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τοῦτό γε κτλ.: sc. τὸ λογίσασθαι κτλ. In λογιστικοῦ there is a glance at the etymology of the word: cf. VII 525 B note τούτῳ δὲ -- ὓμα περὶ ταὐτά κτλ. ‘And to this principle, when it has measured and signifies that some given objects are greater or less than or equal to some others, the contrary appearances are often presented in connexion with the same objects at the same time.’ τἀναντία is ‘the contrary,’ i.e. not ‘opposite appearances’ in general, but the contrary (in any given instance) of the impression formed without the aid of measurement: cf. τἀναντία in V 453 C, Euthyph. 6 A et al. The object A, for example, which is nearer and smaller, appears larger than B: but the λογιστικόν discovers after measurement that A is smaller and reports accordingly. We have thus two contrary coexisting impressions of the same object. One of the two impressions belongs to the λογιστικόν: does the other also? No, because, as shewn in IV 436 A—C, it is impossible for the same faculty to hold contrary opinions simultaneously in relation to the same objects. Hence the part of soul which δοξάζει παρὰ τὰ μέτρα is not the same as that which δοξάζει κατὰ τὰ μέτρα (i.e. the λογιστικόν). And as the part which believes λογισμός (viz. the λογιστικόν, cf. 604 D, 605 B) is the best, that which opposes it will be τῶν φαύλων τι ἐν ἡμῖν. J. and C. follow Schneider in translating τῷ αὐτῷ as instrumental (‘with the same faculty to form opposite opinions at the same time’); but it is much more natural to make it the dative with infinitive after ἀδύνατον, in view especially of τὸ—δοξάζον τῆς ψυχῆς immediately following. Rightly understood, IV 436 B is also in favour of taking τῷ αὐτῷ in this way. For other views of this difficult passage see App. II.
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