  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 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 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 ??t?fas??, 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. The terminology of Aristotle in the Analytica is to a certain extent different from that in the treatise De Interpretatione. The Enunciation (?p?fa???) appears under the new name of ???tas??, Proposition (in the literal sense) or Premiss; while, instead of Noun and Verb, we have the word Term (????), 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 ???? — 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 — d??st?a. (Analyt. Prior. I. xv. p. 35, a. 12. ste??t???? d?ast??t??, &c. See Alexander, Schol. pp. 145-146.) In the Topica Aristotle employs ???? in a very different sense — ????? ? t? t? ?? e??a? s?a???? (Topic. I. v. p. 101, b. 39) — hardly distinguished from ???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.). 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 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. 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. 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 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, 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 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. The rule for the simple conversion of the Universal Negative rests upon the same evidence of Induction, never contradicted. So also Aristot. Analyt. Poster. I. iv. p. 73, b. 32: t? ?a????? d? ?p???e? t?te, ?ta? ?p? t?? t????t?? ?a? p??t?? de?????ta?. 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. 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). 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 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
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:—
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. 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 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 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 He had before said this about the relation of the three terms in the Syllogism, I. iv. p. 25, b. 32: ?ta? ???? t?e?? ??t?? ???s? p??? ???????? ?ste t?? ?s?at?? ?? ??? e??a? t? ?s? ?a? t?? ?s?? ?? ??? t? p??t? ? e??a? ? ? e??a?, ?????? t?? ????? e??a? s??????s?? t??e??? (Dictum de Omni et Nullo). 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, 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, 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 It deserves to be remarked that Aristotle uses the phrase s??????s?? ?? ?p???se??, not s??????s?? ?p??et????. 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; Again, every demonstration is effected by two propositions (an even number) and by three terms (an odd number); though the same proposition may perhaps be demonstrable by more than one pair of premisses, or through more than one middle term;43 that is, by two or more distinct syllogisms. If there be more than three terms and two propositions, either the syllogism will no longer be one but several; or there must be particulars introduced for the purpose of obtaining an universal by induction; or something will be included, superfluous and not essential to the demonstration, perhaps for the purpose of concealing from the respondent the real inference meant.44 In the case (afterwards called Sorites) where the ultimate conclusion is obtained through several mean terms in continuous series, the number of terms will always exceed by one the number of propositions; but the numbers may be odd or even, according to circumstances. As terms are added, the total of intermediate conclusions, if drawn out in form, will come to be far greater than that of the terms or propositions, multiplying as it will do in an increasing ratio to them.45 It will be seen clearly from the foregoing remarks that there is a great difference between one thesis and another as to facility of attack or defence in Dialectic. If the thesis be an Universal Affirmative proposition, it can be demonstrated only in the First figure, and only by one combination of premisses; while, on the In the Analytica Priora, Aristotle proceeds to tell us that he contemplates not only theory, but also practice and art. The reader must be taught, not merely to understand the principles of Syllogism, but likewise where he can find the matter for constructing syllogisms readily, and how he can obtain the principles of demonstration pertinent to each thesis propounded.47 A thesis being propounded in appropriate terms, with subject and predicate, how are you the propounder to seek out arguments for its defence? In the first place, Aristotle reverts to the distinction already laid down at the beginning of the CategoriÆ.48 Individual things or persons are subjects only, never appearing as predicates — this is the lowest extremity of the logical scale: at the opposite extremity of the scale, there are the highest generalities, predicates only, and not subjects of any predication, though sometimes supposed to be such, as matters of dialectic discussion.49 Between the lowest and highest we have intermediate or graduate generalities, appearing sometimes as subjects, sometimes as predicates; and it is among these that the materials both of problems for debate, and of premisses for proof, are usually found.50 You must begin by putting down, along with the matter in hand itself, its definition and its propria; after that, its other predicates; next, those predicates which cannot belong to it; Thus, when the thesis to be maintained is an universal affirmative (e.g. A is predicable of all E), you will survey all the subjects to which A will apply as predicate, and all the predicates applying to E as subject. If these two lists coincide in any point, a middle term will be found for the construction of a good syllogism in the First figure. Let B represent the list of predicates belonging universally to A; D, the list of predicates which cannot belong to it; C, the list of subjects to which A pertains universally as predicate. Likewise, let F represent the The way just pointed out is the only way towards obtaining a suitable middle term. If, for example, you find some predicate applicable both to A and E, this will not conduct you to a valid syllogism; you will only obtain a syllogism in the Second figure with two affirmative premisses, which will not warrant any conclusion. Or if you find some predicate which cannot belong either to A or to E, this again will only give you a syllogism in In the survey prescribed, nothing is gained by looking out for predicates (of A and E) which are different or opposite: we must collect such as are identical, since our purpose is to obtain from them a suitable middle term, which must be the same in both premisses. It is true that if the list B (containing the predicates universally belonging to A) and the list F (containing the predicates universally belonging to E) are incompatible or contrary to each other, you will arrive at a syllogism proving that no A can belong to E. But this syllogism will proceed, not so much from the fact that B and F are incompatible, as from the other fact, distinct though correlative, that B will to a certain extent coincide with H (the list of predicates which cannot belong to E). The middle term and the syllogism constituted thereby, is derived from the coincidence between B and H, not from the opposition between B and F. Those who derive it from the latter, overlook or disregard the real source, and adopt a point of view merely incidental and irrelevant.58 The precept here delivered — That in order to obtain middle terms and good syllogisms, you must study and collect both the predicates and the subjects of the two terms of your thesis — Aristotle declares to be equally applicable to all demonstration, whether direct or by way of Reductio ad Impossibile. In both the process of demonstration is the same — involving two premisses, three terms, and one of the three a suitable middle term. The only difference is, that in the direct demonstration, both premisses are propounded as true, while in the Reductio ad Impossibile, one of the premisses is assumed as true though known to be false, and the conclusion also.59 In the other cases of hypothetical syllogism your attention must be directed, not to the original quÆsitum, but to the condition annexed thereto; yet the search for predicates, subjects, and a middle term, must be conducted in the same manner.60 Sometimes, by the help Aristotle concludes by saying that syllogisms from an hypothesis ought to be reviewed and classified into varieties — ?p?s???as?a? d? de? ?a? d?e?e?? p?sa??? ?? ?? ?p???se?? (b. 20). But it is doubtful whether he himself ever executed this classification. It was done in the Analytica of his successor Theophrastus (Schol. p. 179, a. 6, 24). Compare the note of M. BarthÉlemy St. Hilaire, p. 140. Aristotle has now shown a method of procedure common to all investigations and proper for the solution of all problems, wherever soluble. He has shown, first, all the conditions and varieties of probative Syllogism, two premisses and three terms, with the place required for the middle term in each of the three figures; next, the quarter in which we are to look for all the materials necessary or suitable for constructing valid syllogisms. Having the two terms of the thesis given, we must study the predicates and subjects belonging to both, and must provide a large list of them; out of which list we must make selection according to the purpose of the moment. Our selection will be different, according as we wish to prove or to refute, and according as the conclusion that we wish to prove is an universal or a particular. The lesson here given will be most useful in teaching the reasoner to confine his attention to the sort of materials really promising, so that he may avoid wasting his time upon such as are irrelevant.62 This method of procedure is alike applicable to demonstration in Philosophy or in any of the special sciences,63 and to debate Julius Pacius (p. 257) remarks upon the word d?a?e??a???? as indicating that Aristotle, while alluding to special sciences distinguishable from philosophy on one side, and from dialectic on the other, had in view geometrical demonstrations. What Aristotle says here — of astronomical observation and experience as furnishing the basis for astronomical science — stands in marked contrast with Plato, who rejects this basis, and puts aside, with a sort of contempt, astronomical observation (Republic, vii. pp. 530-531); treating acoustics also in a similar way. Compare Aristot. Metaphys. ?. p. 1073, a. 6, seq., with the commentary of Bonitz, p. 506. Respecting the word ?st???a — investigation and record of matters of fact — the first sentence of Herodotus may be compared with Aristotle, Histor. Animal. p. 491, a. 12; also p. 757, b. 35; Rhetoric. p. 1359, b. 32. For the fuller development of these important principles, the reader is referred to the treatise on Dialectic, entitled Topica, which we shall come to in a future chapter. There is nothing in all Aristotle’s writings more remarkable than the testimony here afforded, how completely he considered all the generalities of demonstrative science and deductive reasoning to rest altogether on experience and inductive observation. We are next introduced to a comparison between the syllogistic method, as above described and systematized, and the process called logical Division into genera and species; a process much relied upon by other philosophers, and especially by Plato. This logical Division, according to Aristotle, is a Thus, they take the subject man, and propose to prove that man is mortal. They begin by laying down that man is an animal, and that every animal is either mortal or immortal. Here, the most universal term, animal, is selected as middle or as medium of proof; while after all, the conclusion demonstrated is, not that man is mortal, but that man is either mortal or immortal. The position that man is mortal, is assumed but not proved.69 Moreover, by this method of logical division, all the steps are affirmative and none negative; there cannot be any refutation of error. Nor can any proof be given thus respecting genus, or proprium, or accidens; the genus is assumed, and the method proceeds from thence to species and differentia. No doubtful matter can be settled, and no unknown point elucidated by this method; nothing can be done except to arrange in a certain order what is already ascertained and unquestionable. To many investigations, accordingly, the method is altogether inapplicable; while even where it is applicable, it leads to no useful conclusion.70 We now come to that which Aristotle indicates as the third section of this First Book of the Analytica Priora. In the first section he explained the construction and constituents of Syllogism, the varieties of figure and mode, and the conditions indispensable to a valid conclusion. In the second section he tells us where we are to look for the premisses of syllogisms, and how we may obtain a stock of materials, apt and ready for use when required. There remains one more task to complete his plan — that he should teach the manner of reducing argumentation as it actually occurs (often invalid, and even when When a piece of reasoning is before us, we must first try to disengage the two syllogistic premisses (which are more easily disengaged than the three terms), and note which of them is universal or particular. The reasoner, however, may not have set out both of them clearly: sometimes he will leave out the major, sometimes the minor, and sometimes, even when enunciating both of them, he will join with them irrelevant matter. In either of these cases we must ourselves supply what is wanting and strike out the irrelevant. Without this aid, reduction to regular syllogism is impracticable; but it is not always easy to see what the exact deficiency is. Sometimes indeed the conclusion may follow necessarily from what is implied in the premisses, while yet the premisses themselves do not form a correct syllogism; for though every such syllogism carries with it necessity, there may be necessity without a syllogism. In the process of reduction, we must first disengage and set down the two premisses, then the three terms; out of which three, that one which appears twice will be the middle term. If we do not find one term twice repeated, we have got no middle and no real syllogism. Whether the syllogism when obtained will be in the first, second, or third figure, will depend upon the place of the middle term in the two premisses. We know by the nature of the conclusion which of the three figures to look for, since we have already seen what conclusions can be demonstrated in each.73 Sometimes we may get premisses which look like those of a true syllogism, but are not so in reality; the major proposition ought to be an universal, but it may happen to be only indefinite, and the syllogism will not in all cases be valid; yet the distinction between the two often passes unnoticed.74 Another source M. B. St. Hilaire observes in his note (p. 155): “L’erreur vient uniquement de ce qu’on confond l’universel et l’indeterminÉ sÉparÉs par une nuance trÈs faible d’expression, qu’on ne doit pas cependant negliger.” Julius Pacius (p. 264) gives the same explanation at greater length; but the example chosen by Aristotle (? ???st????? ?st? d?a???t?? ???st?????) appears open to other objections besides. Again, we must not suppose that we can always find one distinct and separate name belonging to each term. Sometimes one or all of the three terms can only be expressed by an entire phrase or proposition. In such cases it is very difficult to reduce the reasoning into regular syllogism. We may even be deceived into fancying that there are syllogisms without any middle term at all, because there is no single word to express it. For example, let A represent equal to two right angles; B, triangle; C, isosceles. Then we have a regular syllogism, with an explicit and single-worded middle term; A belongs first to B, and then to C through B as middle term (triangle). But how do we know that A belongs to B? We know it by demonstration; for it is a demonstrable truth that every triangle has its three angles equal to two right angles. Yet there is no other more general truth about triangles from which it is a deduction; it belongs to the triangle per se, and follows from the fundamental properties of the figure.77 There is, however, a middle term in the demonstration, though it is not single-worded and explicit; it is a declaratory proposition or a fact. We must not suppose that there can be any demonstration without a middle term, either single-worded or many-worded.
For every syllogism in which the conclusion is qualified, the terms must be set out accordingly.80 We are permitted, and it is often convenient, to exchange one phrase or term for another of equivalent signification, and also one word against any equivalent phrase. By doing this, we often facilitate the setting out of the terms. We must carefully There is no fear of our being misled by setting out a particular case for the purpose of the general demonstration; for we never make reference to the specialties of the particular case, but deal with it as the geometer deals with the diagram that he draws. He calls the line A B, straight, a foot long, and without breadth, but he does not draw any conclusion from these assumptions. All that syllogistic demonstration either requires or employs, is, terms that are related to each other either as whole to part or as part to whole. Without this, no demonstration can be made: the exposition of the particular case is intended as an appeal to the senses, for facilitating the march of the student, but is not essential to demonstration.83 This chapter is a very remarkable statement of the Nominalistic doctrine; perceiving or conceiving all the real specialties of a particular case, but attending to, or reasoning upon, only a portion of them. Plato treats it as a mark of the inferior scientific value of Geometry, as compared with true and pure Dialectic, that the geometer cannot demonstrate through Ideas and Universals alone, but is compelled to help himself by visible particular diagrams or illustrations. (Plato, Repub. vi. pp. 510-511, vii. p. 533, C.) Aristotle reminds us once more of what he had before said, that in the Second and Third figures, not all varieties of conclusion are possible, but only some varieties; accordingly, when we are reducing a piece of reasoning to the syllogistic form, the nature of the conclusion will inform us which of the three No syllogisms from an Hypothesis, however, are reducible to any of the three figures; for they are not proved by syllogism alone: they require besides an extra-syllogistic assumption granted or understood between speaker and hearer. Suppose an hypothetical proposition given, with antecedent and consequent: you may perhaps prove or refute by syllogism either the antecedent separately, or the consequent separately, or both of them separately; but you cannot directly either prove or refute by syllogism the conjunction of the two asserted in the hypothetical. The speaker must ascertain beforehand that this will be granted to him; otherwise he cannot proceed.86 The same is true about the procedure by Reductio ad Absurdum, which involves an hypothesis over and above the syllogism. In employing such Reductio ad Absurdum, you prove syllogistically a certain conclusion from certain premisses; but the conclusion is manifestly false; therefore, one at least of the premisses from which it follows must be false also. But if this reasoning is to have force, the hearer must know aliunde that the conclusion is false; your syllogism has not shown it to be false, but has shown it to be hypothetically true; and unless the hearer is prepared to grant the conclusion to be false, your purpose is not attained. Sometimes he will grant it without being expressly asked, when the falsity is glaring: e.g. you prove that the diagonal of a square is incommensurable with the side, because if it were taken as commensurable, an odd number might be shown to be equal to an even number. Few disputants will hesitate to grant that this conclusion is false, and therefore that its contradictory is true; yet this last (viz. that the contradictory is true) has not been proved syllogistically; you M. BarthÉlemy St. Hilaire remarks in the note to his translation of the Analytica Priora (p. 178): “Ce chapitre suffit À prouver qu’Aristote a distinguÉ trÈs-nettement les syllogismes par l’absurde, des syllogismes hypothÉtiques. Cette derniÈre dÉnomination est tout À fait pour lui ce qu’elle est pour nous.” Of these two statements, I think the latter is more than we can venture to affirm, considering that the general survey of hypothetical syllogisms, which Aristotle intended to draw up, either never was really completed, or at least has perished: the former appears to me incorrect. Aristotle decidedly reckons the Reductio ad Impossibile among hypothetical proofs. But he understands by Reductio ad Impossibile something rather wider than what the moderns understand by it. It now means only, that you take the contradictory of the conclusion together with one of the premisses, and by means of these two demonstrate a conclusion contradictory or contrary to the other premiss. But Aristotle understood by it this, and something more besides, namely, whenever, by taking the contradictory of the conclusion, together with some other incontestable premiss, you demonstrate, by means of the two, some new conclusion notoriously false. What I here say, is illustrated by the very example which he gives in this chapter. The incommensurability of the diagonal (with the side of the square) is demonstrated by Reductio ad Impossibile; because if it be supposed commensurable, you may demonstrate that an odd number is equal to an even number; a conclusion which every one will declare to be inadmissible, but which is not the contradictory of either of the premisses whereby the true proposition was demonstrated. Here Aristotle expressly reserves for separate treatment the general subject of Syllogisms from Hypothesis.88 Syllogisms from Hypothesis were many and various, and Aristotle intended to treat them in a future treatise; but all that concerns the present treatise, in his opinion, is, to show that none of them can be reduced to the three Figures. Among the Syllogisms from Hypothesis, two varieties recognized by Aristotle (besides ?? d?? t?? ?d???t??) were ?? ?at? et?????? and ?? ?at? p???t?ta. The same proposition which Aristotle entitles ?at? et??????, was afterwards designated by the Stoics ?at? p??s????? (Alexander ap. Schol. p. 178, b. 6-24). It seems that Aristotle never realized this intended future treatise on Hypothetical Syllogisms; at least Alexander did not know it. The subject was handled more at large by Theophrastus and EudÊmus after Aristotle (Schol. p. 184, b. 45. Br.; Boethius, De Syllog. Hypothetico, pp. 606-607); and was still farther expanded by Chrysippus and the Stoics. Compare Prantl, Geschichte der Logik, I. pp. 295, 377, seq. He treats the Hypothetical Syllogism as having no logical value, and commends Aristotle for declining to develop or formulate it; while Ritter (Gesch. Phil. iii. p. 93), and, to a certain extent, Ueberweg (System der Logik, sect. 121, p. 326), consider this to be a defect in Aristotle. In the last chapter of the first book of the Analytica Priora, Aristotle returns to the point which we have already considered in the treatise De Interpretatione, viz. what is really a negative proposition; and how the adverb of negation must be placed in order to constitute one. We must place this adverb immediately before the copula and in conjunction with the copula: we must not place it after the copula and in conjunction with the predicate; for, if we do so, the proposition resulting will not be negative but affirmative (?? eta??se??, by transposition, according
No. 1 is affirmative; No. 3 is affirmative (?? eta??se??); Nos. 2 and 4 are negative. Wherever No. 1 is predicable, No. 4 will be predicable also; wherever No. 3 is predicable, No. 2 will be predicable also — but in neither case vice versÂ.89 Mistakes often flow from incorrectly setting out the two contradictories. |
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