c LIBRARY UNIVERSITY OF CALIFORNIA. Received t-^t^s^r^, , i8g 2- ^ccesdoiis No. -4^^ ^ ^/ Class No. d3k Digitized by the Internet Archive in 2007 with funding from IVIicrosoft Corporation http://www.archive.org/details/essayonnewanalytOObaynrich AN ESSAY NEW ANALYTIC OF LOGICAL FORMS. a. ■■/t^ri-ic^- AN" ESSAY NEW AMLYTIC OF LOGICAL FOEMS, BEING THAT WHICH GAINED THE PRIZE PROPOSED BY SIR WILLIAM HAMILTON, IN THE YEAR 1846, FOR THE BEST EXPOSITION OF THE NEW DOCTRINE PROPOUNDED IN HIS LECTURES ; AN HISTORICAL APPENDIX. BY THOMAS SPENCER BAYNES. TaANSLATOR OF THR PORT-ROYAL I.OOIC. '^^ OF THB UHIVBESIIT] ^^^^ EDINBURGH: SUTHERLAND AND KNOX. LONDON : SIMPKIN, MARSHALL, AND CO. MDCCCL. J/$'3S / ED.NBOKGH: T.CONSTABtR, PR.NTRR TO HEH MAJESTY. CONTENTS. PAGE Preface, vii Sir William Hamilton's Requirements for Essay, ... xi Essay on New Analytic op Logical Forms, .... 1 Appendix : — I. Historical Notice touching tlie Explicit Quantification of the Predicate, 81 II. On the Catholic Doctrine touching the Implicit Quantification of the Predicate, 137 III. Of Figure, 147 IV. Of Notation, 160 Note by Sir W. Hamilton, 153 PREFACE. The following Essay was written exclusively from the notes which I took while attending Sir William Hamilton's class in 1845-6, that being the only ses- sion during which I ever attended his logical course as a student. It has thus a certain value, though this be but shght,'''' as evidence that Sm W. Hamilton then taught his new doctrine so clearly, that it might be readily apprehended in all its essential points by an ordinary student. In order, therefore, that it may not lose any accidental value which it pos- sesses on this account, the Essay is pubhshed as it was originally written. I should not, indeed, have felt myself at libert}^ even on other grounds, to have altered it. Accordingly, with the single excep- ■^ Were it necessary, abundant evidence might be at once ob- tained to prove that Sir W. Hamilton taught his new doctrine five years earlier than the above date. viii PREFACE. tion of a somewhat fuller statement of the three canons of figure, (pages 67, 68,) it is printed with- out alteration, as at first given in for competition. A few foot-notes have been added while it was going through the press. These are distinguished from the original notes of the Essay by being placed within brackets. Some historical details, too, touch- ing the doctrine which the Essay expounds, have been given in the form of an Appendix. These de- tails, though presenting little attraction to those who have only a general acquaintance with logical science, will, I hope, be found of some interest to its more advanced students. The few definitions and canons which occur through the Essay, and which are marked with inverted commas, are given from my notes of the lectures. They are, I believe, substantially correct ; but as the notes are in many places brief and im- perfect, I cannot vouch for their verbal accuracy. The requirements originally prescribed for the Essay are reprinted with it, as they to some extent explain the form which it assumed. Although at the time when the prize was awarded Sir William Hamilton had suggested to me that FEEFACE. IX the Essay was of sufficient general interest for pub- lication, still I naturally felt that, in the event of being printed, it would be more satisfactory, both to the public and to myself, to receive some formal con- firmation of this. Indeed, as the Essay is mainly de- voted to the exposition of Sir W. Hamilton's new doctrine, I should not, for a moment, have enter- tained the idea of publishing it at all without his ex- press sanction. Accordingly, when I first thought of doing so, I applied to Sir W. Hamilton for this purpose, and received from him the following note, for the kindness of which, I need scarcely say, I feel personally indebted : — « Edinburgh, 9th March, 1860. " My Dear Mr. Baynes, — So far from having any objection to the publication of your Logical Essay, the intention has my fullest approval. When I first perused that Essay, it seemed to me, not only pre- eminently entitled to the annual prize for which it was written, but well deserving of the attention of logical readers in general. This I probably ex- pressed to you at the time. And having now ob- tained all the highest honours in Philosophy proper which our University offers to her Alumni, I am happy to learn that you propose printing that Essay, in its original form, with the addition of relative X PREFACE. matters furnished by your subsequent thought and reading. I am acquainted with no one who has more zealously — more unexclusively followed out his philosophical and, in particular, his logical researches ; your supplement cannot, therefore, fail to be at once curious and important. I would say, that any infor- mation from me is at your service, were I not aware that you are laudably desirous to limit your addi- tions to your own resources. I shall only request the annexation of a note, principally for the purpose of showing in what respect my present views may differ from those of mine stated by you in the Essay.'"' — Beheve me," &c. I have only further to return my best thanks to Sir W. Hamilton for his constant kindness in facili- tating my inquiries, giving me free access to his library for works which I could not otherwise have obtained, and in various other ways affording me the benefit of his invaluable counsel and assistance. THQs- S. BAYNES. Edinburgh, May 23, 1850. * See Note by Sir William Hamilton, p. 153. REQUIREMENTS FOR ESSAY ON THE NEW ANALYTIC OF LOGICAL FORMS. Without wishing to prescribe any definite order, it is required that there should be stated in the Essays : — 1° What hogic postulates as a condition of its applicability? 2° The reasons why common language makes an ellipsis of the expressed quantity^ frequently of the subject, and more fre- quently of the predicate, though both have always their quanti- ties in thought. 3° Conversion of propositions on the common doctrine. 4° Defects of this. 5° Figure and Mood of Categorical Syllogism and Reduction, — on common doctrine. (General Statement.) 6° Defects of this. (General Statement.) 7° The one Supreme Canon of Categorical syllogisms. 8° The evolution from this canon of all the Species of Syllo- gism. 9° The evolution from this canon of all the General Laws of Categorical Syllogisms. 10° The error of the Special Laws for the several Figures of Categorical Syllogisms. 11° How Many Figures are there ? 12° What are the Canons of the several Figures? 13° How many Moods are there in all the Figures ; showing, in concrete examples, through all the Moods, the unessential va- riation which Figure makes in a Syllogism ? 14° What relation do the figures hold to Extension and Com- prehension f 15° Why have the second and third Figures no determinate mapr and minor premises, and two indifferent conclusions ; while the first Figure has a determinate major and minor premise, and a single proximate conclusion ? 16° What relation do the Figures hold to Deduction and In- duction ? April 15, 1846. >>' OF THR NEW AMLYTIC OF LOGICAL FORMS. The main principle on which the new Analytic of Logical Forms proceeds is that of a thorough-going quantification of the predicate. This principle in its full scientific significance has been totally overlooked by logicians ; and when noticed at all, has for the most part been referred to only to be discarded as useless, if not to be condemned as false * In conse- * Dico, signum esse addendum subjecto, nunquam prsedicato. Si enim in propositione universali affirmativa signura universale addas praedicato, falsa erit propositio : ut " omnis homo est omne animal." Si vero in universali negativa signum universale, itemque particulari, sive affirmativa fuerit sive negativa, signum particulare addideris praedicatis, propositiones non quidem falsae fient, sed tamen efficies redundantiam et ravrcXoylav. Doctrina Propositionum Disputa- tionibus, xii. comprehensa . a M. Daniele Stahlio. Oxon. 1663. (Dis- putatio vi. § 16.) [This is the tradition touching the express quantification of the pre- dicate almost universally prevalent in the Aristotelic schools and commentaries, as will be more fully shown in the Appendix. The Regulce Philosophicce of Stahl, however, from which it is taken, is an acute and valuable work, faithful to its title, and containing more learning and philosophy than could readily, even in the works of his A 2 NEW ANALYTIC OF LOGICAL FORMS. quence of this omission, logic as a formal science has received only a one-sided development — has been deprived of much that is scientifically true — encum- bered with much that is scientifically false ; and time, be found within the same compass. It long maintained a very high reputation, and was often reprinted, both on the continent and in this country. The first edition was published in 1635, the second in 1641, another in 1653. Several others followed, which were re- printed at London and Oxford ; and finally a revised edition appeared in 1676, with notes by the elder Thomasius, who was the master of Leibnitz. From having thus ascertained more accurately the dates of the earlier editions of this work, I can now state with confidence what I had before surmised, viz., that it contains at least a partial anticipation of the distinction which was for the first time fully taken and established in modern philosophy by the Port-Royal logicians, — the distinction, to wit, in notions of the two wholes or quantities, the comprehensive and the extensive. In answer to the question, how the wider predicate comes to be in its narrower subject ; or, what is the same thing, how the whole essence of the genus comes to be in the species ; the author says, " Esse praedicatum in aliquo subject© totum seu universaliter potest dupliciter accipi : primo, inteTisive seu ratione essentice, et sic animal totum inest homini, et sic quodvis prae- dicatum superius inest inferiori : secundo, extensive, seu ratione la- titudinis ut hie accipitur, et sic animal non inest totum seu univer- saliter in homine, quia animal non totum comprehenditur ab ipso, ita ut extra ipsum non sit." {Reg. Phil., p. 687.) The contrasted character of these counter quantities, the intensive, as the quantity of essence, the extensive, as the quantity of extent, is here given with even scientific precision. Many of the older logicians say in general, (after Aristotle,) that in one sense the species is in the genus, and in another the genus in the species ; but I have not found any state- ment of this distinction before the time of the Port-Royalists at all so precise and explicit as that given in the above passage. Daniel Stahl, beside the Regulce, was the author of a number of other works on Logic, Metaphysics, and Ethics. He was professor of Philosophy in the University of Jena, and died in the year 1654, after having occupied the chair thirty-one years. {Witteni Memorice, p. 166.)] NEW ANALYTIC OF LOGICAL FORMS. 3 throughout its entire history exhibited in a per- verted and erroneous form. On the principle of a quantified predicate, however, past evils are cor- rected, past omissions supplied; and logic receives its highest development in the perfection and sim- phcity of its form. To exhibit some of the more immediate improvements thus effected by the appH- cation of this principle is the design of the following essay. In seeking to accomplish this there will be ; — I. A statement and application of the fundamental postulate of logic, from which application there arises the principle of an expressed quantification of the predicate. II. The application of this principle (of a quan- tified predicate) to propositions ; and in particular to the doctrine of their conversion, in which the complexity and incompleteness of the old doctrine will be contrasted with the simplicity and perfection of the new. III. The influence of this principle on the doc- trine of categorical syllogisms, in contributing to effect specially ; the reduction of their general laws to one ; the abolition of their special laws ; and from this new simplicity the amplification of the valid forms of reasoning. [It may, perhaps, be well to state at the outset that in the following essay, when not otherwise stated, we proceed in the whole commonly recognised by logicians — the whole of Extension ; understanding, 4 NEW ANALYTIC OF LOGICAL FORMS. however, that by changing the copula in propositions, and accompanying this change by a transposition of the propositions in syllogisms, what is said of the one whole of Extension is equally applicable to the counter w^hole of Comprehension.] We proceed then — I. To state and apply the fundamental pos- tulate of logic. This postulate is, — " That we be allowed to state in language what is contained in thought." The application of this postulate to the subject of a proposition is not denied. Logicians now universally allow that the subject has a determinate quantity in thought, and this is accordingly expressed in lan- guage. With the subject of a proposition we have here, therefore, nothing to do. It is to the predicate that we have to vindicate an interest in the postulate co-equal with that of the subject. In order to determine this, we must inquire whether a notion holding the place of predicate in a proposi- tion always has a determinate'"' quantity in thought. * In order to obviate mistake, we may say that we use the word *' determinate" in relation to quantity, generically, as including under it definite (universal or individual) and indefinite (particular). In this sense it is simply opposed to that absence of all expressed quantity which logicians have generally represented by the term indefinite. We do not know whether such usage be strictly correct, but adopt it for the sake of convenience. NEW ANALYTIC OF LOGICAL FORMS. 5 If it have, then the postulate has an immediate ap- pKcation, and this quantity must be expressed. In answering this question we shall show — i. That the predicate always has a determinate quantity in thought : and ii., Explain the reason why this quantity is not generally expressed in common language. We proceed to show then — i. That a notion holding the place of predicate in a proposition always has a determinate quantity in thought. That this is the case will appear from a little con- sideration of what a notion is. " A notion or con- cept"'"' is defined to be "the cognition or idea of the * [As this term has fallen out of use, it may be necessary to say a word or two in explanation of its recall. It is employed by Sir W. Hamilton to discriminate in conception the product from the process. The meaning of the term conception, as commonly used, is ambi- guous, since it is employed to denote both the act of conceiving and the product of that act. The correlative term concept removes this ambiguity, since it designates exclusively the product, while the term conception is restricted to denote the act of conceiving. It need scarcely be added that these terms are here employed according to their true etymological and scientific meaning, to denote the acts and products of the comparative, and not those of the representative faculty. We said advisedly that this term had " fallen out of use," inasmuch as it was commonly employed by the older English writers on Logic in its precise scientific significance, to express those generalisations represented by common terms, which are the ultimate elements of logical analysis, and with which the first part of logic has mainly to do. Thus Coke, speaking of first and second notions, says — " Those that primarily imposed names intended to name first the things f) NEW ANALYTIC OF LOGICAL FORMS. general attribute or attributes in which a pluraUty of objects coincide." This obviously involves the per- ception of a number of objects — their comparison — the recognition of their points of similarity — and their subjective union by this common attribute. The themselves, as the word 7nan is to express primarily the conceit which we form of human nature." Again, " Now the second notions do not directly and by themselves shadow out unto us the things them- selves, nor anything accidental or appendant unto them, but point out certain intellectual rules whereby we do with all distinctness and regularity form things, that is the conceits of things." {Art of Logick, p. 11.) And again, speaking of the various relations of words, he says quite explicitly — " The formal is the signification of the word, and by consequence the relation to the conceit of the mind which it giveth knowledge of." (P. J 5.) Fraunce also uses the term con- tinually. Thus, to take a single example in defining an axiom, he says, " It here signifieth any sentence or proposition whatsoever wherein one argument, reason, conceipt, thing, is so conjoined with, or severed from another, as that thereby we judge the one eyther to bee or not to bee, the cause, efiect ; whole, part ; generall, speciall ; subject, adjunct : divers, disparate ; relative, repugnant ; like, unlike; equall more or less to the other." {Lawyer's Logick, fol. 87.) See another instance of this use from Fraunce in the quotation given at p. 2.3 (note). The term is also used by Granger in his " Divine Logick ;" and, if I remember aright, by Wilson in his " Rule of Reason." It is to be found, too, employed in the same sense out of logical works, and is used in this way by writers of authority ; we may sj)ecify as exam- ples among others Dr. Henry More, and Sir W. Raleigh. All that now remains to us of this old use is the restricted sense in which the word conceit is employed — a sense at once so restricted and so esta- blished as to unfit it for scientific use. Concept, as strictly analo- gical in form and precise in meaning, is exactly the term we need to express the simplest products of the comparative faculty ; and, as we have shown, it already exists in the language. What is necessary, therefore, in employing it, is not an apology for its introduction, but simply a vindication of its recall.] NEW ANALYTIC OF LOGICAL FORMS. 7 possibility of this process determines the possibility of knowledge to man. Had he no power of classi- fying in intellect the confused multitude of objects presented in sense, he must remain for ever destitute of anything worthy of the name of knowledge. With no clear recognition even of the individual, since com- parison and discrimination would be impossible, he must for ever abide amidst the obscurity and vague- ness in which knowledge commences — helpless amidst a multiplicity of objects which he could not compre- hend — bewildered by a confusion which there was no possibility of recalling to order. The earliest effort of the mind is accordingly directed to extricate itself from this confusion ; and this determines the exercise of the comparative faculty, and the formation of con- cepts or notions. Amidst the multitude of confused objects presented to the mind in perception, some are found to affect us similarly in certain respects. These objects the mind considers ; by comparison it recognises their resembling qualities ; by attention these are exclu- sively considered, since the concentration of the mind on those qualities in which objects coincide involves of necessity its abstraction from those in which they are severally dissimilar. These various objects, since the resembling attributes which they possess in common cannot, when considered alone, be discri- minated, are, in this restricted point of view, consi- dered as one. In other words, the mind grasps into ^ NEW ANALYTIC OF LOGICAL FORMS. unity a multitude of objects severally distinct by a common point of identity. On this unity thus formed it sets the seal of a name, that it may be enabled ever afterwards at once to discriminate the various ob- jects of its knowledge, commodiously refer each to its own class, and thus be saved the endless labour of enumerating all the particulars by which objects are individually discriminated. A notion is thus a purely ideal or subjective whole, which the mind from the limitation of its powers is necessitated to form, in order to classify in thought and discriminate in language the various objects of its knowledge. This being the case, it is obvious that a concept or notion can afford only a partial knowledge, and has only a relative existence. It can afford only a par- tial knowledge, since it embraces some only of the many marks by which an object is known. It has only a relative existence, since this knowledge is not given absolutely, but only in connexion with some one of the objects to which the concept is related. For a notion, though potentially applicable to all the objects which it contains, can only be truly known on occasion of its being actually applied to some one of these objects. This is at once the test and the evidence of its relative character. And this being its character, it is obviously altogether dependent on the objects from which it is formed. A notion has thus, in its totality, a purely subjective existence, destitute of an}^ objective reality. Being what it is NEW ANALYTIC OF LOGICAL FORMS. 9 — an ideal whole only by relation to the objects whose resembling part it embraces — it is obvious, as we have said, that it can pretend to no independent existence, much less to any independent knowledge. Its existence entirely depends on that of the objects from whence it is derived, and to each of which it is linked by the common resembling attribute w^hich it embraces. Destroy the objects, you destroy the re- sembling attributes in each ; and destroying the re- sembling parts, you annihilate the whole which they together constituted. As, however, a concept has only a subjective being, existence and knowledge are here identical. If no qualities be discriminated in objects as similar, we have no knowledge of a concept — no concept exists. If we cannot assign an object to any class — cannot say it does or does not belong to any notion, we do not comprehend it. We think an object (recognise to be what it is) only as we think it under some notion or concept. This being premised with regard to notions in general, it will be seen, that when we bring an ob- ject under a notion, i.e., wdien we predicate of it that it belongs to such a class, we must know that it oc- cupies a certain place in that class. For if we w^ere uncertain what place the individual object occupied in the class, or whether it occupied any place at all, we should not know the class, and could not, there- fore, bring any object under it ; — e.g., If I do not know whether rose comes under the concept flovjer 1 NEW ANALYTIC OF LOGICAL FORMS. — whether it is equal to some part, or the whole, or superior to it — I do not know the class flower^ and cannot, of course, predicate flower of rose ; in other words, I cannot bring rose under the concept flower^ since I do not know w^hat the concept means, what it contains, and what it does not. This is clear ; for as we have just explained — since a notion, as a factitious unity in thought, is absolutely worthless, and, indeed, not cognisable, out of relation to the in- dividual objects, the aggregate of whose resembling qualities it constitutes ; and since an object is truly known only as it is thought through or under a notion, it follows, that comprehension, in such a case, would be impossible. If, therefore, we understand the object at all, we must fix, in thought, the sphere which it occupies under the class to which, in predi- cation, we have assigned it. In other words — if we comprehend what we utter, every notion holding the "place of predicate in a proposition must have a deter- minate quantity in thought. This, indeed, is always involved in predication. For predication is nothing more or less than the expression of the relation of quantity in which a notion stands to an individual, or two notions to each other. If this relation were indeterminate — if we were uncertain whether it was of part, or whole, or none — there could be no predication. The very fact of predication is thus always evidence that the predicate notion holds a relation of determinate NEW ANALYTIC OF LOGICAL FORMS. 11 quantity to the subject. In other words, we think only as we think under some determinate quantity ; for all thought is comparison of less and more, of part and whole. All predication is but the utterance of thought. All predication must, therefore, have a determinate quantity. Since, therefore, the quantity always exists in thought^ the postulate applies; that is, in logic the quantity must be expressed, on demand, in language. It only remains to remark here, that this quantity of the predicate notion, always determinate, will be definite, (universal or individual,) or indefinite, (par- ticular,) as the subject notion is greater, equal to, or less than the predicate. If the subject notion be less, we attribute to it a part only of the predicate — say it is some part, but not the whole, which that notion comprises, e.g., " all man is some mortal." If the subject be equal to the predicate, we attribute the whole notion to it, e.g., " all man is all rational'^ If the subject be greater, we attribute the whole pre- dicate to it, as a part only of its extension, e.g., " some mortal is all man." Logicians who have occasionally touched upon the quantification of the predicate, seem for the most part to have conceived the possibility of its express quantification only universally ; '"' and because this * Notandura, signum universalitatis non esse apponendum prse- dicato, sed tantum subjecto ; recte enim dicitur, " omnis homo est animal ;'''' sed non recto dicitur, " omnis homo est omne animaV 12 NEW ANALYTIC OF LOGICAL FORMS. cannot be done in a great number of cases — in all those cases, indeed, in which an individual is brought under a concept, or a species under a genus, (as we cannot say, " all man is all mortal,") — since this cannot be done, to have lightly thereupon thrown aside the whole doctrine as of no avail. It is, how- ever, clear, from the nature of a notion as a whole made up of the like characters in a number of ob- jects Avhich thus stand to it in the relation of parts, that we are quite as much at liberty to say of one of these objects, that it forms a part of the notion, as we are to say of all the objects together, that they constitute the whole.* We may, therefore, and in Verum duse limitationes adhiberi possunt : Prima, ut intelligatur de iiniversalibus affirmatis non autem de negatis ; recte enim dicitur " omnis homo est nullus asinus,^^ " nullus asinus est omnis homo ;" Secunda, ut signum universalitatis immediate ponatur ante praedi- catum ; nam si apponatur tantum adjuncto prsedicati, enunciatio non erit falsa, ut " visus percipit omnem colorem" " Christus curabat oimiem morhumj" &c. Quia adjunctum ejusmodi poterit fieri sub- jectum, mutando verbum activum in passivum hoc pacto, ^^ omnis color jjercipitur a visu." Davidis Derodonis Logica Restituta. Geneva, 1659. (Page 573.) On the catholic doctrine held by logicians touching the quantifi- cation of the predicate, see the Appendix. ■^ [The older logicians laid down many rules which were often useless, sometimes false, and at best of only partial and limited ap- plication, about what they termed the regular and irregular order of predication — the natural or unnatural, direct or indirect, consecu- tion of the terms in a proposition. Natural, or regular, or direct predication (^predicatio tiatur'alis, directa, ordinatd) they held to be that in which the genus is predicated of the species, the species of the individual, the attribute of its subject, and in general the exten- NEW ANALYTIC OF LOGICAL FORMS. 13 fact we must continually (][uantify the predicate par- ticularly. sive whole of its part ; and in which, therefore, the subject notion was always of less extent than the predicate notion. Unnatural, in- direct or irregular predication {predicatio non naturalis, indirecta, inordinata) was the reverse of this, that, to wit, in which the species was predicated of the genus, the subject of its attribute, and in general the extensive part of its whole. Language is, however, but the instrument of thought ; and its natural order in the last resort must ever be that in which it best expresses the thought of which it is the vehicle. What this order shall be will thus in great measure be determined by the feeling and purpose of the speaker. If he have a special interest in any parti- cular term, or wishes in any way to make it emphatic, it will occupy the more prominent and important place in the proposition. The order in such a case will be that of interest and emphasis, and this surely is as natural an arrangement as any other. Accordingly, if I wish to direct particular attention to the genus, it may stand first in the proposition, and the species follow as its predicate. Thus, for example, if referring to the Scaligers I were to say, " An acute phi- losopher was the father, an erudite philologer the son," there would be nothing unnatural in this, for by such an arrangement of the terms, I simply direct special attention to the diflferent departments of science in which they respectively excelled. So, again, in the line of the poet, " The proper study of mankind is man ;" and in a num- ber of other examples that will readily suggest themselves. With equal justice, if I wish to make the species (or genus) emphatic, I may pre- dicate the individual of it, e.g., in the expressions, " A philosopher, in- deed, was Socrates ;" " The poet of all time is Shakspeare ;" there is nothing unnatural, but attention is appropriately directed to the high type of poetic and philosophic character which respectively belonged to these great men. It is no valid objection to this form of predication to say, that in all such cases we do in reality so restrict the genus or species, that they become convertible with the species or individual severally pre- dicated of them ; for this objection lies equally against all predication whatever. Thus, in what is termed the regular form, the genus is never 14 NEW ANALYTIC OF LOGICAL FORMS. We proceed then — ii. To explain why the quantity of the predicate is not expressed in common language. We have already explained the nature of a con- taken in the whole of its extent, but only in so much of it as is oc- cupied by the species of which it is predicated, e.g.^ when we say, " All man is animal," we do not of course mean all animal, but simply that part of animal which is convertible with man, or as is sometimes more explicitly stated, " Man is an (one) animal," or, (if referring to the race,) " a species of animal." So, again, in relation to the second division of what is called un- natural predication — that in which the subject is predicated of its attribute — if we wish specially to signalise the attribute that will naturally stand first, e.g., in the exclamation, " Great is Diana of the Ephesians," the greatness of Diana is far more emphatically marked than it would be if the terms were reversed. So, again, in the line, " The fairest of her daughters, Eve," it is to the beauty of Eve that special attention is directed. It may be at once conceded to the logicians, that what they have termed the natural or direct order, is the more common, inasmuch as the concrete terms of a proposition are generally of greater interest than the abstract ones ; but it is unjust to speak of any other order of predication as unnatural or unlawful, while it is quite obvious that, logically considered, either order of consecution in the terms of a proposition is equally valid, for we may indifferently predicate a part of the genus of the whole species, or the whole species of a part of the genus. This liberty, indeed, is not denied, though it is generally, nevertheless, even by late writers, allowed as an exception rather than as a rule. Thus Gassendi, after giving the rule that the species cannot be predicated of the genus, says, in mitigation of its force — " Additur nihilominus, nisi generi limitatio adhiheatur ; dicere enim possumus, ut jam ante insinuatum est, aliquod animal est homo ; certus quidam color est candor ; una qucepiam virtus est justitia. Efficitur nempe, ut particulis hujusmodi limitantibus genus veluti contrahatur, neque amplius pateat, quam species ; ac proinde ut species de eo enunciari, fierive illius attributum reciproce possit." {Logica. Oxon. 1718, p. 367.) ] NEW ANALYTIC OF LOGICAL FORMS. 15 cept — that it is a factitious whole obtained from a number of individual objects, each of which, there- fore, occupies a certain part of the whole concept. We have also shown, that this relation of quantity is always present to the mind, when a concept and an individual (or a part and whole of any kind) are thought together as subject and predicate. Though always thus contained in thought, this quantity is, however, rarely expressed in ordinary language. The name in which the totality of attribute em- braced by the concept is fixed, is usually applied to any one of its individual objects, without any particle of quantification. The explanation of this omission is to be found in the end which com- mon language seeks. The end which it proposes to itself is the clear utterance of meaning ; it seeks to render at once intelligible, by its signs, the thing signified. Asa vehicle for the conveyance of thought, ordinary language is mainly concerned about what is thought — not the manner of thinking it. In other and more technical terms, it is primarily engaged with the matter of thought, and only considers the form incidentally, and as a mean to an end. What- ever, therefore, is not really necessary to the clear comprehension of what is contained in thought, is usually ehded in expression. Thus common lan- guage abounds with abbreviations and elliptical forms of expression. The expression of those phases of thought and feeling which arise in conjunctions of 16 NEW ANALYTIC OF LOGICAL FOIIMS. circumstances which are frequent and familiar, are almost always and conveniently of this description, e.g., — we meet a friend and say, " good morning J^ What is intended here ? A cordial greeting and the expression of friendly feeling. This, however, is not expressed in terms, but the elhptical form above is understood to represent, " I wish you a good morning." So also in " farewell," and a multitude of other cases which might be adduced. In fact, so that what is meant by it be at once clearly intelhgible, an expression, however elliptical, is true and valid for all the purposes of ordinary language. Thus, to facilitate the communication of thought, not only are words omitted, but the steps of the reasoning pro- cess itself are for the most part abbreviated. Com- mon language almost invariably makes an ellipsis of one step of this process. For since the reasoning process is the same in all men — being governed by laws necessary and universal — the mind at once and intuitively supplies the omitted step, and the process is complete. This principle, too, affords the true explanation why, in common language, the overt quantification of the predicate is neglected. It is not necessary for the clear comprehension of a pro- position that the predicate be quantified in terms. And the reason why this is not necessary is to be found in the universality of generalisation or the formation of concepts, and the sameness of the pro- cess in all men. All men must generalise, for the NEW ANALYTIC OF LOGICAL FORMS. 17 necessity which determines this process exists in all. All men must generahse alike, for the faculties which accomplish it are the same in all. In order to com- prehend the many objects by which he is surrounded, he must reduce them to order. To accomplish this, he classifies or groups into unity a number of objects which affect him in the same manner. But as the same objects affect all men in the same manner, it follows, that where the same term exists to express the same mental modification, this term may and will be apphed to all the individual objects which determine such similar impressions. As all men, therefore, know what is meant by a general term, — that it is a name equally applicable to all and each of the individual objects which it em- braces, when one of these objects is brought under it ; that is to say, when it is predicated of this object that it forms a part of the notion which the general term expresses, it is not absolutely necessary overtly to declare that it forms only some part ; for as it is universally known that the concept is of far wider extension, the quantity is immediately supplied in thought, and no mistake arises. Thus, when we say, " Every horse is an animal" — " All men are mortal,'' it is not necessary to say that there are other ani- mals besides horses, or to guard explicitly against the conclusion that man alone is mortal ; for as the extension of the general terms is understood by all, every one knows at once, by a reference to the matter B 18 NEW ANALYTIC OF LOGICAL FORMS. of the thought, that in each case the predicate is affirmed of its subject only in some part of its exten- sion, not in the whole. Since, therefore, it is not necessary, for the clear understanding of a proposition, that the predicate notion be expressly quantified, and as the continual repetition of it would be wearisome, the quantifica- tion is usually omitted ; in other words, since it is not necessary for the purposes which ordinary lan- guage seehs to accomplish, the quantity of the predi- cate, though always contained in thought, is usually elided in expression. All this, however, becomes widely different when in the progress of reflective inquiry a science arises, which seeks, as its express aim, to accomplish a full and final analysis of the form of thought. It will be the office of such a science to supply for its own purposes the omissions of common language — to re- store whatever of the form of thought may have fallen out of expression in ordinary parlance. The procedure of logic and that of common language are thus different, and to some extent opposed ; the former recalling to expression as of scientific value what the latter had thrown aside as of no account. The different nature of their procedure is, as w^e have hinted, determined by the different nature of the ends which they respectively seek to accomplish. Common language, as we have seen, seeks as its elid to exhibit with clearness the m,atter of thought. What- NEW ANALYTIC OF LOGICAL FORMS. 19 ever does not contribute to this is thrown aside as worthless. Logic, on the other hand, seeks as its end to exhibit imth exactness the form of thought. Whatever contributes to this is retained as of scien- tific value. All the elements ^vhich the analysis of the form of thought furnishes must be brought out to view, and explicitly considered. Whatever does not belong to the form of thought must be cast aside as without the province of the science. We have seen, that in thought the predicate notion of a pro- position is always of a given quantity. This quan- tity is not expressed in common language ; because, by a knowledge of, and reference to, the matter of thought, the omission is at once supplied. This pro- cedure is, however, of course incompetent to logic. As a formal science, it knows nothing of the matter of thought ; it makes no elisions ; it can understand nothing ; it can supply nothing ; it can only recog- nise and deal scientifically with what is given for- mally. If, therefore, the predicate has always a cer- tain quantity in thought, (and we have shown it has,) that quantity must be expressed before it can be logically taken into account, and its significance in- vestigated. The recognition of the expressed quan- tity of the predicate is then as imperative in logic as the neglect of such recognition is convenient in com- mon language ; for it is plain that, unless all the elements furnished by analysis be received and con- sidered in their relative influence and importance, the 20 NEW ANALYTIC OF LOGICAL FORMS. science cannot pretend to completeness. Logic, in common with all sciences, seeks perfection ; but, as a formal science, it can only realise scientific perfec- tion as it attains to formal exactness. The condition of its formal exactness is, that its analysis of the form of thought be exhaustive and complete. As soon as this is the case, synthesis may commence, and the science will emerge in its full beauty and true perfection. This explains how it is that logic has remained so imperfect and deformed in the hands of all previous logicians. They were, in the main, right as far as they went ; but they did not go far enough. Their investigation of the form of thought was arrested before it had attained the necessary completeness. Proceeding in their analysis, they correctly recalled to expression what common language had omitted in the reasoning process, and exhibited the three steps of that process in their formal order and complete- ness. Still continuing their analysis, they proceeded to investigate the properties of a proposition in order to determine its scientific capabilities. Here they discovered that the subject has always a certain quantity in thought. This quantity, in conformity with the necessities of their science, they accordingly expressed, and turned to scientific account. But here their analysis was stopped, just at the very point the investigation of which would have conferred upon their science the completeness which it lacked ; NEW ANALYTIC OF LOGICAL FORMS. 21 and, as the natural result of a defective analysis, logic, as a science, has always remained incomplete. Had logicians proceeded further, they would have discovered that a determinate quantity always be- longs, not only to the subject of a proposition, but also to the predicate ; that the recognition of this quantity in logic affords an important principle, the true appKcation of which would relieve it of its many inconsistencies, and confer upon it scientific perfec- tion. It will be our business presently to inquire into some of the improvements thus effected. To recapitulate, then : — We have seen, from the nature of a notion in general, and of a predicate no- tion in particular, that it always has a determinate quantity in thought. We have thus vindicated to it an interest in the fundamental postulate of logic. And from the application of that postulate there has emerged the principle — That the quantity of the pre- dicate notion of a proposition he explicitly noted in logic. We proceed now to show — II. — The application of this principle to propo- sitions, and in particular to the doctrine of their conversion. A proposition is defined to be " the expression in language of the relation of congruence or confliction, in which two notions, two individuals, or an individual 22 NEW ANALYTIC OF LOGICAL FORMS. and a notion, (in a word, two terms,) are recognised, when compared together, to stand to each other." '" This being the nature of a proposition, it is evident that it involves a plurality of ideas,f thought toge- ther in mutual relation and dependence; since it is only by being viewed in relation as subject and pre- dicate, determining and determined, that a plurality of thoughts can be reduced to a mental oneness, and recognised together in the same individual act of consciousness. The terms of a proposition are thus always related, and this relation constitutes their scientific signifi- cance. To investigate this relation fully, and de- termine it exactly, is the office of logical analysis. As a proposition is the expression of the relation of congruence or confliction between two thoughts, it is surely of the highest importance — in fact, a con- dition of its intelligible existence — that the amount of this agreement or difference be known and stated. This can only be done by ascertaining the quantity of both the terms, and thus determining the space of each in relation to the other. Until this be done, the properties of a proposition have not been fully analysed ; its scientific capabilities cannot be fully determined. The quantity of the terms in relation ■**• ^'■Jvdicium est comparatio ideas cum idea ; propositio est judicium terminis expressum." — Ploucquet. t Using the term " idea " generically, to include the products of sense, imagination, and intellect. NEW ANALYTIC OP LOGICAL FORMS. 23 to each other is thus the most important aspect in which a proposition can be considered ; but though thus the most important relation, it is one in the ana- lysis of which logicians in general seem to have been more than commonly unsuccessful. In opposition to the obvious importance and necessity of determining the quantity of the terms in a proposition, they have introduced into logic a class of propositions distin- guished by the absence of all quantity;* and this, too, * [The class of propositions distinguished by the absence of all ex- pressed quantity, and termed by logicians indefinite, (more properly indesigyiate,) affords another curious illustration of how completely the force of authority has in logic prevailed over the most obvious and elementary necessities of the science. Introduced originally by Aristotle, and subsequently reproduced by Boethius, these proposi- tions have continued to occupy a place in the science, and to Be dis- criminated as a separate class, under the division of quantity. In the absence of all expressed quantity, however, it was very difficult to turn them to any logical account. Some kind of quantity was necessary for this purpose ; but the only way in which they could be quantified with any certainty was particularly, according to the caution of Apuleius, who says, referring to the division of proposi- tions under the head of quantity, " Aliae indefinitaa, ut animal spirat, non enim definit utrum omne, an aliquod. Sed tamen pro particu- lari semper valet. Quia tutius est id ex incerto accipere quod minus est." (^De St/Uogismo Categorico. Apideii Opera. Lugd., 1600, p. 415.) Subsequently, however, as stated in the text, rules for determin- ing this quantity were laid down, derived from the object-matter of the propositions themselves. By Ramus, indeed, and some of his followers, indefinites, it would seem, were altogether rejected as of no logical account. His English representative, Fraunce, (I have not Ramus' own works at hand,) says, referring to the indefinite propo* sition, " But Ramus expelleth that uncertaine and indefinite axiom ; for every conceipt of the mind is determinatly eyther generall or 24 NEW ANALYTIC OF LOGICAL FORMS. in opposition to the fundamental postulates of their science : — that in order to deal with a proposition we must know what it means — i.e., understand the quan- tity of its subject and predicate — and that we be speciall, and speciall eyther particular or singular." {Lawyer's Lo- gike, fol. 92.) They continued, nevertheless, to be currently received, and a later English writer but reflects this common acceptance when he makes them the matter of special legislation. " The canons hereof," says Coke, speaking of this class, " are two." He then gives the follow- ing, which are, however, but translations and abridgments from Keckermann : — " 1. The chief force and use of indefinites is in propositions of the idea : that is in such as where the universal subject is taken abso- lutely, as — the Lord's Supper is a sacrament ; man is the noblest creature ; the soul of man is immortal, &c. " 2. There is also a use of indefinites to signifie that the conse- quent is in the antecedent, for the most part, though not always, cog It/ tI (koTJ). As, the Cretians be lyars ; mothers are too much cockerers of their children," &c. {Art of Logick, p. 106.) A recent British writer, however, sees far more clearly into their true logical character and relation, and says of them, (referring to the rules which are given for their reduction,) with far more scien- tific than historic truth, " By reduction here is to be understood that logicians recognise no indefinite propositions, and that if an inde- finite occur, they require it to be expressed in a definite form. In- definite propositions are noticed that the logician should be on his guard against them, and not because they are legitimate, or of any legitimate class." {Thymu's Compendium of Logic. Dublin, 1827, p. 47.) This is far from being historically true, and if it were, it would not avail to defend the position which indefinites occupy ; for the prin- ciple of their reduction is as extra-logical as are the propositions to be reduced. In the statement, that they are not recognised in logic as a separate class, this author has at once overrated the acuteness of his predecessors, and underrated his own ; for it is only within a comparatively short period that their logical position has been seri- NEW ANALYTIC OF LOGICAL FORMS. 25 allowed to express all that is understood. As an antidote to the disorders thus introduced into their science, they have had recourse to an extra-logical remedy ; that is to say, they have laid down a num- ber of rules for determining the quantity of these unquantified propositions, founded on the object- matter of these propositions themselves, according as this matter is possible or impossible, necessary or contingent. In so doing they have, implicitly at least, destroyed their science ; for if logic be com- petent to this discrimination, it can no longer vindi- cate to itself the character of a special science, but must become co-extensive with the whole domain of human knowledge. They have thus also destroyed the possibility of its thorough-going apphcation, since they have tacitly laid down, as a preliminary to such application, the impossible condition that we should ously called in question. They are, however, as need scarcely now be stated, to be rejected from logic as utterly unscientific in their character. They belong, indeed, to the same confusion of the acci- dental with the essential, through which the enthymeme was dis- criminated as a separate form of reasoning. In both the mere co7i- tingencies of speech are identified with the necessities of thought ; and the accidents of expression are received and incorporated with the science as valid elements of form. Science, however, is no longer worthy of the name when it accepts and incorporates, without ex- amination, the rude materials which it is its office to elaborate ; and places among its elements the confused wholes, which a more search- ing analysis would have decomposed into their constituent parts. This is precisely what has happened in relation to the indefinite proposition and the enthymeme ; and accordingly, by a truer scien- tific analysis, they arc finally rejected from logic] 26 NEW ANALYTIC OF LOGICAL FORMS. know all that is possible and all that is impossible, all that is contingent and all that is necessary. To say nothing of its logical inconsistency, — this being impossible is, as a scientific demand, absurd. Logicians have not, however, of course left propo- sitions generally in this unquantified state, or their science would have remained hopelessly crippled. They proceeded, as w^e have already said, to consider the quantity of propositions ; but here their incon- sistency still attends them, and they are destined to be again unsuccessful. For in their analysis, as we have seen, they have only considered the subject, and determined its quantity, while that of the predi- cate, which it was equally necessary to determine, as being of equal scientific value, is altogether neglected. The proximate influence of this omission in intro- ducing complexity and inconsistency into the science will be seen in the common doctrine of conversion ; while some of its remoter consequences will be here- after signalised. It may be well to premise here that we speak of categorical propositions throughout. We go on, then, to notice the conversion of proposi- tions on the common doctrine. When the subject and predicate of a proposition change places, the proposi- tion is said to he converted. This conversion is threefold. 1. Simple conversion. — This takes place when the terms are simply transposed, without any change of quantity or quality in the proposition. This is com- NEW ANALYTIC OF LOGICAL FORMS. 27 petent in universal negative and particular affirma- tive propositions ; e.g., — No man is a stone. Therefore, No stone is a man. Some man is a tinker. Therefore, Some tinker is a man. 2. Conversion 'per accidens. — This takes place when the quality of the two propositions remains the same; but the quantity is altered, the predicate in the one being limited on becoming the subject of the other. This holds true in universal propositions, both affir- mative and negative ;* e.g., — All violets are flowers. ■ Therefore, Some flowers are violets. * [This is the statement of Peter Hispanus, and it has been re- peated by Derodon and some others among the later logicians. The application of this species of conversion to universal negatives is, however, altogether useless, as they are converted simply, and thus retain their universality after conversion. It is, moreover, incompe- tent, inasmuch as, in such a process, an inference of subordination is involved. By the majority of logical writers it is therefore applied exclu- sively and formally to universal affirmative propositions. As so ap- plied, however, the name which it bears is unsuitable, since it no longer truly designates the nature of the process which it is em- ployed to express. Taken in this exclusive application, restrictive or attenvMe conversion (the name given to it by Granger) would be much better. It is worth while noticing, that the logicians in general do not seem to be at all aware by whom this term, per accid£ns, as applied to conversion, was introduced, or what was the kind of process which it was originally employed to denote. None refer to its authorship, while few attempt any explanation of its meaning ; and the few who do are for the most part incorrect. Isenach, whose " E'pitome Dia- 28 NEW ANALYTIC OF LOGICAL FORMS. 3. Conversion by co7itraposition^'- — This takes place when the quahty of the propositions remains un- affected ; but the terms are changed into what is called by logicians infinite, but more appropriately lecticce''' was published about the year 1510, says, in explanation, that this species of conversion was called ^er accidens, because one of the accidents of the proposition (quantity, to wit) was changed. Keckermann gives a somewhat difierent and longer account of it. He says, that this kind of conversion is called per accidens, because the converted is not immediately inferred from the converse, but only mediately through the intervention of another proposition ; e.g., from the proposition " all man is animal," it is not imm^diatdy inferred that " some animal is man," but rather that " some man is animal ;" and as this can be converted simply, it follows that " some animal is man." {Sy sterna Logicce. Francof, 1628, p. 348.) While a recent Oxford writer gives the following not very intelligible explanation : — " Per accidens — putting in the place of the subject the quality, whether proprium or accident, which the predicate implies. By the old logicians the proprium is constantly called accidens proprium." {Moherlfs Lectures on Logic. Oxford, 1848, p. 85.) It is difficult to see how this statement (even supposing it to be just as far as it goes) can be accepted as a full account of the matter, — inasmuch as it can at best only apply to those cases in which the predicate is the property or accident of the subject, and by no means to the generic latitude of possible predication to which the conversion extends. Ploucquet is the only one among modern logicians, so far as my knowledge extends, who seems to have understood the sense in which it was originally employed, and who has accordingly given the true explanation of the term. He says, explaining the significance of the letter P (per accidens) in the mnemonic verses, " No tat universalem in particularem, et particularem in universalem esse convertendam, id quod fit per accidens, ex natura materice.^'' {Fundamenta Phil. Spec. Tubingse, 1758, p. 45.) This is the true explanation of the term, and of the process which it originally designated, as employed *• Some logical writers, it appears, have rejected this species as of no logical value. — Crachanthorpe, Book iii. chap. 10. NEW ANALYTIC OF LOGICAL FORMS. 29 indefinite, by the addition of negative particles ; that is to say, when in the converted proposition, instead of the subject and predicate simply, the contradictory of each is found ; e.g., — PC very raan is mortal. Therefore, Everything which is not mortal is not man. by Boethius,* who was the author of this species of conversion, name, and thing. This term was employed by him to denote the conver- sion of the universal into the particular, and the particular into the universal, from the accident of the matter of the proposition. He says, " Harum (propositionum) igitur, particularis affirmatio, particu- lariter quidem sibi ipsa convertitur, universali autem affirmationi per accidens, et rursus universalis negatio, loco principe sui recepit conversionem, ad particularem vero negationem per accidens converti potest. Nee vero negationis particularis ad seipsam principaliter stabilis ac firma conversio est, sed negationi universali secundo loco atque accidentaliter." Introductio ad Syllogismos. {Opera, Basil. 1546, p. 575.) But the formal conversion, (to be carefully distinguished from the material conversion of Boethius,) which the term was subsequently and exclusively employed to denote, had been discriminated long before the time of Boethius. It was expressly taken by Aristotle, and called by him partial or particular conversion, {dvTiffT^o(pr} sv fMs^si.) Anal. Pr. i. c. 2, § 1.) It was subsequently also signalized by Apuleius under the name of reflex conversion. His words are, " Universalis autem dedicativa et ipsa quidem non est conversibilis, sed particu- lariter tamen potest converti : ut cum sit 07nnis homo animal, non potest ita converti, ut sit omne animal homo; sed particulariter potest, quoddam animal homo. Verum hoc in simplici conversione, quae in conclusionum illationibus refiexio nominatur." {Opera Omnia, Lugd., 1600, p. 419.) It may seem useless to have dwelt so long on this kind of conver- sion in the very act of abolishing it ; but that it is dead is no reason whatever why so venerable a member of the ancient system should not receive decent burial.] * For the reference to Boethius I am indebted to Sir W Hamilton. 30 NEW ANALYTIC OF LOGICAL FORMS. This holds true in universal affirmative and par- ticular negative propositions. The rules "^^ governing conversion are given by different logicians with nu- merical differences ; in substance, however, they are much the same. But as the enumeration would be tedious, and the nature of the process in its different species is evident from the statement of each, we shall not repeat them here. We might remark on this process generally, that it is, for the most part, logically incompetent ; since in the second species we interpolate a quantity not formally given, and in the third create an entirely new proposition by new terms. But w^e pass on to remark, specially, that the whole doctrine of conversion, as commonly understood, is on the principle of the new analytic false and use- less. This inconsistent and cumbrous doctrine resulted, as we have said, from a false analysis by logicians of the elements with which they had to deal. The whole doctrine is founded upon the relation of quantity be- tween the subject and predicate in a proposition ; but if a principal element of that relation be left out, the doctrine will of course be defective. Logicians stand chargeable with this neglect. They commenced to recompose their system before, hy thorough decom- position, they had obtained all the elements requisite * Crackanthorpe gives five, Wallis six, another British writer twelve. — The Port-Royal three, reduced to two. NEW ANALYTIC OF LOGICAL FORMS. 31 for that purpose ; and to remedy the deficiencies thus occasioned, they have had recourse to the com- phcated process briefly stated above ; — thus, with labour and difficulty — and even then imperfectly — by a complex process, and a number of particular rules, effecting that which a simple proximate principle of their science would, if recognised, have spontaneously and perfectly accomplished. As this confusion and complexity has arisen from a faulty analysis, so a perfect analysis at once introduces order and simpli- city. A full decomposition of the elements contained in thought discovers that the predicate is always of a given quantity in relation to the subject ; that this is known and recognised as the condition of predica- tion. It thus reveals that the relation between the terms of a proposition is one not only of similarity, but oiidentity ; that the subject and predicate of a proposition, when the relation of each to the other is recognised, and both are quantified, are always neces- sarily simply convertible ; that the terms of a propo- sition, in short, are of an absolute equality, and all predication an equation of subject and predicate. Quantify the predicate, and two notions of different extension are at once brought into equality ; the sphere of an individual object in a notion is marked out, and that sphere becomes absolutely convertible with the object ; e.g., — All man is some animal. Some animal is all man. 32 NEW ANALYTIC OF LOGICAL FOEMS. Some men are all philosophers. AH philosophers are some men. . Boethius is some Roman. * Some Roman is Boethius. Thus, on the principle of the new analytic the whole doctrine of ordinary conversion, with its com- plex species and its manifold rules, passes away, and the whole process becomes as practically simple as it is scientifically complete. The terms of a pro- position are exhibited in their true relation, and that relation reduces all the species of conversion to one — that of simple conversion. Thus, in the words of an acute writer, whose apt statement in relation to this doctrine is true in a far wider ap- plication than he designed it — to the whole and not to a part alone of the doctrine of conversion. " Omnes conversionum leges pendent a cohaesione, vel potius ab identitate subjecti et attributi : quod si enim subjectum conjungitur et identificatur ut aiunt, cum attribute, necesse est pariter attributum uniri et identificari cum subjecto." '"' We proceed to consider — III. The influence of this principle on the doctrine of categorical syllogisms, in contributing to effect, 1°, the reduction of their general laws to one ; 2°, the * ^^ Philosophia Burgundica'' 1678, torn. i. Institutiones Logicae. Sect. Secunda, cap. 2. (By Du Hamel.) NEW ANALYTIC OF LOGICAL FORMS. 38 abolition of their special laws : and from this new simplicity the amplification of the valid forms of rea- soning. We premise a word or two on the nature of a syllogism in general, and of a categorical syllogism in particular. " A syllogism is the product of that act of mediate comparison, by which we recognise that two notions stand to each other in the relation of whole and part, through the recognition that these notions severally stand in the same relation to a third." A syllogism or reasoning is thus like a concept and a judgment the product of the comparative faculty — the comparison of part and whole. This indeed is the characteristic of all reasoning — alike of the simple syllogism and of the most lengthy and profound argu- ment. All reasoning is but the comparison and deter- mination of wholes and parts. As in concepts various attributes, in judgments various thoughts, are com- pared in order to determine the relation of part and whole subsisting between them; so in reasoning two notions are compared together with a third in order to determine their connexion with each other — the only difference being the higher complexity which in this case the act of comparison assumes. A rea- soning thus differs from a judgment in the superior complexity of the act, in being an immediate act of comparison in which two notions, whose relation c 34 NEW ANALYTIC OF LOGICAL FOKMS. to each other is unknown, are compared together through a third, whose connexion with both is recog- nised. But it agrees with a judgment in being an undivided act of mind ; for as the connexion of part and whole between two notions, enounced by a reason- ing, is determined by the recognition of their mutual relation to a third ; and as relatives are only recog- nised together, it follows that it is an undivided act of consciousness.'"'' A syllogism indeed forms as truly a mental whole as a concept, though each is capable of being subsequently analysed for scientific purposes into its constituent elements. On being subjected to such analysis, every true syllogism or reasoning is found to contain three, and no more than three, pro- positions, each of which has three, and only three, terms. There will be three propositions, since the two notions, touching whose relation the mind is in doubt, are both compared with another whose rela- tion to each is manifest. This affords two proposi- tions, in one of which the third notion is a contained part in relation to one of the doubtful notions ; and in the other a containing whole in relation to the other doubtful notion. And there is of necessity the conclusion in which the doubt is dispelled, and the relation of the tw^o notions themselves determined. This process is in every valid syllogism determined by a law of thought, and the connexion is thus one * Hie modus (per syllogismum) ratiocinandi est ex simplissimis, et intuitive uno actu mentis perspicitur. — Ploucquet. NEW ANALYTIC OF LOGICAL FORMS. 35 of absolute necessity. There will, it is obvious, be as many valid kinds of syllogism as there are different laws of thought on which they may be respectively founded. A categorical syllogism is one in whose major pre- mise the relation of the terms is simple ; whose pro- cedure is determined and whose conclusion is neces- sitated by the laws of identity and contradiction. Having premised thus much about syllogisms in general, and categorical syllogisms in particular, we shall proceed — i. To state with brevity the common doctrine of syllogistic figure, mood, and reduction; and then generally some of the defects by which it is charac- terised. And, ii. To state the one supreme canon of the new analytic, which potentially contains the whole doctrine of categorical syllogisms, and then proceed to develop from it some parts of that doctrine. We proceed then — i. To state the common doctrine of syllogistic figure, mood, and reduction. Figure. — What is commonly termed by logicians syllogistic figure arises from the relation of the middle term as subject or predicate to the extremes. The four possible varieties of position which the middle 36 NEW ANALYTIC OF LOGICAL FORMS. term may occupy in the premises thus determine four syllogistic figures. ^ In iYiQ first figure the middle term is the subject of the major premise, and the predicate of the minor ; e.g,,— All rational is risible. All man is rational. Therefore — All man is risible. The laws of this figure are — 1. That the subsumption'" be affirmative. * [These words sumption and subsumption are new, and may there- fore require a few words of explanation if not of defence. They are introduced and employed by Sir William Hamilton to express the two first propositions of a syllogism, instead of the common designa- tions major and minor premise. We much need apt terms by which to express these members of the syllogism, and for the creation of new or introduction of foreign words in such a relation, may certainly urge the first part of the plea of Lucretius, "propter egestatem linguae,'^ if we cannot go on to add with him, " et rerum novitatem." The members to be named are old enough, but they still have never received precise and discrimi- native epithets, at least in our logical terminology. The terms major and minor premise are objectionable, if for no other reason, from the confusion of terms with propositions likely enough to arise from the omission of the second member of the term, and the consequent in- discriminate use in hasty reference of the epithets major and minor alone. Single, precise, and discriminative words would on every account be far better than these combinations. The term propositio was the designation of the major premise of a syllogism, from the days of Cicero downwards, with few exceptions. Among these excep- tions are Quintilian who uses intentio, Boethius who sometimes uses sumptum, and Rodolphus Agricola who employs expositio. Assump- tio was even more generally the designation of the second proposition of the syllogism — the minor premise. So that, in fact, when the two first members of a syllogism are not (after Aristotle) called the major NEW ANALYTIC OF LOGICAL FORMS. 37 2. That the sumption be universal. In the second figure, the middle term is the pre- dicate in each premise ; e.g., — No liar is to be believed. Every good man is to be believed. Therefore — No good man is a liar. and minor proposition, the terms proposition and assumption are generally found employed to express them. These terms all more or less shadow forth the relation of subordination which exists be- tween these parts of the syllogism. This relation is, however, far more aptly and explicitly denoted by the correlative terms sumption and suhsumption ; and as we have already assume and assumption with other cognate terms, there is no reason why we should not also avail ourselves of the convenient terms subsumption and subsume. Even the terms sumption and subsumption are not, however, equal to the whole extent of the necessity, for between the two first mem- bers of syllogisms, in the second and third figures, there is no such relation of subordination as they express, so that it is only by courtesy to custom that these terms can be applied to them. The same is true of the reasoning from wholes to wholes ; so that we still need terms of generic latitude sufficient to express the two first members of any syllogism. For all ordinary purposes, however, the above are sufficient. They have the great merit of being single and precise epithets ; and after what has been said, their use in the figures can- not be misunderstood. The only thorough-going and consistent attempt ever made, that 1 am aware of, to render the technicalities of logical science into English terms, was that of Ralph Lever, Dean of Durham. In his logical treatise, entitled, " The Art of Reason, rightly termed Wit- craft, teaching a perfect way to argue and dispute," and published in London in the year 1573, he expressly undertakes to accomplish this. He explains and defends his procedure in the preface, {forespeach,) of which the following extract may be taken as a specimen : — " For trial hereof I wish you to aske any English man, who understandeth neither Greek nor Latin, what he conceiveth in his mind when he heareth this word, a backset, and what he doth conceive when he heareth this term, a predicate. And doubtlesse he must confesse, if 38 NEW ANALYTIC OF LOGICAL FORMS. The laws of this figure are — 1. That one of the two premises be negative ; and consequently that the conclusion be so. 2. That the sumption be universal. In the third figure, the middle term is the subject in both premises ; e.g., — All man is risible. All man is capable of science. Therefore — Some capable of science is risible. The laws of this figure are — 1. That the subsumption be affirmative. 2. That the conclusion be particular. In \he fourth figure, the middle term is the predi- cate in the sumption and the subject in the subsump- tion ; e.g., — All oranges are fruit. All fruit is refreshing Therefore — Some refreshing things are oranges. he consider the matter aright, or have any sharpnesse of wit at al, that by a backset he conceiveth a thing that must be set after, and by 21, predicate that he doth understand nothing at all." He accordingly renders every (or certainly almost every) technical term of common use in logic by combinations of purely Saxon words. We will give both as a specimen of his coinage, and as pertinent to the purpose of the present note, the terms which he has used to express the differ- ent members of the syllogism. These are foresay and endsay, first foresay, (major premise,) second foresay, (minor premise,) endsay, (conclusion ;) or, in his own words, — " The two first shewsayes (propo- sitions) that are placed in a reason by rule, are called foresayes — the third may be termed an endsay." {Art of Witcraft, p. 103.) These terms are sufficiently general for any form of syllogism ; but as the technical terms of the science are all of Latin derivation, Saxon com- pounds cannot be accepted.] NEW ANALYTIC OF LOGICAL FORMS. 39 The laws of this figure are — 1. When the sumption is affirmative, the sub- sumption is always universal. 2. When the subsumption is affirmative, the con- clusion is particular. 3. If either of the premises be negative, the sump- tion must be universal."' Mood. — What is called mood is a modification of a syllogism, determined by the quantity and quality of the propositions of which it is composed. All the figures admit of syllogistic variations thus deter- mined. Four kinds of propositions are, according to logicians, afforded by the various possible combi- nations of quantity and quality. Universal affirma- tive, (A) — universal negative, (E) — particular affir- mative, (I) — particular negative, (0.) And as there are three propositions in every syllogism, all the possible combinations of quality and quantity will be sixty-four. Of these sixteen are excluded from hav- ing negative premises ; twelve from having parti- cular premises ; twelve more for having a negative premise with an affirmative conclusion ; eight, from one of the premises being particular, but the con- clusion universal ; and, finally, four more from hav- ing a negative conclusion where both premises were affirmative. There are thus left twelve moods. Of these, however, for various particular reasons, six * These laws are taken from the Fort-Royal Logic, but they are substantially the same in most logical systems. 40 NEW ANALYTIC OF LOGICAL FORMS. only are allowable in each figure, making twenty- four in all. Of these, however, five are again thrown out of account from having a particular conclusion where a universal is competent. There thus re- main in all the figures only nineteen valid moods. These are embodied in the well-known lines — " Bar- bara Celarent" &c. Reduction. — Logicians, though altogether ignor- ant of the true character of figure, strictly so called, seem always, however, to have felt, that the form of reasoning in the first figure was more exact than in the three others, and the conclusion afforded by it more logically direct and satisfactory. They have, accordingly, devised a process for changing syllo- gisms of the three other figures into those of the first. This process is technically/ termed reduction. Reduction is twofold — 1. Reductio ostensiva. — This is effected by the conversion and transposition of propositions ; e.g. — Disamis of the third figure into Darii of the first. Disamis. Some tyrant is unjust. All tyrants are cruel. Some cruel is unjust. Here the major premise and conclusion are first converted, and then the premises transposed ; thus — Darii. All tyrants are cruel. Some unjust are tyrants. Some unjust are cruel. NEW ANALYTIC OF LOGICAL FOKMS. 41 2. Redudio ad impossibile. — This is effected, not directly, but indirectly, by proving that the contra- dictory of the syllogism under examination is some- thing impossible or absurd. This is accomplished by taking the contradictory of the conclusion with one of the premises, and inferring from these the contradictory of the other premise — in the second figure the contradictory of the minor — in the third the contradictory of the major premise. Thus — BaroGO of the second into Barbara of the first figure : Baroco. All rational is risible. Some animals are not risible. Some animals are not rational. Barbara. All rational is risible. All animal is rational. All animal is risible. Such, on the common doctrine, is a short account of syllogistic figure, mood, and reduction. We go on to state generally some of the defects by which this doctrine is characterised. 1. The whole doctrine is cumbrous and unsatis- factory. In addition to the general laws which are laid down by logicians as governing all syllogisms, and the various kinds of syllogism which have been sig- nalised, we have here what are represented as four new essential variations of the syllogistic form, each 42 NEW ANALYTIC OF LOGICAL FORMS. guarded by its own complement of special laws — forming altogether a code of particular rules for the detection of petty offences, which is certainly suffi- ciently intricate and perplexing. Now, the necessity of these empirical laws is not attempted to be vindi- cated on any thorough-going logical principle ; and until it be, we think their minute and multiform cha- racter is valid ground of objection. No right, more- over, has been established, that we know of, for the three last figures themselves, to occupy the place they do as true variations of the syllogistic form, ex- cept a prescriptive right, which in logic, where so many false principles have for so long usurped author- ity, and so many true ones been ignored, w^hen un- accompanied by any other, is rather ground for sus- picion than otherwise. Accordingly, till a better right than this be shown why they should be con- sidered independent syllogistic forms, their want of conformity to recognised law in their procedure, and the indirectness of their conclusions, furnish good ground of dissatisfaction with their logical position. Next we have the moods in each figure — the test- ing of the true moods in each by its particular laws — these laws determining the exclusion in one figure of moods that are held valid in another ; and this exclusion thus necessarily producing a numerical dif- ference of moods in different figures. That what is a true mood in one figure should be a false one in another, is, to say the least, unsatisfactory. NEW ANALYTIC OF LOGICAL FORMS. 43 And, finally, we have the doctrine of eeduction — the most cumbrous and unsatisfactory part of the whole. The first half of the process involves a pro- cedure opposed to other parts of the common doc- trine, since it admits the transposition of premises at will. (Of this, however, more presently.) The procedure of the second half of the process is almost ingeniously perplexed and cumbrous, and when accomplished, so far as we see, answers no end what- ever. We form an entirely new syllogism, which throws no light upon the old ; illustrates no part that was before obscure ; gives us, indeed, no know- ledge at all. The old figure remains with its irregu- larity (if it had any) uncorrected ; with its illegal aspect (if that were its vice) unremoved. It is, in short, cumbrous without the justification of being useful ; and most unsatisfactory it should seem in the mere wantonness of being so, since it cannot urge the extenuating plea of necessity. 2. This doctrine is inconsistent. We begin by noticing the inconsistency displayed in the discrimination of moods on the common doc- trine. We have seen that the principle on which the discrimination of moods proceeds, is the differ- ence in quantity and quality of the propositions which constitute a syllogism. In order to deter- mine a variation of mood in a syllogism, one at least of its propositions must differ, in one or other of these respects, from all other co-ordinate moods. If in 44 NEW ANALYTIC OF LOGICAL FORMS. two syllogisms of a given figure there be no differ- ence, either of quantity or quality in the proposi- tions of which they are composed, these syllogisms are logically reckoned as one and the same : there is in this case no modal variation. This at least is the common doctrine, if that doctrine have any meaning. In opposition thereto, however, we have discriminated as true varieties i/i the second figure — Camestres. All wise men are truly happy. No intemperate man is truly happy . No intemperate man is a wise man. Cesare. No intemperate man is truly happy. All wise men are truly happy. No wise man is an intemperate man. And in the third figure — Datisi. Every true patriot is brave. Some true patriots are persecuted. Some that are persecuted are brave. Disa7nis. Some true patriots are persecuted. Every true patriot is brave. Some who are brave are persecuted. Now, the syllogisms thus given under each figure have no difference w^hatever in the quantity or qua- NEW ANALYTIC OF LOGICAL FOEMS. 45 lity of their propositions ; they are, therefore, accord- ing to the theory of logicians, the same. Practically, however, they have been considered from time im- memorial as distinct ; as constituting valid variations of syllogistic form under each figure. If this be so, the principle on which the discrimination of mood avowedly proceeds is, at least, implicitly given up, and another is introduced — that of the transposition of propositions ; for it is on this principle alone that the syllogisms given above can be vindicated as distinct. If, however, the transposed order of propositions be recognised as affording a w^orthy principle for the dis- crimination of syllogistic difference, then a number of new forms will emerge, of which logicians have taken no account. The catalogue of syllogistic va- riations will immediately be swelled far beyond its present limits, since every existing variety is capable, by the transposition of its propositions, of receiving a fivefold amplification. It is clear, however, that the transposed order of propositions affords no prin- ciple of any scientific value in the discrimination of syllogisms ; and, what is more to the present pur- pose, it is equally clear, that whatever be its value, it is one which has not been recognised by logicians, since they have not even incidentally adverted to it in the exposition of their doctrine. The principle of Aiodal variation on that doctrine is manifestly the difference of propositions in quantity and qua- lity. In obvious inconsistency, however, with this 46 NEW ANALYTIC OF LOGICAL FORMS. recognised law, there remain the syllogisms stated above. But a more important, if not more glaring inconsis- tency remains to be noticed in the correlative doctrines of figures and reduction."^' The opposition here is so great that it seems to us truly marvellous that they should ever have existed together. They appear mu- tually and triumphantly to destroy each other. We will endeavour to explain what we mean in a few words. The inconsistency may he stated thus : In the cogency of formal proof there can be no degrees. Logic, it need hardly be repeated, is a formal science, proceed- ing from, and determined through its whole course by, the laws of thought. This being the case, every true variation of the syllogistic form will afford a conclu- sion equally valid and direct ; since a form of rea- soning is true only as determined by a law of thought, and the laws of thought are equally universal and imperative. If a given syllogistic form be deter- mined as original and essential, it is already self- sufficient ; it needs no help, it can receive none. No process can be devised for bestowing on its con- clusion a higher validity than that which it possesses of its own inherent right. For, as we have said of formal proof, there can be no degrees of cogency. In every case it is alike valid and direct. With * Throughout the ensuing discussion we use ^* figure" simply in the sense of figure strictly so called, i.e., to the exclusion of the first, in order to avoid constant circumlocution. NEW ANALYTIC OF LOGICAL FORMS. 47 respect, therefore, to every true variation of syllogistic form, what is called reduction is useless. On the other hand, if the conclusion afforded by a syllogism be one-sided and indirect, and its form, therefore, apparently imperfect or irregular, there is good ground for suspicion that there is some com- plexity or confusion in its form, which it is necessary to disentangle or clear away before the syllogism emerges in its purity, and before, therefore, its true form can be correctly determined. When this con- fusion or complexity is removed — and it is the office of logical analysis to accomplish this — the syllogism will appear in its real character, and naturally fall under some determinate class of recognised syllogistic form. To such a syllogism what is called reduction is obviously inapplicable, since we cannot talk of re- ducing a syllogism of a given form to itself. Thus, on either alternative, the doctrine of reduction seems alike inapplicable, if not impossible. If the syllo- gistic form he essential, reduction is useless ; if it he accidental, it is absurd. Logicians, however, so far as we can understand the common doctrine, seem to have chosen the former, and perhaps milder alternative, — that the figures are true variations of syllogistic form. If this were not the case, it would have been explicitly stated ; and, in fact, the doctrine of reduction itself is at once the evidence that they so regarded them, and the proof that this is not their true character. If they did not so consider them, 48 NEW ANALYTIC OF LOGICAL FORMS. it is difficult to account for the existence of reduction at all ; for it is introduced expressly to strengthen the position of the figures, and to vindicate to their conclusions the highest validity. Not clearly com- prehending the true nature of the figures, their doc- trine respecting them was imperfect. They deter- mined that they were true syllogistic varieties, but still felt that this was not perfectly satisfactory — that there was something wanting in their doctrine ; and in order to bestow upon it the requisite completeness, they unhappily fell upon the doctrine of reduction — a process which, so far from establishing the truth of that which it was introduced to demonstrate, is the clearest evidence of its falsehood. In fact, we can- not but regard the whole doctrine as the work of a most perverse and suicidal ingenuity. Introduced as a bulwark of strength to the figures, it becomes the very mockery of their weakness, and affords its sus- taining aid only in the moment of their dissolution. For we certainly cannot understand how a doubtful variation of syllogistic form is vindicated as valid and essential through a process which accomplishes its destruction ; in other words, by being changed into a form, of the validity of which there is no question. But even granting that the whole doctrine was, in the main, correct, it is chargeable with inconsis- tencies of detail ; such as the explicit recognition at one time of a determinate major and minor premiss in the figures, and the implicit denial of this at an- NEW ANALYTIC OF LOGICAL FORMS. 49 other. These, however, it is not necessary now to notice. Suffice it to have shown that the related doctrines of figure and reduction are most seriously inconsistent, if not mutually destructive of each other. 3. The common doctrine is destructive of the science itself. We bring this charge against the common doctrine — and it is the gravest of all — that it is destructive of the science, by implicitly, at least, impeaching the veracity of the laws of thought upon which the science is founded. This may be illustrated in one or two ways. In all categorical syllogisms the reasoning is founded upon the law of identity, that a whole is identical with all its parts, a concept with its attributes, &c., and thus that " a part of a part is a part of the whole." The reasoning, therefore, if it be valid, always of necessity involves the subordination of one part to a whole through a larger part.* * [This statement requires some modification ; for, on the law of identity, the reasoning from wholes to wholes is as competent as that from parts to wholes. By this law we are entitled to infer the rela- tion of identity between two wholes, from the perception that they stand in this relation to a common third whole, just as certainly as we are to infer the relation of inclusion between a given whole and part, from the perception that these stand in the same relation to a common third part ; e.g., the reasoning all B is all A, all C is all B, therefore all C is all A, is just as competent as the reasoning all B is A, all C is B, therefore all C is A. Taken absolutely, therefore, the objection stated in the text will not stand the test of criticism. Considered relatively to the common doctrine, it is, however, -valid. On that doctrine all direct formal reasoning from wholes to wholes D 50 NEW ANALYTIC OF LOGICAL FORMS. Now, the common doctrine recognises the figures strictly so called as ultimate varieties of syllogistic form ; implicitly, at least, declares that they are simple forms, and their reasoning valid as it stands. But in these figures there is no subordination of lesser parts to wholes through larger parts. It is therefore clear that if the law, according to which all true reasoning is affirmed to proceed, en- joins this subordination, but that there are found, and recognised as valid, forms in which there is no such subordination — that the law is no longer trust- worthy ; for having proved mendacious once, its character of necessity and universality has departed, and it may deceive again. Thus, implicitly at least, the main foundation of all reasoning is cut away. Again, we have seen in the common doctrine the numerical inequality of the moods under different figures. In each figure some mood is determined as valid which is ignored by the rest. Thus throughout the whole scheme moods are retained as true at one time which are rejected as false at another. Now, what is the test by which the validity and is impossible. For a direct reasoning from wholes to wholes is only possible through the express quantification of the predicate, which the forms and rules of the common doctrine alike forbid. In every reasoning which obeys these rules, the extent of some one term in the syllogism is necessarily left indeterminate, so that its identity of extent with any other term cannot be formally inferred. On the common doctrine, the reasoning is thus necessarily from parts to wholes. Taken with this explanation, therefore, the statement in the text may stand.] NEW ANALYTIC OF LOGICAL FORMS. 51 invalidity of any and every variet}" of syllogism is determined ? No other than its ohedience to, or vio- lation of, the laws of thought. If a syllogism agrees with these laws, it is true ; if not, it is false. This criterion imperatively determines every possible va- riation of syllogistic form. If a syllogism be true to the laws of thought in its formal essence, no acci- dental irregularity or transposition in the arrange- ment of its parts can render it invalid ; if it be not thus true in the inward essence of form, no outward regularity of expression can supply the wanting validity. In other words, what the laws of thought allow, no schematic difference can ever ignore ; w^hat the laws of thought condemn, no schematic variation can ever successfully vindicate. In direct opposition to all this stands the common doctrine. Judged by the above standard, its whole procedure is most illegal. In its admissions and ex- clusions of syllogistic variety it is equally capricious and empirical ; and throughout it is consistently regardless of the great laws by w^hich the whole process, as a logical procedure, must be ever deter- mined. We find a certain mood rejected under a given figure — we presume because it is invalid ; but it can be invalid only because it violates the laws of thought. Presently after we find this same mood reappearing in another figure, and recognised as scientifically true. What the laws of thought ere- while condemned, the schematic difference now tri- 52 NEW ANALYTIC OF LOGICAL FORMS. umphantly interposes to allow. What we had sup- posed to be rejected from the science as logically false, now boldly returns under express scientific sanction. Thus, on the common doctrine, the accidents of arrangement triumph over the essentials of form. : the contingencies of expression are of higher authority than the necessities of thought. In a formal science we have its accidental arrangement actually de- throning its essential principles ; contingent and particular irregularities boldly usurping that author- ity, through the legitimate exercise of w^hich they would have been themselves exiled, — that authority through the recognition of which alone the science can be preserved — the authority of law necessary and universal. If the procedure which issues in these results be competent, then it is evident that logic exists no longer; for the destruction of the science is ob- viously involved in the destruction of the laws on which it is exclusively founded. And this destruc- tion of the laws of thought is, as we have shown, in effect accomplished by the common doctrine, in the practical contradictions of these which its procedure involves. It thus implicitly contains in it principles which, if fully developed, would overthrow logical science. We have thus seen amongst the more general de- fects which belong to the common doctrine, that it is practically cumbrous and unsatisfactory ; that it NEW ANALYTIC OP LOGICAL FORMS. 53 is theoretically inconsistent ; and that it involves principles destructive to the science itself. The special falsity and uselessness of this doctrine will appear more fully in detail under the division to which we now proceed. ii. To state the one supreme canon of the new analytic, which potentially contains the whole doctrine of categorical syllogisms, and then to develop from it some parts of that doctrine. That canon is — " What worse relation of subject and predicate subsists between either of two terms and a common third term, with which both are re- lated, and one at least positively so — that relation subsists between these two terms themselves." This canon, as we have said, involves the whole doctrine of categorical syllogisms ; determines every kind of such syllogisms ; and is to them an all-suffi- cient and exhaustive code of law, observing which none can be formally invalid. We have now to show, in conformity with the pur- pose of this essay in general, and its third division in particular, the influence which the principle of a quantified predicate has in accomplishing the reduc- tion of syllogistic rules to this single canon. And here it is obvious at once, that before any such reduction of the general laws of these syllo- gisms can take place, the special laws which govern particular classes of such syllogisms must be dealt 54 NEW ANALYTIC OF LOGICAL FORMS. with. In fact, the condition of the possibihty of any such reduction of the general laws is the annihila- tion of the special laws. Whatever, therefore, tends to effect the abolition of these special laws, (laws of the several figures, to wit,) tends directly to facilitate that simplification of syllogistic law which emerges in the single canon. The principle of a quantified pre- dicate directly contributes to accompHsh this, by proving the falsity and uselessness of these special rules. For example, the laws of \hQ first figure are — that the sumptio7i be universal, and — that the subsumption be ajffirmatwe. Quantify the predicate, however, and neither of these laws hold. First rule falsified. Some men are some fleet-footed. All rational is all man. Some rational is some fleet-footed. Second rule falsified. All idealists are some philosophers. No sensualist is any idealist- No sensualist is some philosopher. The laws of the second figure are — that one of the premises be negative, and — that the sumption be uni- versal. To these the principle of a quantified predi- cate immediately applies, and falsifies them — First nde falsified. All risible is all man. All philosophers are some men. All philosophers are some risible. NEW ANALYTIC OF LOGICAL FORMS. 55 Second ride falsified. Some mortal is all man. All rational is all man. All rational is some mortal. These examples may serve to illustrate specially the influence which the principle of a quantified pre- dicate has upon these laws. It is to be remarked^ however, generally, that these laws exist only in con- sequence of a defective logical analysis, and apply only to such an imperfection. When the analysis, therefore, is complete, these laws naturally fall away as henceforth useless and inapplicable. The recog- nition of a quantified predicate and of the true nature of figure renders this analysis complete. Thus the principle of a quantified predicate co-operating with the true doctrine of figure, sweeps away for ever from logic, as an encumbrance, all these special laws. The special laws being swept aside, the way is prepared for the reduction of the general. And this reduc- tion effected, there emerges the supreme canon as given above. This canon, it is obvious, may be again easily evolved into those general laws of which it is the compend. We may take as an example the two most general laws of categorical syllogisms. 1. That both premises he not negative. This is ex- pressed in the canon by the clause — " With which both are related, and one at least positively so" 2. That the middle term he distributed in one at least of the premises; and it is of no consequence 56 ^^EW ANALYTIC OP LOGICAL FORMS. whether it be distributed as the subject or the predi- cate. The first part of this law is expressed in the canon by the clause — " related to a common middle term.'' If it be common to both it must be distributed. The latter part of the law is expressed in the clause — " What worse relation of subject and predicate sub- sists between either of two terms and a common third term.'' The common third term may thus be related to the two other terms either as subject or predicate. It would be by no means difiicult to evolve in the same manner the canon into the six more general syllogistic laws commonly given by logicians. The relation of these' laws to the diiferent clauses of the supreme canon is, however, manifest, and therefore need not be formally evolved here. We pass on, then, to consider this canon in another aspect. We said that it determined every kind of categorical syllogism. We shall endeavour to illustrate this; — 1. Syllogisms differ with respect to the wholes in which they proceed. We have already said that all reasoning (deduc- tive, that is, to which, when not otherwise specified, we always refer when speaking of reasoning in general) is from whole to part ; but as there are two kinds of logical wholes and parts, there will naturally be two kinds of reasoning, corresponding severally to these different quantities. These wholes are — the meta- NEW ANALYTIC OP LOGICAL F0KM8. 57 physical or comprehensive whole ; and the logical or extensive whole. A syllogism proceeding in the for- mer is a comprehensive syllogism, in the conclusion of which the subject is the greatest whole, and the predicate the smallest part. A syllogism proceeding in the latter whole is an extensive syllogism, in the conclusion of which the subject is the smallest part, and the predicate the greatest whole. These differ- ent kinds of syllogism, distinguished by the different kinds of wholes in which they proceed, are determined by the first clause of the canon — " What worse rela- tion of subject and predicate subsists between two terms," &c. In the whole of comprehension the predicate is Avorse than the subject, since it is a part in relation to a whole; in the counter whole of extension, the subject is worse than the predicate, since in this quantity the predicate has become the greatest whole, and the subject the smallest part. 2. Syllogisms differ with respect to figure. The variation of figure arises, as we have seen, from the various positions of the middle terra in relation to the extremes. This variation is evidently determined in the canon by the clause — " What relation subsists between either of two terms and a common third term!' But here it is necessary to go somewhat more fully into detail. We shall therefore return to the canon presently. In considering figure somewhat more closely, we shall notice ; — 58 NEW ANALYTIC OP LOGICAL FORMS. a. The true nature of the figures. We have seen that the figures, strictly so called, are regarded by logicians as true and original varia- tions of the sjdlogistic form. We have seen also some of the inconsistencies which such a doctrine involves: enough certainly to beget a suspicion that it could not be the true one. We were thus led to suppose that there must be some confusion or complexity in the figures which it is necessary to remove before the true form of their reasonings could be seen in its ul- timate purity and exactness. What this confusion or complexity is, we now proceed explicitly to show, by stating and illustrating the true nature of the figures.'"' " The figures (strictly so called) are hybrid or mixed reasonings, in which the steps of the process are only partially expressed ; the unexpressed steps are, in general, only conversive inferences w^hich we are en- titled to make from those that are expressed." This being the nature of figure, it follows that, since all the real steps of the process are not expressed in the reasonings, the conclusion does not of necessity follow from the expressed premises ; but the mind at once inferring and interpolating the wanting steps of the process, the conclusion follows in virtue of such inference and interpolation. When this mental in- terpolation is recognised, and the real premise which it constitutes is expressed, the s^dlogism emerges in * Touching the value and history of this exposition, see the Ap- pendix. KEW ANALYTIC OF LOGICAL FORMS. 59 its simple form, and is at once recognised to be (through all the variations of mood and figure, strictly so called) a syllogism of the first figure. All such varieties are therefore thus shown to be, in their com- plex state, only unessential variations of that figure. We shall illustrate this in concrete examples through the moods of the second and third figures. The moods of the second figure are — Gesare, Oa- 7nestres, Fesiino, Baroco. A syllogism in Cesare is — No unreflective man is a philosopher. All idealists are philosophers. No idealist is unreflective. By conversive inference we obtain as the real sump- tion, " no philosopher is unreflective," and this inter- posed in the place of the ostensible sumption, pro- duces a syllogism in Celarent of the first figure, e.g., — No philosopher is unreflective. All idealists are philosophers. No idealist is unreflective. A syllogism in Gamestres is — All animals are sentient. Nothing unorganised is sentient. Nothing unorganised is animal. We have already said, when speaking generally of figure, that the premises are in this syllogism trans- posed ; and that but for such transposition it is exactly the same as Gesare; reversing the premises then, and 60 NEW ANALYTIC OF LOGICAL FORMS. dealing with it in the same manner as the last, it ap- pears in the same form as Celarent of the first figure. Nothing sentient is unorganised. All animals are sentient. No animal is unorganised. A syllogism in Festino is — No truly wise men go to extremes. Some truly religious men go to extremes. Some truly religious men are not truly wise men. Here with the sumption converted the syllogism appears as Ferio of the first. None who go to extremes are truly wise. Some truly religious men go to extremes. Some truly religious men are not truly wise. A syllogism in Baroco^ is — All lilies are fi'agrant. Some flowers are not fragrant. Some flowers are not lilies. Here the subsumption is to be dealt with ; and by conversive influence we obtain " some things not fra- grant are flowers^ Interpolating this, we have a syllogism in Darii of the first. * [The reduction of this mood was one of the standard difficulties of the logicians. It could only at best be done with difficulty, and through the clumsy process designated ad impossihile ; nor does it accommodate itself well to this expository process of Kant's ; since after all it appears as a syllogism of the fourth figure rather than of the^^rs^.] NEW ANALYTIC OF LOGICAL FORMS. 61 All lilies are fragrant. Some things not fragrant are flowers. Some things not lilies are flowers. The third figure. We have seen, in removing the complexity of syl- logisms in the second figure, that it is the sumption for the most part (that is in three out of four moods) which is aftected by conversive inference ; and that that inference is made chiefly from " the absolute negation of the first notion as predicate of the second, to the absolute negation of the second notion as pre- dicate of the first.'' In the third figure, on the other hand, it is the subsumption which is chiefly affected by conversive inference, and that inference is generally made " from the total or partial affirmation of a lesser notion of a greater, to the partial affirmation of the greater notion of the lesser." The moods of the third figure are — Darapti, Felap- ton, Disamis, Datisi, Bocardo, Ferison. A syllogism in Darapti is — Every good man is happy. Every good man fights with himself. Some who fight with themselves are happy. By conversive inference we here obtain as sub- sumption — " some who fight with themselves are good men ;" and the syllogism is then in Darii of the first figure. Every good man is happy. Some who fight with themselves are good men. Some who fight with themselves^ are happy. 62 NEW ANALYTIC OP LOGICAL FORMS. A syllogism in the second mood Felapton is — No man is winged. All men are bipeds. Some bipeds are not winged. By converting the subsumption it becomes Ferio of the first. No man is winged. Some bipeds are men. Some bipeds are not winged. J syllogism in the third mood Disamis is — Some patriots are persecuted. Every patriot is brave. Some that are brave are persecuted. Here the premises are first to be transposed, then converting the subsumption the syllogism appears as Darii of the first— Every patriot is brave. Some that are persecuted are patriots. Some that are persecuted are brave. With respect to the fourth mood in the figure, Datisi, we have already shown that it differs from the third only in the transposition of its premises. Con- verting the subsumption, therefore, it appears in the same form as Darii of the first. We give the syllo- gisms together — All true philosophers are truly noble. Some true philosophers are despised. Some that are truly noble are despised. KEW ANALYTIC OF LOGICAL FORMS. 60 All true philosophers are truly noble. Some that are despised are true philosophers. Some that are despised are truly noble. A syllogism in the fifth mood Bocardo is — Some poets are not philosophers. All poets have genius. Some who have genius are not philosophers. The premises are here transposed, and the sump- tion converted — it then appears as Darii — All poets have genius. Some not philosophers are poets. Some not philosophers have genius. A syllogism in the last mood Ferison is — No hope is unattended with pleasure. Some hopes are delusive. Some delusive things are attended with pleasure. Converting the sumption it becomes Ferio of the first — No hope is without pleasure. Some delusive things are hopes. Some delusive things are attended with pleasure. The syllogisms of the figures, strictly so called, are thus shown to be complex reasonings, which, when cleared of their complexity, and simply ex- pressed, i.e., expressed in that form which all true reasonings must ultimately assume, in which the least part or whole is subordinated to the greatest part or whole, through a lesser part or whole — appear in their true character as syllogisms of the first figure. 64 IS^EW ANALYTIC OF LOGICAL FORMS. We have here taken no notice of the fourth figure, not because the same process of simplification is not equally applicable to its reasonings, as to those of the second and third figures ; but because, as we shall presently explain, we do not consider it properly a figure at all. h. The true number of the figures. We have seen that in the common doctrine four figures are recognised as valid.* The fourth figure * [The validity of the fourth figure as a separate form of reasoning has been often contested, but that of the other three has remained for the most part unassailed. The third, however, has not wholly escaped assault. It was rejected by Laurentius Valla on the ground that such a form of reasoning was never used, that there are no examples of it to be found, and as it would seem, as much as for any other reason, because it offended his eye or ear. He does not, in- deed, so much reason seriously against its validity, but rather de- nounces it at once, and that, too, in the most lively and rhetorical manner, as in the last degree preposterous and absurd. The whole style of the rejection is quite in harmony with his wayward original- ity and independence, as well as with his habit of rash yet fastidious criticism. It is, indeed, but another instance of the fastidious taste of Valla, in whose eyes a sin against purity of style was a moral offence of the gravest kind, who did not scruple to correct the Latinity of Cicero, and of whose criticism it is said the Devil himself stood in such awe, that he was afraid to speak in his presence, being nervous as to the classic purity of his Latin style. The objections of Valla to the third figure were rebutted by Lazarus Schonerus, (as quoted by Fraunce,) who adduced examples of its use from Cicero and Virgil. Its validity and usefulness were also defended against Valla by Melanchthon, who, referring to it, says, " Laurentius Valla non leviter stomachatur hoc loco, et Aristotelem tanquam capitali judicio accusat, qui banc figuram tradiderit. Sed Valla dum nullum rixandi finem facit, saepe etiam incurrit, ut sit ab iracundis, in illos qui nihil peccaverunt. Mihi non tam plumbeo ingenio Aris- NEW ANALYTIC OP LOGICAL FORM55. 65 was, however, introduced into the science later ; and when introduced obtained a footing less secure than the other three. Some logicians, indeed, have omitted it altogether ; while others have expressly redargued its claim to be admitted as a true schematic difference. The reasons of those who opposed its admission have not, however, proved sufficiently strong to eject it finally from the science, since it reappears in the latest systems, and is recognised as logically competent. It is, nevertheless, to be rejected from logic, as being utterly deformed and useless. Deformed, since its premises proceed in the whole of comprehension, and its conclusion in the counter whole of extension- Useless, since the reasoning in both these wholes is scientifically complete without it. The fourth figure being thus rejected, the three first alone remain. These, therefore, are to be considered as exclusively competent in logic. c. The true canons of the figures. We have said that the syllogistic diflference of toteles fuisse yidetur, \xi nulla de causa tertiam figuram tradiderit. Est eniiii reperire exempla ejus figura;, in quibus si mutes disposi- tionem medii, feceris totum syllogismum obseuriorem." {Dialeetica. Paris, 1532, fol. 42.) This extract is taken from Melanchthon's seco7id logical work. His opinion in relation to this matter does not appear to have been always equally decided, since Melchior Adam says, (in his short life of him,) referring to it ; " Edidit Philippus eodem anno (1520) primura sua Pialectica in quibus tertiam jiguram syllogismorum neque recipit, neque rejicit ; quara deinde iterata editione, an. 1528, admisit." ( Vit(S Oermanorvm Philoaophorum. Francof, 1706, p. 88). ] Q6 NEW ANALYTIC OF LOGICAL FORMS. figure is determined in the supreme canon of syllogism bj the clause, " What worse relation of subject and predicate subsists between either of two terms and a common third term," &c. It is obvious, that this clause determines figure ; for the syllogistic differ- ence of figure is discriminated by the position of the middle term in relation to the extremes ; or in other words, by the " relation of subject and predicate subsisting between the two terms and a common third term." But this clause not only determines the difference of figure, it also immediately determines all the possible varieties of such difference. For the relation of sub- ject and predicate, subsisting between two terms and a common third term, must be either that in which the common third term is the subject of one and the predicate of the other, or that in which it is the predicate of both, or that in which it is the subject of both. No other is possible. Now, these three possible variations of relation determine at once the number of the figures and the canons by which they are regulated. For the number of different figures can only answer to the number of different relations ; and the evolution of these different relations in de- tail will be the expression of the laws by which the figures thus discriminated are severally governed. These relations are — I. That in ivhich the common third term is the NEW ANALYTIC OF LOGICAL FORMS. 67 subject of one of the terms, and the predicate of the other. This constitutes the first figure alike in extension and comprehension. First in Extension. B is A Here B, the third term, is the subject of C is B (that is to say, contained under) A, C is A the one term ; and the predicate of (that is to say, contains under it) C, the other term. So in Comprehension. C is B Here B, the middle term, is the predi- B is A cate of (that is to say, comprehended C is A in) C, the one term ; and the subject of (that is to say, comprehends in it) A, the other terra. The canon of this figure is — " In so far as two no- tions or terms are related, either both positively, or one positively and the other negatively, to a common third term, of which the one is subject and the other predicate, — these two notions are related positively or negatively to each other as subject and predi- cate." II. That in which the common third term is the predicate of both the other terms. This constitutes the second figure, and that (con- trary to the logicians) is either of affirmative or of negative syllogisms. 68 NEW ANALYTIC OF LOGICAL FORMS. Affirmative. Negative. All A is all B. All A is all B. All C is some B. No C is any B. All C is some A. No C is any A. The canon of this figure is — " In so far as two terms or notions, both subjects, are either both positively, or the one positively and the other negatively, re- lated to a common predicate, — in so far are they either positively or negatively subject and predicate, and that indifferently of each other." III. That in which the third term is the subject of both the other terms. This constitutes the third figure, and that (also contrary to the logicians) in syllogisms with either universal or particular conclusions. Universal. Particular. All B is some A. All B is all A. All B is all C. Some B is some C. All C is some A. Some C is some A. The canon of this figure is — " In so far as two no- tions or terms, both predicates, are either each posi- tively, or the one positively and the other negatively, related to a common subject, — in so far are they positively or negatively subject and predicate, and this indifferently of each other." The relation of the middle term in the second and third figures explains how it is that these figures have no determinate major or minor premise, and two in- XEW ANALYTIC OF LOGICAL FORMS. 69 different conclusions. There can be no determinate major or minor premise ; for a determinate major premise is one in which the middle term is compared with the greatest notion, and determined by it ; a determinate minor premise, one in which the middle term is compared with the smallest notion, and de- termines it. The middle notion thus always is in the former determined, in the latter determining. Now, in the second and third figures there can be no such relation of determination. In one the middle term determines both the premises, in the other it is deter- mined by both. Hence, since there is no determinate major or minor premise, it is manifest that we may have two indifferent conclusions ; for we may, in the second figure, indifferently predicate one subject of the other, and in third the one predicate of the other. In the first figure, again, it is equally clear that there is such a relation of determination. In it the reasoning is perfect, proceeding from the largest whole through a lesser whole to the least whole. The first figure accordingly has a determinate major and minor premise, and one immediate conclusion. d. The true relations of the figures. We have already spoken of the two wholes in which reasoning proceeds — the whole of comprehension and that of extension — the characteristic of the former being that the predicate is contained in the subject, of the latter that the subject is contained under the predi- cate. This being remembered, it will appear that in 70 NEW ANALYTIC OF LOGICAL FOKMS. the second figure, where the middle term as predicate contains both the subjects under it, extension will pre- dominate. In the third, where the middle term as subject is contained under, and therefore comprehends in it both the predicates, comprehension will prevail. In the first figure, again, where the middle term is both subject and predicate, extension and comprehen- sion balance each other. The first figure is indiffer- ently competent to either. Reasoning, however, proceeds not only in different wholes, but in different aspects of the same whole. We may, it is evident, regard any whole, considered as the complement of its parts, in either of two ways ; for we may, on the one hand, look from the whole to the parts, and reason accordingly downwards ; or, on the other hand, look from the parts to the whole they constitute, and reason accordingly upwards. The former of these reasonings is called deductive, the latter inductive. Deductive reasoning is founded on the maxim — " What belongs to the containing whole belongs also to the contained parts :" Induction on the contrary maxim — " What belongs to the con- stituent parts belongs also to the constituted w^hole." Thus in deductive reasoning the whole is stated first, and what is affirmed of it is affirmed of the parts it contains ; in other words, a general law is laid down, and predicated of the particular instances to which it applies. In inductive reasoning the parts are first stated, and what is predicated of them is also predi- NEW ANALYTIC OF LOGICAL FORMS. 71 cated of the whole they constitute ; in other words, the particular instances are first stated as facts, and then the law they constitute is evolved. This being the nature of these counter and corre- lative reasonings, it appears to us, that though each kind is competent in either whole, (extension or com- prehension,) yet that the reasoning in the whole of extension is more naturally allied to the deductive, and that in comprehension to the inductive. For, in the whole of extension, the reasoning proceeds from the general to the special — from the abstract to the concrete — from general laws to the particulai> in- stances which are contained under them ; while in that of comprehension, on the other hand, the rea- soning proceeds from the special to the general — from the concrete to the abstract — from the parti- cular instances to the general laws, whose operation they exemphf)^ The special adaptation of comprehension for induc- tive, and of extension for deductive reasoning, might be illustrated more fully in detail, and on other grounds ; but it may perhaps suffice to have indi- cated the relation between the two kinds of reasoning, and the two counter wholes in which they proceed. Considering these kinds of reasoning in relation to the figures, it will appear, then, that since extension prevails in the second, that will be so far more suit- able for deductive reasoning ; and since comprehension prevails in the third, that figure will so far be more 72 KEW ANALYTIC OF LOGICAL FOKMS. adapted for inductive reasoning ; while, since exten- sion and comprehension prevail equally in the first, that figure will be equally fitted for either kind of reasoning.* * [The relation of the figures to these different kinds of reasoning will be best illustrated by an example. We will take first the second figure : — ■ Deductive reasoning : Quantity of eictension. / Endowed with reason is all man. J European, Asiatic, African, American, are all man. ^^' ' \ European, Asiatic, African, American, are endowed with V reason. Here the reasoning is deductive, for the law is first enounced, the indiviflual instances are next brought under it, and it is then affirmed of them J it is eMensive, for it proceeds from the wider notion through the narrower to the individual. Let us now take the same terms and treat them inductively, beginning with the individuals. The reason- ing will then be in the whole of comprehension, and will naturally appear in the form of the third figure : — Inductive reasoning : Quantity of comfreheixsion. ( European, Asiatic, African, American, are all man. -,. ^_.^ ) European, Asiatic, African, American, are endowed with Pig. III. \ I reason. ^ Endowed with reason is all man. Here the reasoning is inductive, for beginning with the individual in the premises, we arrive at the law (with which we started in the previous syllogism) in the conclusion ; it is comj^rehensive or inten- sive, for it proceeds from the concrete to the abstract, from a greater totality of attribute to a less. In other words, in either quantity (extensive or intensive) we reason from the greatest whole ; but in the quantity of extension the greatest whole is the most abstract notion, {i.e., the widest law,) whereas in that of comprehension, the greatest whole is the most concrete notion, {i.e., the individual in- stance.) But proceeding thus from the widest law the reasoning is necessary deductive, while on the other hand, proceeding from the individual instance, it is as necessarily inductive. NEW ANALYTIC OF LOGICAL FORMS. 73 The second and third figures are indeed naturally respectively connected with deductive and inductive We may give the same example in the first figure, to illustrate (what will now be quite obvious) that it is indifferently competent Deductive reaso7iing : Qiiantity of extension. All man is endowed with reason. European, Asiatic, African, American, are all man. European, Asiatic, African, American, are endowed \vith reason. Fig. I. Inductive reasoning : Quantity of comprehension^ European, Asiatic, African, American, are all man. All man is endowed with reason. European, Asiatic, African, American, are endowed with reason. I need scarcely say, that at the time of writing the essay, I was quite unaware that any of these special relations of the figures had been noticed by logicians. I find, however, that Wilson, in his " Rule of Reason,''^ (1580,) has, among other remarks tracing the use of the figures, the following : — " Use of the third figure: This figure profiteth much in provoking particular things, and gathering of conjectures in causes that are doubtful, when probability only, and no assured knowledge, boulteth out the truth of a matter. And because several things (individuals) come sonest to our senses, we use suche gather-' ing moste commonly, and by triall of particular causes, assure our- selves of the truthe generally." Again, " when we make an argu- ment and procede from the general word (genus) to the kind (species), it is in the first figure, and even by our reason we learn this, that if the greater bee not, the lesse cannot bee." " When we procede from the kinde to the general, making the con- clusion particular, the argument is in the third figure. And this \6 for ever true, that when the kinde is rehearsed, the generall must needes foUowe." (Fol. 30-1.) This, however, is little beyond a more explicit statement of what is commonly said of the thii'd figure, that it is a reasoning from the special.] 74 NEW ANALYTIC OF LOGICAL FORMS. reasoning ; for in the second we judge the likeness or unHkeness of two parts, as they are contained or not contained by a common whole ; while in the third we judge the likeness or unlikeness of two wholes, as they severally contain or do not contain common parts. 3. Syllogisms differ with respect to mood. The syllogistic variety of mood arises from the different quality and quantity of propositions, and this difference is determined by the supreme canon in the clause — " What worse relation of subject and predicate" kc, since a negative quality is a worse relation than a positive, and a particular quantity a worse relation than a universal. Logicians, however, in their enumeration of moods, as we have seen, have taken into account only one quantity of propositions ; have considered the subject as quantified to the exclusion of the predicate ; and have, in so doing, deformed their science by excluding from it many valid forms of reasoning — forms which logic, if it be an exact science, and its analysis of the form of thought exhaustive, is bound to recognise and vindicate as valid. It is the design of the " new analytic " that its analysis shall be thus exhaustive ; and it vindicates its title by discovering and developing, in its various relations, an element of formal thought which had remained undeveloped, if not unrecognised, in every previous analysis. That element is, as we have NEW ANALYTIC OF LOGICAL FORMS. 75 said, the express quantification of the predicate, the true application of which recalls to the science many true forms of reasoning, the date of whose logical proscription may be reckoned as coeval with that of the science itself We shall proceed briefly to vindicate these forms. Logicians, combining the quality of a proposition with the quantity of its subject, reckon in all four kinds of propositions ; and combining these propositions in every possible way, evolve sixty-four moods. But it is clear, that if the quantity of the predicate be taken into account, the various kinds of propositions discriminated by quantity and quality will be doubled in number, and a proportionate increase effected in the number of possible moods ; for we shall now have eight kinds of propositions, viz., four affirma- tive : — Definite, affirmative, definite. Definite, affirmative, indefinite. Indefinite, affirmative, definite. Indefinite, affirmative, indefinite. And in the same manner, four negative propositions. With this increase in the number of propositions, we need new symbols by which to designate them. We may, however, still retain the old notation A. I. E. 0., and express the new forms by combining the letters into diphthongs, or placing them within brackets, as occasion may require ; e.g.^ — Definite, affirmative, indefinite = (Al.) Indefinite, negative, definite = CE, &c. 76 NEW AKALYl^lC OF LOGICAL FOKMS. This increase in the number of propositions to be combined effects of course a great increase in the number of moods resulting from such combination. Of the possible moods thus given, a number are in- vaKdated by the clause of the canon, " o?ie at least positively so," from having two negative premises. A number more are excluded by the clause, " a com- mon third term" from the middle term being undis- tributed* Throwing these out of account, together with some others having particular conclusions where universal are competent^ there remain in all thirty- six valid moods, (twelve affirmative and twenty-four negative,) and these thirty-six are valid in each figure. We employ the symbolical notation,''^ using the comma (,) to denote " some,^' (indefinite quantity,) and the colon (:) to denote "alV (definite quantity). Valid moods of the first figure. Afiirmative. Negative. i. A : : B : . : C J ' (2. A: ii. A , B ; ,C l.A, 2. A, n.A:-|— iii. A : : B : , C \ (2. A : B B B B B B :C :C ,C ,C * On this system of notation, see the Appendix. NEW ANALYTIC OF LOGICAL FORMS. 77 iv. A , : B : V. A: : B , > vi. A : , B : vii. A , : B , , B : ix. A : -^— : B , X. A , , B : xi. A , ■ > : B , xii. A : , B : :C :C : C ,C ,C :C ,C r (2. A ... (2. A r (2. A r (2. A r (2. A |1.A I2. A I" (2. A r (2. A r h. A B B B B B B B B B B B B B B B B B We have given the above syllogisms in the first figure ; but they may all be easily translated into the two others.* * [The only one of the prescribed requisites which the essay does 78 NEW ANALYTIC OF LOGICAL FORMS. To recapitulate then. We set out with the prin- ciple of a quantified predicate. We have noticed some things by the way not immediately connected therewith ; but recurring to it, we have endeavoured to vindicate that principle. We have indicated its influence on propositions in abolishing the complex doctrine of conversion ; its influence on categorical syllogisms, in reducing their laws to a higher sim- plicity, and amplifying their vaKd forms, — in short, by correcting what was false, and supplying what was wanting ; and thus, by securing to logic a higher degree of formal exactness, realising for it a higher degree of scientific perfection. not to some extent attempt to meet, is that in which it is required " to show in concrete examples, through all the moods, the unessen- tial variation which figure makes in a syllogism." This was omitted at the time of writing through haste ; but it is so obvious, that with a little trouble, each reader may do it for himsel£ As an illustration, however, the following is a concrete example of the first mood, car- ried through all the figures : — ( All man is some animal. Fig* ^' <. Every Celt is some man. ' Every Celt is some animal. i Some animal is all man. Fig. II. J Every Celt is some man. ( Every Celt is some animaL r All man is some animal. Fig. III. ) Some man is every Celt. ( Some animal is every Celt. ( Some animal is all man. Fig. IV. < Some man is every Celt. ' Some animal is every Celt. NEW ANALYTIC OF LOGICAL FORMS. 79 The " neiv analytic^' accomplishes this by being- true to its office, and fully investigating the form of thought. The form, the whole form, and nothing hut the form of thought, is indeed the bannered motto which it bears on its triumphant way. True to its purpose, it advances over the whole region of formal thought, conquering and to conquer ; de- stroying the false landmarks which had been set up by the early discoverers of that territory ; re- pressing the incursions which were continually made into neighbouring kingdoms ; destroying the border ground by determining for ever the frontier line ; dethroning the potentates who had intrenched them- selves in its high places, and long there exer- cised a usurped authority ; recalling from their long exile the true lords of the soil ; re-establish- ing the laws on which their rights were founded, and enforcing strict obedience to these in every province of the empire. Thus, though in some respects its path is as the path of the destroyer, in a higher and truer sense it is the path of peace ; for through its instrumentality there breaks at length upon this long distracted region the golden age of simphcity and order. And anarchy, the result of laws neglected and rights ignored, is for ever abolished in the esta- blishment of perfect harmony — a harmony the result of law clearly expounded and rigidly obeyed through- out the entire empire of formal thought. 80 NEW ANALYTIC OF LOGICAL FOKMS. In conclusion, we are. well aware of the very im- perfect manner in which we have signalised those parts of the new^ discovery on which we have touched. We cannot, however, close without expressing the true joy we feel, (though, w^ere the feeling less strong, we might shrink from the intrusion,) that in our country, and in our time, this discovery has been made. We rejoice to know that one has at length arisen, able to recognise and complete the plan of the mighty builder, Aristotle,— to lay the top-stone on that fabric, the foundations of which were laid more than two thousand years ago by the master hand of the Stagirite, which, after the labours of many gene- rations of workmen, w^ho have from time to time built up one part here and taken down another there — -re- mains substantially as he left it ; but which, when finished, shall be seen to be an edifice of wondrous beauty, harmony, and completeness. APPENDIX. No. I. HISTORICAL NOTICE TOUCHING THE EXPLICIT QUANTIFICATION OF THE PREDICATE. The statement made in the first page of the Essay, that " the principle of a quantified predicate had, in its full scientific sig- nificance, been totally overlooked by logicians, and that, when noticed at all, it had, for the most part, been referred to only to be discarded as useless, if not to be condemned as false" — was made, it scarcely need be said, upon a very limited acquaintance with logical works. It was, however, the conclusion to which my inquiries, so far as they extended, led me. I had examined several logical treatises, and found that the majority made no re- ference at all to a quantified predicate, that the few who noticed it, (two of which are quoted,) treated it in the manner described in the text ; while, so far as my reading extended, I had not found a single instance in which it was admitted in any form. Since writing the Essay, I have naturally been curious in my occasional logical reading to mark any references which might be made to this subject; and as the result of a somewhat fuller knowledge of the historical development of the science, I am able to establish, upon somewhat wider evidence, the general truth of the statement made in the text. The full scientific signifi- cance of the principle certainly never has been appreciated. It has been, *' for the most part, rejected;" that is to say, it has been denounced by the vast majority of logicians as 82 APPENDIX. useless and false. Some exceptions, however, to this sum- mary rejection of the principle are to be found. A few of these I have met with, some of which as in themselves curious, and as lying out of the way of ordinary reading, it may be worth while explicitly to notice. In order, therefore, to place the Essay on a level with my present knowledge, I shall briefly establish historically the common rejection of the principle of a quantified predicate, and then notice some of the exceptions to this treat- ment of it which are to be found. I. The common doctrine which altogether rejects the express quan- tification of the predicate. This doctrine dates substantially, as do most logical truths and heresies, from Aristotle. He refers explicitly in two places* — in his Book on the Doctrine of Enunciation^ and in his Prior Analytics — to the quantification of the predicate, and in both rejects it in a very summary manner. In the Book touching Enunciation he says, — " 'Et^ hs rou xarriyopovfisvou xa^oXo'j TO xa&oKov xciTrjyoosTv, oox sffriv dSrjXsg' ovds/j^ia ya^ za-d B denotat, A non est B. N.A — B denotat, Nullum A est B. A praefixum propositioni significat affirmationem universaliter sumtam. I, affirmationem particulariter sumtam. E, negationem universaliter sumtam. O, negationem particulariter sumtam. Cum seriei cuidam subjungitur signum, &c., denotatur series infinita, vel integra. Cum non subjungitur, denotatur series abrupta." § 36. " Sit primo, propositio universaliter affirmans, O.A — B. Haec in generalibus et in symbolis spectata non infert hanc O.A est O.B sed O.A est q.B, § 24. Ubi vero notandum, quod par- ticularitas nunquam intelligatur exclusiva." § 40. " Sit particulariter negans q.A> B. Omne, quod est B, diversum est a quodam A. Ergo N.B est q.A. Not. Operatio haec vocari solet Conversio propositionum." APPENDIX. 129 § 41. " Apparet hinc, conversionem propositionum nihil aliud esse, quam transpositionera eorundem terminorum logice expres- sorum, nee quidquam in sensu ipso immutari. Si enim in sensu aliquid immutaretur, propositio non amplius esset hsec proposi- tio. Ita nee ex particulari fit universale, nee ex universali fit particulare." § 43. " Sit propositio identica O.A — O.B : Hie A cum B identificatur, adeoque conversio fit O.B — O.A. " Sint enim du£e series : AB. AB. AB. &c. CB. CB. CB. &c. " Hie C non potest esse diversum ab A. Si enim B est idem cum A, tum B non potest inesse tm A& tuj non - A, quia idem est idem. " Cum usitatus loquendi modus non secum ferat banc propo- nendi rationem : O.A est O.B, sed pro bac propositione substi- tuatur O.A est B ; eonvertendo dici potest q.B est A, modo attendatur ad id, quod particulari tas comprebensiva et definita sit intelligenda, quse justo modo extensa potest coincidere cum omnitudine. Si vero exacte loquendum sit, propositio O.A est B distingui debet ab bac O.A est O.B. " Brevius : O.A est O.B. Si est O.B, pr^eter B nibil aliud datur ; adeoque O.B est O.A per conversionem. " Vel sic : O.A est O.B; boc est: Omne quod est in A tarn ratione comprebensionis, quam extensionis, est quoque in B. A et B igitur nullo modo differunt, adeoque A est B, uti A est A." § 44. " Sit propositio particulariter affirmans q.A est B. " Hie aut q.A est q.B, aut q.A est O.B. " Priori casu patet, eonvertendo q.B fore q.A ; posteriori autem, O.B esse q.A. Si enim quoddam A est quoddam B ; patet ex bypotbesi, nee ad omnia A, nee ad omnia B beic respici ; adeoque seriem ita esse concipiendanr AB. AB. CB. CB. AD. AD. &c. loO APPENDIX. " Ex intuitione manifestum est, A et B non semper conjungi, sed tarn A quam B cum diversis signis connecti. Ergo q.A — B idem est cum q.B — A. " Posteriori casu autem sequens apparet facies AB. AB. AB. &c. AC. AD. AE. &c. " Ex hypothesi non datur B, quod non insit rtZ A ; adeoque, OB — A, sed tantum quoddam A est cum B conjunctum. e.g.j Quidam homo est miles. Non datur miles, de quo non prasdi- cari possit, quod sit homo, adeoque omnis miles est homo, sed tantum idea cujusdam hominis connectitur cum idea militis. Sed cum in expressione consueta quantitas priedicati non determi- netur ; utraque propositio facta conversione habebit subjectum particulare." It will be seen from these extracts that Ploucquet compe- tently understood the use of a quantified predicate in relation to propositions. Strangely enough, however, he makes no use of it in his treatment of syllogisms, where especially, or rather exclusively, its higher scientific value rises into view. With the exception of two short sentences, (if I remember aright, for I speak only from past reading,) he gives no intimation of its use in relation to syllogisms. These sentences are both notes. The first, which is appended as an exceptional provision to the rule, that from two particular propositions nothing follows, is this: — "Hie tantum agitur de expressione conclusionis con- sueta, ubi praedicato non addi solet signum quantitativum." The second, which is given at the end of his consideration of the third figure, and is to the effect that valid negative syllo- gisms may be obtained in it, is as follows : — " Si in proposi- tionibus particulariter negantibus adjiciatur praedicato signum particularitatis ; figura hsec procedit in omnibus modis supra recensitis." This want of anything like a scientific application of the prin- ciple to the syllogism may indeed be said to be universal. APPENDIX. 131 None of those to whom we have referred as having partially appreciated it, seem to have been at all aware of its value in relation to the forms of reasoning ; but, after having applied it to some particular detail in the treatment of propositions, or at most to some exceptional case of syllogism, appear to have abandoned it altogether. The promised historical evidence touching the previous partial appreciations of the new doctrine properly terminates here. I cannot, however, close this note without quoting a curious pas- sage from a recent British writer, whose contributions to logical science have not met with the attention which their merit de- serves. This writer is Mr. Thynne, who would probably have become better known, but for the comparatively humble form which he has chosen as the vehicle of his logical discussions. These discussions were published in the form of notes to Walker's Compendium of Logic, which has been for many years, and, for anything I know to the contrary, still is, the text-book of Trinity College, Dublin. These notes evince a careful study of logic, and an acute comprehension of the science, both in its general scope and in its particular detail ; they show also an amount of independent thought in relation to the science, very rarely indeed to be met w^ith in recent logical writings. The para- graph I am about to quote is really a very curious one, on ac- count of its strange prophetic significance. It is given as a note to a passage in the text, in which the author, after giving the common statement that there are in all 64 moods, and that so many are mvalid, adds, '^ accordingly there are 19 concluding 7noods." The note is as follows : — " (8) This, as logicians say, is gratuitously assumed. Suffi- cient has been said by our author to establish that all other modes than these 19 are inconclusive, but nothing has pre- ceded to verify the conclusions in these modes. This may be done by the axioms, even more easily perhaps than by the 132 APPENDIX. famous rules of Aristotle ; but in the application of the axioms the distinction of the premises and extremes into major and minor vanishes. If all men and so7ne animals agree with a third, they agree with each other ; and it is indifferent whether it be understood that all men are certain animals, or certain animals are all mankind. And it is certain, that if syllogistic reasoning had been thus viewed and followed up, it would be more readily brought into practice, or rather — as it is already, in the general practice of reasoning, thus treated, imperceptibly because en- thymematically — it would receive less distinction of mode and figure ; and consequently require less rule, and admit of simpler and readier, although not more ingenious, complete, or elegant verification than as at present treated." This prophetic foresight, however, exercises no beneficial in- fluence on Mr. Thynne's own treatment of the science. Logic certainly receives no simplification at his hands ; on the con- trary, the whole subject of mood and figure is made, if possible, still more intricate by the ingenious involution and evolution of detail which Mr. Thynne has introduced into its treatment. He therefore does not accept the doctrine he foreshadows. He may be said indeed to have explicitly rejected the quantification of the predicate ; for he has propounded a theory of his own in relation to quantification, which is, to say the least of it, of the strangest kind. Mr. Thynne holds that quantification is an af- fection of the copula. This is a confusion so strange and com- plete, that were it advanced by a less able author, it might be rejected at once. Anything, however, which is seriously urged by so careful a student of logic as Mr. Thynne proves himself to be, is entitled to consideration, and if found to be erroneous, merits at least the courtesy of a refutation. Mr. Thynne adverts to his doctrine in several of his notes; but the passage in which he most clearly propounds it is the following : — '' Such marks of quantity, although grammatically qualifying APPENDIX. 133 the subject, do in sense qualify the copula ; intimating the ex- tent in which the agreement or disagreement of the terms is declared — that this extent is implied to be, either general or partial. ' Every man is an animal' implies that ' man is uni- versally an animal :' * No man is a stone,' that ' man — is uni- versally not — a stone:' and ' some men are just' that ' man — is in part of the extension — just.' " On this I would remark generally, that even supposing it to be correct, and that we may thus explain the quantity of the subject, still no account is given of the quantity of the predicate, which on the same principle ought also to be an affection of the copula, and to be represented accordingly in the same manner. Thus, to take the last of the above examples, since this is the only one which is given in the regular form, Mr. Thynne hav- ing in effect quantified the predicate in the others by the articles, — this is, '* some man is just," which, says Mr. Thynne, is truly expressed thus, " man — is in part of the extension — ^just;" but whether is this part of the extension of man equal to all just or some justi This is a pertinent inquiry, for "just^* is indefinite and must be formally limited or amplified in the same way as the subject. On Mr. Thynne's doctrine, therefore, two parts of extension must be expressed by the copula, in some such man- ner as the following: " Man — is in part of its extension — just — in part of its extension." All of this on the doctrine in ques- tion save man and just belongs to the copula; in short, Mr. Thynne maintains that the terms of a proposition are always taken absolutely/, and that all modification of this absolute mean- ing belongs not to themselves but to the copula. The absur- dity of such a doctrine will be better seen from an example in which the modification of the terms is of a more definite numer- ical kind than that usually found in the ordinary examples ; for as all modification of the terms, that is, everything indicating the extent of the agreement or difference existing between them, belongs to the copula, then necessarily numerical modification 184 APPENDIX. of extent belongs to it also. Take then the following proposi- tion : — " ten horses are equal in strength to a hundred men." This expounded according to Mr. Thynne's doctrine will become the following : — " horse (taken absolutely) — is, considered under the relation of ten — equal to man (taken absolutely) considered under the relation of one hundred : or a part equal in extent to ten taken out of the whole concept horse — is equal in a given relation, that of strength, to wit, to a part equal to a hundred taken out of the whole concept man." The former way of stat- ing it is contradictory enough, and in the latter surely no one will seriously say that the true subject is the whole concept horse, and the true predicate the whole concept man. This criticism would be in part inapplicable, did Mr. Thynne, with Aristotle and some of the older logicians, and indeed with the Compendium itself on which he comments, — did he hold, we say, with these, that the copula is included in the predicate. It is manifest, however, partly from his remarks, and abundantly from his practice, that he does not do so, but with logicians in general considers it simply as the bond of connexion between two terms, and in the last resort, therefore, always the substan- tive verb. Mr. Thynne gives the following illustration in confirmation of his doctrine : — " This is not the only respect in which a circumstance gram- matically associated with the subject is logically associated with the copula. In the proposition above, * no man is a stone,' no- man is surely not the subject, for the declaration is intended to be made of man, and not of that which is not man : nor is the copula zs, for the relation is intended to be one of disagreement. The negation therefore is here logically a modification of the copula, although grammatically of the subject." If the principle was unfortunate, the illustration is certainly equally so. " In the proposition, ' no man is a stone,' " says Mr. Thynne, " no-man is surely not the subject, for the declara- APPENDIX. 135 tion is intended to be made of man, and not of that which is not man." Now, what is the declaration made in the above propo- sition ? It must surely be, " is a stone,'' which, says Mr. Thynne, is intended to be made of man, and not of that which is not man, but which I cannot but think notwithstanding is intended to be made of that which is not man, since it cannot be truly said that any stone is a man ; and accordingly that no-man is the subject. Speaking generally, it may be said that in any negative propo- sition the negation may fall on the subject, on the predicate, or on the copula ; but that on whatever member it falls, it becomes truly a part of that member ; falling on the subject therefore in the proposition above, no-man is the true subject. Let us look, however, for a moment more closely into the matter. A proposition is but the reflex in language of a judg- ment; a judgment is the product of a particular mental act. Now, the whole question is determined by ascertaining specifi- -cally ivhat that act is. It is, in brief, one of comparison. Two things (terms of any kind) are compared together in order to ascertain whether they stand to each other in the relation of de- termining and determined, of whole and part ; in a word, to discover what is the extent of their agreement or difi'erence. Now, does this extent of agreement or difference belong to the objects which are compared together, or to the mind which com- pares them ? Surely when two things are compared together, and found to stand to each other in the relation of part and whole, that affection of quantity belongs to the things, and not simply to the mind which perceives them. But on the theory in question the act of judgment is not only cognitive, hut creative ; it not only perceives a relation, but also creates the relation which it perceives. This involves a double confusion ; a confusion in philosophy of the perceiving mind with the things perceived, of the mental act with its object : a confusion in logic of quantity with quality, of the matter with the form of a proposition, of things to be connected with the bond of their connexion. It is 136 APPENDIX. indeed an error so manifest, that we need not dwell upon its re- futation, and one which could only have been committed by so able a logician as Mr. Thynne through great haste, or greater oversight. We have thus the subject of quantification viewed in almost all the aspects in which it can possibly be considered. The common doctrine considers it in relation to the subject-, Sir Wil- liam Hamilton as it aifects the, predicate ; Mr. Thynne maintains that it is an affection of the copula ; while Mr. De Morgan (after Lambert) has elaborated a particular quantification of the middle term. This last scheme, however, as but a trivial and prac- tically useless refinement on a doctrine universally held by logi- cians, may be thrown out of account. Mr. Thynne's notion, we have endeavoured to show, must, as the result of a strange confusion, be at once rejected. The subject and predicate, therefore, alone remain to be considered. The quantification of these terms, as the ultimate constituents of logical analysis, is, we need scarcely say, all-important. Through the working out of quantification in relation to the subject, the existing logic has attained to whatever of perfection of detail it can pretend to ; through its working out in relation to the predicate, it will at- tain to the whole perfection of which, as a science, it is suscep- tible. The former is substantially the work of Aristotle ; the latter is equally so that of Sir William Hamilton. To sum up, then, the evidence we have gained : We have found, on the one hand, that the express quantification of the predicate has been rejected with singular uniformity throughout the entire history of the science. On the other hand, that it has been sometimes partially adopted in theory, and at other times in various ways applied in practice ; only, however, to amend some particular detail of the science — it may be to simplify the process of conversion — it may be to modify, by a problematical excep- tion, some particular rule of syllogism. But we have also seen, that its want has never been signalised as a fundamental de- APPENDIX. 137 feet in the original logical analysis ; — a defect through which the science had been encumbered by unscientific supports — dis- figured by unscientific additions — dwarfed by unscientific re- strictions, and thus shorn of its true beauty of proportion and completeness ; that it has never, therefore, been employed as a principle to reconstruct the whole edifice of the science, and by removing what was useless, rejecting what was false, and sup- plying what was wanting, to restore it to its perfect and har- monious beauty. Despite, therefore, the evidence of partial perception which we have adduced, the original statement, " that the principle in its full scientific significance has been altogether overlooked by logicians," is vindicated. No. II. ON THE CATHOLIC DOCTRINE TOUCHING THE IMPLICIT QUANTIFICATION OF THE PREDICATE. Though logicians have with one consent rejected the eoq)licit, they have, nevertheless, always held an implicit quantification of the predicate. This was indeed absolutely necessary to the existence and working of the science. For since all reasoning is in the last resort but the comparison of two terms with a third term, — but the perception of how far two terms mutually agree or disagree through the perception of how far they agree or dis- agree with a third, — it is obviously not only important but im- perative that the extent of these terms themselves should be taken into account. This is, indeed, the very essence of the reasoning. Apart from this there can be no measurement of extent, and no conclusion of identity or difference of extent ad the result of such measurement. -^^^•^%^'^Ia^^^ V OP TH7? tTiri7EE3lT7l 138 APPENDIX. In other words, in any syllogism the process of the reasoning and the evidence of its validity is the same, and it is the follow- ing : — the middle term is the mean or measure ; in the first place, one extreme is compared with the middle term, and seen to agree with it so far; then the other extreme is compared with the middle, and also seen to agree with it so far ; and thereupon this iden- tity of agreement is affirmed. But, in either case, if I do not know the extent of the term compared, (cannot take it, that is, in some definite extent,) I cannot tell how far it agrees with the middle. Or, again, if I do not know the extent of the middle term, (cannot take it in some definite extent,) I cannot tell whether the term to be compared agrees with it in extent or not, — whether it is part, or whole, or none. The predicate no- tion, however, in every reasoning is one of these terms, and stands in one of these relations. It is, therefore, absolutely necessary, as we have said, to the validity of the reasoning, that it should have a definite quantity ; and a definite quantity, accord- ingly, it always has had in the science. This quantity, how- ever, instead of being left like that of the subject to reflect itself in language according to the whole extent of its possible variation as an element of formal thought, was made the sub- ject of arbitrary legislation. Logical law enacted that the quan- tity of the predicate should always be held particular in affirma- tive propositions, and universal in negative ones. That logicians by this arbitrary and unjust enactment crippled their science, — crippled it too in a useless and preposterous manner — could be very easily shown. The natural scientific action of one of its parts was at once interfered with ; it could only work now under given artificial restrictions — restrictions which had not the slightest shadow of scientific warrant for their imposition. These restrictions are, indeed, not only artificial but capricious, for no reason whatever can be shown why one term of a rela- tion of quantity should be made the subject of arbitrary legis- lation rather than another. It would have been just as wise APPENDIX. 139 and just as scientific to have laid down arbitrary rules for the quantity of the subject as for that of the predicate. Why not, it may be asked of the logicians, place the subject under the same restrictions, and enact, for instance, that its quantity shall always be held universal in affirmative propositions, and parti- cular in negative ones? The whole subject may be illustrated more fully in detail by the use of a figure which we have already partially employed. The subject and the predicate may be said to be the legs on which the syllogism stands. Its free progress, of course, de- pends on the natural unrestricted action of these members. Logicians have, however, crippled one of these — the predicate — by preventing such natural action. In order, however, that the syllogism might work at all after having been thus maimed, it became necessary to provide some support for the crippled limb. This, accordingly, was found, and in the shape of a body of special rules, a crutch which partially supplied the place of the natural support, was realised. Why did not the logicians, we ask, since they had thus endorsed the principle of such a procedure, destroy the natural action of the other limb also, and provide it with the same artificial support? They would thus have solved the problem, how far the syllogism could pro- ceed when altogether deprived of the native strength of its own members, and supported on two crutches instead of one ; — an in- genious experiment enough, certainly, but one which bears exactly the same relation to the natural development of the science, that racing in sacks does to the natural exercise of the limbs in walking. This ingenious problem, however, the logicians have not at- tempted in its integrity. They have remained satisfied with its partial solution in relation to the predicate. The laws which they have laid down for the regulation of its quantity are, as we have said, two, viz. : — 140 APPENDIX. 1°. That in all affirmative propositions the quantity of the predicate is particular. 2°. That in all negative propositions the quantity of the pre- dicate is universal. On these two axioms, as they are commonly, but of course erroneously termed, the whole detail of the existing logic rests. They have determined its peculiar form and necessitated its special rules. These special rules may indeed be appropriately described as a body of provisions to secure that the predicate is always really taken according to the quantity assigned to it in the axioms. This will be at once manifest by an examination of the demonstrations of those special rules which are sometimes given in logical works. These demonstrations contain of course the reasons on which the rules rest, and these will be found in every case to arise from the necessities of the predicate in rela- tion to its implied quantity. We quote the following in illus- tration from the Port Royal Logic, where all the special rules are briefly but adequately explained. " RULES OF THE FIRST FIGURE. " The minor must be affirmative ; " For, if it were negative, the major would be affirmative by the third general rule, and the conclusion negative by the fifth ; therefore the greater term would be taken universally in the con- clusion, since it would be negative, and particularli/ in the major; for it is its attribute in this figure, and would be affirmative, thus violating the second rule, which forbids us to conclude from the particular to the general. This reason holds also in the third figure, where the greater term is also attribute in the major. " I'he major must be universal; " For, the minor being affirmative, by the preceding rule, the middle term, luhich is its attribute, is taken particularly; therefore it must be universal in the major, where it is subject, which renders this proposition universal ; otherwise it will be taken APPENDIX. 141 twice particularly^ contrary to the first general rule." — Pp. 189, 190. " FIRST RULE OF THE SECOND FIGURE. " One of the two propositions mmt he negative^ and consequently the conclusion also, hy the sixth general rule ; " For, if both propositions were affirmative, the middle, which is here always attribute, would he taken twice particularly, contrary to the first general rule." — P. 193. The necessity of these special rules is manifest, for of those of the first figure, if the former were violated, a term which, as the predicate of an affirmative proposition, was particular in the major premise, would, as the predicate of a negative proposition, become universal in the conclusion ; if the latter were violated, the middle term would, as predicate in the affirmative minor premise, be particular there, as well as in the major, and thus remain undistributed. The same reason holds in the first rule of the second figure. The other special rules are susceptible of a similar explanation, as may be readily tested, by taking them and the two axioms of quantity, and working out the relation of determination which exists between them. These axioms are thus, as we have said, operative through the whole detail of formal reasoning, as it stands in the existing logic. Re- jecting the explicit, and accepting only the implicit quantification of the predicate, the question to be determined by logicians was, — how many of the possible forms of reasoning are valid with- out such explicit quantification ? The commonly accredited syllogisms were the result of this examination. The special rules which protect them were generalisations from the causes which rendered the rest invalid. Those which severally, accord- ing to their figure, obeyed the conditions of these rules, were alone accepted. These reasonings were declared, moreover, not only to embrace all the valid syllogisms which could be obtained under such restrictions, but also to exhaust all the possible forms 142 APPENDIX. allowed by the laws of thought. This is, indeed, one of the grounds expressly taken by Pacius in his rejection of a quan- tified predicate. He says, (at the close of the second of the ex- tracts given from him earlier in the Appendix,) that the quan- tification of the predicate is of no use to the syllogism, — that it does not at all aid its validity ; and gives a syllogism in Barbara of the first figure in illustration of his statement. The answer to this is easy. The express quantification of the predicate will not, of course, help the validity of those syllogisms which have been expressly constructed so as to be independent of its aid ; and the cogency of which, therefore, is complete without such quan- tification. All the syllogisms of the existing logic are of this kind ; and their validity, accordingly, is independent of any such expressed quantity. But this does not at all prove, on the one hand, that even these would not possess a higher formal com- pleteness with the quantity of the predicate expressed ; or on the other hand, that there may not be other syllogisms whose validity entirely depends on such expressed quantity. This is indeed the case ; for on the one hand, everything of force in a formal science ought to be formally expressed; and on the other, there are a number of forms of reasoning guaranteed by the laws of thought., whose validity is not only contributed to, but constituted hy^ the expressed quantity of the predicate. Looked at therefore from the lower ground of the axioms of quantity, and the reasonings possible through them, the ex- pressed quantity of the predicate is not absolutely necessary ; but, regarded from the higher ground of the laws of thought, and their scientific development, this quantification is not only imperative, but indispensable. These so-called axioms of quantity, it may be worth while to notice, are but corollaries from the laws laid down touching regular predication ; for if the only lawful predication is that in which a genus is predicated of its species, since the genus is al- ways of wider extent than its species, when so predicated it can APPENDIX. 143 only be taken in some part of its whole extent, that is say par- ticularly. Again, if it be unlawful to affix marks of quantity to the predicate, we cannot deny some part of a genus of one of its species; and all negative propositions must therefore contain repugnant species or genera, which will accordingly be denied of eacU other in their ivhole extent. The rules for predication, and those for the quantity of the predicate, thus at bottom im- ply each other. It would perhaps be difficult to say which were cause and which effect ; or rather, it would probably be nearer the truth to say, that they are both the result of the same defec- tive analysis and want of scientific insight. Out of this original defect have arisen, as we have shown, the complexity, the re- striction, and the disorder, of which the special rules, and the syllogisms they indorse, are at once the evidence and the result. The simplicity of the reasoning process, contrasted with the complexity of the rules devised for its guidance and protection, could hardly, however, fail to arrest the attention of some of the many thinkers who have from time to time undertaken their exposition. They accordingly have, in various ways, betrayed their sense of the want of thorough scientific simplicity and com- pleteness which these rules indicated : some, as we have seen, by falling upon stray syllogisms which violated the rules, but which were nevertheless quite valid, without, however, being able to offer any theoretic explanation of the fact : others, again, by simplifying the syllogistic law in theory, without being able to show how this theoretical simplification could be realised in actual practice. A curious instance of this latter kind occurs in the Logic of Caspar Wyss, which was published at Geneva (where he was for some time Professor) in the year 1669. He reduces all the rules of syllogism, both the general and the special^ to the single one, that every syllogism should have three and only three terms. We subjoin his reduction of the special rules as a specimen of the way in which he accom- plishes this : — 144 APPENDIX. " De Regulis specialibus Syllogismorum. " Regiilse speciales syllogismorum sequentes po- nuntur a Philosophis : I^ Regula specialis est : In prima fig lira, major debet esse universalis. Unde hie syllogismus non valet : Omnis homo non est in hac urbe ; Tu es homo ; E. tu non es in hac urbe. Item : Omne animal non est rationale ; Sed homo est ani- mal ; E. homo non est rationalis. " Verum contra banc regulam dici potest, illam non esse uni- versaliter veram ; si enim toUatur ambiguitas, et sint tantum tres termini in syllogisrao, syllogismi primae figurag, ex majori particulari, sunt legitimi : v. g. Aliquod animal est rationale ; Homo est animal ; E. bomo est rationalis. Quare dicendum est : syllogismos superius allatos esse vitiosos, quia in iis dantur quatuor termini. In priore enim syllogismo, homo aliter su- mitur in majore, quam in minore. In posteriore vero, animal aliter sumitur in majore, quam in minore, ut patet attendenti. " II*. Regula specialis est : In prima figura, minor debet esse affirmata. Unde hie syllogismus non valet : Omnis asinus est animal ; Sed homo non est asinus ; E. homo non est animal. Item : Omne rationale est animal ; Solus homo est rationalis ; E. solus homo est animal. " Verum contra banc regulam, dici potest, illam non esse uni- versaliter veram, eo, quod dentur syllogismi primi© figuras, ex minore negata, qui sunt legitimi : v. g. Qui non credit in Cbris- tum damnabitur; Sed reprobi non credunt in Christum; E. reprobi damnabuntur. Quare dicendum est, syllogismos primas figurse, ex minore negata, esse legitimos, si sint tantum tres ter- I APPENDIX. 145 mini; esse vero vitiosos, si sint quatuor termini, nt patet in syllogismis superius allatis, in quibus sunt duo media, adeoque quatuor termini. Nam in priore, asinus est medium in majore, non asinus vero, est medium in minore. In posteriore vero syllogismo, rationale est medium in majore, non rationale vero, est medium in minore. Adde, quod in ejusmodi syllogismis, ani- mal aliter sumatur in majore, quam in conclusion e, ut patet attendenti. " IIP. Regula specialis est : In secunda figura, major debet esse universalis. " Verum contra banc regulam dici potest, syllogismos se- cundae figurse ex majore particulari esse bonos, si sint tantum tres termini, non aequivoci ; esse vero vitiosos, si sint quatuor termini, sicut diximus, de syllogismis primse figurae ; adeoque haec regula, est superflua. " Iy^ Regula specialis est : In secunda figura, altera prcemissarum debet esse negans ; juxta illud vulgatum : ex puris affirmantibus, in secunda figura, nihil concluditur. Quare hie syllogismus non valet : Asinus est animal; Sed homo est animal; E. homo est asinus. Item : Asinus habet aures ; Tu habes aures ; E. tu es asinus. "Verum, contra banc regulam dici potest, illam non esse uni- versaliter veram, cum multi sint syllogismi recti, in secunda figura, ex puris affirmantibus : v. g. Omne rationale est risibile ; Omnis homo est risibilis ; E. omnis homo est rationalis. Item : Omne brutum est animal ; Omnis asinus est animal ; E. omnis asinus est brutum. Quare dicendum est, ex puris affirmantibus, in secunda figura recte concludi, si sint tantum tres termini; K 146 APPENDIX. male vero concludi, si sint quatuor termini, ut patet in syllo- gismis vitiosis superius allatis. In priore enim, animal aliter suraitnr in majore quam in minore ; in majore enim, sumitur pro animali conlracto ad asinum, in minore vero, pro animali contracto ad hominem. In posteriore vero syllogismo, medium, scilicet aures, aliter sumitur in majore ac in minore. In ma- jore enim, aures sumuntur pro auribus asininis ; in minore vero, pro auribus humanis. " V^. Regula specialis est : In tertia figura, minor debet esse afirmata. " Verum, contra banc regulam dici potest, syllogismos tertise figurse, ex minori negante, posse esse rectos, modo, non sint quatuor termini, et nuUus terminus sit aequivocus, sicuti dixi- mus de syllogismis primae figurse; atque adeo base regula est superflua. " VI^. Regula specialis est : In tertia figura, con- clusio debet esse particularis : Unde hie syllogismus non valet : Omnis homo est rationahs ; Omnis homo est animal ; E. omne animal est rationale. " Yerum, contra banc regulam dici potest, syllogismos tertiee figurae, conclusionem universalem habentes, esse rectos, modo, sint tantum tres termini non a^quivoci. In syllogismo autem, superius allato, sunt quatuor termini, eo, quod animal aliter su- matur in conclusione, aliter vero in minore. Quod si, animal, in conclusione sumatur eodem modo ac in minore, scilicet pro animali identificato cum bomine, conclusio erit vera, omne scilicet animal, identificatum cum bomine, esse rationale; et sic, syllogismus erit rectus, ut patet attendenti. " Ex bis omnibus patet: omnes regulas syllogismorum, esse superliuas, bac unica excepta, in syllogismo, debent esse tantum tres APPENDIX. 147 termini^ nonplures, nee pauciores ; et per consequens, omnia sophis- mata, ad unum posse revocari, scilicet ad sophisma ab gequivo- catione." Logica C. Wyssii. Genevse, 1669, pp. 318-321. All that is here said touching the certainty of obtaining valid syllogisms, if we only avoid having more than three terms, is quite true ; but the question arises, how are we, under the ex- isting syllogistic forms, to avoid having more than three terms ? The answer is more simple than satisfactory. It is, by observ- ing those very precautions which the special rules enjoin ; — in other words, by recalling in practice the code which bad been theoretically abolished. The reduction of the special rules is so far just, but not a single step is thus taken towards relieving the science in its practical working from the necessity which imposed them. This could only have been done by the express quantification of the predicate, without which, indeed, many of the syllogisms given by Wyss in his reduction are formally worthless, but of which he does not seem to have had a glimpse. No. III. ON FIGURE. The opinions which have been from time to time held by logicians touching the nature and value of the Figures, (that is to say, of the second and third,) seem to have been very fluc- tuating, if not inconsistent and even contradictory. Some have maintained their independence as separate forms of reasoning. Others, again, and these are the great majority, have maintain- ed, that whatever value they possess is reflected on them from the first figure, and thus solely derived from their connexion 148 APPENDIX. more or less direct with it. Of the former, some inconsequently retained the doctrine of reduction^ and thus neutralised by a vicious practice their purer faith; a few, again, altogether rejected it ; while Valla (abolishing the third) even proposes to reduce the moods of the first figure to those of the second. Then, again, with regard to the different members of the syllogism in these figures, opinions seem to have been almost equally divided. The great majority maintained that the syllogisms in these figures had a determinate major and minor premise, and consequently a single determinate conclusion. Others, again, could see no grounds sufficiently decisive on which to establish such certainty of premise, and held, as Apuleius and Valla, in certain cases, two conclusions. The former, however, while at one in their belief, are by no means unanimous as to the grounds on which they vindicate a determinate major and minor premise to these figures. They all agree, of course, that the major premise is that in which the major term is found. All the difficulty lay in discovering the major term ; and the ways in which this was attempted to be done are, so far as I have met with them, of the most inconsequent and assumptive kind. Sometimes the major term was held to be the predicate of the question, or rather the term occupying the predicate place in the question, that, to wit, touching which the doubt arises ; for example, if it were inquired whether man were a stone, stone would on this doctrine be the major term. Sometimes, again, the major was held to be the term which was first enounced; and often enough a major term was conveniently postulated through the arbitrary assumption of a major pre- mise. The methods, indeed, by which it has been attempted to vindicate determinate members to the syllogisms of these figures, all resolve themselves, in the last resort, to a beg- ging of the question. This was generally done in one of two ways ; either a determinate conclusion was begged in order to establish determinate premises, or determinate premises were APPENDIX. 149 begged in order to obtain a single determinate conclusion. I had intended to have gone into this whole question at some length historically, and for this purpose had marked a number of references in various logical writers ; but though those are numerous and varied in character, I do not feel that I am by any means in possession of sufficient evidence to determine his- torically what the catholic doctrine in relation to the above points really was. I have accordingly thrown them aside, and must for the present leave the statement given in the body of the work as it stands. One thing, however, is plain ; that from the earliest to the latest times the procedure of the other figures was felt to be less direct, and their conclusion less authoritative than those of the first. To rectify this imperfection two processes have been devised. The first — that of Reduction — is of old date in the sci- ence, and is that usually practised by the logicians ; the second — which may be termed that of Exposition — is comparatively new ; for though anticipated in some of its details, it is substan- tially Kant's.* This process is that briefly expounded and ap- plied in the text. These two processes have this much in com- mon — that they accomplish their end by the transposition and conversion of propositions. In the former, however, the change made in the proposition is accomplished by formal conversion, in the latter by real inference. Of reduction we have already spoken in the Essay, and need not dwell upon it again here. With regard to the process of Kant, it is itself as tedious and involved as are the reasonings which it is employed to explicate. It is at best but a round-about way of accomplishing what per- haps there is no need for doing at all. The new doctrine, in- * Kant first expounded this speculation of his in a tract published in 1762, and entitled, *' The false Subtilty of the Four Syllogistic Figures Demon- strated," (Die falsche Spitzfindigkeit der vier Syllogistischen Figuren er- wiesen.) This is, I believe, republished with Kant's Logic, in the French translation of that work by M. Tissot. 150 APPENDIX. deed, does away with the necessity or usefulness of any such process. On that doctrine this exposition is itself expounded, and this abolition of the figures itself abolished : — figure appears in its true character as an unessential variation of syllogistic form ; the several figures remain in their integrity with what- ever of special value they ever possessed, while the essential form of the reasoning, when fully stated, is manifest through all the accidental positions of its constituent elements. The same reasoning may be given in either of these accidental varieties of position ; but since it obviously appears as essentially one, its cogency remains the same, and reduction and exposition are therefore equally vain and useless. No. ly. ON NOTATION. The notation employed in the text is that of one of the systems devised by Sir W. Hamilton, in order to represent to the eye the various possible forms of reasoning by distinctive sym- bols. It has all the virtues of a perfect notation. It is simple, distinctive,* perspicuous, and complete. It can represent any * That it be distinctive is a virtue of first account in any system of logical notation; for to borrow the accredited signs of any other science is on every account to be avoided. I need scarcely say, therefore, how earnestly I unite with Mr. Mansell in deprecating ** that mathematical method of ex- position," which is, as he truly says, in relation to logic, " alike injurious to the science and repulsive to the learner." The introduction of mathematical symbols and methods of working into logic is indeed, on every account, to be protested against by all who are in- terested in the welfare of the science. The rejection of these is the more to be insisted on, as well-meaning efi'orts still continue to be made to im- prove logic by mathematical treatment, if not indeed to afford it mathe- APPENDIX. 151 relation of the terms, any order of the propositions, any extent of quantity. The letters represent the terms, the points their quantity, and the lines with the letters the propositions. The letters express, by position to the eye, the relation which the terms have in thought, the middle being placed between the ex- tremes. The meaning of the points has been already explained ; the colon denotes universal quantity, " all ;" the comma particu- lar quantity, " some." Of the lines the shorter denote the pre- mises, the longer the conclusion ; the thick end denotes the subject, the thin end the predicate. Thus the first syllogism given in the table would read as follows : — All B is all A. All C is all B. Therefore, All C is all A. Negative propositions are marked by crossing the copulative line on which the negation falls, as shown in the premises of the negative syllogisms given in the table. I may here notice that the cross is absent from the lines of conclusion in these syllo- gisms by accident, and not by design. It was omitted in the cutting of the type ; and I must request the reader to be good enough to supply it by the pen. This system of notation will now probably be generally known to logical students through the exposition of it given by Mr. matical protection. With all such help, however, it can well afford to dis- pense ; if it could not — indeed, if this were not to it hindrance rather than help — it would have no claim to rank as a separate science. The notion of extending the sphere of mathematics so as to include logic, is as theoreti- cally absurd as its realisation is practically impossible. To identify logic with mathematics is to make the whole equal to its part : while to subor- dinate the former to the latter is to increase the marvel, by making the whole less than its part. And those who, without attempting this, display their skill by translating logical forms into mathematical language, accom- plish a work just about as useful and praiseworthy as that " of the two zealous, but thick-headed logicians — Herlinus and Dasypodius by name — who rendered the first six books of Euclid into formal syllogisms." All such endeavours possess the singular merit of making logic as repulsive as possible, without doing the least service to mathematics. 152 APPENDIX. Thomson in his "Outline of the Laws of Thought," where further details respecting it may be found. Mr. Thomson says, in intro- ducing his explanation, that " many of the different elements of the notation are not new." With all respect for the statement of so careful and zealous a student of logic as Mr. Thomson, I must say I cannot but think that this is a mistake. I do not know, of course, what authority Mr. Thomson may have for his statement ; but, with some general knowledge of most of the previous systems of logical notation which have been employed, I cannot recall any which anticipate the present, either in no- tion or detail ; unless^ indeed, the bare use of lines, though in a totally different manner, can be said to do so. I cannot think, however, that this is what Mr. Thomson refers to ; for the linear notation is a separate system, altogether distinct from the one in question. I had intended to have introduced here fuller tables, running the positive and negative syllogisms through all the figures, as well as some specimens of other systems of notation which Sir William Hamilton has kindly placed at my disposal. I am not without hope, however, that Sir William will himself publish them in full before very long ; and I need scarcely say, there- fore, how gladly I relinquish their partial introduction here. I know how earnestly all who are interested in logical science will unite with me in the hope that Sir William Hamilton's health and leisure may be such as to enable him to carry through the press at no distant interval some portion of his promised work. APPENDIX. 15{ NOTE BY SIR WILLIAM HAMILTON. The following note contains a summary of my more matured doctrine of the Syllogism, in so far as it is relative to the pre- ceding Essay. All mediate inference is one — that incorrectly called Categori- cal; for the Conjunctive and Disjunctive forms of Hypothetical reasoning are reducible to immediate inferences. Mentally one, the Categorical Syllogism^ according to its order of enouncement, is either Analytic or Synthetic. Analytic, if (what is inappropriately styled) the conclusion be expressed first, and (what are inappropriately styled) the premises be then stated as its reasons. Synthetic, if the premises precede, and, as it were, eifectuate the conclusion. These general forms of the syllogism can with ease be distinguished by a competent notation ; and every special variety in the one has its corre- sponding variety in the other. Taking the syllogism under the latter form, (which, though perhaps less natural, has been alone cultivated by logicians, and to which, therefore, exclusively all logical nomenclature is rela- tive,) — the syllogism is again divided into the Unjigiired and the Figured, The Unfigured Syllogism is that in which the terms compared do not stand to each other in the reciprocal relation of subject and predicate, being in the same proposition, either both sub- jects or both predicates. Here the dependency of Breadth and Depth, (Extension and Intension,, Extension and Comprehension, &c.,) does not subsist, and the order, accordingly, of the pre- mises is wholly arbitrary. This form has been overlooked by the logicians, though equally worthy of development as .iny L 164 APPENDIX. other ; in fact, it affords a key to the whole mystery of Syllo- gism. And what is curious, the canon by which this syllogism is regulated, (what may be called that of logical Analogy or Proportion,) has, for above five centuries, been commonly stated as the one principle of reasoning, whilst the form of rea- soning itself, to which it properly applies, has never been gene- ralized. This canon, which had been often erroneously, and never adequately enounced, in rules four, three, two, or one, is as follows : — In as far as two notions, (notions proper or indivi- duals,) either both agree, or one agreeing^ the other does not, with a common third notion ; in so far, these notions do or do not agree with each other. — The propositions of this syllogism in no-figure are marked in the scheme of pure logical notation by horizontal lines of uniform breadth. In the Figured Syllogism, the terms compared are severally subject and predicate, consequently, in reference to each other, containing and contained in the counter wholes of Intension and Extension. Its canon is : — What worse relation of subject and predicate subsists between either of two terms and a common third term, with which one, at least, is positively related; that relation subsists be- tween the two terms themselves. — In the scheme of pure logical notation a horizontal tapering line marks this relation ; the sub- ject standing at the broad, the predicate at the pointed end. There are three, and only three. Figures — the same as those of Aristotle ; and in each of these we may distinguish the orders of Breadth and of Depth. The First Figure emerges, when the middle term is subject of the one extreme and predicate of the other ; that is, when we pass from the one extreme to the other, through the middle, in the order whether of Extension or of Intension. In the nota- tion of this Figure, we may of course arbitrarily make either of these orders to proceed from left to right, or from right to left ; that is, two arrangements are competent. — There is here, deter- minately, one direct and one indirect conclusion. APPENDIX. 155 The Second Figure arises, when the middle term is the predi- cate of both extremes ; the order of Breadth proceeding from middle to extremes, the order of Depth from extremes to middle. The Third Figure is determined, when the middle term is the subject of both extremes; the order of Extension proceeding from extremes to middle, the order of Intension from middle to extremes. In the Second and Third Figures there is thus only one ar- rangement possible in logical notation. And as Extension and Intension are here in equilibrium, there is no definite major and minor premise, and consequently no indirect, but two indifferent conclusions. — This is best marked by two crossing lines under the premises, each marking the extreme standing to the other as subject or as predicate. Of course each Figure has its own canon, but these it is not here requisite to state. The First Figure, besides its more general canon, has also two more special, — one for Syllogisms in the order of Extension, and one for Syllogisms in the order of Intension. And what is remarkable, Aristotle's Dictum de Omni, &c., (in the Prior Analytics,) gives that for Extension, whilst his rule — Prcedicatum prcedicati^ &c., (in the Categories,) affords that for Intension, although this last order of Syllogism was not developed by him or the logicians ; — both inadequately. In regard to the notation of Quality and Quantity in the syllo- gisms unfigured and figured : — Negation is marked by a per- pendicular line, which may be applied to the copula, to the term, or to the quantification. — As to Quantity, (for there are subordinate distinctions,) it is sufiicient here to state, that there is denoted — by the sign [ ? or « ] (for the quantity of one term ought to face the other), some; — by the sign [ *.], all; — by the sign [ . ], 05 half; — by the sign [ ' or * ], more than a half The last two are only of use to mark the ultra-total distribution of the middle term of a syllogism, between both the premises, as af- fording a certain inference, valid, but of little utility. This I 156 APPENDIX. once thought had been first generalized by me, but I have since found it fully stated and fairly appreciated by Lambert, to say nothing of Fromraichen. Above (p. 76) is a detail of ray pure logical notation, as ap- plicable to the thirty-six moods of the first figure. The order there is not, however, that which I have adopted. The follow- ing is my final arrangement, and within brackets is its corre- spondence with the numbers of that given above : — The moods are either A) Balanced^ or B) Unbalanced. In the former class both terms and propositions are balanced, and it contains two moods — i ; ii, [=i ; ii.] In the latter class there are two sub- divisions. For either, a) the terms are unbalanced, — iii, iv, [=xi, xii] ; or, b) both the terms and propositions are unbalanced, — V, vi; vii, viii ; ix, x ; xi, xii, [=vii, viii ; iii, iv ; v, vi ; ix, X.] The following equation applies to my table of moods given in Mr. Thomson's Laws of Thought; — i; ii ; xi, xii; vii, viii; iii, iv; v, vi; ix, x. — The present arrangement is also more minutely determined by another principle, but this it is not here requisite to state. If we apply the moods to any matter however abstract, say letters, there will emerge forty -two syllogisms ; for the formal identity of the balanced moods will then be distinguished by a material difference. On the contrary, if we regard the mere form,al equivalence of the moods, these will be reduced to twenty- one reasonings, — seven affirmative, and fourteen negative. Of the balanced moods, i and ii are converted each into itself; of the unbalanced, every odd, and the even number immediately fol- lowing, are convertible ; and in negatives, the first and second moods (a, b) of the corresponding syzygy or jugation, is reduced from or to the second and first moods {b, a) of its reciprocal. There are no exceptions. The canon is thorough -going. Only it must be observed : that the doctrine is erroneous which teaches, that a universal negation is not a woi^se relation than a particular ; and that the identity of a negative with an aflnirma- APPENDIX. 167 five mood, is regulated exclusively by the identity in quantity of the two syzygies or antecedents. The Greeks, in looking to the conjugation of the premises alone, are more accurate than the Latins, who regard all the three propositions of a syllogism in the determination of a mood. It is not to be forgotten, that as the correlation of the logical terms ought to be known only from the expression, (ex facie propositionis aut syllogisrai,) for all other knowledge of the re- ciprocal dependence of notions is contingent, material, and ex- tralogical; and as the employment of letters, following upon each other in alphabetical order, may naturally suggest a cor- responding subordination in the concepts which they denote : I have adopted the signs C and F, which are each the third letter in its respective alphabet, for the extremes ; and the sign M, for the middle term of the syllogism. The scheme is thus eman- cipated from all external associations, and otherwise left free in application. I also transpose the former symbols in the in^ terconvertible moods ; so that whereas in the one stand C M F, in the other stand F M C. W. H. EDINBURGH I T. CONSTABLE, PRINTER TO rfER MAJESTY. ERRATA. Supply brackets to the extracts given from Ploucquet at the foot of pages 22, 23. Page 48, second line from bottom, ./or premiss, read premise. ,, 73, line 17, note, /or tracing, read touching. „ 85, line 20, /or eisque, read ejusque. ^^^Al^ OF THT: :UHIVBIISIT7^ POBT-ROTAL LOGIC, In r2mo, cloth, price 5s. 6d., LOGIC; OR, THE ART OF THINKING. BEING THE PORT-ROYAL LOGIC. Translated from the French. With an Introduction. By THOMAS SPENCER BAYNES. This work possesses nearly every quality desirable in a text-book, and, in this point of view, stands in remarkable contrast to the feeble, superficial, and incomplete compilations with which even our best schools of learning have hitherto been contented. There have been two previous translations into Eng- lish, but the present is the only trustworthy one. We have to express our en- tire satisfaction with the manner in which Mr. Baynes has discharged his duty, and to express the hope that a gentleman who can write so well as he has done in his introductory essay, may soon appear before the public again in the char- acter of author." Critic. " With regard to this translation we need say little. The ability, accomplish- ments, and competence of Mr. Baynes are a suflBcient guarantee for its ac- curacy. We have examined several parts of it, and can freely affirm, that the terseness and precision of the original are admirably preserved in the transla- tion. The manner in which the publishers have performed their part is every way worthy of the work itself. We would, therefore, earnestly recommend this treatise to all who are desirous of acquiring habits of accurate thinking. No modern work in English with which we are acquainted is at all so well adapted to secure this end — Whately not excepted. We trust, then, that the success of this book will be proportioned to its merits ; and, in parting with Mr. Baynes, we would express our hope that he will ere long favour us with other similar contributions from his pen, which are at present so much needed in this country, and for which his acute philosophical powers and great stores of knowledge so well qualify him." Edi7iburgh News. " The Port-Royal Logic, though familiar to the more advanced logicians of this as of other countries, and appreciated by them as it deserves to be, has not hitherto met with an English translator capable of doing, or who has done, any- thing like justice to the work. Mr. Baynes has a freedom and power which could only have been achieved by one fully conversant, not only with both the languages he has to deal with, but the subject of which the book treats ; while we instinctively feel the rendering to be faithful, without necessity for reference to the original, from its reticence of that idiomatic spirit which so pleasantly in- troduces us to the individualities of its authors, without that undue deference to the mere idiomatic form which might have painfully obtruded on us continually that they were men of another country and another era." Caledonian Mercury. " The translation is evidently the product of an accomplished scholar and an expert logician — who has done good service to the cause of mental improve- ment by the publication of this volume, appropriately dedicated to Sir William Hamilton, Professor of Logic and Metaphysics in the University of Edinburgh, to whose kind encouragement the author expresses many obligations." Glasgow Constitutional. " We have much pleasure in commending this volume to the favourable notice of students and intelligent general readers ; and have to bear testimony to the accuracy, spirit, and elegance with which the translation is eifected." Scottish Press. " Mr. Baynes has performed a welcome service to the student. The Port- Royal Logic needs no commendation from us ; and this neatly-printed transla- tion of it will contribute, we trust, to make it better known to the youth of the country." British Quarterly Review. SuTHEi.LAND &. Kxox, Edinburgh ; Simpkin, Marshall, & Co., London. MR. M'COSH ON THE METHOD OF THE DIVINE GOVERNMENT. In the Press, Second Edition of THE METHOD OF THE DIYINE GOYEMMENT, PHYSICAL AND MORAL. By the rev. JAMES M'COSH, A.M. " Aloof from any difference of opinion, and though I have as yet only read the work in part, it appears to me worthy of the highest encomium, not only from the excellence of the intention, but" for the ability with which it is exe- cuted. It is refreshing to read a work so distinguished for originality and soundness of thinking, especially as coming from an author of our own coun- try." Sir William Hamilton, Bart. " In the writer of this work we meet with a man of extraordinary calibre, alike remarkable for the vigour and originality of his thinking — for the fine taste and freshness of his writing — for the extent of his learning, and the breadth and minuteness of his acquaintance with those sciences which, from the circumstance that they are prosecuted with avidity by the greater minds of the age, impart, more than the others, colour and tone to the age's thinking. Mr. Si'Cosh's work is of the compact cast and thought-eliciting complexion, which men do not willingly let die ; and we promise such of our readers as may possess themselves of it much entertainment and instruction of a high order, and a fund of solid thought which they will not soon exhaust." Witness. " To the great task which he has thus set himself, Mr. M'Cosh has brought great powers and ample resources. He is evidently a man of a profoundly plii- losophic spirit, and at the same time a man of extensive and varied culture in science and literature. His philosophic reading seems to have been very ex- tensive, embracing not only all the better authors in theological, metaphysical, and ethical science, but also the most approved writers on the various branches of physical speculation. He combines with this a power of independent think- ing and original speculation which enables him to move easily under the accu- mulated mass of his learning, and at the same time to apply what he has learned from others to purposes of his own. Perhaps, for most readers, less copious- ness of matter, and greater condensation and point in respect of argumentation, would have been an advantage, as it requires considerable previous familiarity with the subject to be able always to find the author's bearings in the extensive field he has selected for exploration. To others, however, who desire not merely to gather a result, but also to witness the process by which the author has himself advanced to it, the plan Mr. M'Cosh has followed will have its charms ; whilst for those who find it somewhat difficult to follow him, the sum- maries he has given at successive stages of his argument will, doubtless, prove serviceable. We may add, that he possesses an enviable power of apt and striking illustration, by which he is enabled both to relieve the attention and facilitate the comprehension of the reader in the abstruse parts of his book." British Quarterly Revieiv. DESCARTES. In the Press, AN ENGLISH TRANSLATION OF DESCARTES ON METHOD. By JOHN VEITCH. Sutherland & Knox, Edinburgh ; Simpkin, Marshall, & Co., Loiidoti. THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO 50 CENTS ON THE FOURTH DAY AND TO $1.00 ON THE SEVENTH DAY OVERDUE. im 1. ? APR l a mu AtiTC.piscJAN29 '87 MAY 9 1944 ISWar'fiSHD MAR 1 9. 1953 LU 27Nov54¥Li jVOVl 8 ■■^- JUL 29 1978 JUL 1 7 1978 JBiMX — ji ii 1 '' 73 '^AN sm r - ^ LD 21-100m-7,'33 jmmi / S3- //f3^7 Win ^ . 1 .' J ^ 3 ^H 2 III ^^^^^^ ^^H