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The Bed-Rock Beneath Pragmaticism

MS (R) 300, pages numbered by Peirce 1-65, dated “about March of 1908” by Roberts (1973, 99); wrongly dated in CP and the Robin catalogue. Some partial (rejected?) drafts are included below. Peirce's page numbers are given in {brackets}.

The present paper is the fourth and antepenultimate of a series. Those that forewent it appeared in Vols. XV and XVI of the Monist. I shall very often have occasion herein to cite one or another passage already so in print.

[long digression here mostly about citation format; most of the citations in what follows are crossed out in the MS and omitted in this transcript.]

I originally announced that this series would consist of three articles,— {5} and I owe you, Reader, an explanation of the change of plan;—an explanation that had better be costumed in an abridged summary of the reflexions that led to the writing of this series of articles in the first place, and afterward to successive modifications of the plan of the series.

It is now, You see, Reader, time, times, and half a time,—which is the inspired way, according to the postils, of describing 3½ years,—since I took pen in hand to write these papers. I was moved thereto by a brace of circumstances; the one exterior, the other interior to my own intellectual experiences. The former was the seeming unconcern of the preachers of Pragmatism, strenuous enough though they surely were, at their dogma’s so long floating in the air as though it were a cloud, unsupported by any solid {6} argument calculated to force conviction upon such impartial truth-seekers as might be plagued with doubts as to whether it be true or false. It almost seemed as if the Pragmatists deemed it sufficient for them to postulate, with Protagoras, that the “truth” is whatsoever seems “satisfactory” to this or that thinker in one or another stage of his reflexions. The adjective “satisfactory” seems to be satisfactory to Schiller, and I am told to James, too. Were the theory not so ancient nor made so familiar by James Mill’s remarkable school, it might have been suggested by a passage where (p. 6-13) I said, “That the settlement of opinion is the sole end of inquiry is a very important proposition,” etc. But the further exposition of my view shows that it {7} was already sharply opposed to Protagoreanism, being that the Truth is the opinion which sufficient inquiry would establish and fix forever. This I later modified by adding that there is no reason whatsoever entitling us to any confidence that the pendulum of opinion will not, in regard to any given question you please, continue to oscillate back and forth forever, in which case, in regard to such question, there is no real truth, at all. Today, however, introducing a distinction to be hereinafter explained, I say that in such a case (if such there be), the real truth would be of an indefinite nature, that is, would in some measure violate the principle of contradiction, being as much pro as con, somewhat the one and somewhat the other. This theory, which runs distinctly counter to the Protagorean notion, is, in advance of all proof, in a good deal better accord with the principles of the only precise logic than the {8} latter is; and I am in possession of a proof of it which I hope sometime to publish. Meantime, it alone accords with the principle of Pragmatism.

[Peirce's footnote on Protagoras omitted here.]

{9} It seems pertinent to mention here, though anachronistically, that I lately saw in a newspaper a brief notice,—to which I have no reason to attribute exceptional accuracy (and the marvel to me is how the great dailies, those, I mean, that are “great” in any sense implying a claim to respect, such as The New York Times, for example, manage to control, as well as they do, that awful bore of the daily reports, which far out-yang-tsuh-kiangs the Bay of Funday),—to the effect that Professor James has declared that the true “basis” of the principle of pragmatism lies in a “practical preference.” I cannot imagine what position Professor James could hold that could so be described with any precision; and though his dialect is peculiar to himself, I venture to surmise that he has been misunderstood by an inattentive or incompetent reporter, {10} or one whom James’s manner of expressing himself deceived. Four different doctrines, indeed, that would be so describable occur to me as possible to be broached by somebody. It will be more to the purpose to glance at each, than to inquire of Prof. James which one, if any, he embraces; and would be so, even if I knew his present address. In the first place, then, it is not incredible that somebody should opine that all human thoughts are so indefinite as to render it a pure question of convenience what we should regard as anybody’s meaning in any utterance; although to hold that vagueness of thought could, of itself, have such a result would involve a confusion of vagueness with generality in depth. (See above.) Yet certainly whosoever should himself have ideas so confused as to think all human ideas to be of that more than miraculous vagueness might well be expected to fall into the confusion in question. In the second place, it is conceivable that a person should hold “man to be the measure of things” in so extreme a sense as to make practical convenience his sole criterion of truth; and he {11} might then easily be led to consider this opinion to be what pragmatism consists in. In the third place, a person of positivistic opinions might admit (or even insist, as Comte himself did), that the meanings of some concepts involve elements other than conceivable practical consequences; while yet, in view of our total ignorance as to what subjects those further elements are true of, and what they are false of, he might think it expedient altogether to ignore those further elements, and might entitle this policy “pragmatism”; and being, like most positivists, a bad logician and a vague thinker, he might be under the delusion that, in doing so, he was basing pragmatism on a “practical preference.” In the fourth place, since Professor James’s definition of pragmatism in Baldwin’s {12} Dictionary of Psychology and Philosophy (Vol. II, p. 321 f.) says that it is “the doctrine that the whole ‘meaning’ of a conception expresses itself in practical consequences” [The Italics are mine. C.S.S.P.], upon which follows a division of the genus ‘practical consequences’ into two species, it is easily imaginable that he holds that some, if not all, concepts are capable of expressing themselves or getting expressed in several ways, thus giving room for a practical preference between these ways. If such be in fact his opinion, my half-century’s acquaintance with him leads me to believe, without doubt, that what he said or meant by whatever he may have said, was that, once a {13} man has come understandingly to assent to pragmatism, as defined by James, “a practical preference” will lead him to translate abstract expressions of concepts into expressions of his conceptions of their essential conditional “practical consequences.” The correctness of the opinion here hypothetically attributed to the famed psychologist will be considered in my next article. At present, it does not concern us. But of course there are such different forms of expression, whether practical convenience alone, or whether, along with convenience, precision [in the sense of definiteness], too recommends the pragmatist’s practice. But this practice,—Have patience, Reader, with a trite remark!—need not be due exclusively to a self-controlled preference: the experiences of his life, together with the native bent of his genius, {14} may tend to constrain the man to that practice, and so cooperate with his Reason. This, I think, is the case with William James. An intense dislike, if not also some inaptitude, for mathematically exact but abstract definitions, so reenforces his rational conviction of the truth of pragmatism that I cannot feel quite sure that he did not inadvertently use the expression that “pragmatism is based on a practical preference”; although that characteristic flair that usually carries him straight at the throat of a question of philosophy is so nearly unerring that I am as sure as, in the nature of things, one ever could be in such a case, that he never meant to say that the controversy which has been waging of late years as to the objective truth of “the doctrine that every conception fully expresses itself in practical consequences” {15} can be logically settled by adducing any subjective preferences, whether they be practical or of any other description. For to say that it could would be to beg the question.

The second circumstance that contributed to my deciding to write these articles, the inner experience that prompted me to it, was that I had myself passed through a doubt of pragmaticism lasting very nearly twenty years.

Do you, Reader, happen to know what “doubt” means? Do you know it, I mean, in that sense of “knowing” in which we say that few boys know what money means? If you do you have attained man’s estate in philosophy, and won’t grow much more, though you will gain strength and maybe, art. Before I made its actual acquaintance, as applied to pragmaticism, I had printed over a score of {16} logical and epistemological papers containing the majority of the original thoughts that I have as yet set forth, and had preached pragmaticism for over ten years. Very ignorant persons confound doubt with disbelief. Many others think simple unbelief constitutes doubt. What “doubt” really denotes is to be insupportably discontent to dispose, for oneself, of the proposition that is said to be “doubted,” in any suggestible way whatever, whether it be to affirm it to oneself, or to deny it, or yet to leave the question of its truth unsettled. The abstract definition is easily apprehended; but an intimate, pillow-sharing acquaintance with the thing must come when living experience brings it.

The result of this protracted self-debate of mine was the same that the sciences that involve exact reasoning show to have been {17} invariable in such cases; namely, that the demonstration I at length discovered was, in its first form, very intricate. It was, indeed, so intricate that it seemed hopeless to expect that philosophical students generally would be able to follow it. Nor was its intricacy the most forbidding feature of my proof. For it was entangled with a multitude of novel distinctions, as well as with some very broad ideas that, at least, presented the appearance of novelty; and some of these distinctions and conceptions were not as sharply cut in my own mind as was desirable, although they were sufficiently so for the purposes they were put to in the proof. It is true that in former times, when a different style of thinking prevailed among students of philosophy, whole dictionaries full of {18} new divisions and definitions have been accepted with gratitude. But in those cases, the words and phrases to which the definitions were attached were themselves familiar, while either none but sensuous schemata of meaning had ever before been attached to them, even much less distinct than the newly framed definitions, or else the old significations had become hazy from disuse. No conscientious thinker of our day, however, could so utterly violate the ethics of terminology. He would be obliged to invent new words to express his new ideas; and that students of philosophy have not yet become sufficiently imbued with the spirit of science to receive such tasteless unancestred new-comer words into their intimacy, has been shown in several instances; as in the case of Avenarius. {19}

The half-dozen years and longer that have elapsed since the proof first stood out clearly in my mind have been employed in patching up its holes and in mitigating its inurbanities and difficulties, with all the ardor of my heart, and with all the diligence that circumstances permitted. But this, like all work of well-studied reform, is a mighty slow process; and this is the sole cause of the two changes that have been made in the plan of the series (first, the introduction of Existential Graphs; and secondly, the addition of an article confirmatory of the main argument); as well as being one of the causes of the delay in the successive publications of the different articles.

Experience shows, as I just now hinted, that the first working embodiments of original inventions, and especially of demonstrations of difficult theorems, are pretty uniformly encumbered with needless complications; and I intend to point out hereinafter, in the convenient {20} context, the logical reason why this almost inevitably follows from the circumstance that the physiological incarnation of the mind absolutely compels it to work logically; so that such fallacies as actually occur, and are not mere figments, are (as various logicians have more or less distinctly perceived, although it more properly belonged to the psychologists to declare it so,) not downright violations of logic, but are only miserably weak, yet nevertheless admissible, inferences, whose strength is exaggerated for a similarly logical, though insufficient, reason. Experience seems further to show, what is credible enough a priori, the needful improvements are more likely to be discerned by some second person, whose mind is free from any deep ruts that may probably have been formed in that of the original inventor, by the particular way in which he has happened long to ponder the problem while it was not yet solved, passing over and over again one roadway of thought; {21} not to speak of the effect of those same ruts in causing the original inventor to regard his way of solving the problem as simpler, that is to say, more facile and natural, than others that, in the absence of such ruts, are far superior to it in that respect. This consideration has encouraged me to publish my proof in the simplest form in which I can present it in a limited time; since I am led to believe that it is not my part, but that of some other thinker, to metamorphose my proof into a new form, in which it shall be vastly superior to its first incorporation, alike in point of evidence and in that of simplicity.

The programme of the remainder of the series stands, at present, thus. In this present paper, you and I, Reader, will examine the general nature of the System of Existential Graphs, and will employ it to develope certain fundamental conceptions which will afterward serve as the building-materials of my argument. In the next paper, the argument itself shall be {22} erected with those materials. Finally, seeing that a certain vague hesitation will inevitably exist, and ought to exist, too, to accepting so complex an argument involving such novel ideas, and the more hesitancy the more ambitious its claim to probative force may be, in the final article the validity of this argument shall be treated as a hypothesis to be experimentally tested by developing the most striking of the consequences which this hypothesis necessitates, and then, on comparing them with the facts, by finding an inductive confirmation of its validity.

The System of Existential Graphs may be characterized with great truth as presenting before our eyes a moving picture of thought. Provided this characterization be taken not as a flatly literal statement, but as a simile, it will, I venture to predict, surprise you to find what a strain of detailed comparison it will bear without snapping. A picture is visual representation of the {23} relations between the parts of its object; a vivid and highly informative representation, rewarding somewhat close examination. Yet from the nature of things it must fall short of perfection, just as a representation of any other kind must. It cannot directly exhibit all the dimensions of its object, be this physical or psychic. It shows this object only under a certain light, and from a single point of view. There is a hind side, too, that it does not show at all. Moreover, it will not bear examination under a high-power microscope; since, so compared with its object, it will be found to represent parts as simple and homogeneous that really are highly complex. It is curious how entirely true each of these statements is of the representations of thought in graphs. On the other hand, whatsoever object is shown in a correct picture as composite, really is composite, and {24} is really composed of such parts as the picture shows; only these are, in reality, only proximate, and not ultimate, parts.

But is this really true of all representations in Existential Graphs? Here is an important question, and a nice one; and to guard ourselves against mistake about it, our closest scrutiny is called for. In order to prepare the ground for this operation, let us begin, as almost every serious inquiry should be begun, by seeking a suitable standpoint for a comprehensive view of the question. Now a “point of view” (not to go into piddling distinctions,) is mostly a comparison; and a “comprehensive” view is one which will {25} be almost certain to suggest to the mind each serious consideration, just when it becomes immediately pressing or pertinent. Well, nobody who sufficiently examines the matter will question that the most interesting term of comparison for Existential Graphs is the system of “rational formulae,” or graphs, that are used in Organic Chemistry.

They are, of course, more or less used in Inorganic Chemistry, where, however, instead of everywhere carrying those potent germs of pragmaticistic truth that they bear in the Organic branch,— veritable spermatozoons of science,—with which they have fertilized their proper field, they have hitherto begotten nothing unless here and there a poor mule. At any rate, in our present business, let us not get entrapped in that dark Mamoth Cave, Inorganic Chemistry.
{26} The general form of structure of the Existential Graph differs from that of the Chemical Graph in but the single particular that its different partial graphs may lie on Areas of different Tinctures. (Convention the Second.) For the roughness of the verso side of the phemic Leaf (Ibid. Convention the Third) is of the same nature as a mode of tincture. But let us make sure of this, before going further.

To this end we must make analytic examinations of four kinds of graph-forms; namely, those shown respectively in Figure 12 of my Prolegomena (Ibid. 539), in Figure 13 (Ib. 530), in Figures 9 and 14 (Ib. 538, 540) and in Figure 5 (Ib. 534), which would have been designed somewhat differently, however, had I bethought me of the present employment {27} of them, for which they can, however, be made to serve. In approaching the subject of Existential Graphs (XVI.iv. 492 et seq.), you and I, Reader, found ourselves, almost at the start, with a bad ford before us, as unavoidable as it was unpleasant. There was absolutely nothing for it but to wade for thirty-two disagreeable pages through deep water, you all the while thinking that I might have led you over an easier course, though I knew better. But when we had once passed over the ford, and stood in an open country, I was so intensely desirous of rendering the interpretation of the Graphs as easy to you and as little tiresome as I possibly could, that I defined the meaning of a ligature in terms of selectives, though I characterized these as a “superfluous device.” But this was dealing with them too leniently. The plain {28} truth is that this device (except for one momentous fault that is its very own), is to be classed along with those departures from logical truth which even mathematicians pardon on account of their convenience and the ease sterilizing their falsity.

[A mathematical footnote omitted here runs from p. 28-30 (Peirce's numbering).]

part of the following published as CP 4.561n and described as “From ‘The Bedrock beneath Pragmaticism’ (2), 1906.”

{31} The essential error, τὸ πρῶτον ψεῦδος, of the Selectives, and their inevitable error, τὸ πρωτὸν ψεῦδος, lies in their putting forth, in a system which aims at giving, in its visible forms, a diagram of the logical structure of assertions, as a representation, for example, of the assertion that Tully and Cicero are the same man, a type of image which does not differ in form from the assertion that Julius Caesar and Louis Seize were both men:

If you attempt any reply to this argument, you can only say, “But a selective is understood to be the proper name of a single {32} object.” Whereupon I inquire, “What kind of a proper name?” You can only answer, “A Selective is a proper noun which is understood to be used, for the first and for the last time, in the Entire Graph in which it occurs.” “Very well,” I reply, “when a proper name occurs without any previous hint as to its denotation, it cannot, to the interpreter’s mind,—and it is the interpreter’s mind and its possible dispositions that can alone give meanings to words,—differ from a common noun. For let a French peasant, for instance, first hear of Tasso in the phrase, ‘la Sophronie du Tasse’, and if it be uttered with an English accent, he will surely think there has been a blunder about the gender. In any case, it cannot suggest any definite individual.” To this your rejoinder will be, “True enough; every language ought to have a distinctive form for proper names. In Existential Graphs, however, there is an express convention as to the distinctive form of Selectives.” Thereupon comes my surrejoinder; that just there Selectives violate the essential idea, or {33} purpose of The System, Existential Graphs, as it is stated in the Prolegomena; namely, to afford a method (1) as simple as possible (that is to say, with as small a number of arbitrary conventions as possible), for representing propositions (2) as iconically, or diagrammatically and (3) as analytically as possible. (The reason for embracing this purpose was developed through the first dozen pages of this paper.) These three essential aims of the system are, every one of them, missed by Selectives. The first, that of the utmost attainable simplicity is so, since a selective cannot be used without being attached to a Ligature, and Ligatures without Selectives will express all that Selectives with Ligatures express. The second aim, to make the representations as iconical as possible, {34} is likewise missed; since Ligatures are far more iconic than Selectives. For the comparison of the above figures shows that a Selective can only serve its purpose through a special habit of interpretation that is otherwise needless in the system, and that makes the Selective a Symbol and not an Icon; while a Ligature expresses the same thing as a necessary consequence regarding each sizeable dot as an Icon of what we call an “individual object”; and it must be such an Icon if we are to regard an invisible mathematical point as an Icon of the strict individual, absolutely determinate in all respects, which imagination cannot realize. Meantime, the fact that a special convention (a clause of the Fourth) is required to distinguish a Selective from an ordinary univalent Spot {35} constitutes a second infraction of the purpose of simplicity. The third item of the idea of the System, that of being as analytical as possible, is infringed by Selectives in no less than three ways. This, at least, is the case if it be true, as I shall endeavour further on to convince the reader that it is, that Concepts are capable of being compounded only in a way differing but in one doubtful particular from that in which the so-called “substances,”— i.e. species,— of Organic Chemistry are compounded, according to the established theory of that science.
{37} That respect is that the different bonds and pegs of the Spots of Graphs are different, while those of chemical atoms are believed to be all alike. But on the one hand, it may possibly be that a more nearly ultimate analysis of Concepts would show, as Kempe's “A Memoir on the theory of Mathematical Form” [Philosophical Transactions of the Royal Society, v. 177, pp. 1-70, 1886] seems to think, that the pegs of simple concepts are all alike. On the other hand, the carbon-atom seems to be the only one for the entire similarity of whose bonds there is much positive evidence. In the case of nitrogen, for example, two of the five valencies seem to be of such different quality from the others as to suggest that the individual bonds may likewise be different; and if there were such difference between different bonds of atoms generally, obvious probable causes would prevent our discovering [them] in the present state of chemistry. Looking at the question from the point of view of Thomson's corpuscles, it seems very unlikely that the looser electrons all fulfill precisely the same function in all cases.
For if this be true, the fact that two or more given concepts can be put together to produce one concept, without either of those that are so put together being separated into parts, is conclusive proof that the concept so produced is a compound of those that were put together. The principle, {36} no doubt, requires to be proved. For it might easily be thought that the concept of a scalar as well as that of a vector (in quaternions) can equally result from putting together the concepts of a tensor and a versor in different ways, while at the same time the concept of a tensor and that of a versor can, in their turn, result from putting together those of a scalar and of a vector in different ways; so that no one of the four concepts is more or less composite than any of the others.

Were such a view borne out by exact analysis, as it certainly is not, a radical disparateness between the composition of concepts and that of chemical species would be revealed. But this could scarcely fail to entail such a serious revolution in accepted doctrines of logic as it would be unwarrantable gratuitously to suppose that further investigation will bring about. It will be found that the available evidence {38} is decidedly that Concepts can only be combined through definite “pegs.” The first respect in which Selectives are not as analytical as they might be, and therefore ought to be, is in representing identity. The identity of the two S's above is only symbolically expressed. Iconically, they appear to be merely coexistent; but by the special convention they are interpreted as identical, though identity is not a matter of interpretation,— that is of logical depth,— but is an assertion of unity of Object, that is, is an assertion regarding logical breadth. The two S's are instances of one symbol, and that of so peculiar a kind that they are interpreted as signifying, and not merely denoting, one individual. There is here no analysis of identity. The suggestion, at least, is, quite decidedly, that identity is a simple relation. The line of identity which may be substituted for the selectives very explicitly {39} represents Identity to belong to the genus Continuity and to the species Linear Continuity. But of what variety of Linear Continuity is the heavy line more especially the Icon in the System of Existential Graphs? In order to ascertain this, let us contrast the Iconicity of the line with that of the surface of the Phemic Sheet. The continuity of this surface being two-dimensional, and so polyadic should represent an external continuity, and especially, a continuity of experiential appearance. Moreover, the Phemic Sheet iconizes the Universe {39½} of Discourse since it more immediately represents a field of Thought, or Mental Experience, which is itself directed to the Universe of Discourse, and considered as a sign, denotes that Universe. Moreover, it [is because it must be understood] as being directed to that Universe, that it is iconized by the Phemic Sheet. So, on the principle that logicians call “the Nota notae” that the sign of a sign of anything, X, is itself a sign of the very same X, the Phemic Sheet, in representing the field of attention, represents the general object of that attention, the Universe of Discourse. This being the case, the continuity of the Phemic Sheet in those places where, nothing being scribed, no particular attention is paid, is the most appropriate Icon possible of the continuity of the Universe of Discourse, where it only receives general attention as that Universe, {39} that is to say, of the continuity in experiential appearance of the Universe, relatively to any objects represented as belonging to it. Therefore since the {40} essence of a graph is to be found in greatest purity in the ideally perfect interpretation of it on a metallic tincture, the continuity in question must of necessity represent the co-being in one universe, which is the continuity of environment. Now for the continuity of the line of identity. This being one-dimensional, or dyadic (i.e. running two ways only,) should represent an internal, or mental continuity; and being definitely marked, should iconize a continuity of attention. But the heavy line is generated by the continuity of the different places of a heavy dot, which is the appropriate icon of an individual object in a Universe of continuous co-being; and therefore the continuity of the line is, best, the Icon of the continuity in attentive observation of an individual object. Now let us turn our back, for a moment, {41} upon the System of Graphs, in order, without being biassed by it, to investigate the true meaning of the term “Identity” in its proper sense of “numerical” (ἀριθμῷ) identity. Our method shall consist in noting exactly what would be proved by an ideally perfect proof of the identity of an object in two unlike phases; and that our conception may be free from our intuitive, unintellectual idea of Space, let us imagine, as an illustrative example, that the psychical researchers were to gain an entirely conclusive proof that the utterer of some of the talk of a woman in a trance is identical with some person who formerly lived physiologically. Since personal identity is only known to us as the continuity of memory with present perception, which involves the continuity {42} of Time, we will suppose that, after some woman in a trance had narrated various insignificant occurrences in the bed-chamber of a defunct person which occurrences had actually happened during his final illness and after everybody had left the chamber except one of the researchers, who had set the little events down, sealing up each record separately, the researchers were still not quite satisfied, because the witness might have disclosed in his sleep what had happened. How could they be imagined to dispose of this doubt? Suppose that they endeavoured to do so by establishing a continuous memory of an utterer of trance discourse back to the death-chamber hours, and (to exclude continuity of time) let us suppose that they found themselves thwarted in this by a blank gap in the apparent utterer’s memory and apparently in his consciousness. {43}

The researchers might still not be at the end of their tether. They might, for example, by the study of the private papers of the deceased, discover that he had a certain combination of small mental characteristics such as, in all probability, no other mortal ever precisely possessed in their entire number, and they might detect these in the trance utterances to be slightly more pronounced; or, if the deceased had been an archeological explorer who had, in Central Africa, in the interior of Borneo, in Thibet, and in the red desert of Arabia, visited spots to no one of which any observer had ever penetrated, nor any barbarian to all of them, then the trance utterances might give minute descriptions, mentioning such things as his having enclosed a gold coin, which he had marked, within the wood of a certain tropical tree, where it would shortly be covered with new growth; and expeditions of the researchers to all the places {44} might verify all these statements, together with supplements in the trance utterances, of facts that the deceased could not have himself known in his life-time, such, e.g., as that on removing from some ancient monument a certain stone too huge for him to have had the means of removing, there would be found, in the upper surface of the subjacent stone, a hollow, containing a diamond worth a fortune, in which diamond a high-power microscope would show a peculiar flaw, minutely described by the lips of the woman in the trance. Even if all such-like disclosures that we could imagine in an hour or a week were actually verified, the verification could not amount to absolute demonstration of the identity of the utterer with the deceased. Still, the example, being well considered, should suffice, I think, to show that “Identity” means a continuity, not necessarily in Place, nor {45} in Date, but in what I may call aspect, i.e. a variety of presentation or representation. In saying that the example “should” suffice for this, I only mean that a thorough consideration of its suggestions will lead to the acceptance of the conclusion just stated. I do not mean that the example of itself renders the conclusion so evident as to justify the slightest reflexion upon the intelligence of the mind that continues to doubt that conclusion. For, on the contrary, I fully admit that an objection suggests itself to this conclusion, and that from the point of view of the ordinary theory of concepts, that the “tree of Porphyry,” as traditionally interpreted, correctly represents the mode of composition, the objection appears conclusive. But I hold and defend below a different theory; namely that in the first place every composition {46} of concepts is built up by the application of relatives, as, for example, that since one may analyze the assertion, ‘M is a cousin of N’, into ‘M has a grandfather that is a grandfather of N’, it follows that the conception of cousinship is a composite of the relation of grandparentage with its converse, and in the second place that this composition takes place, as in chemistry, by units of valency, so that each correlate of a relative term is a single individual, and for example, the relation of being a loving servant is not correctly a mere compound of the relations of loving and serving, but involves besides two relations of co-identity, or, as I usually call it, teridentity; the assertion that M is the loving servant of N being represented by the graph of Figure 25. This theory of the composition of concepts shall be argued below, together {47} with the consideration of the above objection to my analysis of Identity.

The second failure of Selectives to be as analytical as possible lies in their encouraging the idea that negation, or denial, is a relatively simple concept, and that the concept of Consequence, is a special composite of two negations, so that to say, “If in the actual state of things A is true, then B is true,” is correctly analyzed as the assertion, “It is false to say that A is true while B is false.” I fully acknowledge that, for most purposes and in a preliminary explanation, the error of this analysis is altogether insignificant. But when we come to the first analysis the inaccuracy must not be passed over. All my own writings upon formal logic have been based on the belief {48} that the concept of Sequence, alike in reasonings and in judgments, whether the latter be conditional or categorical, could in no wise be replaced by any composition of ideas. For in reasoning, at least, when we first affirm, or affirmatively judge, the conjugate of premisses, the judgment of the conclusion has not yet been performed. There then follows a real movement of thought in the mind, in which that judgment of the conclusion comes to pass. Now surely, speaking of the same A and B as above, it were absurd to say that a real change of A into a sequent B consists in a state of things that should consist in there not being an A without a B. For in such a state of things there would be no change at all. {49} This judgment is, at first, no more than a copy, or “generalized” icon (with a symbolic “legend,” or label, indexically attached to it), of that experience of having been constrained by the supposition of A to join to that the acknowledgment of the truth (in the case supposed), of B. Consequently, since the real sequence, as we have seen, cannot be adequately represented as merely a composite of two negations, no more can the copy [of] it, which is substantially the concept of the sequence. It is but very rarely that a proof as satisfactory as this, that a given concept is not composite, can be obtained. For it is in logical, as in chemical, analysis, tolerably easy to demonstrate compoundness, but next to impossible to make sure of elementality, or elementarity. (I give the Reader his choice between {50} these two words, both obsolete, but both less unpleasant than the modern ‘elementariness.’ And by the way, in speaking above of a “generalized” icon, I used the qualification in a sense of “generalize” common among designers, especially among cartographers, as well as in vernacular talk, though it is not the proper logical sense of the verb, since it does not signify the removal of any constituent of logical depth from a condition, nor confers any liberty on the interpreter, but implies some almost microscopic items that are really falsifications committed in the interest of simplification. Thus, a map “generalizes” its image of a river in representing the latter as not making sundry small windings that it really does make. So recollection may be said to “generalize” the remembered perception in representing this to be without many insignificant details {51} that really did belong to it; and although an icon is not, properly speaking, general, so far as it is a pure icon, yet every icon must “generalize,” more or less, in this peculiar sense. Even a photograph does so.) Indeed, so far is the concept of Sequence from being a composite of two Negations, that, on the contrary, the concept of the Negation of any state of things, X, is, precisely, a composite of which one element is the concept of Sequence. Namely, it is the concept of a sequence from X of the essence of falsity. In order clearly to perceive that this is true, a more minute scrutiny of [the] idea of Sequence is needful. But, before entering upon that, the following remarks will be useful. This concept, without prejudice to its elementality, is nevertheless Janual; that is to say, it cannot be thought without thinking, though as vaguely as you will, that there is something antecedent upon which something {52} is Sequent. The question will here pop up, Why does not this show that the concept of Sequence is a composite of three concepts: that of some antecedent state, that of some consequent state, and between them, that of a state of Heraclitan Flux? It will suggest itself that if a state of motion is Sequent upon a state of rest, then before the instant of starting, there is a state of rest; after that instant, there is a state of motion; but at the very start, there is neither rest without motion, nor motion without rest, but equally or indifferently neither rest nor motion, or else, and likewise, both rest and motion. Your question answers itself, since it proposes an analysis that cannot be stated nor distinctly thought, without absurdity. For, to pass over as unspeakable your “or else, and also,” your supposition assumes that there is what we conceive of as Time; and it would certainly be quite impossible to prove that that conception is not realized; {53} nor, I believe, has anybody yet succeeded in getting a distinct notion of what it would signify to say that that idea is not exactly realized. It is highly improbable that anybody ever will so succeed, since the idea is involved in our very idea of thinking. For we never think at all without reasoning; and if we try to do so, the attempt merely results in our reasoning about reasoning. Now reasoning takes place in Time; and so far as we can understand it, in a Time that embodies our common-sense notion of Time. But this common-sense notion of time implies that every state of things that does not endure through a lapse of time is absolutely definite, that is, that two states, one the negation of the other, cannot exist at the same instant; which, by the way, necessarily follows, if negation be but a particular sort of sequence; though it would be to no purpose to stop to prove this here. Accepting the common-sense {54} notion, then, I say that it conflicts with that to suppose that there is ever any discontinuity in change. That is to say, between any two instantaneous states there must be a lapse of time during which the change is continuous, not merely in that false continuity which the calculus recognizes but in a much stricter sense. Not only must any given instantaneous value, s, implied in the change be itself either absolutely unchanging or else always changing continuously, but also, denoting an instant of time by t, so likewise must, in the language of the calculus, ds/dt, d2s/dt2, d3s/dt3, and so on endlessly, be, each and all of them, either absolutely unchanging or always changing continuously. I know very well that this is contrary to various accepted results of the most scrupulous mathematicians. But having examined such of them as I could come across, I find subtle fallacies in all. The truth is that in the vacuous {55} space of the mathematics of higher transfinite quantity, it is next to impossible always to elude fallacy. Meantime, there are other mathematical theorems, as well established in the minds of mathematicians as those to which I have referred (though some inconsistency must plainly lurk between the two classes), which quite plainly support my view.

[Peirce says to insert a footnote here but left its page number blank.]

I now invite you, Reader, according to promise, to accompany me in a scrutiny of the nature of Sequence. Let us consider the conditional proposition, ‘If A, then B’, where A and B stand for two less complex propositions, no matter what the substance of their meanings may be. But though their substance does not concern us, their modality does. The condition, A, of our conditional proposition is not asserted, even as possible. This {56} is shown in such conditionals as the following. “If the principle of contradiction could be violated, then reasoning would not be exact, and truth generally would be cloudlike and vague”; “If the principle of Excluded Third could be violated, absolutely determinate fact would not underlie all other modes of truth.” Think what you may of the truth of these conditional assertions, they clearly show that if A in our conditional were asserted to have any mode of Being, and thus its assertion were intended to lay distinct claim to Truthfulness, then that part of it could be separated from the rest as a distinct assertion; so that a proposition that is simply conditional and not decomposable in that way does not assert its antecedent in any sense, but only proposes it as an arbitrary hypothesis, or mere {57} idea, such as everybody is free to set up without any responsibility for its answering to any being in that Universe (whatever it may be), which the utterer and the right interpreter of the conditional proposition well understand each other to have in view. To illustrate this matter of the Universe of Discourse (though I hope the Reader is familiar with it, and there is much more he will need to know about it than I can point out just in the present paragraph), I will present the following conditional: “If Hamlet really was insane, then still he might very well also make a pretense of being insane.” This is not intended to make any assertion at all about real insane people, but merely as to the Universe of Shakespeare’s intentional creations in the “Tragedy of Hamlet.” But mark well, Reader, what I am going to say next; and do not skip {58} the remark on account of your being quite familiar already with the concept of the Universe of Discourse. For that may well be, while yet the point I am about to make,—an all-important point it is for our present purpose,—has, very likely, quite eluded your attention hitherto. What I want to say is that while the Universe of Discourse of the above conditional proposition as a whole, as well probably as that of any foregoing context of it that we can naturally imagine that there was, is evidently Shakespeare’s intention as carried out in the scenes of Hamlet, yet for that one sentence and the two clauses of it, two modifications of that Universe are effected by the [antecedent], the one to make the Universe of the substance matter of the antecedent itself,—meaning by its substance matter, Hamlet’s being insane,—the other for the matter of the consequent, this latter change being, loosely speaking, a return to the Universe of the imaginable context, {59} although it is, by no means, exactly that. What is the Universe of Discourse of the matter represented in the antecedent, the matter ‘Hamlet was insane’; in other words, what is well-understood between the utterer and the right interpreter to be the world in which this is one of the items? It is plainly in a loose sense, or, to speak, not in the language of logical terminology, but in the vernacular, it is, “in a general way,” the world of what anybody may suppose to constitute the pertinent truth in addition to what is asserted and admitted in the independent context to the conditional proposition. It is a very odd universe indeed: but I think that, after we shall have noted a single additional feature of it, it will be better to postpone the further consideration of it to that of the Universe of [the] consequent, both of this particular example and of the consequents of other conditional propositions {59[again]}

There appears to be something missing here in the break between two manuscript pages (both numbered 59 by CSP), as the sentence crossing the break does not make grammatical sense as it stands.

mentioned in its antecedent with something mentioned in its consequent. Thus, to say “Whatever phoenix there may be cremates itself,” is the same in meaning as to say, “If there be (in the Universe, of course), a phoenix, then that very same thing cremates itself.”
But sundry others had said, less precisely, the same thing; and I certainly derived the idea from Whately, whose Logic (from the fourth London Edition), gave me my first acquaintance with the science,—and happily before that state of stupefaction had supervened with which the opening to boys of the moral world beclouds the pristine clearness of their intellects for a number of years, at least; often, a good many; sometimes during their whole manhood. Whately’s Logic first appeared as a Part [of] the Encyclopedia Metropolitana (which appeared {60} at first in parts), in 1826. The Third Edition of the separate book came out in 1829; the Fourth in 1831. Later in that year its author became Archbishop of Dublin. Hamilton’s savage and disingenuous attack was made in the Edingburgh Review of April, 1833. §6 of Chapter IV (In the separate publication in Book II), is headed ‘Reduction of Hypotheticals’. It begins as follows: “For this purpose we must consider every Conditional Proposition as a universal affirmative categorical Proposition, of which the Terms are entire Propositions, viz. the antecedent answering to the Subject, and the consequent to the Predicate; e.g. to say, “if Louis is a good King, France is likely to prosper,” is equivalent to saying, “the case of Louis being a good King, is a case of France being likely to prosper.” For Whately, like almost all logicians of his time, takes the view of categorical propositions that is iconized in Euler’s diagrams, and that I formally took when I first called the ordinary copula “the copula of inclusion.” {61} Such error as it involves is, for most purposes, insignificant and is certainly simplificatory. Various Kantian and other logicians, especially Germans, hold that the same categorical proposition, as expressed in ordinary language, is susceptible of different interpretations, usually two in number, the utterer of it not attending to the difference between these, since they all apply to the same state of facts. I formerly insisted, as Mill, if I recollect rightly, does (at any rate, such is his spirit, to which I assent), that since they express the same state of things, their meaning is the same. I sometimes qualified this by saying that logic ought to recognize but one reason, since logic is the science of the application of Symbols to their Objects. Yet in the paper in which I first put forward and defended this definition of Logic (Proc. A.A. 1887 May 14, §15), I noticed but one interpretation of an ordinary proposition, its Subject being taken as denoting the Object of the proposition, this being considered as a Symbol, while the Predicate {62} signifies that non-relative character (which I called the “ground”) which the proposition informatively attributes to that Object. This is what I now call the first pragmatistic interpretation of such a proposition: “pragmatistic,” because the service which the proposition may perform consists in the fact that when its interpreter meets with an object to which the subject applies, he is informed by the proposition that it either has (if it be universal) or may have (if it be particular), the character signified by the predicate; but first only, because if the interpretation is to stop at that point, the information neither contributes toward regulating the interpreter’s conduct, which would be the second, or statically pragmaticistic interpretation, nor does it tend toward bringing about a reconciliation or interadjustment between reason and the facts of experience, which would be a third and evolutionarily pragmaticistic interpretation. Mr. De Morgan16 gives in his Syllabus of Logic (Compare §173 with §206), {63} four different “readings” of categorical propositions, one, at least, and apparent[ly] all of which may be taken in extent, or breadth, or in intent, or depth. The four are 1st, the Mathematical, where both subject and predicate are notions of class, 2nd, the Physical, where the subject is a class, the predicate an attribute, 3rd, the Metaphysical, where both are attributes, 4th, the Contraphysical, where the subject is an attribute and the predicate a class. But thus, like other divisions of De Morgan’s manufacture, seems to me to have the same fault that he so forcibly brings home to Hamilton’s system of propositional forms, where, however, it is present in a far grosser form; I mean the fault of presenting an appearance of systematic perfection due to imperfect study of the nature of the concepts employed. It is hasty machine-made goods, to my apprehension. In §64 of the same book, he distinguishes the exemplar and cumular {64} forms of propositions. But unless the reasoning which has convinced me of the above-described analogy between Logical and Chemical Composition contain some subtle fallacy, as to which the Reader shall soon be placed in a position to form his own judgment, every Subject must denote a single individual object, whether that individual be an electron, a man’s body composed of a million cells, the sum total of mankind, past, present, and future, a single undetermined multitude (maniness), an ens rationis or creature of thought, such as a quality, a relation, or a word, or any object of a given description that the interpreter may choose, or of whatsoever other thing or idea or figment. Mind, Reader; I fully agree that De Morgan’s different “readings” are in truth different forms of judging {65} one and the same fact to be whether in esse or in modo. I only remain unconvinced of two things; first, that they truly fall into the systems he sets forth; and secondly, that the Logician need particularly concern himself with them. For instance, one form under which it would be possible to think the fact that man is an animal would be as possible is this: “Should a man, having just donned an evening suit, fresh from Poole’s and having set off upon a bicycle to go to dinner, meet with an accident, fall, and get covered with mud, then, by logical necessity, it must be that an animal having donned an evening suit, etc. would fall and get covered with mud.” But whether it be needful for logic to treat particularly of evening suits, bicycles, and mud, is another question.

The main draft of R 300 ends here on p. 65. The rest of the MS consists of the partial drafts and fragments listed by Robin in his catalogue as pages numbered “33-40; 38-41; 37-38; 40-43.7; plus 64 pp. of fragments running brokenly from p. 1 to p. 60.” None of these fragments are dated, so we can only guess at the order of composition.

from R 300 alternate draft of pp. 33-40

… in regard to the purpose, or idea, of Existential Graphs, you will not be surprised to hear that, never, from the first, having arrogated to myself the function of being the fabricator of that system, but having all along perceived that I had only stumbled upon it, {34} at first stupidly describing, in place of it, a System that I now call that of ‘Entitative Graphs’, I am half-confident of descrying, deep below the superficies of distinct attention, a far more unitary idea of the system than the somewhat disjointed one that I can alone, as yet, either indicate or even describe, in the sentences following. Loosely, (though, mind you, just in such loose ideas can any rational system be first laid open,) I might say that the system of Existential Graphs is designed to afford a sort of geometrical παρασκευή,— or diagram,— for logical analysis, i.e. for illustrating and facilitating the same. In order that that end should be attained, all students of precise logic will agree that there must be a method of exhibiting rational procedures under high logical magnifications. To this end, it is requisite that, as in mathematics, and as the deepest and most thorough studies of Logic that {35} have hitherto been attained show us to be clearly requisite, there should be (1) illustrations of the logical procedure that shall represent it, (2) not merely by force of any rule or habit of interpretation, and still less by any actual, or dynamical, connexion between the sign, or representamen, and the object signified, but, as far as possible, by and in an analogy, or agreement in the very forms themselves, between a (3) visual, or optico-muscular, presentment and the thought itself. The very nature of reasoning demands such an iconic mode of representation, while the nature of the human mind, to which the representamen is to appeal, strongly recommends the optico-muscular form of the icon. But this plan-germ having been settled upon, there may be different ways of developing it into a definite plan. The Entitative Graphs suggest that such there is, and a good deal more convincingly do Euler's logical diagrams,
[footnote omitted here {36} ]
albeit my utmost efforts to push their sway beyond the narrow province to which Venn's improvements and my own leave it restricted, have come to no fruition. I must, therefore, regard it as additional moments, or [angles?] of the plan, that (4) that visible surface of which the Graph icons are differentiata, is to denote that object (as utterly indefinite as to its signification as it is quite unmistakable in its denotation,) which the Graphs, from their very first, find to have been {37} settled upon, by a previous understanding between the Graphist and the Interpreter, as that to which all their significations must refer. This one and unvarying subject of all discourse whatsoever is the Universe. Yet since the Universe, which force[s] upon us all those enduring thoughts that we call truths, makes its power felt in three ways so utterly different that we may well liken them to a set of three mutually perpendicular directions from which any object may be viewed, we must distinguish, Firstly, those thoughts that come to us, as we may [say], from a star-light aspect of the Universe; suppositions that seem to us to be our own untrammeled fabrications, although we are still bound down in them, somehat loosely, it is true, to the Universe, in ways of which the psychologists have, as yet, only begun to furnish us with some slight and fragmentary accounts, which thoughts {38} refer to what we may entitle the Universe of Real Capacities; then, Secondly, those thoughts which, as we are sensible, are forced upon us by the brute force of experience, the Universe of Actual Fact; and Thirdly, those thoughts into which the growth of our own souls together with the vicissitudes of life give us as insight as features of the ideal destiny toward which the course of events is forever to approach nearer, the Universe of Tendencies. While there are other features of the plan of the System that it matters more than these that you should appreciate, I hope you may be able to see that these three, though they shade into one another, are nevertheless different. Yet the imperative need of the manifold differentiation of each of them is even more apparent. It is clear that their differences are not differences of the predicates, or significations, of the graphs, but of the predetermined objects {39} to which the graphs are intended to refer. Consequently, the Iconic idea of the System requires that they should be represented not by differentiations of the Graphs themselves but by appropriate visible characters of the surfaces upon which the Graphs are marked. There are no sensuous appearances of surfaces that to our knowledge are in plain prose analogous to these subuniverses; so that we are driven to call upon fancies to help us out. If we could fancy or pretend that there were any respective similarities between the three fundamental hues, Vermilion, Emerald and Ultramarine and the three Universes of Capacities, Actualities and Tendencies, then the countless intermediate Hues with their variations in Luminosity and in Chroma would furnish us with all the differences we could ever need. But since the actual {40} coloring would be unhandy and in print impracticable, I have suggested that we resort to the heraldic tinctures; to wit, to color for the Universe of Capacities, to metal for the Universe of actuality, and to fur for the universe of tendencies; further differentiations separating each of these subuniverses into “provinces.” The whole differentiation of the surface remains, however, the most unsatisfactory feature of the system. . . . .

from R 300 alternate draft of pp. 37-38, partly published in CP 4.553n. It begins the same as p.37 of the draft above:

{37} settled upon, by a previous understanding between the Graphist and the Interpreter, as that to which all their significations must refer. It was the genius of my gifted student O. H. Mitchell that first opened our eyes to the identity of the subject of all assertions, although in another sense one assertion may have several individual subjects, which may even belong to what Mitchell called (quite justifiably, notwithstanding a certain condemnatory remark, as superficial as it was supercilious), different dimensions of the logical Universe. The entire Phemic Sheet and indeed the whole Leaf [see 555] is an image of the universal field of interconnected Thought (for, of course, all thoughts are interconnected). The field of Thought, {38} in its turn, is in every thought, confessed to be a sign of that great external power, that Universe, the Truth. We all agree that we refer to the same real thing when we speak of the truth, whether we think aright of it, or not. But we have no cognition of its essence that can, in strictness, be called a concept of it: we only have a direct perception of having the matter of our Thought forced upon it from outside our own control. It is thus, neither by immediate feeling, as we gaze at a red color, that we mean what we mean by the Truth; for Feeling tells of nothing but itself. Nor is it by the persuasion of reason, since reason, as I shall show, always refers to two other things than itself. But it is by what I call a dyadic consciousness, since …

From “The Bedrock beneath Pragmaticism,” MS 300, alternate draft of pp. 40-43.7. It begins where p. 39 leaves off above: “Therefore since the”

{40} essence of any graph is to be found in greatest purity in the ideally perfect interpretation of it as it is instanced on a metallic surface, where it is free from modality, the continuity in question must, of necessity, represent co-being in one universe, since this consists in continuity of environment. But I must not be misunderstood as holding Continuity to be an indecomposable idea.

The following, intended as a footnote to the above sentence, is reprinted in CP 6.174-6 and PM 208-10.

174. I feel that I ought to make amends for my blundering treatment of Continuity in a paper entitled “The Law of Mind,” in Vol. II of The Monist, by here redefining it after close and long study of the question. Whatever is continuous has material parts.

I begin by defining these thus: The material parts of a thing or other object, W, that is composed of such parts, are whatever things are, firstly, each and every one of them, other than W; secondly, are all of some one internal nature (for example, are all places, or all times, or all spatial realities, or are all spiritual realities, or are all ideas, or are all characters, or are all relations, or are all external representations, etc.); thirdly, form together a collection of objects in which no one occurs twice over and, fourthly, are such that the Being of each of them together with the modes of connexion between all subcollections of them, constitute the being of W. Almost everything which has material parts has different sets of such parts, often various ad libitum. Nothing which has an Essence (such as an essential purpose or use, like the jackknife of the celebrated poser) has any material parts in the strict sense just defined. But the term “material parts” may, without confusion (if a little care be exerted), be used in a somewhat looser sense. Namely, if the Being (generally, a Concept) of an object, T, essentially involves something C which prevents it from having any material parts in the strict sense, and if there be something, W, which differs from T only in the absence of C and of any other such hindrances, so that W has material parts, then the material parts of W may loosely be termed material parts of T; but in such case the concept of W so derived from T is nearly or quite always somewhat vague, so that either the material parts will be so too, or else they must be conceived as merely the parts of some state of it, and very likely of an instantaneous state that is an ens rationis closely approximating to the nature of a fiction. It will be seen that the definition of Material Parts involves the concept of Connexion, even if there be no other connexion between them than co-being; and in case no other connexion be essential to the concept of W, this latter is called a Collection, concerning which I have merely to say that my reflexions on Mr. Alfred Bray Kempe's invaluable, very profound, and marvellously strong contribution to the science of Logic in the Philosophical Transactions [of the Royal Society, v. 177] for 1886 (which, by the way, seems to have proved too strong food for the mewling, etc., creatures who write the treatises on the science) have led me to believe it to be indecomposable. But I dare not be positive thereanent.

175. I must here give the substance of a far-famed definition of equality in multitude which was originally due to Bernardo Bolzano. This writer was a Catholic priest in Buda-Pesth who published a treatise on Logic in four volumes, and a work entitled Paradoxes of the Infinite. In one or other of these he certainly laid the foundations of the great modern exact logic of quantity, which has so far been developed under the lead of the immortal Dr. Georg Cantor. Though I have never seen either work I do not hesitate to say that Bolzano put Human Reason under an eternal debt by laying the foundation of this science, since his definition of equality sufficed of itself to do that; and I need hardly say that the Catholic Church, which carries consistency as far as is consistent with any life at all, visited condign punishment upon the priest for such outrageous violation of loyalty to Her as the giving of aid and comfort to Human Reason — and most traitorous of all to Reasoning about Infinity! — was felt to be by Her and by all the world except the poor simple soul who committed that foul offense. I gave the substance of the definition in a former paper, going on to other matters of importance which I need not here touch upon. But owing to my having then a very imperfect understanding of graphs, I expressed the definition in the insufficiently analytic language of my Algebra of Dyadic Relations (the same that is mainly employed in Schröder's third volume). I am continually obliged to make elementary explanations owing to the disgracefully unscientific state of Logic, which is quite as much behind its condition six centuries ago in some particulars as it is in advance of that state in others. As for contemporary text-books in our language, they are the merest rubbish on the whole. The very best that can be said of them is that a few have merits in particular directions. They are all amateurish and encourage amateurish views of the universe and of life. In comparison with the state of all the non-philosophical sciences, they are downright puerile; and a green scum grows over them year by year. If our people were at all aware of this blot upon our civilization, it would be possible for a scientific student of the subject of some real strength to put forth at least a primer of the science. But it is a condition of the success of any such student in penetrating to the true science that he should make himself a recluse. He is thus out of the swim, and is crowded out of all opportunities to be of much service; whereby Spencerism, Agnosticism, and other amateurisms, whose professors lose precious little time in arduous research, are able to gain the exclusive ear of the ignorant persons whom they court. In the fourteenth century Nominalism was rendered a respectable opinion by the halting realism of Scotus and by the extravagant unpragmatism of his followers. But after physical science has discovered so many general principles in Nature, nominalism becomes a disgraceful habitude of thought.

176. But now I define a pseudo-continuum as that which modern writers on the theory of functions call a continuum. But this is fully represented by, and according to G. Cantor stands in one-to-one correspondence with, the totality of real values, rational and irrational; and these are iconized, in their turn, according to these writers [by the] entire body of decimal expressions carried out to the right to all finite powers of 1/10 without going on to Cantor's ωth place of decimals.

For it is a principle continually employed in the reasoning of the universally accepted “doctrine of limits” that two values, that differ at all, differ by a finite value, which would not be true if the ωth place of decimals were supposed to be included in their exact expressions; and indeed the whole purpose of the doctrine of limits is to avoid acknowledging that that place is concerned. Consequently the denumeral rows of figures which, by virtue of a simple general principle, are in one-to-one correspondence with the values, have relations among themselves, quite regardless of their denoting those values that perfectly agree in form with the relations between the values; and consequently these unlimited decimal fractions themselves, apart from their significations, constitute a pseudo-continuum. This consideration renders it easy to define a pseudo-continuum. It is in the first place a collection of objects absolutely distinct from one another. Now from the fact that Cantor and others call it a “continuum,” as well as from other things they say about it, I am led to suspect that they do not regard the pseudo-continuum of unlimited decimal expressions as [having members] all absolutely distinct from any other, for the reason that, taking any one of them, it does not possess any one elementary and definite non-relative character which is not possessed by any other of them. But this is not what I mean, nor what is generally meant, by a collection of absolutely independent members. What I mean by that expression is that every member is distinguished from every other by possessing some one or another elementary and definite non-relative character which that other does not possess; and that this is the usual acceptation of the expression is evidenced by the fact that the majority of logicians are in the habit of conceiving of a universe of absolutely distinct individual objects, by which they only mean that every individual is in every respect, of a certain universe of respects, determined in one or other of two ways and that every individual is differently determined from every other in some of those respects; and they do not generally conceive that every individual object has a determination in any one elementary and definite respect, while all the other individuals are determined in the opposite way.