What follows explains some things that seem clear about A's theories of the soul, but does not attempt to solve all the problems and questions that arise.

First are some notes about the soul, followed by notes from Everson's article 'Psychology' in CCTA.

Works treating psychology:
de Anima
Parva Naturalia "Small works on nature" including de Sensu  and de Memoria.
Scientific writings: especially de Motu Animalium.

In talking about soul, Aristotle is simply saying that every living thing has a certain FORM, part of which explains the thing's being alive: soul is the “first actuality of a natural organic body” De Anima 412b5–6; “substance as form of a natural body which has life in potentiality” De Anima 412a20–21; and “is a first actuality of a natural body which has life in potentiality” De Anima 412a27–8.
Aristotle's "psychology" has different aspects: sometimes he is talking as if he were a natural scientist examining a hylomorphic substance that undergoes change, but he also thinks that thought has no organ, no bodily component!

soul : body : : form : matter : : statue structure : bronze
Materialists hold that all mental states are also physical states; substance dualists deny this, because they hold that the soul is a subject of mental states which can exist alone, when separated from the body.
Perhaps the closest we get to Aristotle's take on this: It is not necessary to ask whether soul and body are one, just as it is not necessary to ask whether the wax and its shape are one, nor generally whether the matter of each thing and that of which it is the matter are one. For even if one and being are spoken of in several ways, what is properly so spoken of is the actuality De Anima 412b6–9
we have no particular reason to ask about some special relation between a soul and its matter: it is the same relation as that between any substance's form and matter.
It is not unclear that the soul – or certain parts of it, if it naturally has parts – is not separable from the body De Anima 413a3–5
Intellect (nous), however, has no bodily component, and so may be separable (413a6-7)

so Aristotle is neither a reductive materialist nor a dualist about the mind-matter problem. He thinks there is a middle course.

souls have capacities:
nutritive
perceptual
intellectual

imagination (aka appearance) and desire are two further soul capacities.
imagination integrates perception, intellect, and nutritive capacities
desire only occurs in living beings that have perception: explains purposive behavior

Capacities can be told apart by their objects: different capacities have different objects, and that is what makes them different capacities.

Aristotle has a typical procedure in treating capacities of the soul: he accounts for the relevant phainomena. He also looks to see whether reductive accounts of them work. Can they be reduced to a materialistic explanation?

First, Aristotle treats nutritive capacity. Why?
1) all life forms have it, and so 2) it's explanation is prior to all the higher life forms in A's order.

nothing is nourished which does not have a share in life ...since nothing is nourished which does not partake of life, what is nourished will be the ensouled body insofar as it is ensouled, with the result that nourishment (i.e. food) is related to the ensouled, and not coincidentally...  that which is nourished, that by which it is nourished, and what nourishes (i.e. that which engages in nutrition)....what nourishes is the primary soul; what is nourished is the body which has this soul; and that by which it is nourished is nourishment. de Anima 415-16.

life does not equal "capacity to be nourished". Aristotle seems not to want to reduce life to some one thing that is basic, even though all living things (on earth at least?) share nutritive capacity.

growth is not merely accrual of material, as might happen with a volcano.
rather, it is end-directed activity (plus, it occurs in a complex non-uniform, non-linear pattern)
if it were merely materialistic, then animals and plants would not cease to grow at a certain size, or change growth patterns after a certain stage. Fire just grows haphazardly, but life forms do not.
perhaps a more nuanced materialism could avoid any reference to life (i.e. to form). It does not seem that there is one, however.

Next, Aristotle treats perceptive capacity, the senses
the differentia of animals (from plants) is having perception: de Sensu 436b10.
In order to reach the goal (being a mature animal and procreating), animals need perception: it is essential to them teleologically speaking (de Anima 414b6, etc.)

Material explanation of perception
it is a kind of change: there is a substrate and a form that activates a potentiality
the thing perceived must be of a suitable sort: in other words, it must be potentially perceivable
perception receives the sensible form of what is perceived: so when I see you, my sight is activated by your perceptible form.
that perceptible form might affect non-living entities too (I cast a shadow on a rock for instance, or light energy causes heating), but that should not be confused with perception: just because sound can cause an avalanche does not mean the mountain hears the sound.
what's the difference?
first off, Aristotle's account of perception involves the faculty of perception becoming like the thing perceived: sight receives the sensible form and becomes like that form. The mountain does not "become like" the sound.
but what about garlic? when you store it with something else, that other thing will smell of garlic: the other thing has become like the perceptible (smellable) form of the garlic. Does it perceive the garlic? No, but how can Aristotle's analysis say why the one is perception, the other not?

the perceptive faculty is in potentiality such as the object of perception already is in actuality: when a perceptible object is encountered by a sense of perception, the perceptive organ is made like it and is such as that thing is De Anima 418a3–6.

SO, perception occurs if the sensing animal has the capacity to receive a sense object's sensible form and the sense object enforms the sense capacity: in that case, the sense capacity acquires the same form as the objects' sensible form.
Do non-living things that absorb garlic odor have the capacity to receive the sensible form of the sense object (garlic)? Aristotle should want to say no: what is his argument?
A related concern is the medium thru which sensible forms must travel: visible, smellable, and hearable forms must pass thru air (or water)...does the intervening medium also take on the form?
And what about that "acquires the same form as the object's sensible form"? How does that work? Does the eye really become red, green, and gold when it sees a Christmas tree? Or is it more like a schematic drawing, a blueprint of a house, where it does not have the SAME form, but has something that specifies what the form is?

PHANTASIA
Perhaps a separate capacity, phantasia is most often translated as "imagination," and sometimes as "appearance"
Phantasia is that in virtue of which an image occurs in us De Anima 428aa1-2 in thoughts, dreams, and memories
Phantasia is distinct from perception and intellect: it occurs in dreams when there is no perception occurring, some animals lack it but have perception, and perception is always true, but imagination can be false 428a5-16.
imagination produces, stores, and recalls images used in a variety of cognitive activities, including those which motivate and guide action De Anima 429a4-7, De Memoria 1450a22-25

whenever one contemplates, one necessarily at the same time contemplates in images De Anima 432a8-9, 431a16-17; De Memoria 449b31-450a1.
but Aristotle accepts the existence of a god whose thinking is not plausibly regarded as imagistic (Metaphysics xii 7,1072b26-30).

INTELLECT

nous, often rendered as "intellect" or "reason" or "mind": the part of the soul by which it knows and understands De Anima 429a9–10
explored in de Anima 3.4 and 3.5
because we are the rational animal, having an intellect is essential to being human.
we understand, know, plan, deliberate, etc. with our intellect
we have a "theoretical" and a "practical" aspect to our intellect.

intellect receives intelligible form and becomes isomorphic with it: this is parallel to his account of perception.

so, when I think of a stone, my mind become like in form to a stone, but only to the form, not the matter, of a stone: for it is not the stone which is in the soul, but its form De Anima 431b29–432a1.
so it seems that thought can only be of universals (forms), not particulars (matter or form-matter compounds), but he speaks of having knowledge of particulars sometimes. It seems that Aristotle can navigate this seeming contradiction.

we don't become what we think of, so there must be some difference between the intelligible form we receive and the form of the object, or some difference between the way our mind receives the form and the way the matter of the object receives it, perhaps something like the relation of a blueprint to an actual building: it may be some representational way of receiving the intelligible form.

A big problem is that the intellect has no material organ: what is it that receives the intelligible form? What is the intellect? How can it be such as to receive these forms? It must itself have no qualities (else they would interfere with reception of certain forms: we would have to have a mental "blind spot"). See De Anima 429a18).  If the intellect is not actually anything before it thinks of something, then how could it "change"? How does the hylomorphic analysis apply?

Aristotle apparently thinks that the intellect is like my ability to speak French: I've got in me all the things necessary to speak French already (vocabulary, rules of syntax, accent, etc.) and so all it takes is my actually speaking French to speak French. Aristotle wants the intellect to be potentially every thought that is thinkable. When I start speaking French, I do not actually change, although I realize a potentiality. Just so, Aristotle says that when I think of something, I do not actually change: I just realize a potentiality. See De Anima 417b6–16.

There is still a problem, for the intellect is then required to be a sort of parallel to prime matter: capable of receiving any form, but having no form itself, no qualities. Maybe Aristotle could say that the intellect is always already thinking SOMETHING, and so the intellect is never prime mental stuff actually, just as the basic elements are always there, and so prime matter is never prime matter actually, but rather is always already combined with a form.

DESIRE:
orexis (desire)

perception and intellect are somewhat passive, but not entirely: they involve discerning differences, selecting things to pay attention to.
why are animals active? what in their soul makes them act?

desire alone is not enough for action: there are people who are self-controlled, who have desires but do not act on them.

Even so, Aristotle says: It is manifest, therefore, that what is called desire is the sort of faculty in the soul which initiates movement De Anima 433a31-b1.
Perhaps he thinks that controlled desire is simply a case where two desires conflict and the superordinate one prevents acting on the other. Does he talk about that?

Desire works with practical reason to produce action. In non-humans, desire works with imagination/appearance.

Desire is always for something which seems to be good to the animal.

He is exceedingly careful and hesitant in his discussion about desire: but he appears committed to the claim that desire is what is responsible for the initial spur to action.

End of Bailly's independent notes

More on Aristotle's de Anima
partially notes from Everson's article 'Psychology' in CCTA.

When we discuss the concept Aristotle labels 'ψυχη,' or 'psyche,' we usually translate it as 'soul.' That has some problems: bear in mind that Aristotle thinks that every living being has a soul. A soul is just the name for the form of a living being (form in the hylomorphic sense of form v. matter): every particular substance can be thought of as a composite of form and matter. Those substances that are living beings have a form that is called 'soul' and are composites of soul and matter. In other words, do not think of the soul as something confined to humans, nor as something that has to do with heaven or hell (Aristotle did not believe in an afterlife in anything like the way some of you perhaps do).

Current philosophy is concerned to explore and explain consciousness and intentionality and to formulate a theory of mind. Aristotle was not concerned about consciousness as such: he takes consciousness for granted.

What Aristotle is concerned with is to develop an account of soul in terms of capacities: growth and nutrition, perception, desire, imagination, belief, understanding.

So Aristotle has no separate concept for what we call "mind." Does that mean he has no theory of the mind? Not necessarily. Aristotle has a theory of what thought and understanding are, and they are what minds do, and so, he has the central components of what could be called a theory of the mind.

412a15-16: every living thing is a substance, a composite of form and matter.

412a19-21: the soul just is the form of a living thing.

A body is not identical to its matter (412a7-8).

The matter of a human (or any other living thing) is its body, which is made up of organs, but the organs are only organs when they are part of a living thing. The body too is only a body when it is part of a living thing. They are potentially alive in a sense, but there is no such thing, for Aristotle, as an organ or a body that is not alive: thus every organ is always already alive and so actual.

The matter of the organs may be something like flesh, bone, arteries, etc. That too may only be flesh, bone, etc. when it is actually alive, and so there is nothing that is merely potentially flesh, bone, arteries, etc.

But at some level, we get to something that is matter that is not alive, but is potentially part of a living thing. There is a fundamental difference between food on the one hand and sperms and eggs on the other. Food is potentially a living thing, but not in the same way that sperm and eggs are.

Some forms can be pretty simple or more complex: in the case of a chunk of pure gold, the form is merely the shape that is compounded with the matter (gold). A separate question is whether he matter  of that chunk (gold), has a form, a definition. But what is pointed out here is that the chunk's 'form' is simply chunkness. Living bodies, however, are complex forms.

The soul is described at 412a20 as "the form of a natural body which potentially has life"--why only potentially? Do such things exist on their own? No.

The soul is also "the first actuality of a body which has organs" at 412b5

Aristotle uses sight as an example: sight is the first actuality of the eye. Sight is the form of the eye. 412b27

Thus "form" is not just arrangement and shape and ingredients: it includes capacities.

Every capacity of a body has to have an organ (GA 766b35): for something to be an organ just is to have a capacity. That's what organ means: Greek ὀργανον means "tool." It is cognate with English "work."

One capacity which has no organ, however, is intellect (more later).

Thus when we come to explain what the soul is, we really need to explain what the soul's capacities are.

414b20-22: there is no soul apart from individual living beings' souls, and there is no such thing as a really informative definition of a generic soul (i.e. of something that is the soul of no particular sort of living thing but is still a soul). Souls are a collection of capacities, and each animal or plant species has its own specific collection of capacities.

For Aristotle, these capacities are nested or built upon each other: plants have nutritive and reproductive capacities; animals have sensible, locomotive, nutritive, and reproductive capacities; humans have intellectual, sensible, locomotive, nutritive, and reproductive capacities. Each one adds more to what came before, and that puts humans at the top, the most complex end.

order of explanation: remember the triangle passage: it's not just about how to explain, but about causal direction of what causes what
If one is to say what each of these [capacities] is, for instance what the capacity for thought is, or for perception or nutrition, one should first say what thinking and perceiving is; for activities and actions are prior in account to capacities [for those actions and activities]. And if this is so, and if, even before these, one should have investigated their correlative objects, then for the same reason one should first determine these, i.e. about food and the objects of perception and thought. 415a16-22

In the de Anima passage just quoted, 415a16-22, Aristotle suggests that in order to explore capacities, we must first say what they are capacities for: in other words, we should explore the actions that the capacities lead to before we look at the capacities, for the actions are prior in account. One can only understand what the ability to X is if one understands first what X-ing is.

Aristotle also says in that passage that we need to look at the objects of the capacities' actions even before we look at their actions. That seems strange: why do we need to know about things we see in order to know about sight? It becomes less strange when we discover that Aristotle means that we are to explore which things are suitable to actualize a capacity. Food actualizes the nutritive capacity. Visible things actualize the visual capacity, etc.

415B8 says that the soul is the cause of a body: it is easy to see how it is the formal cause. It is also the final cause, as I think should be clear.

415b21-26 At DA 2.5, Aristotle suggests that we think about flammable objects: they can never set themselves on fire. Perception, a capacity of some souls, is like that: it is dependent on external things to set it moving. Everson P 176: "The force of A's claim is that how a substance can be affected by other objects depends on the nature of that substance."

An aside I don't want to pursue: what about fire? It seems to feed and grow and propagate. Is it alive?

Since nothing except what is alive can be fed, what is fed is the animate body and just because it is animate. Hence food is essentially related to what has psyche. Food has a power which is other than the power to increase the bulk of what is fed by it; for in so far as what is animate has bulk this is increased, but in so far as it is a "something in particular," that is a substance, the food acts as food; for it preserves the substance. 416b9-15

The above passage, 416b9-15, explains what food is. It does not just increase bulk of the living body: living things are nourished by food, which is more than simply increasing bulk. So food is a causal thing: food is what has the capacity to set the nutritive capacity in motion.
That is why one needs to know the objects of each capacity first: the nutritive capacity is that by which the living thing engages in certain activities called nourishing (growth, preservation, recovering, energizing, etc.) which are not just increasing in bulk by the addition of material. And the activity of nourishment has an object, namely food. So to understand the nutritive capacity, we need to look at the activity, nourishing, as well as its object, food. Food is a causal notion: it is what causes the nutritive capacity to activate.
To explore the capacity of sight, we have to first see that all objects of vision have the quality of acting on what is transparent. Color is what causes the sight capacity to activate. Each sense has its own proper sense object (see deAnima 3.1 partial explanation then come back here).

Sight sees color (and some common objects), (but can only accidentally see things like Jacques or Cleisenthene?). In other words, the capacity of sight is properly limited to its proper objects and certain common objects (such as unity, shape, ...). In order to "see" Jacques, some other capacity must exist which can put together what sight sees with other things to lead to "seeing" Jacques. Surely memory is one of the capacities that is called on in such cases. Perhaps thought too. But Aristotle does seem to want it to be a matter of perception. Why? perhaps because thought belongs to humans, whereas animals can see Jacques too: if seeing Jacques required thought, then how do the thoughtless animals do it?

MATERIAL EXPLANATION

So, souls have capacities that have activities that have proper objects. How does this work?

Well, every capacity (except intellect) needs an organ, a tool.

Aristotle uses a concept called "hypothetical necessity": For just as there is a necessity that the axe be hard, since one must cut with it, and, if hard, that it be of bronze or iron, so too since the body is an instrument [organon] (for each of its parts is for the sake of something, and so is the body as a whole), therefore there is a necessity that it be such a thing and made of such things if that end is to be.
Think of Gary Larson's "Boneless Chicken Ranch"

Aristotle proceeds via hypothetical necessity to ask how the organs must be composed: in the case of sight, the material components of the eye must be transparent (it must admit of every color: it it itself were colored, that would interfere with seeing other colors). Water is the best candidate (air would not work because it is less easily contained in the appropriate configuration: another hypothetical necessity constraint). So if there is to be an eye, it must be made of a transparent substance. Water fits that best. Thus the capacity (form) must include to some extent specification of the matter: it seems that Aristotle's division of matter and form (even if it is only conceptual) is falling apart to some degree. See the middle paragraph of Everson P 181.

Each sense organ has to be made up of matter that can take on the range of qualities that the sense organ perceives. Because of that, in order to have a sense of touch, the animal (every animal has at least the sense of touch says A) must be comprised of more than just earth (earth cannot become hot because it is dry and cold, and touch detects heat) in order to have the tactile capacity.

403a5 Living things have specifications of the kind of matter they must be composed of in their forms!! Why? Because a thing's form is its essence, and an essence specifies the things which a thing must have in order to be that thing. What animals must have in order to be animals is nutritive capacity, reproductive capacity, sensory capacity, and locomotive capacity. In order to be sight, for instance, a thing must have something transparent. That is, it must have transparent matter. Aristotle calls the accounts of the activities of the psyche "enmattered accounts."
The form of man is always found in flesh and bones and parts of this kind; are these then also parts of the form and the account? No, they are matter; but because man is not found also in other materials we are unable to effect the severance. Metaphysics 1036b3-7
In other words, it is only because humans happen to be always flesh and bone that we cannot effect the severance. "Affecting the severance" would be like what we can do with bronze triangles and chalk triangles and stick triangles and paper triangles. With those triangles, we can effect the severance, we can separate triangularity from bronzen triangularity or sticky triangularity. But the situation with humans is really no different than with triangles seems to be the suggestion. We should simply get over it and effect the severance. Perhaps if we had access to AI robots or intelligent aliens, we could effect the severance. Maybe when Christianity comes along with its angels, who are intelligent, they could more easily 'effect the severance.'

So how does Aristotle connect mechanical physical material states with psychological states? An answer can be found in the following passage:
Sometimes when there are violent and striking occurrences, one is neither excited nor afraid, whilst at other times one is affected by slight and feeble things--when the body is angry, that is when it is in the same condition as when one is angry. Here is a still clearer case: in the absence of any external cause of fear, we find ourselves in the state of someone frightened. If this is so, it is obvious that the affections of the psyche are enmattered accounts. DA 403a19-25

While the physicist defines anger as boiling of blood around the heart (we might call it activation of a certain neural module in the brain by certain chemicals), most of us call anger something like "the desire for revenge." 403b1-9: the physicist's "definition" gives the matter/material account, the psychological account gives the form and account.

I cannot pretend that that is a clear and plausible account of how physical material states connect to psychological states. But then again, I am not able to really get too far in fathoming this issue via modern science (note that I am limited: perhaps other do much better).

The form/matter distinction seems to be in danger. One weak way for A to keep the form/matter distinction is to say that although living beings have accounts that must include material specifications, they do not specify any particular bits of matter. Thus our bodies change their atoms every so and so many days completely, but our form remains the same. A is not satisfied by that: in the Metaphysics, he says that even when some particular form is instantiated only in one particular batch of matter and does not change its matter ever, it is still the case that the form and the matter are distinct.

"Aristotle's claim is that it is only by treating the substance as conceptually prior to its matter that one will be able to achieve a properly integrated explanation of what the matter and its properties contribute to the substance it initiates." Everson P 183

Even though bodies are composed of organs, it would be impossible to understand an organ without the body. The body is thus prior.

403a 19ff
sometimes one is angry or afraid although there is no anger-causing or fear-causing external thing present. At other times, when there is a (normally) anger or fear-causing thing present, one is not angry or afraid.

Animal movement:
So how do animals move? Aristotle's answer is the "practical syllogism":
"I want to drink," says appetite. "That's drink," says perception or phantasia or thought (depending on what type of animal we're talking about: not all have all of those): immediately the animal drinks. In this way animals are impelled to move and to act, and desire is the last cause of movement. de Motu Animalium 701a32-35

Common Affections

In on the Senses 436a7-8, Aristotle says that perception and desire are common to the psyche and the body. He also says that in de Anima 1.1. The only psychic capacity that is not common to the psyche and the body is thought.
It is apparent that all the affections of the psyche are with body ... in all these the body undergoes some affection. DA 403a15-19.
But body and psyche are matter and form, not two separate substances that form a system.
SO how should we understand what Aristotle means here? Perhaps there is one substance that can be described in two ways: 1) a material thing that has certain material affections, and 2) an organ whose capacity is actualized.
But Aristotle seems to reject that way of understanding what happens when we perceive something.
Suppose an animal desires drink and sees drink, as in the example of a practical syllogism above. Aristotle wants to say that seeing the drink is not IDENTICAL with the material alterations of the sense organ, and also that the action of drinking is not identical with the material physical movements of the limbs to put the drink into the mouth. BUT he does want to say that such material alterations of the sense organs and such limb movements are sufficient for drinking to take place. WHAT IS GOING ON? This seems confusing.
Aristotle wants to say that the material alterations and limb movements are the matter, while the psychological events (seeing and satisfying desire) are the form. WHY? and how can he show that the psychological events are causally relevant?
He might want to say that for there to be an action, there must be psychological states. Otherwise, it will just be an event like the wind pushing a leaf off the sidewalk. No action, just event.
For action, you need more than material mechanical events. You need things like desire and perception.Moderns speak of 'agency' in this regard.
You might think that the psychological is caused by the material, but Aristotle thinks the reverse is the case: the only reason why there is a drinking or other action at all is because there is a form animal, which is the efficient, formal, and final cause of there being an animal and of its actions.
This is not a distinction between mental events and physical events such as Descartes posits: Aristotle's distinction is between events which are actualizations of capacities of living things and those which are just alterations.
Everson thinks that "In recognizing [the idea] that one does not have to attempt an identification of mental events with physical events in order to integrate the psychological within the physical world view, Aristotle at least shows a greater sophistication than is perhaps common even now."