discursive thought, intention, meaning
Διάνοια in Plato is the discursive thinking of the soul (ψυχή), occupying the place between opinion (δόξα) and knowledge proper (ἐπιστήμη). In the Republic it is placed on the third segment of the "divided line": the soul investigates here not the Forms themselves but mathematical objects, starting from hypotheses (ὑποθέσεις) that are accepted as clear without any account being given of them, and moves from them "to a conclusion," using sensible images as supports (Resp. VI 510b–511a). Such cognition is "clearer than opinion but dimmer than knowledge" (Resp. VII 533d): the soul has already turned away from what becomes, but has not yet reached the unhypothetical principle to which dialectic (διαλεκτική) leads.
Διάνοια is itself a form of inner speech: in the Theaetetus and the Sophist, Plato defines it as "the discussion that the soul holds with itself" without voice, in the form of questions and answers (Tht. 189e–190a; Soph. 263e). Διάνοια is thus the movement of λόγος within ψυχή, and its completion is opinion (δόξα) — the affirmation or denial of that voiceless conversation. With respect to νοῦς, διάνοια remains preparatory: it is the task of dialectic to lift its hypotheses and to convert the discursive movement into the immediate intellection (νόησις), which alone gives a full account of being (τὸ ὄν) and partakes of truth (ἀλήθεια).