A Budget of Paradoxes Volume I Part 28

which is the "conclusion or proof." We learn, also, that "sometimes the first is called the premises (_sic_), and sometimes the first premiss"; as also that "the first is sometimes called the proposition, or subject, or affirmative, and the next the predicate, and sometimes the middle term." To which is added, with a mark of exclamation at the end, "but in a.n.a.lyzing the syllogism, there is a middle term, and a predicate too, in each of the lines!" It is clear that Aristotle never enslaved this mind.

I have said that logic has no paradoxers, but I was speaking of old time.

This science has slept until our own day: Hamilton[707] says there has been "no progress made in {332} the _general_ development of the syllogism since the time of Aristotle; and in regard to the few _partial_ improvements, the professed historians seem altogether ignorant." But in our time, the paradoxer, the opponent of common opinion, has appeared in this field. I do not refer to Prof. Boole,[708] who is not a _paradoxer_, but a _discoverer_: his system could neither oppose nor support common opinion, for its grounds were not in the conception of any one. I speak especially of two others, who fought like cat and dog: one was dogmatical, the other categorical. The first was Hamilton himself--Sir William Hamilton of Edinburgh, the metaphysician, not Sir William _Rowan_ Hamilton[709] of Dublin, the mathematician, a combination of peculiar genius with unprecedented learning, erudite in all he could want except mathematics, for which he had no turn, and in which he had not even a schoolboy's knowledge, thanks to the Oxford of his younger day. The other was the author of this work, so fully described in Hamilton's writings that there is no occasion to describe him here. I shall try to say a few words in common language about the paradoxers.

Hamilton's great paradox was the _quantification of the predicate_; a fearful phrase, easily explained. We all know that when we say "Men are animals," a form wholly unquantified in phrase, we speak of _all_ men, but not of all animals: it is _some or all_, some may be all for aught the proposition says. This some-may-be-all-for-aught-we-say, or _not-none,_ is the logician's _some_. One would suppose {333} that "all men are some animals," would have been the logical phrase in all time: but the predicate never was quantified. The few who alluded to the possibility of such a thing found reasons for not adopting it over and above the great reason, that Aristotle did not adopt it. For Aristotle never ruled in physics or metaphysics _in the old time_ with near so much of absolute sway as he has ruled in logic _down to our own time_. The logicians knew that in the proposition "all men are animals" the "animal" is not _universal_, but _particular_ yet no one dared to say that _all_ men are _some_ animals, and to invent the phrase, "_some_ animals are _all_ men" until Hamilton leaped the ditch, and not only completed a system of enunciation, but applied it to syllogism.

My own case is as peculiar as his: I have proposed to introduce mathematical _thought_ into logic to an extent which makes the old stagers cry:

"St. Aristotle! what wild notions!

Serve a _ne exeat regno_[710] on him!"

Hard upon twenty years ago, a friend and opponent who stands high in these matters, and who is not nearly such a sectary of Aristotle and establishment as most, wrote to me as follows: "It is said that next to the man who forms the taste of the nation, the greatest genius is the man who corrupts it. I mean therefore no disrespect, but very much the reverse, when I say that I have hitherto always considered you as a great logical heresiarch." Coleridge says he thinks that it was Sir Joshua Reynolds who made the remark: which, to copy a bull I once heard, I cannot deny, because I was not there when he said it. My friend did not call me to repentance and reconciliation with the church: I think he had a guess that I was a reprobate sinner. My offences at that time were but small: I went on spinning syllogism systems, all alien from the common logic, until I had six, the initial letters of which, put together, from the {334} names I gave before I saw what they would make, bar all repentance by the words

RUE NOT!

leaving to the followers of the old school the comfortable option of placing the letters thus:

TRUE? NO!

It should however be stated that the question is not about absolute truth or falsehood. No one denies that anything I call an inference is an inference: they say that my alterations are _extra-logical_; that they are _material_, not _formal_; and that logic is a _formal_ science.

The distinction between material and formal is easily made, where the usual perversions are not required. A _form_ is an empty machine, such as "Every X is Y"; it may be supplied with _matter_, as in "Every _man_ is _animal_."

The logicians will not see that their _formal_ proposition, "Every X is Y,"

is material in three points, the degree of a.s.sertion, the quant.i.ty of the proposition, and the copula. The purely formal proposition is "There is the probability [alpha] that X stands in the relation L to Y." The time will come when it will be regretted that logic went without paradoxers for two thousand years: and when much that has been said on the distinction of form and matter will breed jokes.

I give one instance of one mood of each of the systems, in the order of the letters first written above.

_Relative._--In this system the formal relation is taken, that is, the copula may be any whatever. As a material instance, in which the _relations_ are those of consanguinity (of men understood), take the following: X is the brother of Y; X is not the uncle of Z; therefore, Z is not the child of Y. The discussion of relation, and of the objections to the extension, is in the _Cambridge Transactions_, Vol. X, Part 2; a crabbed conglomerate.

_Undecided._--In this system one premise, and want of power over another, infer want of power over a conclusion. {335} As "Some men are not capable of tracing consequences; we cannot be sure that there are beings responsible for consequences who are incapable of tracing consequences; therefore, we cannot be sure that all men are responsible for the consequences of their actions."

_Exemplar._--This, long after it suggested itself to me as a means of correcting a defect in Hamilton's system, I saw to be the very system of Aristotle himself, though his followers have drifted into another. It makes its subject and predicate examples, thus: Any one man is an animal; any one animal is a mortal; therefore, any one man is a mortal.

_Numerical._--Suppose 100 Ys to exist: then if 70 Xs be Ys, and 40 Zs be Ys, it follows that 10 Xs (at least) are Zs. Hamilton, whose mind could not generalize on symbols, saw that the word _most_ would come under this system, and admitted, as valid, such a syllogism as "most Ys are Xs; most Ys are Zs; therefore, some Xs are Zs."

_Onymatic._--This is the ordinary system much enlarged in propositional forms. It is fully discussed in my _Syllabus of Logic_.

_Transposed._--In this syllogism the quant.i.ty in one premise is transposed into the other. As, some Xs are not Ys; for every X there is a Y which is Z; therefore, some Zs are not Xs.

Sir William Hamilton of Edinburgh was one of the best friends and allies I ever had. When I first began to publish speculation on this subject, he introduced me to the logical world as having plagiarized from him. This drew their attention: a mathematician might have written about logic under forms which had something of mathematical look long enough before the Aristotelians would have troubled themselves with him: as was done by John Bernoulli,[711] {336} James Bernoulli,[712] Lambert,[713] and Gergonne;[714] who, when our discussion began, were not known even to omnilegent Hamilton. He retracted his accusation of _wilful_ theft in a manly way when he found it untenable; but on this point he wavered a little, and was convinced to the last that I had taken his principle unconsciously. He thought I had done the same with Ploucquet[715] and Lambert. It was his pet notion that I did not understand the commonest principles of logic, that I did not always know the difference between the middle term of a syllogism and its conclusion. It went against his grain to imagine that a mathematician could be a logician. So long as he took me to be riding my own hobby, he laughed consumedly: but when he thought he could make out that I was mounted behind Ploucquet or Lambert, the current ran thus: "It would indeed have been little short of a miracle had he, ignorant even of the common principles of logic, been able of himself to rise to generalization so lofty and so accurate as are supposed in the peculiar doctrines of both the rival logicians, Lambert and Ploucquet--how useless soever these may in practice prove to be." All this has been sufficiently discussed elsewhere: "but, masters, remember that I am an a.s.s."

I know that I never saw Lambert's work until after all Hamilton supposed me to have taken was written: he himself, who read almost everything, knew nothing about it until after I did. I cannot prove what I say about my knowledge of Lambert: but the means of doing it may turn up. For, by the casual turning up of an old letter, I _have_ {337} found the means of clearing myself as to Ploucquet. Hamilton a.s.sumed that (unconsciously) I took from Ploucquet the notion of a logical notation in which the symbol of the conclusion is seen in the joint symbols of the premises. For example, in my own fashion I write down ( . ) ( . ), two symbols of premises. By these symbols I see that there is a valid conclusion, and that it may be written in symbol by striking out the two middle parentheses, which gives ( . . ) and reading the two negative dots as an affirmative. And so I see in ( . ) ( . ) that ( ) is the conclusion. This, in full, is the perception that "all are either Xs or Ys" and "all are either Ys or Zs" necessitates "some Xs are Zs." Now in Ploucquet's book of 1763, is found, "Deleatur in praemissis medius; id quod restat indicat conclusionem."[716] In the paper in which I explain my symbols--which are altogether different from Ploucquet's--there is found "Erase the symbols of the middle term; the remaining symbols show the inference." There is very great likeness: and I would have excused Hamilton for his notion if he had fairly given reference to the part of the book in which his quotation was found. For I had shown in my _Formal Logic_ what part of Ploucquet's book I had used: and a fair disputant would either have strengthened his point by showing that I had been at his part of the book, or allowed me the advantage of it being apparent that I had not given evidence of having seen that part of the book. My good friend, though an honest man, was sometimes unwilling to allow due advantage to controversial opponents.

But to my point. The only work of Ploucquet I ever saw was lent me by my friend Dr. Logan,[717] with whom I have often corresponded on logic, etc. I chanced (in 1865) {338} to turn up the letter which he sent me (Sept. 12, 1847) _with the book_. Part of it runs thus: "I congratulate you on your success in your logical researches [that is, in asking for the book, I had described some results]. Since the reading of your first paper I have been satisfied as to the possibility of inventing a logical notation in which the rationale of the inference is contained in the symbol, though I never attempted to verify it [what I communicated, then, satisfied the writer that I had done and communicated what he, from my previous paper, suspected to be practicable]. I send you Ploucquet's dissertation....'

It now being manifest that I cannot be souring grapes which have been taken from me, I will say what I never said in print before. There is not the slightest merit in making the symbols of the premises yield that of the conclusion by erasure: _the thing must do itself in every system which symbolises quant.i.ties_. For in every syllogism (except the inverted _Bramantip_ of the Aristotelians) the conclusion is manifest in this way without symbols. This _Bramantip_ destroys system in the Aristotelian lot: and circ.u.mstances which I have pointed out destroy it in Hamilton's own collection. But in that enlargement of the reputed Aristotelian system which I have called _onymatic_, and in that correction of Hamilton's system which I have called _exemplar_, the rule of erasure is universal, and may be seen without symbols.

Our first controversy was in 1846. In 1847, in my _Formal Logic_, I gave him back a little satire for satire, just to show, as I stated, that I could employ ridicule if I pleased. He was so offended with the appendix in which this was contained, that he would not accept the copy of the book I sent him, but returned it. Copies of controversial works, sent from opponent to opponent, are not _presents_, in the usual sense: it was a marked success to make him angry enough to forget this. It had some effect however: during the rest of his life I wished to avoid provocation; for I {339} could not feel sure that excitement might not produce consequences. I allowed his slashing account of me in the _Discussions_ to pa.s.s unanswered: and before that, when he proposed to open a controversy in the _Athenaeum_ upon my second Cambridge paper, I merely deferred the dispute until the next edition of my _Formal Logic_. I cannot expect the account in the _Discussions_ to amuse an unconcerned reader as much as it amused myself: but for a cut-and-thrust, might-and-main, tooth-and-nail, hammer-and-tongs a.s.sault, I can particularly recommend it. I never knew, until I read it, how much I should enjoy a thundering onslought on myself, done with racy insolence by a master hand, to whom my good genius had whispered _Ita feri ut se sentiat emori_.[718] Since that time I have, as the Irishman said, become "dry moulded for want of a bating." Some of my paradoxers have done their best: but theirs is mere twopenny--"small swipes," as Peter Peebles said. Brandy for heroes! I hope a reviewer or two will have mercy on me, and will give me as good discipline as Strafford would have given Hampden and his set: "much beholden," said he, "should they be to any one that should thoroughly take pains with them in that kind"--meaning _objective_ flagellation. And I shall be the same to any one who will serve me so--but in a literary and periodical sense: my corporeal cuticle is as thin as my neighbors'.

Sir W. H. was suffering under local paralysis before our controversy commenced: and though his mind was quite unaffected, a retort of as downright a character as the attack might have produced serious effect upon a person who had shown himself sensible of ridicule. Had a second attack of his disorder followed an answer from me, I should have been held to have caused it: though, looking at Hamilton's genial love of combat, I strongly suspected that a retort in kind

{340}

"Would cheer his heart, and warm his blood, And make him fight, and do him good."

But I could not venture to risk it. So all I did, in reply to the article in the _Discussions_, was to write to him the following note: which, as ill.u.s.trating an etiquette of controversy, I insert.

"I beg to acknowledge and thank you for.... It is necessary that I should say a word on my retention of this work, with reference to your return of the copy of my _Formal Logic_, which I presented to you on its publication: a return made on the ground of your disapproval of the account of our controversy which that work contained. According to my view of the subject, any one whose dealing with the author of a book is specially attacked in it, has a right to expect from the author that part of the book in which the attack is made, together with so much of the remaining part as is fairly context. And I hold that the acceptance by the party a.s.sailed of such work or part of a work does not imply any amount of approval of the contents, or of want of disapproval. On this principle (though I am not prepared to add the word _alone_) I forwarded to you the whole of my work on _Formal Logic_ and my second Cambridge Memoir. And on this principle I should have held you wanting in due regard to my literary rights if you had not forwarded to me your asterisked pages, with all else that was necessary to a full understanding of their scope and meaning, so far as the contents of the book would furnish it. For the remaining portion, which it would be a hundred pities to separate from the pages in which I am directly concerned, I am your debtor on another principle; and shall be glad to remain so if you will allow me to make a feint of balancing the account by the offer of two small works on subjects as little connected with our discussion as the _Epistolae Obscurorum Virorum_, or the Lutheran dispute. I trust that by accepting my _Opuscula_ you will enable me to avoid the {341} use of the knife, and leave me to cut you up with the pen as occasion shall serve, I remain, etc. (April 21, 1852)."

I received polite thanks, but not a word about the body of the letter: my argument, I suppose, was admitted.

SOME DOGGEREL AND COUNTER DOGGEREL.

I find among my miscellaneous papers the following _jeu d'esprit_, or _jeu de betise_,[719] whichever the reader pleases--I care not--intended, before I saw ground for abstaining, to have, as the phrase is, come in somehow. I think I could manage to bring anything into anything: certainly into a Budget of Paradoxes. Sir W. H. rather piqued himself upon some caniculars, or doggerel verses, which he had put together _in memoriam_ [_technicam_]

of the way in which A E I O are used in logic: he added U, Y, for the addition of _meet_, etc., to the system. I took the liberty of concocting some counter-doggerel, just to show that a mathematician may have architectonic power as well as a metaphysician.

DOGGEREL.

BY SIR W. HAMILTON.

A it affirms of _this_, _these_, _all_, Whilst E denies of _any_; I it affirms (whilst O denies) Of some (or few, or many).

Thus A affirms, as E denies, And definitely either; Thus I affirms, as O denies, And definitely neither.

A half, left semidefinite, Is worthy of its score; U, then, affirms, as Y denies, This, neither less nor more.

Indefinito-definites, I, UI, YO, last we come; {342} And this affirms, as that denies Of _more_, _most_ (_half_, _plus_, _some_).

COUNTER DOGGEREL.

BY PROF. DE MORGAN.

(1847.) Great A affirms of all; Sir William does so too: When the subject is "my suspicion,"

And the predicate "must be true."

Great E denies of all; Sir William of all but one: When he speaks about this present time, And of those who in logic have done.

Great I takes up but _some_; Sir William! my dear soul!

Why then in all your writings, Does "Great I" fill[720] the whole!

Great O says some are not; Sir William's readers catch, That some (modern) Athens is not without An Aristotle to match.

"A half, left semi-definite, Is worthy of its score:"

This looked very much like balderdash, And neither less nor more.

It puzzled me like anything; In fact, it puzzled me worse: Isn't schoolman's logic hard enough, Without being in Sibyl's verse?

{343} At last, thinks I, 'tis German; And I'll try it with some beer!

The landlord asked what bothered me so, And at once he made it clear.