Some one after having heard with patience what we have just said up to this point, will lose it after all and will say that we are amusing ourselves with frivolous statements, and that all identical truths are useless. But he will make this judgment for want of having thought sufficiently upon these matters. The deductions of logic, for example, are demon strated by identical principles; and geometers require the principle of contradiction in their demonstrations which re duce to the impossible.1 Let us be content here to show the use of identicals in the demonstrations of rational deduction. I say, then, that the principle of contradiction alone suffices to demonstrate the second and the third figures of the syllogism by means of the first. For example, we may conclude in the first figure, in Barbara: All B is C, All A is B, Then All A is C.
Suppose that the conclusion is false (or that it is true that some A is not C), then one or the other of the premises will be false also. Suppose that the second is true, the first must then be false, which maintains that all B is C. Then its con tradictory will be true, i.e. some B will not be C. And this will be the conclusion of a new argument, drawn from the 1 I.e. the so-called indirect proof, which provisionally assumes the truth of the contradictory opposite of the proposition to he proved, and then, having discovered the impossibility of this assumption, concludes, hy the aid of the principle of contradiction, that the original proposition is correct. — TR.
CH. n] ON HUMAN UNDERSTANDING 407 falsity of the conclusion and the truth of one of the premises of the preceding argument. Here is this new argument: Some A is not C. This is opposed to the preceding conclusion supposed false.
All A is B. This is the preceding premise supposed true.
Then some B is not C.
This is the present true conclusion, opposed to the preced ing false premise.
This argument is in the mode Disamis of the third figure, which is thus plainly demonstrated and at once from the mode Barbara of the first figure by employing simply the prin ciple of contradiction. And I noticed in my youth, when I examined minutely these things, that all the modes of the second and third figure may be drawn from the first by this method alone, by supposing that the mode of the first is valid, and consequently that the conclusion being false, or its contra dictory being taken as true, and one of the premises being taken as true also, the contradictory of the other premise is true. It is true that in the schools of logic they prefer to make use of conversions to draw the less principal figures from the first which is the principal, because this method appears better suited to the scholars. But for those who seek demon strative reasons, in which the least possible suppositions must be employed, we shall not demonstrate by the supposition of conversion what may be demonstrated by the primitive princi ple alone, which is that of contradiction and which assumes nothing. I have also made this apparently remarkable obser vation, that only the less principal figures which are called direct, viz. the second and the third, can be demonstrated by the principle of contradiction, by itself: but the less original indirect Jigure, the fourth, whose invention the Arabs attribute to Galen,1 although we found nothing concerning it in the 1 Claudius Galonus, c. 130-c. 201, a very celebrated physician and medical writer, who also wrote a large number of philosophical and logical works, the greater part of which are now lost. His medical and scientific treatises contain considerable philosophical and logical discussion, and his De usu partiurn 408 LEIBNITZ'S CRITIQUE OF LOCKE [BK. iv works of his remaining to us, nor in the other Greek authors, the fourth, I say, has this disadvantage, that it cannot be derived from the first or principal figure by this method alone, and it is necessary besides to employ another supposi tion, viz. conversions, so that it is farther removed by one degree than the second and the third, which are on a level and equally removed from the first; while the fourth needs also the second and the third for its demonstration. For it is found very opportunely that the conversions required are demonstrated by the second or third figure, demonstrable independently of the conversions, as I have just shown. It is Peter Kamus 1 who already made this remark concerning the demonstrability of conversion by these figures; and (if I am not mistaken) he reproached the logicians, who make use of conversion to demonstrate these figures, with arguing in a circle, although it was not so much the circle that he found it necessary to reproach them with (for they did not use these corp. hum. is, says Janet, " an apology for and a continual application of the principle of final causes." The most complete ed. of his works containing the Greek text and a Latin version is the Opera Omnia, cur. C. G. Kiihn, Leipzig, 1821-33, 20 vols., 8vo. For his philosophical views, cf. K. Sprengel, Beitr.z. Gesch. d. Medicin, 1, 117-195, Halle, 1794-6; on his logic, cf. Prautl, Gesch. d. Loyik, 1, 559-577. A hrief account of his philosophy is given hy Zeller, Philos. d. Griech., III., 1 [Vol. 5], 823 sq., 3d ed., Leipzig, 1880.
The invention of the fourth syllogistic figure was ascribed to Galen by Averroes, 1105-1198, but without adequate foundation. Galen was led through the additions to the first figure already made by Theophrastus, c. 373-c. 288 B.C., to transpose the premises in the same, and only by this means indirectly to the fourth arrangement of the middle term. Cf. Prantl, (fcsch. d. Logik im Abendlnnde, 1, 570-574; also Sir W. Hamilton, Lects. on Loyic, Boston, 1873, Lect. XX., If LXXIIL, pp. 285-G; Lect. XXI., f LXXIV., p. 302-3, and notes.
1 Petrus Ramus — Pierre de la Ramee — 1515-1572, murdered during St. Bartholomew's Night, was a determined opponent of Aristotelian scholasticism, and especially of its logic or dialectic, in the place of which he attempted to set up a new, simpler, and better grounded dialectic. For this purpose he wrote and published his two works, Animadversiones Aristotelicse, Paris, 1534, etc., and Institutiones dialectic fe, Paris, 1543. Following Cicero and Quintilian, his scheme was a blending of logic and rhetoric. For a long time after him, logicians were divided into Kamists and Anti-Ramists, while the Semi-Ramists, among whom were Alsted and Goclen (cf. ante, p. 311, note 2), sought to mediate between the Aristotelic dialectic, as set forth by Melauch- thon, and that of Ranms. The remarks to which Leibnitz here refers are found, according to Schaarschmidt, in Animad. Aristotel., Lutetia?, 1548, pp. 388 sq. For a good account of Ramus, cf. Stockl, Gesch. d. Philos. d. Mittplalters,lll. [Vol. 4], pp. 29(! sq.; also Ueberweg-Heinze, Gesch. d. Philos., en. n] ON HUMAN UNDERSTANDING 409 figures in their turn to justify the conversions) as the hysteron proteron or the reversal (le rebours); because conversions need rather to be demonstrated by these figures, than these figures by the conversions. But as this demonstration of conversions shows also the use of the identical affirmatives, which many take as altogether frivolous, it will be so much more to the purpose to introduce them here. I wish to speak only of con versions without contraposition, which suffice me here, and which are simple or per accidens, as they are called. Simple conversions are of two kinds: that of the universal negative, as:?io square is obtuse-angled, then no obtuse-angled figure is a square; and that of the particular affirmative, as: some tri angles are obtuse-angled, then some obtuse-angled figures are triangles. But conversion per accidens, as it is called, concerns the universal affirmative, as: every square is a rectangle, then some rectangles are squares. A rectangle is here always under stood to be a figure all of whose angles are right angles, and by the square is understood a regular quadrilateral. Now the question is to demonstrate these three kinds of conversion, which are: (1) No A is B, then no B is A.
(2) Some A is B, then some B is A.
(3) All A is B, then some B is A.
Demonstration of the first conversion in Cesare, which belongs to the second figure.
No A is B.
All B is B.
Then no B is A.
Demonstration of the second conversion in Datisi, which be longs to the third figure.
All A is A.
Some A is B.
Then some B is A.
Demonstration of the third conversion in Darapti, which belongs to the third figure.
All A is A.
All A is B.
Then some B is A.
410 LEIBNITZ'S CRITIQUE OF LOCKE [RK. TV This shows that the purest and apparently most useless identical propositions are of considerable use in the abstract and general; and that may teach us that we should not despise any truth. As for this proposition, that three is a,s much as two and one, which you, sir, still adduce as an example of intuitive knowledge, I have to say that it is only the defini tion of the term three, for the -simplest definitions of numbers are formed in this way: T^vo is one plus one, three is two plus one, four is three plus one, etc. It is true that there is therein a concealed statement, which I have already spoken of, viz. that these ideas are possible: and this is here known intuitively, so that it may be said, that an intuitive knowledge is comprised in definitions when their possibility appears at once. And in this way all adequate definitions contain primitive truths of reason and consequently intuitive knowledge. In short, you can say in general that all primitive truths of reason are imme diate with respect to an immediateness of ideas.
As for the primitive truths of fact, they are the immediate in ternal experiences of an immediateness of feeling. And here it is that the first truth of the Cartesians or of St. Augustine: I think, therefore I am, i.e. / am a thing ivhich thinks, holds good.1 But we must know, that as the identicals are general or par ticular, and as one is as clear as the other (since the statement that A is A is as clear as the statement that a thing is what it is), so is it also with the first truths of fact. For not only is it immediately clear to me that / think, but it is wholly as clear to me that / have different thoughts, that sometimes / think of A, and sometimes of B, etc. Thus the Cartesian principle is valid, but it is not the only one of its kind. You see by this that all primitive truths of reason or of fact have this in common, that they cannot be proved by anything more cer tain.
§ 2. Ph. I am very glad, sir, that you have carried forward 1 Cf. Augustine, Solil. II., 1: " Tn qui scis te nosse, scis te csse? Scio! Unde scis? Nescio! Simplicem te sentis an multiplicem? Nescio! Cogitare te scis? Scio!" in Opera, Vol. 1, p. 3(59, Benedictine ed., Paris, Franciscus Muguet, 1679; Vol. 1, p. 885, ed. Migne, Paris, 1841. Augustine, 354-430, thus anticipated the principle of Descartes, 1596-1650, " Cogito ergo sum," a fact unknown however to Descartes, who was not one of the class of reading philosophers, until brought to his knowledge by Arnauld, 1612-1694, and Mersenne, 1588-1648, iu their criticism of bis philosophy. — TK.
CH. n] ON HUMAN UNDERSTANDING 411 farther than I had done that which relates to intuitive knowl edge. Now demonstrative knowledge is only a concatenation of intuitive knowledge in all the connections of mediate ideas. For often the mind cannot unite, compare, or apply immedi ately the ideas one to the other, and this compels it to make use of other ideas (one or more) as means to the discovery of the agreement or disagreement it seeks, and this is what we call reasoning. As in demonstrating the three angles of a tri angle to be equal to two right angles, it finds some other angles which are seen to be equal both to the three angles of the triangle and to two right angles. § 3. These intervening ideas are called proofs, and the disposition of the mind to dis cover them is called sagacity. § 4. And even when found, this knowledge cannot be acquired without pains and atten tion and by more than a single passing view; for the mind must enter upon a progression of ideas, made gradually and by degrees, § 5. and there is doubt before the demonstra tion. § 6. It is not so clear as the intuitive knowledge, as the image reflected by several mirrors from one to another grows more and more faint with each reflection, and is no longer at once so recognizable especially by weak eyes. It is the same with knowledge produced by a long train of proof. § 7. And although each step taken by reason in the demon stration is intuitive knowledge or simple sight, nevertheless as in this long train of proofs the memory does not so exactly preserve this connection of ideas, men often take fallacies for demonstrations.
Th. Besides natural sagacity or that acquired by exercise, there is an art of finding mediate ideas (the medium), and this art is analysis. Now it is well to consider here, that the ques tion is sometimes to find the truth or falsehood of a given proposition, which is nothing else than an answer to the ques tion An? i.e. whether it is or is not. Sometimes it concerns an answer to a more difficult (cceteris paribus) question, where it is asked for example by whom and hotv f and where there is more to be supplied. And it is these questions alone, which leave a part of the proposition blank, which the mathemati cians call problems. As, when we are asked to find a mirror which collects all the rays of the sun in one point, we are asked for its form, or how it is made. As for the first ques- 412 LEIBNITZ'S CRITIQUE OF LOCKE [inc. iv tions in which the point at stake is only truth and falsehood and where there is nothing to be supplied in the subject or predicate, there is less invention, yet there is some, and the judgment alone is not sufficient. It is true that a man of judg ment, i.e. one who is capable of attention and reserve, and who has the leisure, the patience, and the necessary freedom of mind, may understand the most difficult demonstration if properly set before him. But the most judicious man in the world, without other aid, will not always be capable of discov ering this demonstration. Thus there is still some invention therein; and with geometers there was more of it formerly than now. For when analysis was less cultivated, more sagac ity was necessary to attain it, and it is on this account that some geometers still of the old school,1 or others who have not yet sufficient aptness in the new methods, think they have done something wonderful when they discover the demonstra tion of some theorem that others have invented. But those who are versed in the art of invention know when this is esti mable or not; for example, if some one sets forth the quadra ture of a space comprised within a curved and a straight line, which is successful in all its segments and which I call general, it is always within our power according to our methods to dis cover its demonstration, provided we are willing to take the trouble. But there are some particular quadratures of certain portions, where the thing may be so involved, that it will not always be possible (in potestate) thus far to develop it. It happens also that induction presents us with truths in num bers and in figures whose general reason is not yet discovered. For much is needed in order to attain perfection of analysis in geometry and in numbers, as many are becoming conceited upon the basis of the boasts of some men otherwise excellent, but a little too hasty or too ambitious.
But it is much more difficult to discover important truths, and still more to discover means of producing what is sought, when it is justly sought, than to discover the demonstration of truths which another has discovered. Beautiful truths are often attained by synthesis, by passing from the simple to the complex; but when it is a question of discovering exactly the means of producing what is proposed, synthesis is ordinarily 1 Gerharrtt reads: " roche "; Erdmann and Jacques: "race." — TR, CH. n] ON HUMAN UNDERSTANDING 413 not sufficient, and often to be willing to make all the requisite combinations would be an endless task, although one might often be aided therein by the method of exclusion,1 Avhich cuts off a good portion of useless combinations, and often nature does not admit any other method. But the means are not al ways at hand for the proper pursuit of this method. Analysis then must give us a thread in this labyrinth, when it is possi ble, for there are cases where the nature itself of the question demands that we grope about, short cuts not being always possible.
§ 8. Ph. Now as in demonstration intuitive knowledge is always supposed, it has, I think, given occasion for this maxim: that all reasoning springs from things already knoivn and agreed to (ex prvecognitis et prceconcessis),2 But we shall have occasion to speak of the falsity of this axiom when we speak of the maxims which are improperly taken as the foundation of our reasoning.
Th. I am curious to learn what falsehood you can find in an axiom apparently so reasonable. If it were always necessary to reduce everything to intuitive knowledge, demonstration would often be insufferably prolix. This is why mathema ticians have had the cleverness to divide the difficulties and to demonstrate separately the intervening propositions. And there is art also in this; for as the mediate truths (which are called lemmas, since they appear to be a digression) may be assigned in many ways, it is well, in order to aid the under standing and the memory, to choose those of them which greatly shorten the process, and appear memorable and worthy in themselves of being demonstrated. But there is another 1 The "method of exclusion " or elimination, says Schaarschmidt, proceeds from a disjunctive judgment, the predicate of which embraces in the sum of its divisional members all possible determinations of the subject. After it has been shown that individual divisional members cannot be united with the subject in a categorical judgment, that one alone of the divisional members which cannot be separated from the subject remains as the actual predicate for the valid determination of the subject. For example: A is B, or C, or D, or E. In this formula, B, C, D, E must include all thinkable predicate- determinations of A. In the question: Is A, B, or C, or D, or E, it is then proved that A is not C, D, E, in which case A must be B; or that A is not B, D, E, in which case A must be C, and so on. — TR.
414 LEIBNITZ'S CRITIQUE OF LOCKE [UK. iv obstacle, viz.: that it is not easy to demonstrate all the axioms, and to reduce demonstration wholly to intuitive knowledge. And if we had chosen to wait for that, perhaps we should not yet have the science of geometry. But we have already spoken of this in our former conversations, and we shall have occasion to speak of it again.
§ 9. Ph. We shall come to that presently; now I shall re mark again what I have already touched upon more than once, that it is a common opinion that only the mathematical sciences are capable of a demonstrative certainty; but as the agreement and disagreement which may be known intuitively is not a privilege belonging only to the ideas of numbers and figures, it is perhaps for want of application on our part that mathematics alone have attained to demonstrations. § 10. Many reasons conspired to this end. The mathematical sciences are very generally useful; the least difference therein is very easily recognized. § II.1 These other simple ideas, which are appearances or situations produced in us, have no exact measure of their different degrees. § 12. 2 But when the difference of these visible qualities, for example, is sufficiently great to excite in the mind clearly distinct ideas, as those of blue or red, they are as capable of demonstration as those of number and extension.
Th. There are notable examples enough of demonstration outside of mathematics, and it may be said that Aristotle has already given some in his " Prior Analytics." In fact logic is as susceptible of demonstrations as geometry, and it may be said that the logic of the geometers, or the methods of argu mentation explained and established by Euclid in reasoning upon propositions, are a particular extension or promotion of general logic. Archimedes 3 is the first, whose works we have, who has practised the art of demonstration upon an occasion where he is treating of physics, as he has done in his book on 1 § 11, as also § 12, is § 17 in the texts of Erdmann and Jacques. — TK.
3 Archimedes, 287-212 B.C., the greatest mathematician among the Greeks, distinguished also for his discoveries in hydrostatics and hydraulics, and for his ingenious inventions. He first placed the science of engineering upon a sound mathematical hasis. The most complete and magnificent edition of his extant works is that edited by Torelli and published at Oxford, at the Claren don Press, 1792, fol. — TR.
en. n] ON HUMAN UNDERSTANDING 415 Equilibrium. Furthermore, jurists may be said to have many good demonstrations; especially the ancient Roman jurists, whose fragments have been preserved to us in the Pandects. I am wholly of the opinion of Laurentius Valla,1 who cannot enough admire these authors among others, because they all speak in a manner so just and so clear and in fact reason in a way closely approaching the demonstrative, and often it is wholly demonstrative.2 Indeed, I do not know any science outside that of law and that of arms, in which the Romans have made any considerable addition to what they received from the Greeks.
Tu regere imperio populos Romane memento: Hte tibi erunt artes pacique impouere morem, Parcere subjectis, et debellare superbos.3 This precise manner of expressing themselves is the reason that all the jurists of the Pandects, though sometimes quite 1 Laurentius Valla — Lorenzo della Valle — c. 1407-1457, a humanist and philologian of the earlier Italian Renaissance, was an earnest opponent of the scholastic dialectic, a determined foe of tradition and authority, and the initi ator and champion of a bold and unbiassed criticism which he applied to language, historical documents, and ethical opinions. He was eminent as a Latinist, and his treatise Elerjantise latinos linyuiB, c. 1431, in six books, — the Preface to the third book of which Schaarschmidt thinks Leibnitz probably had in mind in referring to Valla's admiration for the style of the Roman jurists, therein very highly praised, — subjected the forms of Latin grammar and rhetoric to critical investigation and analysis, and established upon a scientific foundation the principles of Latin style. His De /also credita et ementita Constantini Donatione, 1440, destroyed the claims of the Papacy to temporal power based upon this alleged "Donation," by proving its docu mentary foundations to be forgeries. His principal philosophical writings are: De voluptate et vcro bono, 1431, in which he boldly defended the Epicurean doctrine of pleasure as the true and only good; DC libcro arbifrio; and the Dialectic-SB disputationes contra Aristotelicos, 1409, of which Prantl, Gesch.
'1. Lnf/ik im Abcndlande, 4, 1(51-107, gives some account with citations. Valla's Opera Omnia, Basilic, 14<>5 and 1540-1543. Leibnitz refers to him and his DC lib. arb/t. and De rnlnptatc in the Tbe'odicee, Pt. III., §§ 405 *q. For accounts of his life and works, cf. G. Tiraboschi, Storin della Letteratura Italiana, Vol. 0, Pt. II., pp. 339-340, Rome, 1784; Symonds, Renaissance in Italy, Pt. II., The Rer-iral of Learn inf/, p. 258 sq., New York, H. Holt & Co., 1881. For his philosophy, cf. Stockl, Gesch. d. Philos. d. Mittelalters, III. [Vol.4], 279-283. Mancini published at Florence, 1891, a brilliant and exhaus tive monograph investigating and settling disputed points in Valla's life.— TR.
2 Cf. Leibnitz's letter to Kestner, No. 15, in Ch. Kortholt, Leibnit. epist. ad diremos, Lipsia?, 1734-1742, Vol. 3, p. 256, Dutens, Leibnit. opera omnia, Vol. 4, Pt. III., p. 207, where he expresses himself similarly as here. Also, Guhrauer, Leibniz, eine Bior/raphie, Pt. I., pp. 36, 37. — TR.
416 LEIBNITZ'S CRITIQUE OF LOCKE [UK. iv distant from one another in time, seem to be a single author, and there would be much difficulty in distinguishing them, if the names of the writers were not at the beginning of the ex tracts; as it would be difficult to distinguish Euclid,1 Archi medes and Apollonius 2 in reading their demonstrations upon matters which the one as well as the other has touched upon. It must be admitted that the Greeks have reasoned with all possible accuracy in mathematics, and that they have left the human race models in the art of demonstration: for if the Babylonians and the Egyptians had anything more than an empirical geometry, nothing of it at least remains; but it is astonishing that these same Greeks lost it to such an extent3 at once as soon as they turned aside ever so little from num bers and figures in order to proceed to philosophy. For it is strange that we do not see a shadow of demonstration in Plato and in Aristotle (his "Prior Analytics" excepted) and in all the other ancient philosophers. Proclus 4 was an excellent geometer, but he seems another man when he speaks on phi losophy. What has made it easier to reason demonstrably in mathematics is largely the fact that experience can there guar antee the reasoning at every moment, as is also the case in the syllogistic figures. But in metaphysics and ethics this par allelism of reason and experience is no longer found; and in