The irrational (ἄλογον) — that which lacks λόγος, that is, reason and rational ground. Plato thus designates the non-rational parts of the soul (ψυχή) — the appetitive (ἐπιθυμητικόν) and the spirited (θυμοειδές), which do not themselves reason and must obey the calculation of reason, in opposition to the rational principle (λογιστικόν) (Resp. IV 439d–441a; X 604d–605b). In the "Timaeus," ἄλογον denotes the "mortal," irrational kind of soul, separated from the immortal λογιστικόν (Tim. 69c–d, 77b).
In knowledge, ἄλογον is that of which no λόγος (account) can be given, and which therefore escapes knowing. In Socrates' "dream," the primary elements of things turn out to be "ἄλογα καὶ ἄγνωστα": of themselves no account can be given, they are accessible only to sensation, and only the syllables composed from them admit of a λόγος (Tht. 201d–202b). This binds ἄλογον to the demand of διδόναι λόγον — to give an account — as the condition of knowledge (ἐπιστήμη) and of participation in truth (ἀλήθεια) (Resp. VII 534b). In mathematics, ἄλογος means the irrational, the incommensurable — without ratio or measure (to be distinguished from ἄρρητον, the unutterable). Thus ἄλογον outlines λόγος by contrast — as reason, account, and measure — marking the boundary of the rational.