Chapter V: Analytica Priora I (1)

Reviewing the treatise De Interpretatione, we have followed Aristotle in his first attempt to define what a Proposition is, to point out its constituent elements, and to specify some of its leading varieties. The characteristic feature of the Proposition he stated to be--That it declares, in the first instance, the mental state of the speaker as to belief or disbelief, and, in its ulterior or final bearing, a state of facts to which such belief or disbelief corresponds. It is thus significant of truth or falsehood; and this is its logical character (belonging to Analytic and Dialectic), as distinguished from its rhetorical character, with other aspects besides. Aristotle farther indicated the two principal discriminative attributes of propositions as logically regarded, passing under the names of quantity and quality. He took great pains, in regard to the quality, to explain what was the special negative proposition in true contradictory antithesis to each affirmative. He stated and enforced the important separation of contradictory propositions from contrary; and he even parted off (which the Greek and Latin languages admit, though the French and English will hardly do so) the true negative from the indeterminate affirmative. He touched also upon equipollent propositions, though he did not go far into them. Thus commenced with Aristotle the systematic study of propositions, classified according to their meaning and their various interdependences with each other as to truth and falsehood--their mutual consistency or incompatibility. Men who had long been talking good Greek fluently and familiarly, were taught to reflect upon the conjunctions of words that they habitually employed, and to pay heed to the conditions of correct speech in reference to its primary purpose of affirmation and denial, for the interchange of beliefs and disbeliefs, the communication of truth, and the rectification of falsehood. To many of Aristotle's contemporaries this first attempt to theorize upon the forms of locution familiar to every one would probably appear hardly less strange than the interrogative dialectic of Sokrates, when he declared himself not to know what was meant by justice, virtue, piety, temperance, government, &c.; when he astonished his hearers by asking them to rescue him from this state of ignorance, and to communicate to him some portion of their supposed plenitude of knowledge.

Aristotle tells us expressly that the theory of the Syllogism, both demonstrative and dialectic, on which we are now about to enter, was his own work altogether and from the beginning; that no one had ever attempted it before; that he therefore found no basis to work upon, but was obliged to elaborate his own theory, from the very rudiments, by long and laborious application. In this point of view, he contrasts Logic pointedly with Rhetoric, on which there had been a series of writers and teachers, each profiting by the labours of his predecessors.[1] There is no reason to contest the claim to originality here advanced by Aristotle. He was the first who endeavoured, by careful study and multiplied comparison of propositions, to elicit general truths respecting their ratiocinative interdependence, and to found thereupon precepts for regulating the conduct of demonstration and dialectic.[2]

[Footnote 1: See the remarkable passage at the close of the Sophistici Elenchi, p. 183, b. 34-p. 184, b. 9: [Greek: tau/tês de\ tê=s pragmatei/as ou) to\ me\n ê)=n to\ de\ ou)k ê)=n proexeirgasme/non, a)ll' ou)de\n pantelô=s u(pê=rche--kai\ peri\ me\n tô=n r(êtorikô=n u(pê=rche polla\ kai\ palaia\ ta\ lego/mena, peri\ de\ tou= sullogi/zesthai pantelô=s ou)de\n ei)/chomen pro/teron a)/llo le/gein, a)ll' ê)\ tribê=| zêtou=ntes polu\n chro/non e)ponou=men.]]

[Footnote 2: Sir Wm. Hamilton, Lectures on Logic, Lect. v. pp. 87-91, vol. III.:--"The principles of Contradiction and Excluded Middle can both be traced back to Plato, by whom they were enounced and frequently applied; though it was not till long after, that either of them obtained a distinctive appellation. To take the principle of Contradiction first. This law Plato frequently employs, but the most remarkable passages are found in the Phædo (p. 103), in the Sophista (p. 252), and in the Republic (iv. 436, vii. 525). This law was however more distinctively and emphatically enounced by Aristotle. . . . . Following Aristotle, the Peripatetics established this law as the highest principle of knowledge. From the Greek Aristotelians it obtained the name by which it has subsequently been denominated, the _principle_, or _law_, or _axiom_, of _Contradiction_ ([Greek: a)xi/ôma tê=s a)ntipha/seôs]). . . . . The law of Excluded Middle between two contradictories remounts, as I have said, also to Plato; though the Second Alcibiades, in which it is most clearly expressed (p. 139; also Sophista, p. 250), must be admitted to be spurious. . . . . This law, though universally recognized as a principle in the Greek Peripatetic school, and in the schools of the middle ages, only received the distinctive appellation by which it is now known at a comparatively modern date."

The passages of Plato, to which Sir W. Hamilton here refers, will not be found to bear out his assertion that Plato "enounced and frequently applied the principles of Contradiction and Excluded Middle." These two principles are both of them enunciated, denominated, and distinctly explained by Aristotle, but by no one before him, as far as our knowledge extends. The conception of the two maxims, in their generality, depends upon the clear distinction between Contradictory Opposition and Contrary Opposition; which is fully brought out by Aristotle, but not adverted to, or at least never broadly and generally set forth, by Plato. Indeed it is remarkable that the word [Greek: A)nti/phasis], the technical term for Contradiction, never occurs in Plato; at least it is not recognized in the _Lexicon Platonicum_. Aristotle puts it in the foreground of his logical exposition; for, without it, he could not have explained what he meant by Contradictory Opposition. See Categoriæ, pp. 13-14, and elsewhere in the treatise De Interpretatione and in the Metaphysica. Respecting the idea of the Negative as put forth by Plato in the Sophistes (not coinciding either with Contradictory Opposition or with Contrary Opposition), see 'Plato and the Other Companions of Sokrates,' vol. II. ch. xxvii. pp. 449-459. I have remarked in that chapter, and the reader ought to recollect, that the philosophical views set out by Plato in the Sophistes differ on many points from what we read in other Platonic dialogues.]

He begins the Analytica Priora by setting forth his general purpose, and defining his principal terms and phrases. His manner is one of geometrical plainness and strictness. It may perhaps have been common to him with various contemporary geometers, whose works are now lost; but it presents an entire novelty in Grecian philosophy and literature. It departed not merely from the manner of the rhetoricians and the physical philosophers (as far as we know them, not excluding even Demokritus), but also from Sokrates and the Sokratic school. For though Sokrates and Plato were perpetually calling for definitions, and did much to make others feel the want of such, they neither of them evinced aptitude or readiness to supply the want. The new manner of Aristotle is adapted to an undertaking which he himself describes as original, in which he has no predecessors, and is compelled to dig his own foundations. It is essentially didactic and expository, and contrasts strikingly with the mixture of dramatic liveliness and dialectical subtlety which we find in Plato.

The terminology of Aristotle in the Analytica is to a certain extent different from that in the treatise De Interpretatione. The Enunciation ([Greek: A)po/phanis]) appears under the new name of [Greek: Pro/tasis], _Proposition_ (in the literal sense) or _Premiss_; while, instead of Noun and Verb, we have the word _Term_ ([Greek: O(/ros]), applied alike both to Subject and to Predicate.[3] We pass now from the region of _declared_ truth, into that of _inferential_ or _reasoned_ truth. We find the proposition looked at, not merely as communicating truth in itself, but as generating and helping to guarantee certain ulterior propositions, which communicate something additional or different. The primary purpose of the Analytica is announced to be, to treat of Demonstration and demonstrative Science; but the secondary purpose, running parallel with it and serving as illustrative counterpart, is, to treat also of Dialectic; both of them[4] being applications of the inferential or ratiocinative process, the theory of which Aristotle intends to unfold.

[Footnote 3: Aristot. Analyt. Prior. I. i. p. 24, b. 16: [Greek: o(/ron de\ kalô= ei)s o(\n dialu/etai ê( pro/tasis, oi(=on to/ te katêgorou/menon kai\ to\ kath' ou(= katêgorei=tai], &c.

[Greek: O(/ros]--_Terminus_--seems to have been a technical word first employed by Aristotle himself to designate subject and predicate as the _extremes_ of a proposition, which latter he conceives as the _interval_ between the _termini_--[Greek: _dia/stêma_]. (Analyt. Prior. I. xv. p. 35, a. 12. [Greek: sterêtikô=n diastêma/tôn], &c. See Alexander, Schol. pp. 145-146.)

In the Topica Aristotle employs [Greek: o(/ros] in a very different sense--[Greek: lo/gos o( to\ ti/ ê)=n ei)=nai sêmai/nôn] (Topic. I. v. p. 101, b. 39)--hardly distinguished from [Greek: o(rismo/s]. The Scholia take little notice of this remarkable variation of meaning, as between two treatises of the Organon so intimately connected (pp. 256-257, Br.).]

[Footnote 4: Analyt. Prior. I. i. p. 24, a. 25.]

The three treatises--1, Analytica Priora, 2, Analytica Posteriora, 3, Topica with Sophistici Elenchi--thus belong all to one general scheme; to the theory of the Syllogism, with its distinct applications, first, to demonstrative or didactic science, and, next, to dialectical debate. The scheme is plainly announced at the commencement of the Analytica Priora; which treatise discusses the Syllogism generally, while the Analytica Posteriora deals with Demonstration, and the Topica with Dialectic. The first chapter of the Analytica Priora and the last chapter of the Sophistici Elenchi (closing the Topica), form a preface and a conclusion to the whole. The exposition of the Syllogism, Aristotle distinctly announces, precedes that of Demonstration (and for the same reason also precedes that of Dialectic), because it is more general: every demonstration is a sort of syllogism, but every syllogism is not a demonstration.[5]

[Footnote 5: Ibid. I. iv. p. 25, b. 30.]

As a foundation for the syllogistic theory, propositions are classified according to their quantity (more formally than in the treatise De Interpretatione) into Universal, Particular, and Indefinite or Indeterminate;[6] Aristotle does not recognize the Singular Proposition as a distinct variety. In regard to the Universal Proposition, he introduces a different phraseology according as it is looked at from the side of the Subject, or from that of the Predicate. The Subject is, or is not, in the whole Predicate; the Predicate is affirmed or denied respecting all or every one of the Subject.[7] The minor term of the Syllogism (in the first mode of the first figure) is declared to be in the whole middle term; the major is declared to belong to, or to be predicable of, all and every the middle term. Aristotle says that the two are the same; we ought rather to say that each is the concomitant and correlate of the other, though his phraseology is such as to obscure the correlation.

[Footnote 6: Ibid. I. i. p. 24, a. 17. The Particular ([Greek: e)n me/rei]), here for the first time expressly distinguished by Aristotle, is thus defined:--[Greek: e)n me/rei de\ to\ tini\ ê)\ mê\ tini\ ê)\ mê\ panti\ u(pa/rchein.]]

[Footnote 7: Ibid. b. 26: [Greek: to\ d' e)n o(/lô| ei)nai e(/teron e(te/rô|, kai\ to\ kata\ panto\s katêgorei=sthai thate/rou tha/teron, tau)to/n e)sti--tau)to\n], _i.e._ [Greek: _a)ntestramme/nôs_], as Waitz remarks in note. Julius Pacius says:--"Idem re, sed ratione differunt ut ascensus et descensus; nam subjectum dicitur esse vel non esse in toto attributo, quia attributum dicitur de omni vel de nullo subjecto" (p. 128).]

The definition given of a Syllogism is very clear and remarkable:--"It is a speech in which, some positions having been laid down, something different from these positions follows as a necessary consequence from their being laid down." In a _perfect_ Syllogism nothing additional is required to make the necessity of the consequence obvious as well as complete. But there are also _imperfect_ Syllogisms, in which such necessity, though equally complete, is not so obviously conveyed in the premisses, but requires some change to be effected in the position of the terms in order to render it conspicuous.[8]

[Footnote 8: Aristot. Anal. Prior. I. i. p. 24, b. 18-26. The same, with a little difference of wording, at the commencement of Topica, p. 100, a. 25. Compare also Analyt. Poster. I. x. p. 76, b. 38: [Greek: o(/sôn o)/ntôn tô=| e)kei=na ei)=nai gi/netai to\ sumpe/rasma.]]

The term Syllogism has acquired, through the influence of Aristotle, a meaning so definite and technical, that we do not easily conceive it in any other meaning. But in Plato and other contemporaries it bears a much wider sense, being equivalent to reasoning generally, to the process of comparison, abstraction, generalization.[9] It was Aristotle who consecrated the word, so as to mean exclusively the reasoning embodied in propositions of definite form and number. Having already analysed propositions separately taken, and discriminated them into various classes according to their constituent elements, he now proceeds to consider propositions in combination. Two propositions, if properly framed, will conduct to a third, different from themselves, but which will be necessarily true if they are true. Aristotle calls the three together a Syllogism.[10] He undertakes to shew how it must be framed in order that its conclusion shall be necessarily true, if the premisses are true. He furnishes schemes whereby the cast and arrangement of premisses, proper for attaining truth, may be recognized; together with the nature of the conclusion, warrantable under each arrangement.

[Footnote 9: See especially Plato, Theætêt. p. 186, B-D., where [Greek: o( sullogismo\s] and [Greek: ta\ a)nalogi/smata] are equivalents.]

[Footnote 10: Julius Pacius (ad Analyt. Prior. I. i.) says that it is a mistake on the part of most logicians to treat the Syllogism as including three propositions (ut vulgus logicorum putat). He considers the premisses alone as constituting the Syllogism; the conclusion is not a part thereof, but something distinct and superadded. It appears to me that the _vulgus logicorum_ are here in the right.]

In the Analytica Priora, we find ourselves involved, from and after the second chapter, in the distinction of Modal propositions, the necessary and the possible. The rules respecting the simple Assertory propositions are thus, even from the beginning, given in conjunction and contrast with those respecting the Modals. This is one among many causes of the difficulty and obscurity with which the treatise is beset. Theophrastus and Eudemus seem also to have followed their master by giving prominence to the Modals:[11] recent expositors avoid the difficulty, some by omitting them altogether, others by deferring them until the simple assertory propositions have been first made clear. I shall follow the example of these last; but it deserves to be kept in mind, as illustrating Aristotle's point of view, that he regards the Modals as principal varieties of the proposition, co-ordinate in logical position with the simple assertory.

[Footnote 11: Eudemi Fragmenta, cii.-ciii. p. 145, ed. Spengel.]

Before entering on combinations of propositions, Aristotle begins by shewing what can be done with single propositions, in view to the investigation or proving of truth. A single proposition may be _converted_; that is, its subject and predicate may be made to change places. If a proposition be true, will it be true when thus converted, or (in other words) will its converse be true? If false, will its converse be false? If this be not always the case, what are the conditions and limits under which (assuming the proposition to be true) the process of conversion leads to assured truth, in each variety of propositions, affirmative or negative, universal or particular? As far as we know, Aristotle was the first person that ever put to himself this question; though the answer to it is indispensable to any theory of the process of proving or disproving. He answers it before he enters upon the Syllogism.

The rules which he lays down on the subject have passed into all logical treatises. They are now familiar; and readers are apt to fancy that there never was any novelty in them--that every one knows them without being told. Such fancy would be illusory. These rules are very far from being self-evident, any more than the maxims of Contradiction and of the Excluded Middle. Not one of the rules could have been laid down with its proper limits, until the discrimination of propositions, both as to quality (affirmative or negative), and as to quantity (universal or particular), had been put prominently forward and appreciated in all its bearings. The rule for trustworthy conversion is different for each variety of propositions. The Universal Negative may be converted simply; that is, the predicate may become subject, and the subject may become predicate--the proposition being true after conversion, if it was true before. But the Universal Affirmative cannot be thus converted simply. It admits of conversion only in the manner called by logicians _per accidens_: if the predicate change places with the subject, we cannot be sure that the proposition thus changed will be true, unless the new subject be lowered in quantity from universal to particular; _e.g._ the proposition, All men are animals, has for its legitimate converse not, _All_ animals are men, but only, _Some_ animals are men. The Particular Affirmative may be converted simply: if it be true that Some animals are men, it will also be true that Some men are animals. But, lastly, if the true proposition to be converted be a Particular Negative, it cannot be converted at all, so as to make sure that the converse will be true also.[12]

[Footnote 12: Aristot. Analyt. Prior. I. ii. p. 25, a. 1-26.]

Here then are four separate rules laid down, one for each variety of propositions. The rules for the second and third variety are proved by the rule for the first (the Universal Negative), which is thus the basis of all. But how does Aristotle prove the rule for the Universal Negative itself? He proceeds as follows: "If A cannot be predicated of any one among the B's, neither can B be predicated of any one among the A's. For if it could be predicated of any one among them (say C), the proposition that A cannot be predicated of any B would not be true; since C is one among the B's."[13] Here we have a proof given which is no proof at all. If I disbelieved or doubted the proposition to be proved, I should equally disbelieve or doubt the proposition given to prove it. The proof only becomes valid, when you add a farther assumption which Aristotle has not distinctly enunciated, viz.: That if some A (_e.g._ C) is B, then some B must also be A; which would be contrary to the fundamental supposition. But this farther assumption cannot be granted here, because it would imply that we already know the rule respecting the convertibility of Particular Affirmatives, viz., that they admit of being converted simply. Now the rule about Particular Affirmatives is afterwards itself proved by help of the preceding demonstration respecting the Universal Negative. As the proof stands, therefore, Aristotle demonstrates each of these by means of the other; which is not admissible.[14]

[Footnote 13: Ibid. p. 25, a. 15: [Greek: ei) ou)=n mêdeni\ tô=n B to\ A u(pa/rchei, ou)de\ tô=n A ou)deni\ u(pa/rxei to\ B. ei) ga\r tini, oi(=on tô=| G, ou)k a)lêthe\s e)/stai to\ mêdeni\ tô=n B to\ A u(pa/rchein; to\ ga\r G tô=n B ti/ e)stin.]

Julius Pacius (p. 129) proves the Universal Negative to be convertible _simpliciter_, by a _Reductio ad Absurdum_ cast into a syllogism in the First figure. But it is surely unphilosophical to employ the rules of Syllogism as a means of proving the legitimacy of Conversion, seeing that we are forced to assume conversion in our process for distinguishing valid from invalid syllogisms. Moreover the _Reductio ad Absurdum_ assumes the two fundamental Maxims of Contradiction and Excluded Middle, though these are less obvious, and stand more in need of proof than the simple conversion of the Universal Negative, the point that they are brought to establish.]

[Footnote 14: Waitz, in his note (p. 374), endeavours, but I think without success, to show that Aristotle's proof is not open to the criticism here advanced. He admits that it is obscurely indicated, but the amplification of it given by himself still remains exposed to the same objection.]

Even the friends and companions of Aristotle were not satisfied with his manner of establishing this fundamental rule as to the conversion of propositions. Eudêmus is said to have given a different proof; and Theophrastus assumed as self-evident, without any proof, that the Universal Negative might always be converted simply.[15] It appears to me that no other or better evidence of it can be offered, than the trial upon particular cases, that is to say, Induction.[16] Nothing is gained by dividing (as Aristotle does) the whole A into parts, one of which is C; nor can I agree with Theophrastus in thinking that every learner would assent to it at first hearing, especially at a time when no universal maxims respecting the logical value of propositions had ever been proclaimed. Still less would a Megaric dialectician, if he had never heard the maxim before, be satisfied to stand upon an alleged _à priori_ necessity without asking for evidence. Now there is no other evidence except by exemplifying the formula, No A is B, in separate propositions already known to the learner as true or false, and by challenging him to produce any one case, in which, when it is true to say No A is B, it is not equally true to say, No B is A; the universality of the maxim being liable to be overthrown by any one contradictory instance.[17] If this proof does not convince him, no better can be produced. In a short time, doubtless, he will acquiesce in the general formula at first hearing, and he may even come to regard it as self-evident. It will recall to his memory an aggregate of separate cases each individually forgotten, summing up their united effect under the same aspect, and thus impressing upon him the general truth as if it were not only authoritative but self-authorized.

[Footnote 15: See the Scholia of Alexander on this passage, p. 148, a. 30-45, Brandis; Eudemi Fragm. ci.-cv. pp. 145-149, ed. Spengel.]

[Footnote 16: We find Aristotle declaring in Topica, II. viii. p. 113, b. 15, that in converting a true Universal Affirmative proposition, the negative of the Subject of the convertend is always true of the negative of the Predicate of the convertend; _e.g._ If every man is an animal, every thing which is not an animal is not a man. This is to be assumed (he says) upon the evidence of Induction--uncontradicted iteration of particular cases, extended to all cases universally--[Greek: lamba/nein d' e)x e)pagôgê=s, oi(=on ei) o( a)/nthrôpos zô=|on, to\ mê\ zô=|on ou)k a)/nthrôpos; o(moi/ôs de\ kai\ e)pi\ tô=n a)/llôn. . . . . e)pi\ pa/ntôn ou)=n to\ toiou=ton a)xiôte/on.]

The rule for the simple conversion of the Universal Negative rests upon the same evidence of Induction, never contradicted.]

[Footnote 17: Dr. Wallis, in one of his acute controversial treatises against Hobbes, remarks upon this as the process pursued by Euclid in his demonstrations:--"You tell us next that an Induction, without enumeration of all the particulars, is not sufficient to infer a conclusion. Yes, Sir, if after the enumeration of some particulars, there comes a general clause, _and the like in other cases_ (as here it doth), this may pass for a proofe till there be a possibility of giving some instance to the contrary, which here you will never be able to doe. And if such an Induction may not pass for proofe, there is never a proposition in Euclid demonstrated. For all along he takes no other course, or at least grounds his Demonstrations on Propositions no otherwise demonstrated. As, for instance, he proposeth it in general (i. c. 1.)--_To make an equilateral triangle on a line given_. And then he shows you how to do it upon the line A B, which he there shows you, and leaves you to supply: _And the same, by the like means, may be done upon any other strait line_; and then infers his general conclusion. Yet I have not heard any man object that the Induction was not sufficient, because he did not actually performe it in all lines possible."--(Wallis, Due Correction to Mr. Hobbes, Oxon. 1656, sect. v. p. 42.) This is induction by _parity of reasoning_.

So also Aristot. Analyt. Poster. I. iv. p. 73, b. 32: [Greek: to\ katho/lou de\ u(pa/rchei to/te, o(/tan e)pi\ tou= tucho/ntos kai\ prô/tou deiknu/êtai.]]

Aristotle passes next to Affirmatives, both Universal and Particular. First, if A can be predicated of all B, then B can be predicated of _some_ A; for if B cannot be predicated of any A, then (by the rule for the Universal Negative) neither can A be predicated of any B. Again, if A can be predicated of some B, in this case also, and for the same reason, B can be predicated of some A.[18] Here the rule for the Universal Negative, supposed already established, is applied legitimately to prove the rules for Affirmatives. But in the first case, that of the Universal, it fails to prove _some_ in the sense of _not-all_ or _some-at-most_, which is required; whereas, the rules for both cases can be proved by Induction, like the formula about the Universal Negative. When we come to the Particular Negative, Aristotle lays down the position, that it does not admit of being necessarily converted in any way. He gives no proof of this, beyond one single exemplification: If some animal is not a man, you are not thereby warranted in asserting the converse, that some man is not an animal.[19] It is plain that such an exemplification is only an appeal to Induction: you produce one particular example, which is entering on the track of Induction; and one example alone is sufficient to establish the negative of an universal proposition.[20] The converse of a Particular Negative is not in all cases true, though it may be true in many cases.

[Footnote 18: Aristot. Analyt. Prior. I. ii. p. 25, a. 17-22.]

[Footnote 19: Ibid. p. 25, a. 22-26.]

[Footnote 20: Though some may fancy that the rule for converting the Universal Negative is intuitively known, yet every one must see that the rule for converting the Universal Affirmative is not thus self-evident, or derived from natural intuition. In fact, I believe that every learner at first hears it with great surprise. Some are apt to fancy that the Universal Affirmative (like the Particular Affirmative) may be converted _simply_. Indeed this error is not unfrequently committed in actual reasoning; all the more easily, because there is a class of cases (with subject and predicate co-extensive) where the converse of the Universal Affirmative _is_ really true. Also, in the case of the Particular Negative, there are many true propositions in which the simple converse is true. A novice might incautiously generalize upon those instances, and conclude that both were convertible simply. Nor could you convince him of his error except by producing examples in which, when a true proposition of this kind is converted simply, the resulting converse is notoriously false. The appeal to various separate cases is the only basis on which we can rest for testing the correctness or incorrectness of all these maxims proclaimed as universal.]

From one proposition taken singly, no new proposition can be inferred; for purposes of inference, two propositions at least are required.[21] This brings us to the rules of the Syllogism, where two propositions as premisses conduct us to a third which necessarily follows from them; and we are introduced to the well-known three Figures with their various Modes.[22] To form a valid Syllogism, there must be three terms and no more; the two, which appear as Subject and Predicate of the conclusion, are called the _minor_ term (or minor extreme) and the _major_ term (or major extreme) respectively; while the third or _middle_ term must appear in each of the premisses, but not in the conclusion. These terms are called _extremes_ and _middle_, from the position which they occupy in every perfect Syllogism--that is in what Aristotle ranks as the First among the three figures. In _his_ way of enunciating the Syllogism, this middle position formed a conspicuous feature; whereas the modern arrangement disguises it, though the denomination _middle_ term is still retained. Aristotle usually employs letters of the alphabet, which he was the first to select as abbreviations for exposition;[23] and he has two ways (conforming to what he had said in the first chapter of the present treatise) of enunciating the modes of the First figure. In one way, he begins with the major extreme (Predicate of the conclusion): A may be predicated of all B, B may be predicated of all C; therefore, A may be predicated of all C (Universal Affirmative). Again, A cannot be predicated of any B, B can be predicated of all C; therefore, A cannot be predicated of any C (Universal Negative). In the other way, he begins with the minor term (Subject of the conclusion): C is in the whole B, B is in the whole A; therefore, C is in the whole A (Universal Affirmative). And, C is in the whole B, B is not in the whole A; therefore, C is not in the whole A (Universal Negative). We see thus that in Aristotle's way of enunciating the First figure, the middle term is really placed between the two extremes,[24] though this is not so in the Second and Third figures. In the modern way of enunciating these figures, the middle term is never placed between the two extremes; yet the denomination _middle_ still remains.

[Footnote 21: Analyt. Prior. I. xv. p. 34, a. 17; xxiii. p. 40, b. 35; Analyt. Poster. I. iii. p. 73, a. 7.]

[Footnote 22: Aristot. Analyt. Prior. I. iv. p. 25, b. 26, seq.]

[Footnote 23: M. Barthélemy St. Hilaire (Logique d'Aristote, vol. ii. p. 7, n.), referring to the examples of Conversion in chap. ii., observes:--"Voici le prémier usage des lettres représentant des idées; c'est un procédé tout à fait algébrique, c'est à dire, de généralisation. Déjà, dans l'Herméneia, ch. 13, § 1 et suiv., Aristote a fait usage de tableaux pour représenter sa pensée relativement à la consécution des modales. Il parle encore spécialement de figures explicatives, liv. 2. des Derniers Analytiques, ch. 17, § 7. Vingt passages de l'Histoire des Animaux attestent qu'il joignait des dessins à ses observations et à ses théories zoologiques. Les illustrations pittoresques datent donc de fort loin. L'emploi symbolique des lettres a été appliqué aussi par Aristote à la Physique. Il l'avait emprunté, sans doute, aux procédés des mathématiciens."

We may remark, however, that when Aristotle proceeds to specify those combinations of propositions which _do not_ give a valid conclusion, he is not satisfied with giving letters of the alphabet; he superadds special illustrative examples (Analyt. Prior. I. v. p. 27, a. 7, 12, 34, 38).]

[Footnote 24: Aristot. Analyt. Prior. I. iv. p. 25, b. 35: [Greek: kalô= de\ _me/son_, o(\ kai\ au)to\ e)n a)/llô| kai\ a)/llo e)n tou/tô| e)sti/n, o(\ kai\ tê=| the/sei gi/netai me/son.]]

The Modes of each figure are distinguished by the different character and relation of the two premisses, according as these are either affirmative or negative, either universal or particular. Accordingly, there are four possible varieties of each, and sixteen possible modes or varieties of combinations between the two. Aristotle goes through most of the sixteen modes, and shows that in the first Figure there are only four among them that are legitimate, carrying with them a necessary conclusion. He shows, farther, that in all the four there are two conditions observed, and that both these conditions are indispensable in the First figure:--(1) The major proposition must be universal, either affirmative or negative; (2) The minor proposition must be affirmative, either universal or particular or indefinite. Such must be the character of the premisses, in the first Figure, wherever the conclusion is valid and necessary; and _vice versâ_, the conclusion will be valid and necessary, when such is the character of the premisses.[25]

[Footnote 25: Aristot. Analyt. Prior. I. iv. p. 26, b. 26, et sup.]

In regard to the four valid modes (_Barbara_, _Celarent_, _Darii_, _Ferio_, as we read in the scholastic Logic) Aristotle declares at once in general language that the conclusion follows necessarily; which he illustrates by setting down in alphabetical letters the skeleton of a syllogism in _Barbara_. If A is predicated of all B, and B of all C, A must necessarily be predicated of all C. But he does not justify it by any real example; he produces no special syllogism with real terms, and with a conclusion known beforehand to be true. He seems to think that the general doctrine will be accepted as evident without any such corroboration. He counts upon the learner's memory and phantasy for supplying, out of the past discourse of common life, propositions conforming to the conditions in which the symbolical letters have been placed, and for not supplying any contradictory examples. This might suffice for a treatise; but we may reasonably believe that Aristotle, when teaching in his school, would superadd illustrative examples; for the doctrine was then novel, and he is not unmindful of the errors into which learners often fall spontaneously.[26]

[Footnote 26: Analyt. Poster. I. xxiv. p. 85, b. 21.]

When he deals with the remaining or invalid modes of the First figure, his manner of showing their invalidity is different, and in itself somewhat curious. "If (he says) the major term is affirmed of all the middle, while the middle is denied of all the minor, no necessary consequence follows from such being the fact, nor will there be any syllogism of the two extremes; for it is equally possible, either that the major term may be affirmed of all the minor, or that it may be denied of all the minor; so that no conclusion, either universal or particular, is necessary in all cases."[27] Examples of such double possibility are then exhibited: first, of three terms arranged in two propositions (A and E), in which, from the terms specially chosen, the major happens to be truly affirmable of all the minor; so that the third proposition is an universal Affirmative:--

Major and }
} Animal is predicable of every Man;
Middle. }

Middle and }
} Man is not predicable of any Horse;
Minor }

Major and }
} Animal is predicable of every Horse.
Minor }

Next, a second example is set out with new terms, in which the major happens not to be truly predicable of any of the minor; thus exhibiting as third proposition an universal Negative:--

Major and }
} Animal is predicable of every Man;
Middle. }

Middle and }
} Man is not predicable of any Stone;
Minor }

Major and }
} Animal is not predicable of any Stone.
Minor }

Here we see that the full exposition of a syllogism is indicated with real terms common and familiar to every one; alphabetical symbols would not have sufficed, for the learner must himself recognize the one conclusion as true, the other as false. Hence we are taught that, after two premisses thus conditioned, if we venture to join together the major and minor so as to form a pretended conclusion, we may in some cases obtain a true proposition universally Affirmative, in other cases a true proposition universally Negative. Therefore (Aristotle argues) there is no one necessary conclusion, the same in all cases, derivable from such premisses; in other words, this mode of syllogism is invalid and proves nothing. He applies the like reasoning to all the other invalid modes of the first Figure; setting them aside in the same way, and producing examples wherein double and opposite conclusions (improperly so called), both true, are obtained in different cases from the like arrangement of premisses.

[Footnote 27: Analyt. Prior. I. iv. p. 26, a. 2, seq.]

This mode of reasoning plainly depends upon an appeal to prior experience. The validity or invalidity of each mode of the First figure is tested by applying it to different particular cases, each of which is familiar and known to the learner _aliunde_; in one case, the conjunction of the major and minor terms in the third proposition makes an universal Affirmative which he knows to be true; in another case, the like conjunction makes an universal Negative, which he also knows to be true; so that there is no one _necessary_ (_i.e._ no one uniform and trustworthy) conclusion derivable from such premisses.[28] In other words, these modes of the First figure are not valid or available in form; the negation being sufficiently proved by one single undisputed example.

[Footnote 28: Though M. Barthélemy St. Hilaire (note, p. 19) declares Aristotle's exposition to be a model of analysis, it appears to me that the grounds for disallowing this invalid mode of the First figure (A--E--A, or A--E--E) are not clearly set forth by Aristotle himself, while they are rendered still darker by some of his best commentators. Thus Waitz says (p. 381): "Per exempla allata probat (Aristoteles) quod demonstrare debebat ex ipsâ ratione quam singuli termini inter se habeant: est enim proprium artis logicæ, ut terminorum rationem cognoscat, dum res ignoret. Num de Caio prædicetur animal nescit, scit de Caio prædicari animal, si animal de homine et homo de Caio prædicetur."

This comment of Waitz appears to me founded in error. Aristotle had no means of shewing the invalidity of the mode A E in the First figure, except by an appeal to particular examples. The invalidity of the invalid modes, and the validity of the valid modes, rest alike upon this ultimate reference to examples of propositions known to be true or false, by prior experience of the learner. The valid modes are those which will stand this trial and verification; the invalid modes are those which will not stand it. Not till such verification has been made, is one warranted in generalizing the result, and enunciating a formula applicable to unknown particulars (rationem terminorum cognoscere, dum res ignoret). It was impossible for Aristotle to do what Waitz requires of him. I take the opposite ground, and regret that he did not set forth the fundamental test of appeal to example and experience, in a more emphatic and unmistakeable manner.

M. Barthélemy St. Hilaire (in the note to his translation, p. 14) does not lend any additional clearness, when he talks of the "_conclusion_" from the propositions A and E in the First figure. Julius Pacius says (p. 134): "Si tamen _conclusio_ dici debet, quæ non colligitur ex propositionibus," &c. Moreover, M. St. Hilaire (p. 19) slurs over the legitimate foundation, the appeal to experience, much as Aristotle himself does: "Puis prenant des exemples où la _conclusion est de toute évidence_, Aristote les applique successivement à chacune de ces combinaisons; celles qui donnent la _conclusion fournie d'ailleurs par le bon sens_, sont concluantes ou syllogistiques, les autres sont asyllogistiques."]

We are now introduced to the Second figure, in which each of the two premisses has the middle term as Predicate.[29] To give a legitimate conclusion in this figure, one or other of the premisses must be negative, and the major premiss must be universal; moreover no affirmative conclusions can ever be obtained in it--none but negative conclusions, universal or particular. In this Second figure too, Aristotle recognizes four valid modes; setting aside the other possible modes as invalid[30] (in the same way as he had done in the First figure), because the third proposition or conjunction of the major term with the minor, might in some cases be a true universal affirmative, in other cases a true universal negative. As to the third and fourth of the valid modes, he demonstrates them by assuming the contradictory of the conclusion, together with the major premiss, and then showing that these two premisses form a new syllogism, which leads to a conclusion contradicting the minor premiss. This method, called _Reductio ad Impossibile_, is here employed for the first time; and employed without being ushered in or defined, as if it were familiarly known.[31]

[Footnote 29: Analyt. Prior. I. v. p. 26, b. 34. As Aristotle enunciates a proposition by putting the predicate before the subject, he says that in this Second figure the middle term comes [Greek: prô=ton tê=| the/sei]. In the Third figure, for the same reason, he calls it [Greek: e)/schaton tê=| the/sei], vi. p. 28, a. 15.]

[Footnote 30: Analyt. Prior. I. v. p. 27, a. 18. In these invalid modes, Aristotle says there is no _syllogism_; therefore we cannot properly speak of a _conclusion_, but only of a third proposition, conjoining the major with the minor.]

[Footnote 31: Ibid. p. 27, a. 15, 26, seq. It is said to involve [Greek: u(po/thesis], p. 28, a. 7; to be [Greek: e)x u(pothe/seôs] xxiii. p. 41, a. 25; to be [Greek: tou= e)x u(pothe/seôs], as opposed to [Greek: deiktiko/s], xxiii. p. 40, b. 25.

M. B. St. Hilaire remarks justly, that Aristotle might be expected to define or explain what it is, on first mentioning it (note, p. 22).]

Lastly, we have the Third figure, wherein the middle term is the Subject in both premisses. Here one at least of the premisses must be universal, either affirmative or negative. But no universal conclusions can be obtained in this figure; all the conclusions are particular. Aristotle recognizes six legitimate modes; in all of which the conclusions are particular, four of them being affirmative, two negative. The other possible modes he sets aside as in the two preceding figures.[32]

[Footnote 32: Ibid. I. vi. p. 28, a. 10-p. 29, a. 18.]

But Aristotle assigns to the First figure a marked superiority as compared with the Second and Third. It is the only one that yields perfect syllogisms; those furnished by the other two are all imperfect. The cardinal principle of syllogistic proof, as he conceives it, is--That whatever can be affirmed or denied of a whole, can be affirmed or denied of any part thereof.[33] The major proposition affirms or denies something universally respecting a certain whole; the minor proposition declares a certain part to be included in that whole. To this principle the four modes of the First figure manifestly and unmistakably conform, without any transformation of their premisses. But in the other figures such conformity does not obviously appear, and must be demonstrated by reducing their syllogisms to the First figure; either ostensively by exposition of a particular case, and conversion of the premisses, or by _Reductio ad Impossibile_. Aristotle, accordingly, claims authority for the Second and Third figures only so far as they can be reduced to the First.[34] We must, however, observe that in this process of reduction no new evidence is taken in; the matter of evidence remains unchanged, and the form alone is altered, according to laws of logical conversion which Aristotle has already laid down and justified. Another ground of the superiority and perfection which he claims for the First figure, is, that it is the only one in which every variety of conclusion can be proved; and especially the only one in which the Universal Affirmative can be proved--the great aim of scientific research. Whereas, in the Second figure we can prove only _negative_ conclusions, universal or particular; and in the Third figure only _particular_ conclusions, affirmative or negative.[35]

[Footnote 33: Ibid. I. xli. p. 49, b. 37: [Greek: o(/lôs ga\r o(\ mê/ e)stin ô(s o(/lon pro\s me/ros kai\ a)/llo pro\s tou=to ô(s me/ros pro\s o(/lon, e)x ou)deno\s tô=n toiou/tôn dei/knusin o( deiknu/ôn, ô(/ste ou)de\ gi/netai sullogismo/s.]

He had before said this about the relation of the three terms in the Syllogism, I. iv. p. 25, b. 32: [Greek: o(/tan o(/roi trei=s ou(/tôs e)/chôsi pro\s a)llê/lous ô(/ste to\n e)/schaton e)n o(/lô| ei)=nai tô=| me/sô| kai\ to\n me/son e)n o(/lô| tô=| prô/tô| ê)\ ei)=nai ê)\ mê\ ei)=nai, a)na/gkê tô=n a)/krôn ei)=nai sullogismo\n te/leion] (_Dictum de Omni et Nullo_).]

[Footnote 34: Analyt. Prior. I. vii. p. 29, a. 30-b. 25.]

[Footnote 35: Ibid. I. iv. p. 26, b. 30, p. 27, a. 1, p. 28, a. 9, p. 29, a. 15. An admissible syllogism in the Second or Third figure is sometimes called [Greek: dunato\s] as opposed to [Greek: te/leios], p. 41, b. 33. Compare Kampe, Die Erkenntniss-Theorie des Aristoteles, p. 245, Leipzig, 1870.]

Such are the main principles of syllogistic inference and rules for syllogistic reasoning, as laid down by Aristotle. During the mediæval period, they were allowed to ramify into endless subtle technicalities, and to absorb the attention of teachers and studious men, long after the time when other useful branches of science and literature were pressing for attention. Through such prolonged monopoly--which Aristotle, among the most encyclopedical of all writers, never thought of claiming for them--they have become so discredited, that it is difficult to call back attention to them as they stood in the Aristotelian age. We have to remind the reader, again, that though language was then used with great ability for rhetorical and dialectical purposes, there existed as yet hardly any systematic or scientific study of it in either of these branches. The scheme and the terminology of any such science were alike unknown, and Aristotle was obliged to construct it himself from the foundation. The rhetorical and dialectical teaching as then given (he tells us) was mere unscientific routine, prescribing specimens of art to be committed to memory: respecting syllogism (or the conditions of legitimate deductive inference) absolutely nothing had been said.[36] Under these circumstances, his theory of names, notions, and propositions as employed for purposes of exposition and ratiocination, is a remarkable example of original inventive power. He had to work it out by patient and laborious research. No way was open to him except the diligent comparison and analysis of propositions. And though all students have now become familiar with the various classes of terms and propositions, together with their principal characteristics and relations, yet to frame and designate such classes for the first time without any precedent to follow, to determine for each the rules and conditions of logical convertibility, to put together the constituents of the Syllogism, with its graduation of Figures and difference of Modes, and with a selection, justified by reasons given, between the valid and the invalid modes--all this implies a high order of original systematizing genius, and must have required the most laborious and multiplied comparisons between propositions in detail.

[Footnote 36: Aristot. Sophist. Elench. p. 184, a. 1, b. 2: [Greek: dio/per tachei=a me\n a)/technos d' ê)=n ê( didaskali/a toi=s mantha/nousi par' au)tô=n; ou) ga\r te/chnên a)lla\ ta\ a)po\ tê=s te/chnês dido/ntes paideu/ein u(pela/mbanon . . . . _peri\ de\ tou= sullogi/zesthai pantelô=s ou)de\n ei)/chomen pro/teron a)/llo le/gein, a)ll' ê)\ tribê=| zêtou=ntes polu\n chro/non e)ponou=men_.]]

The preceding abridgment of Aristotle's exposition of the Syllogism applies only to propositions simply affirmative or simply negative. But Aristotle himself, as already remarked, complicates the exposition by putting the Modal propositions (Possible, Necessary) upon the same line as the above-mentioned Simple propositions. I have noticed, in dealing with the treatise De Interpretatione, the confusion that has arisen from thus elevating the Modals into a line of classification co-ordinate with propositions simply Assertory. In the Analytica, this confusion is still more sensibly felt, from the introduction of syllogisms in which one of the premisses is necessary, while the other is only possible. We may remark, however, that, in the Analytica, Aristotle is stricter in defining the Possible than he has been in the De Interpretatione; for he now disjoins the Possible altogether from the Necessary, making it equivalent to the Problematical (not merely _may be_, but _may be or may not be_).[37] In the middle, too, of his diffuse exposition of the Modals, he inserts one important remark, respecting universal propositions generally, which belongs quite as much to the preceding exposition about propositions simply assertory. He observes that universal propositions have nothing to do with time, present, past, or future; but are to be understood in a sense absolute and unqualified.[38]

[Footnote 37: Analyt. Prior. I. viii. p. 29, a. 32; xiii. p. 32, a. 20-36: [Greek: to\ ga\r a)nagkai=on o(mônu/môs e)nde/chesthai le/gomen]. In xiv. p. 33, b. 22, he excludes this equivocal meaning of [Greek: to\ e)ndecho/menon--dei= de\ to\ e)nde/chestha lamba/nein mê\ e)n toi=s a)nagkai/ois, a)lla\ kata\ to\n ei)rême/non diorismo/n.] See xiii. p. 32, a. 33, where [Greek: to\ e)nde/chesthai u(pa/rchein] is asserted to be equivalent to or convertible with [Greek: to\ e)nde/chesthai mê\ u(pa/rchein]; and xix. p. 38, a. 35: [Greek: to\ e)x a)na/gkês ou)k ê)=n _e)ndecho/menon_]. Theophrastus and Eudemus differed from Aristotle about his theory of the Modals in several points (Scholia ad Analyt. Priora, pp. 161, b. 30; 162, b. 23; 166, a. 12, b. 15, Brand.). Respecting the want of clearness in Aristotle about [Greek: to\ e)ndecho/menon], see Waitz's note **ad p. 32, b. 16. Moreover, he sometimes uses [Greek: u(pa/rchon] in the widest sense, including [Greek: e)ndecho/menon] and [Greek: a)nagkai=on], xxiii. p. 40, b. 24.]

[Footnote 38: Analyt. Prior. I. xv. p. 34, b. 7.]

Having finished with the Modals, Aristotle proceeds to lay it down, that all demonstration must fall under one or other of the three figures just described; and therefore that all may be reduced ultimately to the two first modes of the First figure. You cannot proceed a step with two terms only and one proposition only. You must have two propositions including three terms; the middle term occupying the place assigned to it in one or other of the three figures.[39] This is obviously true when you demonstrate by direct or ostensive syllogism; and it is no less true when you proceed by _Reductio ad Impossibile_. This last is one mode of syllogizing from an hypothesis or assumption:[40] your conclusion being disputed, you prove it indirectly, by assuming its contradictory to be true, and constructing a new syllogism by means of that contradictory together with a second premiss admitted to be true; the conclusion of this new syllogism being a proposition obviously false or known beforehand to be false. Your demonstration must be conducted by a regular syllogism, as it is when you proceed directly and ostensively. The difference is, that the conclusion which you obtain is not that which you wish ultimately to arrive at, but something notoriously false. But as this false conclusion arises from your assumption or hypothesis that the contradictory of the conclusion originally disputed was true, you have indirectly made out your case that this contradictory must have been false, and therefore that the conclusion originally disputed was true. All this, however, has been demonstration by regular syllogism, but starting from an hypothesis assumed and admitted as one of the premisses.[41]

[Footnote 39: Ibid. xxiii. p. 40, b. 20, p. 41, a. 4-20.]

[Footnote 40: Ibid. p. 40, b. 25: [Greek: e)/ti ê)\ deiktikô=s ê)\ e)x u(pothe/seôs; tou= d' _e)x u(pothe/seôs_ me/ros to\ dia\ tou= a)duna/tou.]]

[Footnote 41: Ibid. p. 41, b. 23: [Greek: pa/ntes ga\r oi( dia\ tou= a)duna/tou perai/nontes to\ me\n pseu=dos sullogi/zontai, to\ d' e)x a)rchê=s _e)x u(pothe/seôs_ deiknu/ousin, o(/tan a)du/nato/n ti sumbai/nê| tê=s a)ntipha/seôs tethei/sês.]

It deserves to be remarked that Aristotle uses the phrase [Greek: sullogismo\s _e)x u(pothe/seôs_], not [Greek: sullogismo\s u(pothetiko/s]. This bears upon the question as to his views upon what subsequently received the title of _hypothetical syllogisms_; a subject to which I shall advert in a future note.]

Aristotle here again enforces what he had before urged--that in every valid syllogism, one premiss at least must be affirmative, and one premiss at least must be universal. If the conclusion be universal, both premisses must be so likewise; if it be particular, one of the premisses may not be universal. But without one universal premiss at least, there can be no syllogistic proof. If you have a thesis to support, you cannot assume (or ask to be conceded to you) that very thesis, without committing _petitio principii,_ (_i.e._ _quæsiti_ or _probandi_); you must assume (or ask to have conceded to you) some universal proposition containing it and more besides; under which universal you may bring the subject of your thesis as a minor, and thus the premisses necessary for supporting it will be completed. Aristotle illustrates this by giving a demonstration that the angles at the base of an isosceles triangle are equal; justifying every step in the reasoning by an appeal to some universal proposition.[42]

[Footnote 42: Analyt. Prior. I. xxiv. p. 41, b. 6-31. The demonstration given (b. 13-22) is different from that which we read in Euclid, and is not easy to follow. It is more clearly explained by Waitz (p. 434) than either by Julius Pacius or by M. Barth. St. Hilaire (p. 108).]