LOGIC 787 from having a higher value than mere likelihood, but does not ati ect the chain of inference, which proceeds on assumptions iden tical with these involved in apodictic. Aristotle is chary of any examples of dialectic syllogism, and indeed, if one considers that all forms of modality are investigated in the general analysis of syllogism, it becomes difficult to see what specially distinguishes dialectic inference. It is not to be denied, however, that the investigation of the grounds for the coexistence of dialectic and apodictic is incomplete in Aristotle, as it confessedly is in Plato. Unless, then, it can be shown beyond possibility of question that Aristotle does lay down purely formal rules for syllogism, rules de- ducible simply from the fundamental axiom of thought and the evidence on which such a view is based will be examined later we do not obtain much light from the opposition between dialectic and apodictic. More important results, however, are gained when we consider the Aristotelian doctrine of genuine knowledge, of d7rd5ei|is, for, among the numerous elements that here fall to be noted, some are of quite general import, and apply to the whole process of the formation of knowledge. 13. Apodictic knowledge generally is definable through the special marks of its content. It deals with the universal and necessary, that which is now and always, that which cannot be other than it is, that which is what it is simply through its own nature. It is the expression of the true universal in thought and things, -rb KaOoov. Further, as a method, dird5ei|is is characterized by the nature of its starting point, and of the connecting link in volved, as well as by the peculiarity of its result. It rests upon the first, simplest, best known, unprovable elements of thought, the jrpuTo. Kal a,uf<ra, which are not themselves in the strict sense matters of apodictic science, which are avawoSfiKra. In all the inter mediate processes of scientific proof there is involved generally this dependence upon previously established principles, and, when apo dictic is taken in its ultimate abstraction, these previously estab lished principles are seen to be the prior, ultimate elements, assumptions in thought about things, as one may provisionally describe them. The peculiar connexion involved is simply what we understand by the principle, of syllogism. No syllogism is pos sible without the universalizing element, the Kad6ov, and know ledge in its essence is syllogistic. 1 The conclusion of the syllogism in which essential attributes are attached to a subject is the con cretion or closing together of the two aspects of all thought and being, the universal and particular. 2 The fuller explanation of apodictic thus refers us to three points of extreme importance in the Aristotelian theory of knowledge, the precise nature of the Ka06ov, which presents itself as the character istic feature of a.ir65fiis, the relation of fundamental and universal in things on which the possibility of airoSet^ts is founded, and the forms of thought through which the universal and particular factors are subjectively realized. The three are most closely connected, and as they involve the main difficulties of the Aristotelian philo sophy as a whole, a general treatment of them is indispensable. First then of -rti Ka6oov, the characteristic term in the explanation of knowledge. This notion is essentially double-sided. On the one side it is the universal of empirical knowledge, the generic or class universal it is rb Kara iravrds ; ou the other hand, it is the root or ground of the empirical universal it is rb KO.& avrb KO.I fj avro, s that which is in, for, and through itself, the essential. Now the essential, Kaff abro, is, in the first place, either that which enters into the being and notion of a thing as a necessary prerequisite (for example, line is a necessary element in the being and notion of triangle), or that which is the necessary basis of an attribute (e.g., line in reference to straight and curved}, or in the second place tliat which is as subject only and not as predicate, or finally that which is per se the cause or ground of a fact or event. 4 Thus the function of thought (of apodictic) is the exposition with reference to a deter mined class of objects of all that necessarily inheres in them, on account of the elementary factors which determine their existence and nature. Real things, individual objects, are the basis of all knowledge, but in these individuals the elementary parts, causally connected, and leading to ulterior consequences, form the general element about which there may be demonstrative science. Thought which operates upon them does so, as we have already seen, under the peculiar restriction of its very nature, as the subjective reali- 1 Cf. Topics, pp. 164a 10. 2 See specially Anal. />/-., 67a, 39 sq., and compare the elaborate note of Kampp, Erkenntnisstheorie (Its Aris.,i>. 220 (also p. 84). Grote (Aristotle, i. p. 263a) remarks : " Complete cognition (TO tvepyeiv, according to the view here set forth) consists of one mental act corresponding to the major premiss, another corre sponding to the minor, and a third including both the two in conscious juxta position. The third implies both the first and the second." The connexion between this ami the Aristotelian doctrines of vovs in its relation to ai<r#j)<ns will not escape attention. 3 Anal. Post., 73b. l (J, Ka06ov 6e Ae ycu o av Kara Travro<; re vrrdpriicai Ka.6 avrb Kal rj avro See Jn<le.r Aristotelicut, s. r., pp. 356-57, and on Kad avro compare Ileyder, 3MhJ. J. Arift., 310 n., and Bonitz, Com. in Mtt., pp. 265-66. Ou the distinction between Kad6 v and yeVo?, see Bonitz, Com. in Met., p. 299, 300: Zeller. Ph. d. Gr., ii. 1, p. 20.-,. 2<)6. 4 Cf. Prantl, Ges. d. Loyii; i. 121, 122, who has rightly placed the function of KaSoAov in the foreground. zation of the notion of things, and the principles expressing this restriction, the logical axioms, maybe appealed to if demonstration be opposed groundlessly, but these axioms do not enter into the process of demonstration. " When the apodictic process has attained its end, that is, when all the universal propositions relat ing to a given class, with insight iuto the necessary character of the predication in each case, have been gathered up, then the Ka.06ov of knowledge in respect to that class has been realized." 6 14. Probably the example of apodictic which Aristotle bears chiefly in mind is mathematical science, and in his treatment of the characteristic marks of this doctrine most of the peculiarities of apodictic occur. In mathematical science abstraction is made of the material qualities of the things considered, of those qualities which give to them a place as physical facts, but the abstracta are not to be conceived as entities, self-existing. They are not even to be conceived as existing only in mind, as ideal types ; they truly exist in things, but are considered separately (e| o4>a(peVea>s). The first principles of mathematical science are few and definite, and the procedure is continuously from the simple and absolutely more known to the concrete and relatively more known. As in proof generally, so in mathematical demonstration, an essential quality ((Tv/j.Q(priK6s naff aurb) may be proved of a subject, and yet such quality may be still accidental, i.e., not predicated of the subject on account of its generic constituent marks, but capable of being deduced from the constituent mark of that which enters into the subject, as, e.g. , a given figure s exterior angles are equal to four right angles. Why ? Because it is an isosceles triangle. Why has an isosceles this property ? Because it is a triangle. Why has a triangle ? Because it is a rectilineal figure. If this reason is ultimate, it completes our knowledge, /ecu Ka66ov 8e roVe. 6 Thus the range of mathematical proof extends from the IT puna, the original definitions, which at the same time assume the existence of the things defined, through the determinations nad avrd to the qualities (crvuflepriKOTa), which can be shown to attacli to their subjects, to be in a sense xaff aura, while a continuous series of middle notions, concerning which there cannot be much ambiguity, effects the transi tion. Moreover, in mathematical science, one can see with the utmost evidence the correlation of reason and sense, which will pre sently appear as a fundamental factor in Aristotle s general theory of knowledge. The irpSira are not to be conceived as innate or as possessed before experience. They are seen or envisaged, intuited in perception by vovs, and induction here as elsewhere is the pro cess by which perceptions are gathered together for the reflective and intuiting action of vovs. In the mathematical individual, more evidently than in any other case, is visible the union of thought and sense. The demonstration which employs a diagram does not turn upon any properties of the diagram which are there for sense only, not for reason, but upon the general elementary relations contemplated in thought. 7 In mathematical development, that which is potentially contained in the Z
vorjTr) on which
mathematical thinking operates is brought forward into actuality by the constructive processes through which the proof is mediated, and the potential knowledge contained in the intuition of mathe matical elements becomes actual through the process of construc tive thought. 8 Finally, the relation of pure mathematical reasoning to that found in sciences generically one with mathematics, e.g. , optics, astronomy, harmonics, &c. , furnishes an interesting example of the relation between reasoning based on fact and on causal ground. 9 15. The process of &ir65eiis generally and of mathematical demon stration in particular has brought into clear light the prominent characteristic of knowledge according to the Aristotelian view. Knowledge must always be regarded from two sides, as having rela tion to the universal, and as bearing upon the particular. 10 It is in itself the union of the general and the particular, of the universal and the individual. This fundamental notion of knowledge is not only the integral element in the Aristotelian theory of science, but also the guiding principle in his scientific method. 11 In all cases we require to keep in mind the necessary correlation of the particular facts and the general grounds, the multiplicity of effects and the unity of cause. The one element is not apart from the other. Universal^ as such are of no avail either as explanations of knowledge or as grounds of existence. Particulars as such are infinite, indefinite, 5 I rantl, i. 126. Anal. Post., 1. 24, 86s, 2. 7 Cf. the passage from De AJemor., p. 450. quoted by Brandis, Ariftotele$, p. 1133, (rvufiaivfi yap rb avro Tratfos eV ria votiv ical iv ria Siaypatfxiv exec -re yap bvOfv rrpo^xptafievoi, rta rb TTOVOV wpKr^fvov eu ai rb rpiyiovov, OM<D? ypa^ojuci uipicriifvov Kara rb rtoaov Kal 6 vouiv ajo-avrios, KO.V fir) rrotrbv i-ojT, riOcrai rrpb omj-driav iroaor, voei S ov 7? Jror6r. av Se ri tftvcru; jj riar TTOCTCOI , ndpiaTOf Sf, riOero.!. iifv Ttotrov wpiafiivov, vot I Se jj rtoaov i^ovov. Cf. also Met., vii. 10 and 11. Aristotle s view strongly resembles, in this point at least, that of Kant. 8 See Afetaph., ix. c. 9, p. 1051s. Some interesting remarks on the process of mathematical construction and its relation to syllogistic proof will be. found in Ueberweg s Si/stem der Logik, J 101. p. 273. 9 See generally Anal. Post., chap. 13. Of Aristotle s views on mathematics the best expositions seem to be those of Biese (Ph. d. Arist., ii. 216-34), Brandis (Anttoteles, pp. 13.5-39. and Aristot. Lehrgebiiude, 7-11), and Eucken (Af -Jtiodt ^,. Arift. Forschung, pp. 56-66) 10 Cf. specially Anal. Pr., ii. 21. 11 This is excellently put by Eucken, op. tit., pp. 44-55.
