Does Aristotle have a clear stance on the question of the immortality of the soul?
The human intellect / "intellective soul" is immortal
As St. Thomas Aquinas writes in Sententia De anima a. 14 co., Aristotle
proves in the De anima III, 4, 429b3 [that] intellection is not an act executed by any bodily organ.
ἀλλ' ὁ νοῦς ὅταν τι νοήσῃ σφόδρα νοητόν, οὐχ ἧττον νοεῖ τὰ ὑποδεέστερα, ἀλλὰ καὶ μᾶλλον· τὸ μὲν γὰρ αἰσθητικὸν οὐκ ἄνευ σώματος, ὁ δὲ χωριστός.
But when the intellect understands something highly intelligible, it does not understand what is inferior to these less than before, but more so. For whereas the sensitive faculty is not found apart from the body, the intellect is separate.
and Aristotle proves
that the intellect is something divine and everlasting
χωρισθεὶς δ' ἐστὶ μόνον τοῦθ' ὅπερ ἐστί, καὶ τοῦτο μόνον ἀθάνατον καὶ ἀΐδιον
Only separated, however, is it what it really is. And this alone is immortal and perpetual.
J. A. Smith's translation:
When mind is set free from its present conditions it appears as just what it is and nothing more: this alone is immortal and eternal
One soul in man
Plato held that there are several souls in man:
- the intellective soul (intellect),
- nutritive soul, and
- sensitive soul,
cf. St. Thomas's Summa Contra Gentiles II cap. 58
but Aristotle refutes this in De Anima I, 5, by a reducio ad absurdam:
τί οὖν δή ποτε συνέχει τὴν ψυχήν, εἰ μεριστὴ πέφυκεν; οὐ γὰρ δὴ τό γε σῶμα· δοκεῖ γὰρ τοὐναντίον μᾶλλον ἡ ψυχὴ τὸ σῶμα συνέχειν· ἐξελθούσης γοῦν διαπνεῖται καὶ σήπεται. εἰ οὖν ἕτερόν τι μίαν αὐτὴν ποιεῖ, ἐκεῖνο μάλιστ' ἂν εἴη ψυχή. δεήσει δὲ πάλιν κἀκεῖνο ζητεῖν πότερον ἓν ἢ πολυμερές. εἰ μὲν γὰρ ἕν, διὰ τί οὐκ εὐθέως καὶ ἡ ψυχὴ ἕν; εἰ δὲ μεριστόν, πάλιν ὁ λόγος ζητήσει τί τὸ συνέχον ἐκεῖνο, καὶ οὕτω δὴ πρόεισιν ἐπὶ τὸ ἄπειρον.
If then the soul is of its very nature divisible, what holds it together? Not the body, certainly: much rather the contrary seems to be true, that the soul holds the body together; for when it departs, the body expires and decomposes. If there is some other thing which makes it one, this other is rather the soul. One would then have to ask, concerning this other, whether it be one or of many parts. If it is one, why not call it the soul straightway? But if it is divisible, reason again demands, what it is that holds this together? And so on ad infinitum.
I basically agree with Geremia's answer. Here is similar perspective.
According to Dominic J. O'Meara, who is writing about the philosophical influences Plotinus had to address claimed Aristotle did not believe in an immortal soul, but he did consider the intellect to be immortal (page 14):
One of Plotinus' first writings, Ennead IV. 7 , is devoted to showing that the soul is immortal. Plato had argued for this in the Phaedo and in the Phaedrus (245ce), Plato's claim that soul is an incorporeal, non-composite reality not suject to desctruction is rejected by Aristotle. For Aristotle, soul, as the structure (or 'form') responsible for the various functions of a living body, cannot escape death. Yet one living function, intellect, seems to be an exception: in Aristotle's view thinking is not the function of a particular bodily organ. Intellect thus seems to have a claim to immortality (De anima, 2. 2. 413b24-7; 3. 4-5). However, Aristotle is at his most obscure here and in any case the question of immortality lies far from his primarily biological interests in the De anima.
This seems to agree with D. W. Hamlyn's translation of De anima (On the Soul) (page 168) although I don't know if I have the exact location O'Meara referenced.
For the present let it be enough to say only that the soul is the source of the things above mentioned and is determined by them--by the faculties of nutrition, perception, thought, and movement. Whether each of these is a soul or a part of a soul, and if a part, whether it is such as to be distinct in definition only or also in place, are questions to which it is not hard to find answers in some cases, although others present difficulty.
And as for the intellect:
Concerning the intellect and the potentiality for contemplation the situation is not so far clear, but it seems to be a different kind of soul, and this alone can exist separately, as the everlasting from the perishable.
O'Meara, D. J. (1995). Plotinus: an introduction to the Enneads. Oxford University Press on Demand.
Hamlyn, D. W. (1989). A New Aristotle Reader Edited by JL Ackrill Oxford: Clarendon Press, 1987