LOGIC 787
from having a higher value than mere likelihood, but does not affect the chain of inference, which proceeds on assumptions identical 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 deducible 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 (
Greek characters), 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, (Greek characters). Further, as a method, (Greek characters) is characterized by the nature of its starting point, and of the connecting link involved, as well as by the peculiarity of its result. It rests upon the first, simplest, best known, unprovable elements of thought, the (Greek characters) which are not themselves in the strict sense matters of apodictic science, which are (Greek characters). In all the intermediate processes of scientific proof there is involved generally this dependence upon previously established principles, and, when apodictic is taken in its ultimate abstraction, these previously established 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 possible without the universalizing element, the (Greek characters), and knowledge in its essence is syllogistic.[1] The conclusion of the syllogism in which essential attributes are attached to a subject is the concretion 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 (Greek characters), which presents itself as the characteristic feature of (Greek characters), the relation of fundamental and universal in things on which the possibility of (Greek characters) 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 philosophy as a whole, a general treatment of them is indispensable. First then of (Greek characters), 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 (Greek characters); on the other hand, it is the root or ground of the empirical universal – it is (Greek characters),[3] that which is in, for, and through itself, the essential. Now the essential, (Greek characters), 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 that 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 determined 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 realization 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 relating to a given class, with insight into the necessary character of the predication in each case, have been gathered up, then the (Greek characters) of knowledge in respect to that class has been realized."[4]
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 ((Greek characters)). 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 ((Greek characters)) 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, (Greek characters).[5] Thus the range of mathematical proof extends from the (Greek characters), the original definitions, which at the same time assume the existence of the things defined, through the determinations (Greek characters) to the qualities ((Greek characters)), which can be shown to attach to their subjects, to be in a sense (Greek characters), while a continuous series of middle notions, concerning which there cannot be much ambiguity, effects the transition. Moreover, in mathematical science, one can see with the utmost evidence the correlation of reason and sense, which will presently appear as a fundamental factor in Aristotle's general theory of knowledge. The (Greek characters) are not to be conceived as innate or as possessed before experience. They are seen or envisaged, intuited in perception by (Greek characters), and induction here as elsewhere is the process by which perceptions are gathered together for the reflective and intuiting action of (Greek characters). 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.[6] In mathematical development, that which is potentially contained in the (Greek characters) 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 mathematical elements becomes actual through the process of constructive thought.[7]
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.[8]
15. The process of (Greek characters) generally and of mathematical demonstration 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 relation to the universal, and as bearing upon the particular.[9] 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.[10] 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. Universals as such are of no avail either as explanations of knowledge or as grounds of existence. Particulars as such are infinite, indefinite,
2 See specially Anal. Pr., 67a, 39 sq., and compare the elaborate note of Kampe,
Erkenntnisstheorie des Aris., p. 220 (also p. 84). Grote (Aristotle, i. p. 263a)
remarks: "Complete cognition ((Greek characters), according to the view here set forth) consists of one mental act corresponding to the major premiss, another corresponding to the minor, and a third including both the two in conscious juxtaposition. The third implies both the first and the second." The connexion between this and the Aristotelian doctrines of (Greek characters) in its relation to (Greek characters) will not escape attention.
3 Anal. Post., 73b, 26, (Greek characters) See Index Aristotelicus, s. v., pp. 356-57, and on (Greek characters) compare Heyder, Method. d. Arist., 310 n., and Bonitz, Com. in Met., pp. 265-66. On the distinction between (Greek characters) and (Greek characters), see Bonitz, Com. in Met., p. 299, 300; Zeller. Ph. d. Gr., ii. 1, p. 205, 206.
4 Cf. Prantl, Ges. d. Logik, i. 121, 122, who has rightly placed the function of (Greek characters) in the foreground.
7 Cf. the passage from De Memor., p. 450. quoted by Brandis, Aristoteles, p. 1133, – (Greek characters). Cf. also Met., vii. 10 and 11. Aristotle's view strongly resembles, in this point at least, that of Kant.
- ↑ Cf. Topics, pp. 164a, 10.
- ↑ 2
- ↑ 3
- ↑ Prantl, i. 126.
- ↑ Anal. Post., i. 24, 86a, 2.
- ↑ 7
- ↑ See Metaph., ix. c. 9, p. 1051a. Some interesting remarks on the process of mathematical construction and its relation to syllogistic proof will be. found in Ueberweg's System der Logik, § 101. p. 273.
- ↑ 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 (Aristoteles, pp. 135-39. and Aristot. Lehrgebäude, 7-11), and Eucken (Methode d. Arist. Forschung, pp. 56-66)
- ↑ Cf. specially Anal. Pr., ii. 21.
- ↑ This is excellently put by Eucken, op. cit., pp. 44-55.
