter so seriously? It is that if we try to analyze what is meant by saying that a present idea "resembles" one that is past and gone, we cannot find that anything else is meant but that there is this feeling connected with it. From which it seems to result that for this feeling to be mendacious would be a self-contradictory state of things; in which case we might well say that the bottom had fallen out of Truth itself. Now a symbol which should by logical necessity signify what it does would obviously have nothing but its own application, or predication, for its own signification as a predicate; or, to express the matter in abbreviation, would signify itself alone. For anything else that might be supposed to be signified by it might, by logical possibility, not be so signified. That such a symbol should be false would indeed involve a contradiction. Thus, the feeling of amazement at the "sense of pre-existence," as it has been called, amounts virtually to nothing more than the natural confusion of that which is necessary by virtue of the constitution of the mind with that which is logically necessary. The feeling under discussion is necessary only in the latter sense. Consequently, its falsity is not absurd but only abnormal. Nor it it its own sole signification, but only the sole signification recognizable without /320/ transcendental thought; because to say that a present idea is really similar to an idea really experienced in the past means that a being sufficiently informed would know that the effect of the later idea would be a revivification of the effect of the earlier idea, in respect to its quality of feeling. Yet it must be remarked that the only effect of a quality of feeling is to produce a memory, itself a quality of feeling; and that to say that two of those are similar is, after all only to say that the feeling which is the symbol of similarity will attach to them. Thus the feeling of recognition of a present idea as having been experienced has for its signification the applicability of a part of itself. The general occurrences of the feeling of similarity are recognized as themselves similar, by the application to them of the same symbol of similarity. It is Kant's "I think," which he considers to be an act of thought, that is, to be of the nature of a symbol. But his introduction of the ego into it was due to his confusion of this with another element.
This feeling is not peculiar among feelings in signifying itself alone; for the same is true of the feeling "blue" or any other. It must not be forgotten that a feeling is not a psychosis, or state of mind, but is merely a quality of a psychosis, with which is associated a degree of vividness, or relative disturbance, or prominence, of the quality in the psychosis, as measured chiefly by after-effects. These psychoses are icons; and it is in being a symbol that the feeling of similarity is distinguished from other feelings. But the significance of the psychosis as a sign is that the percept to which it ultimately refers has the same quality, as determined by the symbol-feeling of similarity.
My principal object in drawing attention to this symbol of similarity is to show that the significations of symbols have various grades of directness up to the limit of being themselves their own significations. An icon is significant with absolute directness of a character which it embodies; and every symbol refers more or less indirectly to an icon.
An index is directly denotative of a real object with which it is in reaction. Every symbol refers more or less indirectly to a real object through an index. One goes into a shop and asks for a yard of silk. He wants a piece of silk, which has been placed (either by measurement or estimation depending on habits involving indices) into reactive comparison with a yardstick, which itself by successive reactions has been put into reaction with a certain real bar in Westminster. The word "yard" is an example of a symbol which is denotative in a high degree of directness. When we consider the successive comparisons more scientifically, we have to admit that each is subject to a probably error. The more pains we take to make the reaction significant, the more we are forced to recognize that each single act of comparison gives its own result; and general inference which we make from a large number of these represents what each one would be if it could be performed more accurately. It is by the intervention of the inferential symbol that we virtually obtain a more intimate reaction. When a biologist labels a specimen, he has performed a /321/ comparison, as truly reactive in its nature as that of two standards of length, with an original "type-specimen," as he calls the prototype. His label thus involves something of the nature of an index, although less prominently than does the word "yard." It would be more scientific if in place of a single prototype, comparisons were made with twenty-five different bars of different material and kept under different conditions, and that were called a "yard" which agreed with the mean of them all; and so in biology, there ought to be twenty-five type-specimens, exhibiting the allowable range of variation as well as the normal mean character. This more scientific proceeding is that of common sense in regard to ordinary names, except that instead of twenty-five instances, there are many more. I go into a furniture shop and say I want a "table." I rely upon my presumption that the shopkeeper and I have undergone reactional experiences which though different have been so connected by reactional experiences as to make them virtually the same, in consequence of which "table" suggests to him, as it does to me, a movable piece of furniture with a flat top of about such a height that one might conveniently sit down to work at it. This convenient height, although not measured, is of the same nature as the yard of silk already considered. It means convenient for men of ordinary stature; and his reactive experience presumably agrees with mine as to what the ordinary stature is. I go into a shop and ask for butter. I am shown something, and I ask, "Is this butter, or is it oleomargarine of something of the sort?" "Oh, I assure you that is is, in chemical strictness, butter." "In chemical strictness, eh? Well, you know what the breed of neat cattle is, as well as I do. It is an individual object, of which we have both seen parts. Now I want to know whether this substance has been churned from milk drawn from the breed of neat cattle." That breed is known to us only by real reactional experience. What is gold? It is an elementary substance having an atomic weight of about 197¼. In saying that it is elementary, we mean undecomposable in the present state of chemistry, which can only be recognized by real reactional experience. In saying that its atomic weight is 197¼, we mean that it is so compared with hydrogen. What, then, is hydrogen? It is an elementary gas 14¼ times as light as air. And what is air? Why, it is this with which we have reactional experience about us. The reader may try instances of his own until no doubt remains in regard to symbols of things experienced, that they are always denotative through indices; such proof will be far surer than any apodictic demonstration. As to symbols of things not experienced it is clear that these must describe their objects by means of their differences from things experienced. It is plain that in the directness of their denotation, symbols vary through all degrees. It is, of course, quite possible for a symbol to represent itself, at least in the only sense in which a thing that has no real being but only being represented, and which exists in replica, can be said to be identical with a real and therefore individual object. A map may be a map of itself; that is to say one replica of it may be the object mapped. But this does not make the denotation extraordinarily direct. As an example of a symbol of /322/ that character, we may rather take the symbol which is expressed in words as "the Truth," or "Universe of Being." Every symbol whatever must denote what this symbol denotes; s that any symbol considered as denoting the Truth necessarily denotes that which it denotes; and in denoting it, it is that very thing, or a fragment of it taken for the whole. It is the whole taken so far as it need be taken for the purpose of denotation; for denotation essentially takes a part for its whole.
But the most characteristic aspect of a symbol is its aspect as related to its interpretant; because a symbol is distinguished as a sign which becomes such by virtue of determining its interpretant. An interpretant of a symbol is an outgrowth of the symbol. We have used the phrase, a symbol determines its interpretant. Determination implies a determinandum, a subject to be determined. What is that? We must suppose that there is something like a sheet of paper, blank or with a blank space upon it upon which an interpretant sign may be written. What is the nature of this blank? In affording room for the writing of a symbol, it is ipso facto itself a symbol, although a wholly vague one. In affording room for an interpretant of that particular symbol, it is already an interpretant of that symbol, although only a partial one. An entire interpretant should involve a replica of the original symbol. In fact, the interpretant symbol, so far as it is no more than an interpretant is the original symbol, although perhaps in a more developed state. But the interpretant symbol may be at the same time an interpretant of an independent symbol. A symbol is something which has the power of reproducing itself, and that essentially, since it is constituted a symbol only by the interpretation. This interpretation involves a power of the symbol to cause a real fact; and although I desire to avoid metaphysics, yet when a false metaphysics invades the province of logic, I am forced to say that nothing can be more futile than to attempt to form a conception of the universe which shall overlook the power of representations to cause real facts. What is the purpose of trying to form a conception of the universe if it is not to render things intelligible? But if this is to be done, we necessarily defeat ourselves if we insist upon reducing everything to a norm which renders everything that happens, essentially and ipso facto unintelligible. That, however, is what we do, if we do not admit the power of representations to cause real facts. If we are to explain the universe, we must assume that there was in the beginning a state of things in which there was nothing, no reaction and no quality, no matter, no consciousness, no space and no time, but just nothing at all. Not determinately nothing. For that which is determinately not A supposes the being of A in some mode. Utter indetermination. But a symbol alone is indeterminate. Therefore, Nothing, the indeterminate of the absolute beginning, is a symbol. That is the way in which the beginning of things can alone be understood. What logically follows? We are not to content ourselves with our instinctive sense of logicality. That is logical which comes from the essential nature of a symbol. Now it is of the essential nature of a symbol that it determines an interpretant, which is itself /323/ a symbol. A symbol, therefore, produces an endless series of interpretants. Does anybody suspect all this of being sheer nonsense? Distinguo. There can, it is true, be no positive information about what antedated the entire Universe of being; because, to begin with, there was nothing to have information about. But the universe is intelligible; and therefore it is possible to give a general account of it and its origin. This general account is a symbol; and from the nature of a symbol, it must begin with the formal assertion that there was an indeterminate nothing of the nature of a symbol. This would be false if it conveyed any information. But it is the correct and logical manner of beginning an account of the universe. As a symbol it produced its infinite series of interpretants, which in the beginning were absolutely vague like itself. But the direct interpretant of any symbol must in the first stage of it be merely the tabula rasa for an interpretant. Hence the immediate interpretant of this vague Nothing was not even determinately vague, but only vaguely hovering between determinacy and vagueness; and its immediate interpretant was vaguely hovering between vaguely hovering between vagueness and determinacy and determinate vagueness or determinacy, and so on, ad infinitum. But every endless series must logically have a limit.
Leaving that line of thought unfinished for the present owing to the feeling of insecurity it provokes, let us note, first, that it is of the nature of a symbol to create a tabula rasa and therefore an endless series of tabulae rasae, since such creation is merely representation, the tabulae rasae being entirely indeterminate except to be representative. Herein is a real effect; but a symbol could not be without that power of producing a real effect. The symbol represents itself to be represented; and that representedness is real owing to its utter vagueness. For all that is represented must be thoroughly borne out.
For reality is compulsive. But the compulsiveness is absolutely hic et nunc. It is for an instant and it is gone. Let it be no more and it is absolutely nothing. The reality only exists as an element of the regularity. And the regularity is the symbol. Reality, therefore, can only be regarded as the limit of the endless series of symbols.
A symbol is essentially a purpose, that is to say, is a representation that seeks to make itself definite, or seeks to produce an interpretant more definite than itself. For its whole signification consists in its determining an interpretant; so that it is from its interpretant that it derives the actuality of its signification.
A tabula rasa having been determined as representative of the symbol that determines it, that tabula rasa tends to become determinate. The vague always tends to become determinate, simply because its vagueness does not determine it to be vague (as the limit of an endless series). In so far as the interpretant is the symbol, as it is in some measure, the determination agrees with that of the symbol. But in so far as it fails to be its better self, it is liable to depart from the meaning of the symbol. Its purpose, however, is to represent the symbol in its representation of its object; and therefore, the determination is /324/ followed by a further development, in which it becomes corrected. It is of the nature of a sign to be an individual replica and to be in that replica a living general. By virtue of this, the interpretant is animated by the original replica, or by the sign it contains, with the power of representing the true character of the object. That the object has at all a character can only consist in a representation that it has so,—a representation having power to live down all opposition. In these two steps, of determination and of correction, the interpretant aims at the object more than at the original replica and may be truer and fuller than the latter. The very entelechy of being lies in being representable. A sign cannot even be false without being a sign and so far as it is a sign it must be true. A symbol is an embryonic reality endowed with power of growth into the very truth, the very entelechy of reality. This appears mystical and mysterious simply because we insist on remaining blind to what is plain, that there can be no reality which has not the life of a symbol.
How could such an idea as that of red arise? It can only have been by gradual determination from pure indeterminacy. A vagueness not determined to be vague, by its nature begins at once to determine itself. Apparently we can come no nearer than that to understanding the universe.
That is not necessarily logical which strikes me today as logical; still less, as mathematics amply exemplifies, is nothing logical except what appears to me so. That is logical which it is necessary to admit in order to render the universe intelligible. And the first of all logical principles is that the indeterminate should determine itself as best it may.
A chaos of reactions utterly without any approach to law is absolutely nothing; and therefore pure nothing was such a chaos. Then pure indeterminacy having developed determinate possibilities, creation consisted in mediating between the lawless reactions and the general possibilities by the influx of a symbol. This symbol was the purpose of creation. Its object was the entelechy of being which is the ultimate representation.
We can now see what judgment and assertion are. The man is a symbol. Different men, so far as they can have any ideas in common, are the same symbol. Judgment is the determination of the man-symbol to have whatever interpretant the judged proposition has. Assertion is the determination of the man-symbol to determining the interpreter, so far as he is interpreter, in the same way.
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