STACK ANNJiX SYLLABUS A PROPOSED SYSTEM OF LOGIC. AUGUSTUS DE MORGAN, F.R.A.S. F.C.P.S. OF TKINITY COLLEGE, CAMBRIDGE ; PROFESSOR OF MATHEMATICS IN UNIVERSITY COLLEGE, LONDON. Ai'-rXnv- 'ifiuffiv 01 ftaQivri; y oa./j,uocra,' 'O 'ygapifiaTvii etTtigof au liXixit /3Xf(Tv. LONDON: WALTON AND MABERLY, 28 UPPER GOWER STREET, AND 2T IVY LANE, PATERNOSTER ROW. 1860. SEPARATE WORKS THE AUTHOR OF THIS SYLLABUS. Book of Almanacs, with Index of Reference, obi. 8vo *Connexion of Number and Magnitude, post 8vo. . Differential and Integral Calculus, 8vo. *Elements of Algebra, post 8 vo. .... Arithmetic, post 8vo. * Trigonometry, post 8vo. fEssay on Probabilities and Life Contingencies, 12mo. *First Notions of Logic, 12mo. .... Formal Logic ; or, the Calculus of Inference, 8vo. Notices of Arithmetical Books and Authors, post 8vo. JTreatise on Calculus of Functions, 4to. J Theory of Probabilities, 4to. the Globes, Celestial and Terrestrial, 8vo. Trigonometry and Double Algebra, post 8vo. Syllabus of a Proposed System of Logic, 8vo. *. d. 5 Walton Sf Maberly 4 10 Baldwin 9 Walton if Maberly 5 9 3 6 Lonyman 1 6 Walton &f Maberly 6 6 2 6 3 Griffin 3 6 5 Malby 7 6 Walton if Maberly 1 Out of print. t Cabinet Cyclopaedia. Encyclopaedia Metropolitans. PREFACE. THE matters collected in this Syllabus will be found in those of my writings* of which the titles follow: I. On the Structure of the Syllogism .... (Cambridge Transactions, vol. viii, part 3, 1847). II. Formal Logic, or the Calculus of Inference, necessary and probable (London, Taylor and Walton, 1847, 8vo.). III. On the Symbols of Logic, the Theory of the Syllogism, and in particular of the Copula, .... (Cambridge Transactions, vol. ix, part 1, 1850). IV. On the Syllogism, No. iii, and on Logic in general (Cambridge Transactions, vol. x, part 1, 1858). Of these works the formulae and notation of the first are entirely superseded ; the notation only of the second (the Formal Logic] may be advantageously replaced (see 24) by that of the third and fourth and of the present tract. There is very little in the first three writings on which my opinion has varied ; but of all three it is to be said that they are entirely based on what I now call the arithmetical view 6f the proposition and syllogism ( 8, 173, 174), extending this term not merely to the numerically definite syllogism, but to the ordinary form, to my own extension of it, and to Sir W. Hamilton's departure from it. * The writings which oppose any of my views at length are the following, so far as my memory serves. I. A letter to Augustus De Morgan, Esq. . . . on his claim to an independent rediscovery of a new principle in the theory of syllogism, from Sir William Hamilton, Bart London and Edinburgh, Longman and Co., Mach- lachlan and Co., 1847, 8vo. II. Review of my Formal Logic, signed J. S., in the Biblical Review, 1848. III. Review of my Formal Logic (since acknowledged by Mr. Mansel, the author of the Prolegomena Lpgica) in the North British Review/or May 1851, No. 29. IV. Discussions on Philosophy by Sir W. Hamilton; London, Longman and Co.; Edinburgh, Machlachlan and Co. (1st edition, 1852, 8vo. pp. 621*-652* ; 2nd ed. 1853, 8vo. pp. 676-707) ; in which a letter is reprinted which first appeared in the Athoneettm of August 24, 1850. 4 PBEFACE. The relations of my work on Formal Logic to the present syllabus are as follows. Chapter I, First Notions, may afford previous knowledge to the student who has hitherto paid no attention to the subject. Chapter III, On the abstract form of the Proposition may be consulted at 93 of this syllabus. Chapters IV, On Propositions, and V and VI, On the Syllogism, are rendered more easy by the nota- tion of this syllabus, and are partially superseded. Chapter XIV, On the verbal Description of the Syllogism, is entirely superseded. rest of the work may be read as the titles of the chapters suggest. A syllabus deals neither in development nor in diversified ex- ample : and does not make the space occupied by any detail a measure of its importance as a part of the whole. I have omitted many subjects which are to be found in all the books, or dwelt lightly upon them : partly because more detail is contained in my Formal Logic, partly because any one who masters this tract will be able to judge for himself what I should have written on the omitted subjects. I have also endeavoured to remember that as a work of this kind proceeds, less detail of explanation is necessary. I should suppose that a student of ordinary logic would find no great difficulty in understanding my meaning : and that those who are accustomed to symbolic expression, mathematical or not, would, even though unused to logical study, find no more difficulty than an ordinary student finds in Aldrich's Compendium. Either of these classes, I should think, would not fail to come to the point of under- standing at which a reflecting mind can allow itself to meditate acceptance or rejection without latent fear of over-confidence. Whether a beginner who is conversant neither with ordinary logic nor with symbolic language will comprehend me is another question : and one on which those who try will divide into more than two classes. Such a reader, making concrete examples for himself as he goes on, and never leaving any article until he has done this, will either get through the whole tract, or will stop at the precise point at which he ought to stop, upon the principle of the next paragraph. Every spoon has some mouths that it can feed ; and some that it cannot. Every writer has some readers who are made for him, and he for them ; and some between whom and himself there is a great gulph. I might prove this, in my own case, by a chain of discordant PREFACE. testimonies running through thirty years, if I had leisure and liking to hunt up extracts from reviews. I will content myself with a couple which are at hand : observing that I have no acquaintance with the authors. In 1830, I published my treatise on Arithmetic, and the following sentences speedily appeared in reviews of it : This book appears to us to mystify a very simple science. It is as clear as Cobbett in bis lucid intervals. In 1847, I published my Formal Logic, and two opponents of- my views wrote as follows : Mr. De Morgan I This is an undeniably long extract, and yet we would, did our is certainly not a j limSts allow, continue it .... "We beg the reader's notice to lucid writer. the exquisite precision of its language ; to the definiteness of every line in the picture ; for though it is a description of a profound mental process, still it is a luminous picture, the light of which does not interfuse its lineaments .... We are at very solemn issue with Mr. De Morgan upon this argument . . . These antagonisms remind us of the stork and the fox, and of the failure of their attempts to entertain each other at dinner. When an author's dish and a reader's beak do not match, they must either divide the blame, or agree to throw it on that exquisite piece of atheism, the nature of things. The points on which I differ from writers on logic are so many and so fundamental, that I am among them as Hobbes among the geometers, and, mutato nomine, may say with him of Malmesbury : In magno quidem periculo versari video escistimationem meam, qui a logicis fere omnibus dissentio. Eorum enim qui de iisdem rebus mecum aliquid ediderunt, aut solus insanio ego aut solus non insanio ; tertium enim non est, nisi (quod dicet forte aliquis) insaniamus omnes. Hobbes was in the wrong : the question of parallel or contrast must be left to time. But though some writers on logic explicitly renounce me and all my works, yet one point of those same works is adopted here and one there ; so that in time it may possibly be said that Mahometans eat up the hog. All these differences are not about the truth or falsehood of my neologisms, but about the legitimacy of their adoption into logic, the study of the laws of thought. And I cannot imitate Hobbes so far 6 PREFACE. as to write contra fastum logicorum, seeing that, oblitis obliviscendis, I have personally nothing but courtesy to acknowledge from all the writers of known name who have done me the honour of alluding to my speculations. In return, I endeavour to tilt at their shields and not at their faces. Should I make any one feel that I have missed my attempt, I will pray the introduction into these lists of the old practice of the playground. When I was a boy, any offence against the rules of the game was held to be nullified if the offender, before notice taken, could cry Slips! as I now do over all, to meet con- tingencies. As to the matter of our differences, I neither give nor take quarter : it is I will lay on for Tusculum, Do thou lay on for Rome ! as will sufficiently appear in my notes. Now to another topic. I produce a fragment of a well-known conversation : those who choose may fill up the chasms. "... I therefore dressed up three paradoxes with some ingenuity .... The whole learned world, I made no doubt, would rise to oppose my systems : but then I was prepared to oppose the whole learned world. Like the porcupine, I sat self- collected, with a quill pointed against every opposer. Well said, my boy, cried I, .... you published your paradoxes; well, and what did the learned world say to your paradoxes ? Sir, replied my son, the learned world said nothing to my para- doxes ; nothing at all, sir. Every man of them was employed in praising his friends and himself, or condemning his enemies ; and unfortunately, as I had neither, I suffered the cruellest mortification, neglect." A friend of the author just quoted remarked that a shuttlecock cannot be kept up unless it be struck from both ends. I was spared all the mortification of neglect by the eminence of the player who took up the other battledore. Of this celebrated opponent I can truly say that, so far as I myself was concerned, I never looked with any- thing but satisfaction upon certain points of procedure to which I shall only make distant allusion. For I saw from the beginning that he was playing my game, and raising the wind which was to blow about the seeds of my plant. The mathematicians who have written on logic in the last two centuries have been wholly unknown to even the far-searching inquirers of the Aristotelian world : to the late Sir William Hamilton of Edinburgh I owe it that I can present this tract to the moderately well informed elementary student of logic, as con- taining matters of which he is likely enough to have heard something, and may possibly be curious to hear more. PREFACE. 7 In controversy and controversy was to him an element of life and a spring of action Sir William Hamilton was too much the fencer of the moment, too much the firer of to-morrow's article : his impulses sometimes leap him over the barrier which divides philosophy from philosophism. Hoot the proofs of this out of his pages, and we have before us a man both learned and ingenious, profound and acute, weighty and flexible, displaying a most instructive machinery of thought as well when judged right as when judged wrong, save only when cause of regret arises that Oxford did not demand of his youth two books of Euclid and simple equations. In describing a character some points of which are at tug of war especially when liking for quotation was one of them Greek may meet Greek. He showed more fondness than was politic in one so plainly destined to survive the grave for a too literal version of the motto But on the other hand, though too much inclined to rule the house, he was d/xoSss-jroTJi? oW(y X)(Y X)-)y , x(-(Y , *()y *))y X)-)Y , X)(y x()Y . x C(y 16 RELATIONS OF PROPOSITIONS. [26-30. verted, change both quantities, and preserve the quality : thus X(-)Yisx>(y. 27. The rule of conversion* making the names change places, without altering the import of the proposition is, Write or read the proposition backwards. Thus X))Y is Y((X; or X))Y may be read backwards, Some Ys are all Xs. That is, make both the terms and their quantities change places. * Writers on logic have nearly always meant by conversion merely the change of place in the terms, without change of place in the quantities. Ac- cordingly, when the quantities are different, (common) logical conversion is illegitimate. Thus X))Y and Y))X are not the same : but X()Y and YQX are the same. There is this difficulty in the way of using the word conversion in the sense proposed in the text : namely, that common logic has rooted it in common language that 'Every X is Y' is the converse (true or false as the case may be) of 'Every Y is X.' Leaving the common idioms for the student to do as he likes with, I shall, if I have occasion to speak of a proposition in which terms only are converted, and not quantities, call it a term- converse. 28. Each universal is inconsistent with the universals of dif- ferent qualities, and indifferent to the universal of different quan- tities. Thus X))Y is inconsistent with X) g (Y and X(-)Y, and neither affirms nor denies X((Y. Each universal affirms the particulars of the same quality, contradicts the particular of different quantities, and is indifferent to the particular of the same quantities. Thus X))Y affirms X()Y and X)(Y, contradicts X(-(Y, and neither affirms nor denies X)-)Y. Is inconsistent Neither affirms with nor denies )( Is neither affirmed nor denied by (0 )( ('( )') )( )') C( 29. Contrary names, in identical propositions, always appear with different quantities. We cannot speak of some Xs without speaking about all xs ; nor of all Xs without speaking about some xs. 30. A particular proposition is strengthened into a universal which affirms it (and more, may be) by altering one of the quan- tities : thus )) is affirmed in () and in )( Remember 16. Affirms Contradicts () )( C( )') C( o )( )) Is affirmed by Contradicts )'( C) )) (( )) X )) (( (0 C) )'( (( 31-36.] SYLLOGISM. 17 31. In a universal proposition, if one term be partial, it has the amount, not the character, of the quantity of the other: if both, the quantities of the two terms together make up the whole uni- verse, with the part common to both, if any, repeated twice. 32. In a particular proposition, the quantity of a partial term is vague, but remains the same through all forms. And when both terms are total, the partial quantity still remains expressed : as in X)(Y, or Some things are neither Xs nor Ys ; which some things are as many as the xs or ys in the equivalents x()y, X)-)y,andx(-(Y. ' 33. If a proposition containing X and Y be joined with a proposition containing Y and Z, a third proposition containing X and Z may necessarily follow. In this case the two first pro- positions (premises) and the proposition which follows from them (conclusion) form a syllogism. 34. If an X be a Y, if that same Y be a Z, then the X is the Z. This is the unit-syllogism from collections of which all the syllogisms of this mode of treating propositions must be formed. At first sight it seems as if there were another : if an X be a Y, if that same Y be not any Z, then the X is not any Z. But this comes under the first, as follows : the X is a Y, that Y is a z, therefore the X is a z, that is, is not any Z. The introduction of contraries brings all denials under assertions. 35. Two premises have a valid conclusion when, and only when, they necessarily contain unit-syllogisms; and the con- clusion has one item of quantity for every unit-syllogism so necessarily contained. 36. And all syllogisms may be derived from the following combinations : )) )) or X))Y Y))Z, or All Xs are Ys and all Ys are Zs. The conclusion is X))Z, All Xs are Zs : there is the unit-syllogism, This X is a Y, that same Y is a Z, repeated as often as there are Xs in existence in the universe. Or, X))Y))Z gives X))Z, or )) )) gives )). () )) or X()Y Y))Z, or Some Xs are Ys, all Ys are Zs. The conclusion is X( )Z, Some Xs are Zs : there is the unit- syllogism so often as there are Xs in the first premise. Or, X()Y))Z gives X()Z, or () )) gives (). (( () or X((Y Y()Z, or Some Xs are all Ys, some Ys are Zs. The conclusion is X( )Z, Some Xs are Zs : this case is, as to form, c 18 YAHIETIES OF SYLLOGISM. [36-40. nothing but the last form inverted. Or, X((Y()Z gives X()Z or (( )) or X((Y Y))Z or Some Xs are all Ys, All Ys are Zs. The conclusion is X()Z, Some Xs are Zs, as many as there are Ys in the universe. Or, X((Y))Z gives XQZ, or (( )) gives ( ). But this case gives no stronger conclusion than () )) or than (( (), though it has both premises universal. These are all the ways in which affirmative premises produce a conclusion in a manner which has no need to take cognisance of the existence of contrary terms. And since all negations are con- tained among affirmations about contraries, we may expect that application of these cases to all combinations of direct and contrary will produce all possible valid syllogisms. 37. Apply the form )) )) to the eight varieties XYZ, xYZ, xyZ, xyz, xYz, XYz, Xyz, XyZ, and contravert x, y, z, when- ever they appear. Thus )) )) applied to x y Z is x))y))Z, or X((Y combined with Y(-)Z or X((Y(-)Z or (( (). The con- clusion is x))Z or X(-)Z. That is, X((Y(-)Z gives X(-)Z; or, If some Xs be all [make up all the] Ys, and everything be either Y or Z, then everything is either X or Z. This process applied to the eight varieties gives the following eight forms of universal syllogism, that is, universal premises with universal conclusion. . )) )) (0 (( () (( (( () X . )) X X (( X () Here are all the ways in which two universals give different quan- tities to the middle term. 38. Apply () )) to the eight varieties and we have eight minor-particular syllogisms, particular conclusion with the minor (or first) premise particular, ())) )))) )(().)((( ))>( OX (((( ('(() Here are all the ways in which a particular followed by a uni- versal give different quantities to the middle term. 39. Apply (( (), and we have eight major-particular syllo- gisms, particular conclusion with the major (or second) premise particular. ((() XO )))). )))( )(((. (((( ())( () Here are all the ways in which a universal followed by a par- ticular gives different quantities to the middle term. 40. Apply (( )) and we have eight strengthened particular syllogisms, universal premises with particular conclusion. (()) >()) ))O ))(( XX ((>( ()(( ()() 40-46.] TEST OF VALIDITY, RULE OF INFERENCE. 19 Here are all the ways in which two universals give the same quantity to the middle term. 41. There are 64 possible combinations, of which the 32 enumerated give inference. The remaining 32 may be found by applying the eight varieties to ( ) (( , )) ( ) , ( ) )( and ( ) ) ) : and in no case does any inference follow. Thus X()Y and Y((Z are consistent with any of the eight relations between X and Z, which should be ascertained by trial. 42. The test of validity and the rule of inference are as follows, There is inference 1. When both the premises are universal ; 2. When, one premise only being particular, the middle term has different quantities in the two premises. The conclusion is found by erasing the middle term and its quantities. Thus )( () gives )) or )) ( 21). That is ' No X is Y, and Everything is either Y or Z ' gives ' Every X is Z '. 43. Premises of like quality give an affirmative conclusion : of different quality, a negative. A universal conclusion can only follow from universals with the middle term differently quan- tified in the two. From two particular premises nothing can follow. 44. A particular premise having the concluding term strength- ened, the conclusion is also strengthened, and the syllogism is converted into a universal : having the middle term strengthened, the conclusion is not strengthened, and the syllogism is converted into a strengthened particular syllogism. Thus if () )), with conclusion (), have the premise () strengthened into )), the syllo- gism becomes )) )) and yields )). But if () be strengthened into ((, the syllogism becomes (( )) and yields only (), as before. 45. A universal conclusion affirms two particulars : if either of these be substituted in the conclusion" of the universal syllogism, the syllogism may be called a universal of weakened conclusion or a weakened universal. Thus X))Y))Z, made to yield only X( )Z or X)(Z, instead of X))Z, is a universal of weakened conclusion. No further notice need be taken of this case. 46. Of the 24 syllogisms of particular conclusion, the con- clusions are equally divided among (), X> )*)' anc ^ (*( ^^ e following table is one of many modes of arrangement of the whole. 20 ARRANGEMENTS OF SYLLOGISM. [46-48. Premises Affirmative Negative Affirmative Minor Affirmative Major Universal ^Particular (( )) (( X C) () X )) X X (( )) X The middle column contains the universals : and each universal stands horizontally between the two particulars into which it may be weakened, by weakening one of the concluding terms. And each strengthened particular stands vertically between the two particulars from which it may be formed by altering the quantity of the middle term in the particular premise only. 47. If two propositions give a third, say A and B give C ; then, a, b, c, meaning the contrary propositions of A, B, C, it follows that A, B, c, cannot all be true together. Hence if A, c, be true, B must be false, or b true : that is, if A, B, give C, then A, c, give b. Or, either premise joined with the contrary of the conclusion, gives the contrary of the other premise. And thus each form of syllogism has two opponent forms . But the order of terms will not be correct, unless the premise which is retained be converted. If the order of the terms in the syllogism be XY YZ XZ, we shall have in one opponent XY XZ YZ, which in our mode of arrangement must be YX XZ YZ, the retained premise changing the order of its terms. Thus the opponent forms of )) )), which gives )), are as follows. First, ((, retained premise converted; ((, contrary of conclusion; ((, contrary of other premise; giving (( (( and con- clusion ((. Secondly, ((, contrary of conclusion; ((, retained premise converted ; (( , contrary of other premise ; giving (( (( and conclusion ((. 48. The universal and particular syllogisms can be grouped by threes, each one of any three having the other two for its 48-53.] OPPONENTS. QUANTITY OF CONCLUSION. 21 opponents. And these groups can be collected in the following zodiac, as it may be called. )) J O w V:^ The universal propositions at the cardinal points are so placed that any two contiguous give a universal syllogism, whether read forwards or backwards, as )( ((, )) )(. Join each of these universals with its contiguous external particular, so as to read in a contrary direction to that in which the two universals were read, and a triad is formed each member of which has the other two members for its opponent forms. As in >((( >(() ())); or as in () )) X (( (OX- 49. The strengthened particulars have weakened universals ( 45) for their opponent forms. Thus (( )) with the conclusion () has )) )( with the conclusion (( and )( (( with the conclusion )), for its opponents. And (( (( with the conclusion () has )) )( with the conclusion ).) and )( )) with the conclusion )) for its opponents. 50. The partial terms of the conclusion take quantity in the following manner, In universal syllogisms. If one term of the conclusion be partial, its quantity is that of the other term : if both, one has at least the quantity of the whole middle term, and the other of the whole contrary of the middle term. 51. In fundamental particular syllogisms. The partial term or terms of the conclusion take quantity from the particular premise. 52. In strengthened particular syllogisms. The partial term or terms take quantity from the whole middle term or its whole contrary, according to which is universal in both of the premises. 53. These rules run through every form of the conclusion in which there is a particular term. Thus X))Y))Z gives 1. X))Z in which Z has the quantity of X 2. x((z in which x has the quantity of z 3. x(-)Z, xs as many as ys, and Zs as many as Ys 22 SORITES. COMPLEX PROPOSITION. [53-57. Again, X(-)Y)(Z gives X(-(Z, X()z, and x)-)z, in which the quantities of X( and of )z are the number of instances in the ' some things ' of Y)(Z. Thirdly, X>(Y>(Z gives X)(Z, x(-(Z, x()z, and X, in which the quantities of x( and )z are the number of instances in Y. 54. A sorites is a collection of propositions in which the major term of each is the minor term of the next, as in X))Y>(Z(-)T))U)(V((W or All Xs are Ys, and No'Y is Z, and everything is either Z or T, and every T is TJ, and No U is V, and Some Vs are all Ws. 55. A sorites gives a valid inference, 1. Universal, when all the premises are universal, and each intermediate term enters once totally and once partially ; 2. Particular, when one (and one only) of the two conditions just named is broken once, whether by con- tiguous universals having an intermediate of one quantity in both, or by occurrence of one particular without breach of the rule of quantity. 56. The inference is obtained by erasing all the intermediate terms and their quantities, and allowing an even number of dots to indicate affirmation, and an odd number of dots to indicate negation. Thus X))Y>(Z()T>(U(-)V))W g i v e s X)-)W X(-)Y((Z((T(.)U)(V((W g i v es X(-(W X((Y(.)Z>(T(-)U>(V gives X((V 57. We have seen that each universal may coexist with either the universal of altered quantities or with its contrary : which is a species of terminal ambiguity. Thus X))Y may have either X((Y or X)-)Y true at the same time. All these coexistences may be arranged and symbolised as follows ; giving propositions which, with reference to the ambiguity aforesaid, have terminal precision. 1. X))Y or both X))Y and XY All Xs and some things besides are Ys 2. X| | Y or both X))Y and X((Y All Xs are Ys, and all Ys are Xs 3. X((Y or both X((Y and X(-(Y Among Xs are all the Ys and some things besides 4. X)o(Y or both X)-(Y and X)(Y Nothing both X and Y and some things neither 5. X|-|Y or both X)-(Y and X(-)Y Nothing both X and Y and every thing one or the other 6. X(o)Y or both X(-)Y and X()Y Every thing either X or Y and some things both. 58-63.] COMPLEX SYLLOGISM. 23 58. If any two be joined, each of which is 1, 3, 4, or 6, with the middle term of different quantities, these premises yield a conclusion of the same kind, obtained by erasing the symbols of the middle term and one of the symbols {o}. Thus X)o(Y(o)Z gives X)o)Z : or if nothing be both X and Y and some things neither, and if every thing be either Y or Z and some things both, it follows that all Xs and two lots of other things are Zs. 59. In any one of these syllogisms, it follows that || may be written for )o) or (( in one place, or |*j for either )( or (o) in one place, without any alteration of the conclusion, except reducing the two lots to one. But if this be done in both places, the con- clusion is reduced to || or |-|, and both lots disappear. Let the reader examine for himself the cases in which one of the premises is cut down to a simple universal. 60. The rules of contraversion remain unaltered: thus X(o)Y)o(Z is the same as X(o(y(o(Z &c. 61. The following exercises will exemplify what precedes. Letters written under one another are names of the same object. Here is a universe of 12 instances of which 3 are Xs and the remainder Ps ; 5 are Ys and the remainder Qs ; 7 are Zs and the remainder Rs. P P P .P P Q Q Q Q Q R R R R R We can thus verify the eight complex syllogisms X)')Y))Z P()Y))Z P(=(Q()Z P( XXX P P P P Y Y Y Y Y Q Q Z Z Z Z Z Z Z In every case it will be seen that the two lots in the middle form the quantity of the particular proposition of the conclusion. 62. The contraries of the complex propositions are as follows : Contraries. X(,(Y X;-;Y X),)Y X))Y X| |Y X((Y X)"(Y XI-IY X()Y Both X))Y and X)-)Y X))Y - X((Y X((Y - X(-(Y X)-(Y - X)(Y X)-(Y - X(-)Y XQY -XQY X(-(Y or X((Y or both X(-(Y - X)-)Y X)-)Y - X))Y X()Y - X(-)Y X()Y - X)(Y X)(Y - X)-(Y X;;Y X),(Y 63. The propositions hitherto enunciated are cumular : each one is a collection of individual propositions, or of propositions about individuals; X.)~)Y is ' All Xs are some Ys '. This pro- position is an aggregate of singular propositions. 24 EXEMPLAR PROPOSITION. [64. 64. There is a choice between this cumular mode of con- ception and one which may* be called exemplar in which each proposition is the premise of a unit-syllogism: as 'this X is one Y ', f this X is not any Y '. The distinction is seen in ' A II men are animals ' and ' Every man is an animal ', propositions of the same import, of which the first sums up, the second tells off instance by instance. In the second, every is synonymous with each and with any. * The late Sir William Hamilton entertained the idea of completing the system of enunciation by making the words all (or when grammatically necessary, any) and some do every kind of duty. He thus put forward, as the system, the following collection : Affirmative 1. All X is All Y 2. All X is some Y X))Y 3. Some X is all Y X((Y 4. Some X is some Y XQY Negative 5. Any X is not any Y X)'(Y 6. Any X is not some Y X)-)Y 7. Some X is not any Y X(-(Y 8. Some X is not some Y Of the two propositions which are not in the common system (1 and 8) the first ( 14, note f) is X| | Y, compounded of X))Y and X((Y : it is contradicted by X(-(Y and X)')Y, either or both. The second (8) is true in all cases in which either X or Y has two or more instances in existence : its contrary is ' X and Y are singular and identical ; there is but one X, there is but one Y, and X is Y '. A system of propositions which mixes the simple and the complex, which compounds two of its own set to make a third in one case and one only, 57, and which offers an assertion and denial which cannot, be contradicted in the system, seems to me to carry its own condemnation written on its own forehead. From this system I was led to the exemplar system in the text. For Sir W. Hamilton's defence of his own views, and objections to mine, see his Discussions on Philosophy, &c. Appen- dix B. In making this reference, however, it is due to myself to warn the reader who has not access to the paper criticised that Sir W. Hamilton did not read with sufficient attention, partly no doubt from ill health. The consequence is that I must not be held answerable for all that is represented by him as coming from me. For example, speaking of my Table of exemplar propositions, he says " And mark in what terms it [the table of exemplars] is ushered in : as ' a system . . . .' Nay, so lucid does it seem to its inventor, that, after the notation is detailed, we are told that it ' needs no explanation? " The paragraph here criticised had two notations, one of which I called the detailed notation, because there is more detail in it than in the other : the other is the old notation, augmented. The first had been sufficiently ex- plained in what preceded ; the second was, as to the augmentations, new to the reader. Accordingly, the table being finished, I proceeded thus " The detailed notation needs no explanation. The form given to the old notation may be explained thus " Sir W. Hamilton represented me as saying that after the notation [all the notation, I suppose] is detailed, it [table or notation, I know not which] needs no explanation. I select this small point 64-69.] EXEMPLAK PROPOSITION. 25 as one that can be briefly dealt with : there are many more, which I shall probably never notice, unless it be one at a time as occasion of illustration arises. A very decisive case is exposed in the postscript of my third paper in the Cambridge Transactions. 65. Quantity is now replaced by mode of selection. There is unlimited selection, expressed by the word any one : vaguely limited selection, expressed by some one. When we say some one we mean that we do not know it may be any one. 66. Let (X and X) now mean any one X: let )X and X( mean some one X. 67. The propositions are as follows : the first of each pair being a universal, the second its contrary particular. Exemplar form. Cumular form. X)(Y Any one X is any one Y X and Y singular and identical X(-)Y Some one X is not some one EitherX not singular, or Y not singu- Y lar ; or if both singular, not identical X))Y Any one X is some one Y All Xs are some Ys X(- (Y Some one X is not any one Y Some Xs are not (all) Ys X((Y Some one X is any one Y Some Xs are all Ys X) ) Y Any one X is not some one Y All Xs are not some Ys X)-(Y Any one X is not any one Y All Xs are not (all) Ys X()Y Some one X is some one Y Some Xs are some Ys. Six of the forms of this exemplar system are identical with six of the forms of the cumular system. And these six forms are the forms of the old logic, if we take care always to read X((Y and X)-)Y backwards, and to count X)-(Y and X()Y as each a pair of propositions, by distinguishing the reading forwards from the reading backwards. 68. The two new forms of the exemplar system (the first and second above) come under the same symbols as the two new forms of the cumular system, () and )(: but the meanings are widely different. Both systems contain every possible combi- nation of quantities, as well in universal as in particular pro- positions. 69. If the above propositions be applied to contraries, we have a more extensive system of propositions. I shall not enter on this enlargement, because the peculiar proposition of this system, X)(Y, is of infrequent* use in thought as connected with the consideration of X and Y in opposition to their contraries. * All necessary laws of thought are part of the subject of logic : but a small syllabus cannot contain everything. The rejection from logic, and the rejection from a book of logic, are two very different things. It has not been D 26 EXEMPLAR SYLLOGISM. [69-71. uncommon to repudiate rare and unusual forms from the science itself, by calling them subtleties, or the like. This ( 73) is not reasonable : but as to the contents of a work, especially of a syllabus, the time must come at which any one who asks for more Inust be answered by Cum tibi sufficiant cyathi, cur dolia quaeris ? As another example : I have, 16, required that no term shall be intro- duced which fills the whole universe. In common logic, with an unlimited universe, there is really no name as extensive as the universe except object of thought. But it is otherwise in the limited universes which I suppose. A short and easy chapter on names as extensive as the universe might be needed in a full work on logic, but not in a syllabus. 70. To make a syllogism of valid inference, it is enough that there be at least one unlimited selection of the middle term, and at least one affirmative proposition. And the inference is obtained by dropping all the symbols of the middle term. Thus X((Y(-)Z shows premises which give the conclusion X(')Z: or ' Some one X is any one Y and Some one Y is not some one Z ' giving ' Some one X is not some one Z '. 71. There are 36 valid* forms of syllogism, as follows, read- ing each symbol both backwards and forwards, but not counting it twice when it reads backwards and forwards the same, as in XX, (()) Fifteen in which X is joined with itself or another, XX )()) )((( XX )()) )((*( )(() )((') Fifteen in which the syllogism is but an exemplar reading of a cumular syllogism, )))) 0)) (()) ))>(. ((>( OX )))') )'))) Six which give the conclusion (), (((') (OO 0)0 * If Sir William Hamilton's system be taken, there are also 36 valid forms of syllogism, the same as in the text : but the law of inference is slightly modified, as follows. When both the middle spicula? turn one way, as in )) and ((, then any spicula of universal quantity which turns the other way must itself be turned, unless it be protected by a negative point. Thus )( (), which in the exemplar system gives )), in the cumular system gives (). Exemplar system. Any one X is any one Y Some one Y is some one Z Therefore Any one X is some one Z Cumular system. All Xs are all Ys. Some Ys are some Zs. Therefore Some Xs are some Zs. This distinction will afford useful study. The minor premise of the exem- plar instance implies that there is but one X and one Y. 72-75.] NUMERICALLY DEFINITE SYLLOGISM. 27 72. The exemplar proposition is not unknown. It is of very frequent use in complete demonstration. When Euclid proves that Every triangle has angles together equal to two right angles, he selects, or allows his reader to select, a triangle, and shows that any triangle has -angles equal to two right angles : and the force of demonstration is for those who can see that the selection* is not limited by anything in the reasoning. The exemplar form of enunciation, then, is of at least as frequent use in purely deductive reasoning as any other ; and is therefore fitly intro- duced even into a short syllabus. In any case it is a subject of logical consideration, as being an actual fonn of thought. * The limitation of the selection by some detail of process is one of the errors against which the geometer has especially to guard. I remember an asserted trisection of the angle which I examined again and again and again, without being able to detect a single offence against Euclid's conditions. At last, in the details of a very complex construction, I found two requirements which were only possible togtther on the supposition of a certain triangle having its vertex upon the base. Now it happened that one of the angles at the base of this triangle was the very angle to be trisected : so that the author had indeed trisected an angle, but not any angle ; he had most satis- factorily, and by no help but Euclid's geometry, divided the angle into three equal parts, 0, 0, 0. A modification of his process would have been equally successful with 180, which Euclid himself had trisected. 73. The following passage, written by Sir Wrlliam Hamilton himself, should be quoted in every logical treatise: for it ought to be said, and cannot be said better. " Whatever is operative in thought, must be taken into account, and consequently be overtly expressible in logic ; for logic must be, as to be it professes, an unexclusive reflex of thought, and not merely an arbitrary selec- tion a series of elegant extracts- out of the forms of thinking. Whether the form that it exhibits as legitimate be stronger or weaker, be more or less frequently applied; that, as a material and contingent consideration, is beyond its purview." 74. The heads* of the numerically definite proposition and syllogism are as follows, Let u be the whole number of individuals in the universe. Let x, y, z, be the numbers of Xs, Ys, and Zs. Then u x, uy, u z are the numbers of xs, ys, and zs. * On this subject I have given only heads of result, the demonstrations of which will be found in my Formal Logic. . 75. Let mXY mean that m or more Xs are Ys. Then mXy means that m or more Xs are ys, or not Ys. And wiYX and myX have the same meanings as TnXY and wXy. 28 NUMERICAL PROPOSITION AND SYLLOGISM. [76-82. 76. Let a proposition be called spurious when it must of necessity be true, by the constitution of the universe. Thus, in a universe of 100 instances, of which 70 are Xs and 50 are Ys, the proposition 20 XY is spurious : for at least 20 Xs must be Ys, and 20 XY cannot be denied, and need not be affirmed as that which might be denied. 77. Let every negative quantity be interpreted as : thus (6-10) XY means that none or more Xs are Ys, a spurious pro- position. 78. The quantification of the predicate is useless. To say that mXs are to be found among nYs, is no more than is said in raXY. To say that raXs are not any one to be found among any lot of nYs is a spurious proposition, unless m + n be greater than both x and y, in which case it is merely equivalent to both of the following, (m + n y) Xy, and (m + n #) Yx, which are equi- valent to each other. 79. In raXY, the spurious part, if any, is (x + y w)XY; the part which is not spurious is (m + u x ?/)XY. For each instance in the last there must be an x which is y. The follow- ing pairs of propositions are identical. m XY and (m+u x y) xy mXy and (m+y T) xY m xY and (m +x y) Xy wixy and (m+x+y M) XY 80. Their contraries. (x+l-m) Xy (y+l_ m ) xY (x+l TW) XY (u+ly-m)xy (u+l ar-m) xy (y + l-wi) XY Identical propositions. TO XY (m+u x y) xy m Xy (m+y x) xY m xY (m+x y) Xy 77i xy (m+x+y u) XY 81. From mXY and wYZ we infer (in + ny) XZ, or its equivalent (m + n + u a: y z)xz. The four following forms include all the cases of syllogism : the first two columns show the premises, the second two the identical conclusions, m XY n YZ (m+n y) XZ (m+n+u xyz) xz m Xy 7i YZ (m+n x) xZ (m+n-z) Xz m XY n yZ (m+n z) Xz (m+n x) xZ . m Xy n yZ (m+n+y x z) xz (m+n+y w)XZ 82. When either of the concluding terms is changed into its contrary, the corresponding changes are made in the forms of inference. Thus to find the inference from mxy and wyz, we 82-86.] NUMERICALLY DEFINITE PROPOSITION. 29 must, in the fourth form, write x for X, z for Z, X for x, Z for z, u x for x, and u z for z. 83. A spurious premise gives a spurious conclusion: and premises neither of which is spurious may give a spurious con- clusion. A proposition is only spurious as it is known to be spurious : hence when u, x, y, z are not known, there are no spurious propositions. 84. Every proposition has two forms, one of names contrary to the other, both spurious, or neither. Whenever X()Y is true in a manner which, by the constitution of the universe, might have been false, then x()y, or X)(Y is also true in the same manner. The ordinary syllogism would have two such contra- nominal forms of one conclusion, and, properly speaking, has two such forms. When the conclusion is universal, we know it has them: for X))Z is x((z, X)-(Z is x(')z, &c. These we may see to be the contranominal conclusions of the numerical syllogism. For X))Y is #XY, and Y))Z is yYZ, whence O+?/-r/)XZ and (x+y y + u x ^)xz, or #XZ, which'is X))Z, and (u z) xz, which is x((z. Again, let X()Y be wXY, then, Y))Z being ?/YZ, we have (m+y ?/)XZ, or mXZ, and (m + u x z) xz, its equivalent. If x, z, u, be known, then if m XZ be any thing except what must be, we have m + u~>x + z, and (m + ux z) xz is x( )z or X)(Z. As it is, x, y } u, being unknown, we have raXZ certainly true, be it spurious or not, and we can say nothing of (m+u x z) xz. 85 . Syllogisms with numerically definite quantity rarely occur, if ever, in common thought. But syllogisms of transposed quan- tity occur, in which the number of instances of one term is the whole possible number of instances of another term. For example; c For every Z there is an X which is Y; some Zs are not Ys'. Here we have zXY and wyZ ; whence (z + n ^)Xz and (z-\-n #)xZ. The first is wXz, a case of X('(Z ; some Xs are not Zs. Thus, ' For every man in the house there is a person who is aged; some of the men are not aged': it follows, and easily, that some persons in the house are not men ; but not by any common form of syllogism. 86. Of terms in common use the only -one which can give the syllogisms of this chapter is e most '. As in Most Ys are Xs ; most Ys are Zs ; therefore some Xs are Zs. Most Ys are Xs ; most Ys are not Zs ; therefore some Xs are not Zs. 30 FIGURE. CONVERTIBILITY. TRANSITIVENESS. [86-92. Most Ys are not Xs; most Ys are not Zs; therefore some things are neither Xs nor Zs. 87. Each one of our syllogisms may be stated in eight different ways, each premise and the conclusion admitting two different orders. Thus X))Y, Y))Z, giving X))Z may be stated as Y((X Y))Z giving Z((X, or as X))Y, Y))Z, giving Z((X, &c. All the orders are as follows I. XY YZ XZ YX ZY ZX II. XY ZY XZ XY ZY ZX III. YX YZ XZ YX YZ ZX IV. YX ZY XZ XY YZ ZX 88. Whenever there is a first and a second, let them be called minor and major. Write the premises so that the minor premise shall contain the minor term of the conclusion (though it has long been most common to write the major premise first), and we have I. XY YZ XZ XY II. ZY XZ III. YX YZ XZ IV. YX ZY XZ These orders are called the four figures. Thus X))Y Y))Z giving X))Z is stated in the first figure; X))Y Z((Y giving X))Z is stated in the second figure ; Y((X Y))Z giving X))Z is stated in the third figure ; Y((X Z((Y giving X))Z is stated in the fourth figure. 89. The .first figure may be called the figure of direct transi- tion : the fourth, which is nothing but the first with a converted conclusion, the figure of inverted transition ; the second, the figure of reference to (the middle term) ; the third, the figure of reference from (the middle term). 90. The first figure is the one which has been used in our symbols; and it is the most convenient. The distinction of figure is wholly useless in this tract, so far as we have yet gone: it becomes necessary when we take a wider view of the copula. 91. A convertible copula is one in which the copular relation exists between two names both ways : thus ' is fastened to ' ' is joined by a road with ' ' is equal to, ' ' is in habit of conversation with/ &c. are convertible copulse. If ' X is equal to Y ' then ' Y is equal to X ' &c. 92. A transitive* copula is one in which the copular relation joins X with Z whenever it joins X with Y and Y with Z. Thus * is fastened to ' is usually understood as a transitive copula : ' X 92-93.] EXTENSION OF COPULA. 31 is fastened to Y ' and ' Y is fastened to Z ' give ' X is fastened to Z'. * All the copulas used in this syllabus are transitive. The intransitive copula cannot be treated without more extensive 'consideration of the combi- nation of relations than I have now opportunity to give : a second part of this syllabus, or an augmented edition, may contain something on this subject. 93. The junction of names by appiirtenance to one object, the copula hitherto used, is both convertible and transitive : and from these qualities, and from these alone, it derives the whole of its functional power in syllogism. Any copula which is both transi- tive and convertible will give precisely the syllogisms* of our system, and no others : provided always that if contrary names be introduced, no instance of a name can, either directly or by transition, be joined by the copula with any instance of the con- trary name. For example, let the copula be some transitive and convertible mode of joining or fastening together, whether of objects in space or notions in the rnind, &c. : so that no X is ever joined with any x, &c. The following are two instances of syllogism. X))Y)'(Z. Every X is joined to a Y; no Y is joined to a Z ; therefore no X is joined to a Z. For if any X were joined to a Z, that Z would be joined to an X, and that X to a Y, whence that Z would be joined to a Y, which no Z is. X(-)Y)(Z. Everything is joined either to an X or to a Y; Some things are joined neither to Ys nor to Zs ; therefore Some Xs are not joined to Zs. For if every X were joined to a Z, then every thing being (by the first premise) joined either to an X or to a Y, is joined either to a Z or to a Y, which contradicts the second premise. * The logicians are aware that many cases exist in which inference about two terras by comparison with a third is not reducible to their syllogism. As ' A equals B ; B equals C ; therefore A equals C.' This is not an instance of common syllogism : the premises are ' A is an equal of B ; B is an equal of C.' So far as common syllogism is concerned, that 'an equal of B' is as good for the argument as 'B' is a material accident of the meaning of ' equal.' The logicians accordingly, to reduce this to a common syllogism, state the effect of composition of relation in a major premise, and declare that the case before them is an example of that composition in a minor premise. As in, A is an equal of an equal (of C) : Every equal of an equal is an equal ; therefore A is an equal of C. This I treat as a mere evasion. Among various sufficient answers this one is enough : men do not think as above. When A = B, B = C, is made to give A = C, the word equals is a 32 EXTENSION OF COPULA. [93-97. copula in thought, and not a notion attaeJied to a predicate. There are processes which are not those of common syllogism in the logician's major premise above : but waiving this, logic is an analysis of the form of thought, possible and actual, and the logician has no right to declare that other than the actual is actual. 94. The convertibility of the copula renders the inference altogether independent of figure. 95. Let the copula be inconvertible, as in 'X precedes Y' from which we cannot say that ' Y precedes X '. We must now introduce the converse relation 'Y follows X', and the conversion of a proposition requires the introduction of the converse copula. 96. This extension, when contraries are also introduced, is almost unknown in the common run of thought : but it may serve for exercise, and also to give an idea of one of those innumerable systems of relation with which thought unassisted by systematic * analysis would probably never become familiar. * The uneducated acquire easy and accurate use of the very simplest cases of transformation of propositions and of syllogism. The educated, by a higher kind of practice, arrive at equally easy and accurate use of some more complicated cases : but not of all those which are treated in ordinary logic. Euclid may have been ignorant of the identity of " Every X is Y" and " Every not-Y is not-X," for any thing that appears in his writings : he makes the one follow from the other by new proof each time. The followers of Aristotle worked Aristotle's syllogism into the habits of the educated world, giving, not indeed anything that demonstrably could not have been acquired without system, but much that very probably would not. The modern logician appeals to the existing state of thought in proof of the completeness of the ordinary system : he cannot see anything in an extension except what he calls a subtlety. In the same manner a country whose school of arith- metical teachers had never got beyond counting with pebbles would be able to bring powerful arguments against pen, ink, and paper, the Arabic numerals, and the decimal system. They would point to society at large getting on well enough with pebbles, and able to do all their work with such means : for it is an ascertained fact that all which is done by those to whom pebbles are the highest resource, is done either with pebbles or something inferior. I have long been of opinion that the reason why common logic is lightly thought of by the mass of the educated world is that the educated world has, in a rough way, arrived at some use of those higher developments of thought which that same common logic has never taken into its compass. Kant said that the study of a legitimate subtlety (necessary but infrequent law of though*,) sharpens the intellect, but is of no practical use. Sharpen the intellect with it until it is familiar, and it will then become of practical use. A law of thought, a necessary part of the machinery of our minds, of no practical use ! Whose fault is that ? 97. Let any two names be connected by transitive converse relations, for an example say gives to and receives from (under- 97-100.] TRANSITIVE AND INCONVEETIBLE COPULA. 33 standing that when X gives to Y and Y gives to Z, X gives to Z) in the following way, No X gives to another X, either directly or transitively, &c. Every X either gives to a Y or receives from a y, but not both Every x either gives to a Y or receives from a y, but not both Every X which gives to a Y, receives from no other Y, &c. The same of all combinations of names, as Y with X and x, &c. 98. The following are the propositions used, with their symbols ; and in a corresponding way for any other copula which may be used, X)')Y Every X gives to a Y X('-(Y Some Xs give to no Ys X)'-(Y No X gives to a Y X(')Y Some Xs give to Ys X('-) Y In every relation, something either gives to an X or re- ceives from a Y (or both) X)'(Y In some relations, nothing gives to any X nor receives from any Y X('(Y Some Xs give to all the Ys X)'-)Y All Xs do not give to some Ys X))'Y Every X receives from a Y X(-('Y Some Xs receive from no Ys X)-('Y No X receives from a Y XQ'Y Some Xs receive from Ys X(')'Y In every relation, something either receives from an X or gives to a Y (or both) X)('Y In some relations, nothing receives from any X nor gives to any Y X((TT Some Xs receive from all Ys XyyY All Xs do not receive from some Ys 99. Propositions are changed into others identical with them by this addition to the rule in 26 : When one term is contraverted, the relation is also converted: when both, the relation remains. In the following lists the four in each line are identical, X)')Y X)'-(Y X))'y X('-)Y X(('y X('(Y X(-)'y x(('Y x))'Y x)-CY x ('(y x)')y XQY X)'(Y X()'y X)('y x)('Y x)'-)y x)-)'Y x)'(y x(-CY x(')y The relations may be converted throughout. 100. To prove an instance, how do we know that X)')Y is identical with x(')'Y? If every X give to a Y, the remaining Ys, if any, do not give to any Xs, by the assigned conditions of meaning : consequently those remaining Ys receive from xs. As to ys, none of them can give to Xs, for then they would give to Ys : therefore all receive from xs. Conversely, if x(-/Y, no X can receive from y, for then neither could that y receive from x, nor could that X give to Y : so that there would be a relation in which neither does any thing give to Y, nor receive from x. Consequently, every X gives to Y. 34 SYLLOGISM OF INCONVEETIBLE COPULA. [101-105. 101. Let the phases of a figure depend on the quality of the premises in the following manner: + meaning affirmative, and negative, remember the phases in the following order, 102. For the four figures, let these four phases be the first or primary phases: thus H is the primary phase of the third figure. To put the other phases in order, read backwards from the primary phases, and then forwards. 1 2 34 Figure I. + + + + II. - + ++ + - III. +- - + + + - - Thus + is the third phase of the second figure. 103. In the primary phases, the direct copula may be used throughout. When one premise departs from the primary phase in quality, the converse copula must be used in the other ; when both, in the conclusion. This addition is all that is required in the treatment of the syllogism of inconvertible copula. 104. Thus,* the premises being X)-(Y Z))Y, we have the primary phase of the second figure, whence X)-(Z with the direct copula. That is, if no X give to a Y, and every Z give to a Y, no X gives to a Z. For if any X gave to a Z, that Z giving to a Y, that X would give to a Y, which no X does. Now con- travert the middle term, and we have X))y Z)-(y, the phase of the second figure in which both premises differ from the primary phase. Hence Every X gives y, No Z gives y, yields no X is given by Z. For if any X were given by Z, a y would be given by that Z, which is given by no Z. But 'no X gives Z' will not do. * The reader may exercise himself in the formation of more examples. The use of such a developement as the one before him is this. Every study of a generalisation or extension gives additional power over the particular form by which the generalisation is suggested. Nobody who has ever returned to quadratic equations after the study of equations of all degrees, or who has done the like, will deny my assertion that O l ^i-rti ftxi^nav may be predicated of any one who studies a branch or a case, without afterwards making it part of a larger whole. Accordingly, it is always worth while to generalise, were it only to give power over the particular. This principle, of daily familiarity to the mathematician, is almost unknown to the logician. 105. The common system of syllogism, which being nearly complete in the writings of Aristotle may be called Aristotelian, 105-108.] AEISTOTELIAN SYLLOGISM. 35 is as much as may be collected out of the preceding system by the following modifications, 1 . The exclusion of all idea of a limited universe, of contrary names, and of the propositions () and )(. 2. The exclusion of all right to convert a proposition, except when its two terms have like quantities, as in )( and (). Thus X))Y must not be read as * Some Ys are all Xs '. But X))Y may undergo what is called the conversion per accidens : that is, X))Y affirming X( )Y, which is Y()X, X))Y may be made to give YQX. 3. The exclusion of every copula except the transitive and convertible copula. 4. The addition of the consideration of the identical pairs X)'(Y and Y)-(X, X()Y and Y()X, as perfectly distinct propositions. 5. The introduction of the distinction of figure. 6. The writing of the major and minor propositions first and second, instead of second and first: thus X))Y))Z is written 'Y))Z, X))Y, whence 106. There are four forms of proposition : A, or X))Y or Y))X, not identical; E, or X>(Y or Y)-(X, identical; I, or X()Y or Y()X, identical; O, or X(-(Y or Y(-(X, not identical. 107. There are four fundamental syllogisms in the first figure, each of which has an opponent in the second, and an opponent in the third. There are three fundamental syllogisms in the fourth figure, each of which has the other two for opponents. Alto- gether, fifteen fundamental syllogisms. There are three strength- ened particular syllogisms, two in the third figure, and one in the fourth : and one weakened universal, in the fourth figure. In all, nineteen forms. 108. Every syllogism has a word attached to it, the vowels of which are those of its premises and conclusion. In the first figure the consonants are all unmeaning; in the other figures some of the consonants give direction as to the manner of converting into the first figure. Thus K denotes that the syllogism cannot be directly converted into the first figure, though its opponent in the first figure may be used to force its conclusion. S means that the premise whose vowel precedes is to be simply converted. P, which occurs in all the strengthened particulars and the weakened universal, means that the conversion per accidens is to be employed on the preceding member. M means that the premises must be transposed in order. Each syllogism converts into that syllogism of the first figure which has the same initial letter. G is an addition of my own, presently described ; it must be left out when 36 ARISTOTELIAN SYLLOGISM. [108-111. the old system is to be just represented. R, N, T, have no signi- fication. The following are the names put together in memorial* verse. * The best attainable exposition of logic in the older form, with modern criticism, is Mr. Mansel's edition of Aldrich's compendium. Should a reader of this work desire more copious specimens of old discussion, he may perhaps succeed in obtaining Crackanthorpe's Logica Libri quinque (4to, 1622 and 1677). Sanderson's Logic is highly scholastic in character. For a compen- dium of mediaeval logic, ethics, physics, and metaphysics, I have never found anything combining brevity and completeness at all to compare with the Precepta Doctrince Logicce, Ethicce, &c. of John Stierius, of which seven or eight editions were published in the seventeenth century (from 1630 to 1689, or thereabouts) and several of them in London. There is a large system of the older logic in the lustitutiones Logicce of Burgersdicius, and a great quantity of the metaphysical discussion connected with the old logic in Brerewood de Predicabilibus et Predicamentis. As all these books were printed in England, there is more chance of getting them than the foreign logical works, which are very scarce in this country. For more than usual infor- mation on parts of the history of logical quantity, a subject now exciting much attention, see Mr. Baynes's New Analytic, in which will be found much valuable history so completely forgotten that it is as new as if he had invented it himself. 109. Barbara, Cela^rent, Darii, Feri^oque prioris: Cesare^r, Camestres, Festino Species; )), but not the greatest possible. (( ((, Genus; ((, but not the least possible. )o( )( , External ; )( , but not the greatest possible, (o) (), Complement; (), but not the least possible. 190-195.] METAPHYSICAL RELATIONS. 55 1 90. I now proceed to the metaphysical relations * between attribute and attribute. * The terms of metaphysical relation are picked up without difficulty in our common language : but those of mathematical relation had in several instances to be forged. This means that the world at large has more of the metaphysical than of the mathematical notion in its usual form of thought. But though the unconnected words essential, dependent, repugnant, alternative, are constantly on the tongues of educated people, the combinations of these relations are not made with any security, and when thought of at all, enter under a cloud of words : while the analysis by which precision of speech and habit of security might be gained is treated with contempt, as being logic. A whole drawing-room of educated men may be without a single person who can expose the falsehood of the assertion that the essential of an incompatible must be incompatible ; a proposition which I have heard maintained, though not in those words, by persons of more than respectable acquirements ; sometimes by actual error, sometimes by confusion between the essential of an incompatible, and that to which an incompatible is essential. But even of the persons who are not thus taken in, very few indeed, when told that the answer to ' the essential of an incompatible is incompatible' is 'not so much; only independent', will be puzzled by the juxtaposition of incompatibility and independence as viewed in a relation of degree. In making these remarks, it will be remembered that I am not speaking of any words of my own, nor of any meanings of my own. The words are common, and. I take them in their common meanings ; but it is not generally seen that these common words, used in their common senses, are sufficient, in conjunction with their contraries, to express all the relations which occur in a completely quantified system of onymatic enunciation. 191. "When X))Y let the attribute Y be called an essential of the attribute X, and X a dependent of Y. In the contrary case, X(- ( Y, let Y be called an inessential of X, and X an independent of Y. Remember that dependent on does not mean dependent wholly on, or dependent only on. 192. When X)-(Y, let each attribute be called a repugnant of the other. When X( )Y, let each be an irrepugnant of the other. 193. When X(-)Y let each attribute be called an alternative of the other. When X)( Y let each be called an inalternative of the other. 194. When difference of symbols is desired, the square bracket may be used instead of the parenthesis : thus ] ] may denote dependent when read forwards, and essential when read back- wards, &c. 195. Essential and dependent are converse relations; as are also inessential and independent. Of repugnant, irrepugnant, alter- 56 METAPHYSICAL SYLLOGISM. [ 1 95-200. native, inalternative, each is its own converse. Essential and inessential are contrary relations ; as are dependent and inde- pendent, repugnant and irrepugnant, alternative and inalter- native. Compare 181. 196. The essential is both irrepugnant and inalternative: as also is the dependent. The repugnant is both independent and inessential: as also is the alternative. Compare 182. 197. The essential has the utmost irrepugnance, and may have the utmost inalternativeness. The dependent has the utmost inalternativeness, and may have the utmost irrepugnance. The repugnant has the utmost inessential ity, and may have the utmost independence. The alternative has the utmost inde- pendence, and may have the utmost inessentiality. Compare 183. 198. These relations also have terminal ambiguity (Com- pare 184). Essential is either dependent or independent Dependent is either essential or inessential Repugnant is either alternative or inalternative Alternative is either repugnant or irrepugnant. 199. Read the identities in 25 into this language, as in Dependent is repugnant of contrary, contrary of dependent is alternative, contraries of dependent and essential are essential and dependent, &c. 200. The following are the combinations* in syllogism, ar- ranged as in 186. )) )) Dependent of dependent is dependent (( ( ( Essential of independent is independent ( ( (( Independent of essential is independent (( (( Essential of essential is essential )) )) Dependent of inessential is inessential )) )) Inessential of dependent is inessential )( () Repugnant of alternative is dependent )( ( ( Repugnant of independent is inalternative ( ( () Independent of alternative is irrepugnant () )( Alternative of repugnant is essential () )) Alternative of inessential is irrepugnant )) )( Inessential of repugnant is inalternative 200-203.] METAPHYSICAL SYLLOGISM. 57 )) )( Dependent of repugnant is repugnant (( ( ) Essential of irrepugnant is irrepugnant ( ) )( Irrepugnant of repugnant is independent (( () Essential of alternative is alternative )) )( Dependent of inalternative is inalternative )( () Inalternative of alternative is inessential Repugnant of essential is repugnant Repugnant of irrepugnant is inessential Irrepugnant of dependent is irrepugnant Alternative of dependent is alternative Alternative of inalternative is independent Inalternative of essential is inalternative. * Note that when, and only when, one of the combining words is either essential or dependent, the other two words are the same ; and this throughout the fundamental or unstrengthened syllogisms. What law of thought does this represent ? And except when one of these words so occurs, the three words of relation are all different. 201. (( )) Essential of dependent is irrepugnant )) (( Dependent of essential is inalternative () () Alternative of alternative is irrepugnant )( )( Repugnant of repugnant is inalternative (( )( Essential of repugnant is independent )) () Dependent of alternative is inessential () (( Alternative of essential is independent )( )) Repugnant of dependent is inessential. 202. I now proceed to form metaphysical* terms expressing relations of terminal precision (compare 188). Let an inherent be an attribute asserted; let an excludent be an attribute denied; let an accident, which is also non-accident, be an attribute affirmed of part and denied of the rest. Thus of man, life is an inherent, vegetation an excludent, wisdom an accident and a non-accident. * This new formation cannot be overlooked, since it is the extension of the Aristotelian system of predicables, genus and species (used in the old sense) and accident, to the system in which contrary terms are permitted. Otherwise, the relations of terminal ambiguity, compounded, might serve the purpose. 203. Each of these relations may be either generic or specific. Either is generic when it applies in as large or a larger degree to a larger genus : specific, when it does not so apply to any larger 58 PREDICABLES. [203-205. genus. This being premised, the following relations will be found correctly stated, >. v T ,. i j i . f Specific accident )o) Inessential dependent is < *, J . {^Generic non-accident Dependent essential is Specific inherent (o( Independent essential is Generic inherent )o( Inalternative repugnant is Generic excludent |-| Repugnant alternative is Specific excludent , N i, ,. f Generic accident (o ) Irrepugnant alternative is < _ . . (.specific non-accident. 204. The following are examples of each of these terms, the universe being terrestrial animal, Specific accident ; generic non-accident. Lawyer is in this rela- tion to man: accident and non-accident, because an attribute of some men, and not of others ; specific accident, because not found in the additional extent of any genus larger than man ; generic non-accident, for the same reason. Specific inherent. Rational is in this relation to man: inhe- rent, because an attribute of all ; specific, because no attribute of the additional extent in a larger genus. Generic inherent Biped is in this relation to man; inherent, because an attribute of all ; generic, because an attribute of the additional extent of a larger genus. Generic excludent. Oviparous is in this relation to man; ex- cludent, because an attribute to be denied of man ; generic, because to be also denied of the additional extent of some larger genera. Specific excludent. Dumb (wanting articulate language with meaning) is in this relation to man; excludent, because to be denied of man ; specific, because not to be denied of the additional extent of any larger genus. Generic accident; specific non-accident. Naked (not artificially clothed) is in this relation to man; accident and non-accident, because some are and some are not; generic accident, because an accident of the additional extent of larger genera; specific non- accident, because not non-accident of any such additional extent. 205. When either of the relations belongs equally to a term and its contrary, it may be called universal. Thus an attribute of both term and contrary is a universal inherent; an accident and non-accident of both term and contrary is a universal accident and non-accident; an excludent of both term and contrary is a uni- versal excludent. But the first and third of these terms are chiefly 205-209.] EXTENT AND INTENT. of use in defining the universe : the second is that relation which we suppose until some contradiction is affirmed. 206. With the arithmetical reading in extension may be joined that in intension, 115, 131. In extension, the unit of enume- ration is one of the objects all of which aggregate into the class : in intension, the unit of enumeration is one of the qualities all of which compose the object. The following is the system of arith- metical reading in intension: naturally connected ( 129) with the metaphysical mode of viewing objects of thought. The inversion of the quantities, presently further described, will be easily seen ; namely, that )X and X( now indicate that X is taken completely, *in all its qualities; while X) and (X indicate that X is taken incompletely, in some (some or all, not known which) of its qualities. The term any ( 22) is here introduced when grammatically desirable. Arithmetical reading in intension All qualities of Y are some qualities of X Some qualities of Y not any qualities of X All things want either some qualities of X, or some qualities of Y Some things want neither any quality of X, nor any quality of Y All things have either all the qualities of X or all the qualities of Y Some things want either some of the quali- ties of X, or some of the qualities of Y Some qualities of Y are all the qualities of X Any qualities of Y are not some qualities of X. Symbol Metaphysical reading X))Y , X dependent of Y X independent of Y X repugnant of Y X(-(Y X)-(Y XQY X irrepugnant of Y X(-)Y X)(Y X((Y X)-)Y X alternative of Y X inalternative of Y X essential of Y X inessential of Y 207. I now proceed to further consideration of the subject of quantity. No new results can appear, but it will be necessary both to adapt the old results to the more subjective view of logical process, and also to consider the distinctions of quantity from new points of view. 208. The distinction of the two tensions, extension and inten- sion ( 131), or, for brevity, extent and intent, may for clearness ( 129) be applied only to classes and attributes. The extent of a class embraces all the classes of which it is aggregated : the intent of an attribute embraces all the attributes of which it is compounded. 209. A class may be subdivided down to the distinct and non- 60 QUANTITY OF EXTENT AND INTENT. [209-211. interfering individual objects of thought of which it is composed : and here subdivision must stop. But it is not for human reason to say what are the simple attributes into which an attribute may be decomposed: the decomposition of the notion rational, for example, into distinct and non-interfering component notions, is the subject of an old controversy which will perhaps never be settled. But this difficulty is of no logical importance. 210. The relation of quantity as exhibited in the arithmetical view of the proposition ( 13, 14), giving the distinction of univer- sal and particular quantity, as it is commonly expressed, or of total and partial quantity, as I have expressed it, may be in this part of the subject most conveniently attached to other names. Let the terms full extent* and vague extent be used to replace total extension and partial extension : and let full intent and vague intent replace total intension and partial intension. * These terms are convenient from their brevity : full extent is shorter than universal extension. But they are still more useful as avoiding the ambiguity of the words some, particular, partial, which, as we have seen ( 14, note f) misleads even the highest writers. The logical opposition of quantity is not quantity universal and quantity not universal, but quantity asserted to be universal and quantity not asserted to be universal. Two words cannot be found which express the opposition of undertaking to assert and not undertaking to assert universality. We may therefore be content with full and vague, which, if they do not express opposition, at least do not, like universal and particular, express the wrong opposition. 211. Additional extent can only be gained by a new aggregate containing extent which is not in the collective extent of the others : additional intent only by a new component which is not in the joint intent of the others. Thus the extent of the class animal is not augmented by the aggregation of the class having volition, if the universe be the visible earth. Again, the intent* of the notion plane triangle is not augmented by the junction of the notion capable of inscription in a circle. The distinction between these real and apparent augmentations is of the matter, not of the form: and is of no logical import except this, that when we say that a new aggregant increases extent, and a new component increases intent, we must be prepared, with the mathematicians, to reckon among the cases of quantity. * There is a remarkable difference between extent and intent, which, though logically nothing at all, is psychologically very striking. Say we discover extent hitherto unknown, without the necessity of reducing intent to include it within a class thought of. Columbus did this when he first was able to add the class American to the classes then known under man. 211-214.] OPPOSITION OF EXTENT AND INTENT. 61 Here is nothing beyond what was possible in previous thought, which could people the seas to any extent. But when we add intent without diminishing extent, which knowledge is doing every day, we cannot conceive beforehand what kind of additions we shall make. A beginner in geometry gradually adds to the intent of triangle, which at first is only rectilinear three-sided figure, the components can be circumscribed by a circle has bisectors of sides meeting in a point has sum of angles equal to two right angles and other properties, by the score. The distinction is that class aggregation joins similars* but that composition of attributes joins things perfectly distinct, of which no one can predicate anything merely by what he knows of another thing. When the old logicians threw the notion of intent out of logic into metaphysics, they were guided by the material differences of qualities, and did not apprehend their similarity of properties a* qualities. 212. The distinction of extent and intent has found its way into common language, in the words scope* and force, which I shall sometimes use. Thus in * every man is animal' the term man is used in all its scope, but not in all its force ; a person incognisant of some of the components of the notion man, that is, of the whole force of the term, might have the means of knowing this proposition. But animal is used in all its force, and not in all its scope. This answers to saying that in 'Every X is Y', the term X is of full extent and vague intent ; the term Y is of full intent and vague extent * The logicians, until our own day, have considered the extent of a term as the only object of logic, under the name of the logical whole : the intent was called by them the metaphysical whole, and was excluded from logic. In our own time the English logical writers, and Sir William Hamilton among the foremost, have contended for the introduction of the distinction into logic, under the names of extension and comprehension : Hamilton uses breadth and depth. Now I say that in the perception of the distinction between scope and force, as well as in other things, the world, which always runs after quack preparations, has ventured for itself out of the logical pharmacopoeia. This certainly in a rude and imperfect way : and without apprehension of any theorems. I have not found, though I have looked for it, any such amount of recognition that the greater the scope the less the force as I could present without suspicion of the aut inveniam out faciam bias. But I think it likely enough that some of my readers may casually pick up passages which show a. feeling of this theorem. 213. The quantity considered in the arithmetical view of logic ( 5-111) was entirely quantity of extent. I now proceed to the comparison of extent and intent. 214. In every use of a term, one of the tensions* is full, and the other vague: the full extent and the full intent cannot be used at one and the same time ; and the same of the vague extent and the vague intent. Thus X) and (X must stand for X used 62 OPPOSITION OF EXTENT AND INTENT. [214. in full extent and vague intent : and )X and X( for vague extent and full intent. The proof of this proposition is as follows. When a term is full in extent, we can abandon or dismiss any aggregant of that extent we please : the proposition, though reduced or crippled by the dismissal, is true of what is left : but we may not annex an aggregant at pleasure. When a term is vague in extent, we cannot dismiss any aggregant whatever: for we know not by what aggregant the proposition is made true : but we may annex any aggregant at pleasure : for we do not thereby throw out what makes the proposition true, even if we annex no additional truth ; and we do not, when speaking vaguely, affirm or deny of any one selected aggregant. And as the extent must be full or vague, and we must be either competent or incompetent to dismiss an aggregant taken at pleasure, and must be either competent or incompetent to annex one, the converses follow ( 151), namely, that when we are competent to dismiss, the extent is full, and when we are incompetent the extent is vague : and also that when we are competent to annex, the extent is vague, and when not, the extent is full. Precisely the same proposition may be established upon the intent of a term, and its components. Now let a term be of full extent. In diminishing the extent, which we may do, we can so do it as to augment the intent : and if we be competent to augment the intent, that intent must be vague, as just proved. Similarly, if a term be of vague extent, we are competent to annex an aggregant, that is, to diminish the intent ; whence the intent must be full. And the same may be proved in like manner when either kind of intent is first supposed instead of extent; though by use of 151 this case may be seen to be contained in that already treated. And the learner may gather the whole from instances. Thus A, B))PQ gives A))PQ, andA))P; but not A,B,C))PQ, nor A,B))PQR. But PQ))A,B does not give P)))A, B, nor PQ))A, though it does give PQR))A,B and PQ))A,B,C; and so on. And further, from 133, this proposition can be made good of all universals when it is known of one : and the same of all particulars. * The logicians who have recently introduced the distinction of extension and comprehension, have altogether missed this opposition of the quantities, and have imagined that the quantities remain the same. Thus, according to Sir W. Hamilton 'All X is some Y' is a proposition of comprehension, but ' Some Y is all X' is a proposition of extension. In this the logicians have abandoned both Aristotle and the laws of thought from which he drew the 214-216.] DISMISSAL AND TRANSPOSITION OF ELEMENTS. 63 few clear words of his dictum : ' the genus is said to be part of the species ; but in another point of view (jai) the species is part of the genus'. All animal is in man, notion in notion : all man is in animal, class in class. In the first, all the notion animal part of the notion man: in the second, all the class man part of the class animal. Here is the opposition of the quantities. 215. It appears then that the elements of a tension (aggregants of an extent, components of an intent) may be dismissed from the term used fully, but cannot be introduced ; may be introduced into a term used vaguely, but cannot be dismissed. The dis- missible is inadmissible : the indismissible is admissible. 216. Elements of either tension may, under the limitations of a rule to be shown, be transposed from one term of a proposition to the other, either directly, or by contraversion, without either loss or gain of import to the proposition. Thus AB)-(Y is the same proposition as A)-(BY, and X))A,B is the same as Xa))B. The demonstration of this may best be seen by observing that every universal is a declaration of incompossibility,* and every particular a corresponding declaration of compossibility . Thus X)-(Y is an assertion that X and Y, as names of one object, are incompossible ; and X( )Y that they are compossible. Again, X))Y declares X and y to be incompossible ; and so on. Now it will be seen that AB)-(Y is merely a statement that the three names A, B, Y, are incompossible ; and so is A)'(BY. Hence AB)(Y and A>(BY are identical. Similarly AB( )Y and A()BY are identical, both declaring the compossibility of A,B, Y: or thus, if two propositions be identical, their contraries must be identical. Hence we learn that in Y))a, b we have YB))a &c. Carrying this through all transformations, we arrive at the following rules: 1. In universal propositions, vague elements (the elements of terms of either vague tension) are transposible ; directly in nega- tives, by contraversion in affirmatives. But full elements are intransposible. 2. In particular propositions, full elements (the elements of terms of either full tension) are transposible ; directly in affirm- atives, by contraversion in negatives. But vague elements are intransposible. Thus in X))Y, Y is of vague extent ; if it be (A, B), its aggregant A is transposible, the proposition being affirmative, by contraversion: that is X))A, B is identical with Xa))B. The rules are for comparison and generalisation, not for use. Nothing 64 DISMISSAL AND TRANSPOSITION OF ELEMENTS. [216-219. can be more evident than that if every X be either A or B, every X which is not A is B. * These good words are Sir William Hamilton's (see 14, note t), to whom, in matters of language, I am under what he would have called obligations general and obligations special. His occasional writing of the adjective after the substantive is a useful revival of an old practice, tending much to clearness. As to my obligations special, he, finding the word parenthesis not enough to erect his reader's hair, described my notation as "horrent with mysterious spiculae". This was the very word I wanted, 21 : for parenthesis has come to mean, not the punctuating sign, but the matter which it includes : and parenthetic notation would have been ambiguous. 217. It has in effect been noticed that for every full term in a proposition a term of as much or less tension may be substituted ; and for every vague term a term of as much or more tension. This is the whole principle of onymatic syllogism, or rather may be made so: for the varieties of principle upon which all onymatic inference may be systematically introduced are numerous. Thus in X()Y)-(Z, giving X(-(Z, all we do is to substitute for Y used vaguely in extent, the as extensive or more extensive term z. Or thus; for Y of full intent, we substitute the as intensive or less intensive term z. For Y)*(Z or Y))z, shows that z, if anything, is of greater extent and less intent than Y. 218. There are processes which appear like transpositions, but are not so in reality. Thus X))PQ certainly gives XP))Q : here is a universal proposition, in which the element of a full tension is transposible. But not transposible within the description in 216, in which it is affirmed that the 'proposition after transposition is identical with the proposition before transposition. This is not the case here; for though X))PQ gives XP))Q, yet XP))Q does not give X))PQ. Here, since X))PQ gives X))Q and X))P, the term XP is really X. And further, since X))PQ gives X))Q from whence XR))Q, be R what it may so long as XR has existence, the deduction of XP))Q from X))PQ is a case of something different from mere transposition : for P, in XP))Q, may be changed into anything else. 219. The dismissal of the elements of terms comes under what may be called the decomposition of propositions. When the elements of both terms are of the full tension, the proposition is a compound of m x n propositions, if m and n be the numbers of elements in the two terms. Thus A, B))CD gives and is given by the four propositions A))C, A))D, B))C, B))D. 220-222.] YAGUE QUANTITY IN CONCLUSION. 65 220. Species, external, deficient, coinadequate Dependent, repugnant, inessential, inalternative must carry the notion of full extent and vague intent. For example, the universe being England, farmer is a deficient of landowner : of all the class * farmer, no part is identical with a certain part of the class landowner. To know this by extent I must know the whole class farmer : but to know it by intent, I need not know all the attributes of the notion farmer. Let there be but one of these attributes which is not an essential of land- owner, and the proposition is established. Genus, complement, exient, partient Essential, alternative, independent, irrepugnant must carry the notion of vague extent and full intent. The symbols will here help the memory of those who have fully connected them with the words. * The student must, in any one proposition, be on his guard against thinking inconsistently of class and of attribute. Either of these modes of thought may be chosen, but not both together, unless the attribute be made to distinguish the class, without exceptions. For a remarkable instance, take the word gentleman : what different things people usually mean, according as they are speaking by notion of class or of attribute ; the common attribute excludes a percentage of the class, and admits many who are not of the class. The reader may be puzzled to make out the text, unless his character of landowner correspond to his class. 221. The rules of 50-52 must be translated as follows. A vague term in the conclusion takes extent or intent (scope or force) as follows. 1. In universal syllogisms, if one term of conclusion be of vague scope or force, it has the scope or force of the other ; if both, one has the scope or force of the whole middle term, the other of its whole contrary. 2. In fundamental particular syllogisms, the vague term or terms of the conclusion take scope or force from the vague premise. 3. In strengthened particular syllogisms, the vague term or terms of conclusion take scope or force from the whole middle term or from its whole contrary, according to which is of full scope or force in both premises. 222. For example, (actual) farming depends on occupation of land (see the caution in 191, often wanted in reference to this i 66 HYPOTHETICAL SYLLOGISM, ETC. [222-225. very instance) ; and occupation of land is an essential of county respectability : therefore farming and such respectability are in- alternative. Here the terms of conclusion are both of vague intent or force, and the middle term of full intent : the force is precisely so much as is contained in the notion of occupying land. Any component either of actual farming or of county respect- ability which can be possessed by a non-occupier of land is of no import in the conclusion as from the above premises. Take the mathematical form of the above: Farmers are a species of occupiers of land ; the county respectables are a species of occupiers ; whence the farmer is a coinadequate of the county respectable, or both together do not make up the whole universe (that is, as implied, the population of the county). Here the con- trary of the middle term, the class of non-occupiers of land, forms the extent of coinadequacy of the terms of conclusion implied in the premises. 223. The admission of complex terms, and of copular relations more general than the word of identification is, enable us to include in common syllogism all the cases known as hypothetical syllogisms, conditional syllogisms, disjunctive syllogisms, dilem- mas, &c. I shall merely take a few cases of these. If P be true, Q is true; but P is true, therefore Q is true. This is an hypothetical syllogism, so called. To reduce it to a common syllogism between * Q', 'true', and a middle term, we have 'Q' is l a proposition true when P is true'; 'A propo- sition true when P is true ' is ' true ' (because P is true) ; there- fore Q is true. Many other ways might be given. In truth, though the reduction is possible, the law of thought connecting- hypothesis with necessary consequence is of a character which may claim to stand before syllogism, and to be employed in it, rather than the converse. But the discussion of this subject is not for a syllabus : see 226. In a similar way may be treated ' If P be true, Q is true ; Q is not true : therefore P is not true '. 224. Say that P is either A, or B, or C ; A is not X, B is not X, C is not X ; then P is not X. This is the syllogism t P))A,B,C>(X giving P>(X a common syllogism with the middle term an aggregate. 225. Either P is true, or Q ; if P be true, X is true ; if Q be true, Y is true ; therefore either X or Y is true. The truth is the alternative of the truth of P or Q ; which is the alternative 225-228.] BELIEF. PROBABILITY. 67 of the truth of X or Y ; therefore the truth is the truth of the alternative of X or Y. Various other instances will be found in my formal Logic, pp. 115-125. 226. In all syllogisms the existence of the middle term is a datum. If the conclusion be false, the syllogism being logically valid, and the premises true if the terms exist, then the non- existence of one of the terms is the error. And if the terms which remain in the conclusion be existent, the nonexistence of the middle term may be inferred. When the syllogism is sub- jective in character, the transition into the objective syllogism frequently hinges on this point. Suppose success in a certain undertaking, such success being conceivable, depends both upon X and upon Z : then X and Z are not subjectively repugnant. Suppose that in objective reality they are repugnant : their coexistence being a thing wholly unknown and incredible. It follows then that success is objectively unattainable; impossible as things are, people say. The metaphysical premises X((Y))Z, X, Y, Z, being conceivable, give X( )Z : and if X and Z have objective existence, and X)'(Z, it follows that Y does not exist ; for if it did, the premises X((Y))Z would give X()Z. Suppose a qualification which depends both upon natural talent and early training; and suppose the talent to be one which cannot be developed early, as things go ; then, as things go, the qualification is unattainable. 227. The remaining logical whole of which we have to con- sider the parts is belief. This feeling is one the magnitude of which ranges between two extremes ; certainty for, such as we have as to the proposition * Two and two make four '; and cer- tainty against, such as we have as to the proposition ' Two and two make five '. The first has the whole belief, or no unbelief; the second has no belief, and the whole unbelief. These extremes are represented by 1 and 0, on the scale of belief: and would be represented by and 1, if we chose (which is not necessary) to have a scale of unbelief. 228. That which may be or may not be claims* a portion of belief and a portion of unbelief: that is, we partly believe in the " may " and partly in the " may not" Thus if an iirn contain 1 3 white balls and 7 black balls, and nothing else, and I am going to draw a ball without knowing which, and without more belief in 68 TESTIMONY AND ARGUMENT. [228-231. one ball than in another ; then my belief in the drawing of white is to my belief in the drawing of black as 13 to 7, that is, 1 re- presenting certainty, I have $ of belief in a white ball, and -^ in a black ball. This is usually expressed by saying that the odds in favour of a white ball are 13 to 7, and the chances, or proba- bilities, of the drawing of white or black ball, are ^ and -/^. I shall call 13:7 the ratio of belief in a white ball, or of unbelief in a black ball; and 7:13 the ratio of belief in a black ball, or of unbelief in a white ball ; and 1 3 and 7 the favourable and unfa- vourable terms for the white ball. * Here, as in all other things, there are portions which are too small to be of perceptible effect. Csesar may not have died in the manner stated : he mat/, if there were such a person, which may not be true, have been captured by the Britons, and detained in captivity for the rest of his life. But the received history absorbs so much of our belief that we have but a mere atom to divide among all the different ways in which that story may be wrong. There are two opposite fallacious methods of thinking : first, the confusion of high moral certainty with absolute knowledge in right of the nearness of the quantities of belief in the two ; secondly, the confusion of high moral certainty with matters of practical uncertainty, in right of the want of absolute knowledge in both. 229. Referring to my Formal Logic, for full explanation on the subject, I shall here only digest a few rules relative to the measures of belief and unbelief, in questions especially relating to logic. 230. Any alteration of our minds with respect to belief or unbelief of a proposition is derived from two sources, 1. Testimony, assertion for or against by those of whose know- ledge we have some opinion. This, when absolutely unimpeachable, is authority; though this word is used loosely for testimony of high value. Testimony speaks to the thing asserted, to its truth or falsehood ; it turns out good if the proposition be true, and bad if the proposition be false. 2. Argument, reasoning for or against, addressed to the mind on its own fprce. This, when absolutely unimpeachable, is de- monstration; though this word is used loosely for reasoning of great force. Reasoning speaks, not simply to the truth or false- hood, but to the truth as proved in one particular way. If an argument be invalid, it does not follow that the proposition is false, but only that it cannot be established in that one way. 231. When the proposer of an argument believes in its con- 231-233.] TESTIMONY AND ARGUMENT. 69 elusion, he is one of the testimonies in favour of the conclusion, independently of his argument. 232. Among the testimonies to a conclusion must be counted the receiver himself, whose initial state of mind enters as the testimony* of a witness into the mathematical formulas, though a thing of a very different kind. Suppose that, all circumstances duly considered so far as he is able, the receiver begins with an impression that the proposition in hand has 7 to 3 against it, or (3 : 7) is his ratio of belief at the outset. Upon this belief the future testimonies and arguments are to act : and the mathe- matical effect is the same as if the first witness bore testimony (7 : 3) against the proposition. * This is the point on which the mathematical study of this theory throws most light. Simple as the thing may appear, there is not one writer in a thousand who seems to know that the legitimate result of argument and testimony depends upon the initial state of the receiver's mind. They request him to begin without any bias ; to make himself something which he is not by an act of his own will. Judges request juries to dismiss all that they know about the case beforehand : and this when the juries know, and the judges know that they know it, that the mere fact of the prisoner's appearance at the bar is itself three or four to one in favour of his guilt. Now the jury do not dismiss this presumption, because they cannot : and they need not, because the sound remedy against the presumption lurks in their own minds, and is ready to act. It would not be advisable to discuss in a short note the method in which common honesty manages to hit the truth, in spite of prepossession. But I may state my conviction that if the juryman were consciously to aim at being somebody else, that is, a person without any preconceived notion, he would give a wrong verdict far more often than he does. I should recommend him not to think about himself at all, but to forget himself altogether, or at least not to be active in bringing himself before himself; and to listen to the evidence. And further, to remember that the inquiry does not terminate in the jury-box ; that the trial of the evidence commences when the jury retire ; that the evidence of eleven other men to the character of the evidence is itself part of the evidence ; and that the demand for unanimity on the part of the jury is the expression of the determination of the law that the juryman shall be forced, if needful, to take other opinions into account. I trust this necessity for unanimity will never be done away with. 233. In assigning numerical value to degrees of belief, we are supposing cases which are nearly as unusual in human affairs as numerically definite propositions ( 13). But by the study of accurate data, supposed attainable, we analyse the sources of error to which our minds are subject in the rough processes which our state of knowledge obliges us to use. 70 CALCULATION OF TESTIMONY. [234-236. 234. The method of compounding testimonies is by multiply- ing together all the favourable numbers for a favourable number, and all the unfavourable numbers for an unfavourable number. For instance, a person thinks it 10 to 3 against an assertion. Two witnesses affirm it, for whose accuracy it is in his mind 7 to 4 and 8 to 3 : two witnesses deny it, for whose accuracy it is in his mind 11 to 5 and 3 to 1. What ought to be his state of belief after the testimony ? The several ratios for the assertion are 3: 10, 7:4, 8:3, 5: 11, 1:3 And 3x7x8x5x1 : 10 x 4 x 3 x 11 x 3, or 7 : 33 is the ratio of belief as it should be after the whole testimony is taken into account : or 33 to 7 against the assertion. 235. When several arguments are advanced on one side of a question, of which the several chances of validity are given, the chance that the side taken is proved, that is, that one or more of the arguments are valid, is as follows. Take the product of the unfavourable numbers for the unfavourable number, and subtract it from the product of the several totals for the favourable number. Thus if three arguments be advanced on one side, the ratios of belief in which are (4 : 3), (2 : 1), (3 : 7), the unfavourable num- ber is 3x1x7, which subtracted from the product of 4 + 3, 2 + 1, 3 + 7, gives the favourable number. Hence (189 : 21) or (9 : 1) is the chance of the side being established by one or more of the arguments. 236. Every argument, however weak, lends some force to its conclusion : for it may be valid, and if invalid does not disprove the conclusion. But it must be remembered that this conclusion is modified by the argument on the other side which arises from the production of weak arguments, or none but weak arguments. Weak arguments from a strong person themselves furnish an argument. If an assertion be true, it is next to certain that very strong arguments exist for it ; if such arguments exist, it is highly probable that such and such a person could find them : but he cannot find them ; whence there is strong presumption that the arguments do not exist, and from thence that the assertion is not true. This kind of reasoning really prevails, and leads to a rational conclusion that the production of none but weak argu- ments is a strong presumption against the truth of their con- clusion. But when weak arguments are mixed with strong ones, 236-241.] CALCULATIC A 000 094 069 they may rather tend to reinforce the conclusion, though the general impression is that they only weaken their stronger companions. 237. If ever an argument be of such nature that according as it is valid or invalid the conclusion is true or false, that argument is of the nature of a testimony, and must be combined with the rest as in 234. 238. When testimony and arguments on both sides are to be combined, the result is obtained as follows. Combine all the testimony into one result, as in 234, all the arguments for as in 235, and all the arguments against in the same way. Then form the favourable and unfavourable numbers in the ratio of belief required, as follows: Favourable number. Unfavourable number. Multiply together Multiply together The favourable number of the testi- | The unfavourable number of the mony testimony The unfavourable number of the ; The unfavourable number of the argument against argument for The total of the argument for i The total of the argument against For instance, testimony giving (7 : 3), argument for, (5:2) and argument against, (8 : 1), the ratio of belief for the truth of the assertion should be (7 x 1 x 7 : 3 x 2 x 9) or (49 : 54), that is, it is 54 : 49 against the assertion being true. 239. When testimony is evenly balanced, (1:1), it may be altogether omitted. When the arguments for and against are evenly balanced, the arguments may be omitted. When the arguments on both sides are very strong, even though not evenly balanced, the mind may be presumed unable to compare the two very small quantities which they want of certain validity, and the arguments may be treated as evenly balanced. 240. When no argument is offered for, let (0 : 1) represent the ratio of belief which is to be used in the above rule : and the same when no argument is offered against. 241. When testimony is evenly balanced, and argument for is (m : n), there being no argument against, we have (1 x 1 x m + n : 1 x n x 1), or m + n : n for the truth of the assertion. Thus, on a matter on which our minds have no bias, an argument which has only an even chance of validity gives 2 to 1 for the truth of the conclusion. 72 TESTIMONY AND ARGUMENT. [242-244. 242. Any one may wisely try a few cases, setting down in each, to the best of his judgment, or rather feeling, his ratios of belief as to testimony, argument for, argument against, and final conclusion. If the last do not agree with the calculation made from the first three, he does not agree with himself. This he may very easily fail to do, for, in such matters of appreciation, one element may have more than justice done to it at the expense of the rest, on the principle laid down in the Gospel of St. Matthew, xxv. 29. 243. The distinction of aggregation and composition occurs in the two leading rules of application of the theory of probabilities. When events are mutually exclusive, that is, when only one of them can happen, the chance that one or other shall happen is found from the separate chances of happening by a rule of aggregation, namely, by addition. But when events are entirely independent, so that any two or more of them may happen together, the chance of all happening is found by applying to the separate chances a rule of composition, namely, multiplication. The connexion of the formulae of probability with those of logic in general has been most strikingly illustrated by Professor Boole, in his Mathematical Analysis of Logic, Cambridge, 1847, 8vo., and subsequently in his Investigation of the Laws of Tliought, London, 1854, 8vo. In these works the author has made it manifest that the symbolic language of algebra, framed wholly on notions of number and quantity, is adequate, by what is certainly not an accident, to the representation of all the laws of thought. 244. I end with a word on the new symbols which I have employed. Most writers on logic strongly object to all symbols except the venerable Barbara, Celarent, &c. in 109. I should advise the reader not to make up his mind on this point until he has well weighed two facts which nobody disputes, both separately and in connexion. First, logic is the only science which has made no progress since the revival of letters : secondly, logic is the only science which has produced no growth of symbols. Erratum. Page 55, line 20, for will be puzzled read will not be puzzled. LONDON : Printed by G. BARCLAY, Castle St. Leicester Sq.