Does Plato view a just person as someone who three parts of their soul are equal?
Or would reason win over appetite and spirit?
For Plato, all the higher virtues stem from letting the better part of your nature rule the worse part. So the just person --as is true for the wise person or the good person-- must have reason ruling over appetite and spirit. The idea that there should be some fairness or equality between the parts of the soul isn't really a Platonic idea.
The only virtue reserved for the spirited person is honor (or at least the love of honor), which accordingly must be lower in the Platonic hierarchy.
▻ TRIPARTITE SOUL
1 The just soul is tripartite : this you already know. The three parts (mere) are reason (to logistikon) by which we learn, know, calculate, the spirited part (to thumos) which is the emotional part and in particular the seat of anger and indignation and motivates self-defence, and the appetitive part (to epithumetikon) to which a miscellany of sense-based desires and cravings belong.
▻ ORDER IN THE SOUL
2 The just soul is more specifically the soul in which the three parts are correctly ordered, with reason (the rational part) directing the appetitive part with the assistance of the spirited part. In more detail reason rules; the spirited part is angry or indignant at whatever disturbs or threatens reason's rule; and the appetites are kept in check, exercised with a moderation induced or exerted by reason.
3 In correctly ordered souls different parts may provide the strongest motivation. In some souls, reason provides the strongest, in others the spirited part, and in yet others the appetitive part. But they all are governed by reason in the measure to which reason has been apportioned to them by nature. Reason in the souls in which the appetitive part provides the strongest motivation takes the controlling form of moderation. Where the spirited part most strongly motivates, reason controls in the form of courage or bravery as opposed to foolhardiness or quarrelsomeness.
▻ SUPREMACY OF REASON
4 The three parts are not 'equal' : and this is so in a dual sense. First, reason must be in overall control and, secondly, the spirited part is 'nobler' than the appetitive
▻ VIRTUES IN THE SOUL
5 The soul is in a just state when the correct order obtains. Under this correct ordering the soul will exhibit three virtues; and their harmony will constitute justice. When reason is in control, the soul has the virtue of wisdom (sophia). When the spirited part assists the rational part, it exhibits the virtue of courage (andreia). When the appetitive part exercises moderation, the soul has the virtue of temperance or moderation (sophrosune). Reason's control is justified in terms of its 'knowing ... what is best for each of the three elements [parts] and for the whole made up of them' (Rep. IV.442e : Lee, Penguin tr.).
6 When each part of the soul (psuche) harmonises with the others in this way, it performs its proper work (ergon). It fulfils its proper role or function in the economy of the soul.
▻ JUSTICE AS THE HEALTH OF THE SOUL
7 A soul which is correctly ordered, and therefore just, is also in its natural state. Justice (dikaiosune) is the health of the soul (Rep.IV 444c-e).
A correctly ordered soul is in other words in a condition of well-being or eudaimonia - of human flourishing. These terms, 'well-being' and 'flourishing' are only approximate to the Greek but they are more accurate than the traditional translation, 'happiness'.
▻ PROVISIONALITY OF TRIPARTITION
8 We should note a provisionality about Plato's view of the just soul in the Republic. The idea of the three parts of the soul had been prefigured mythically in the Phaedrus (246) and readily appears to be a firm analysis in Rep. IV but in Rep. X Plato expresses doubt whether the analysis of the soul as composed of parts would survive a deeper inquiry (611b).
▻ TRIPARTITION NOT PURE DIVISION
9 Plato recognises that each part of the soul will also have elements of the others. For instance, the rational part is motivated by a desire to know and will will under the education described in Rep. III & VIII lead them up from an epistemological state of mere ordinary belief (doxa) through the higher intellectual stages of dianoia (mathematical reasoning) to dialectic (noesis) at the end of which they will apprehend the ultimate source of reality, the Form of the Good (to auto to agathon). So reason is not purely separate from desire.
▻ CITY/ SOUL ANALOGY
10 Plato adopts the investigative device of looking first for justice in the city (polis). If we can define what makes or would make a city just we may have a clue to what makes or would make the individual just (Rep. II.368c-369a). We need to be careful about the sense in which an analogy holds here. It is not that justice in the city is or might be analogous to justice in the individual. Justice in the individual is identical with justice in the city. It is one and the same quality. The analogy will turn turn on the tripartition of classes in the city and the tripartition of parts of the soul. ('Class' is an anachronistic term in some respects, too suggestive of modern sociology; and Plato uses in any case a variety of terms which can be translated as e.g.'groups' or 'kinds'.
▻ JUSTICE : PSYCHIC AND VULGAR
11 It is not immediately obvious that Plato's psychic justice (justice as the correct ordering of the three parts of the psuche or soul) will equate with ordinary or vulgar justice. Ordinary justice requires us not to embezzle money or to steal - examples Plato gives. The details cannot be gone into here but Plato makes an attempt to show that there is at least a close resemblance between the requirements of psychic and ordinary justice (Rep.IV.441c-443b).
Any standard translation of Plato's Republic.
Terence Irwin, 'Plato's Ethics', Oxford : Oxford University Press, 1995.
B. Mitchell & J.R. Lucas, 'An Engagement with Plato's Republic', Aldershot : Ashgate, 2003.