"…I take it that our 'mental' states are nothing other than certain kinds of  states of our brains. … the welter of information that we possess concerning neural deficits, and the nature of various kinds of physical impacts on our 'minds,' provides very strong reason for  denying that we are essentially nonphysical spooks who are only contingently wired up to our bodies."
I shall first reply to the second of two claims I have indicated:
(2) Good thing, too, since this is not what classical anthropology espouses. Aristotle and St. Thomas, i.a., are explicit that we are essentially embodied creatures, not contingently. That's all that "the soul" means: it is just the way in which we exist bodily. If there are no formally intelligible ways in which diverse beings exist, then there is no hope of science describing them intelligibly. Form just means "the way a thing predictably acts by nature." If there is no such thing as form––and form which orders a things parts to its proper function––then science has literally nothing to say about the world.
Sections 2 & 8 of Quaestiones Disputatae de Anima deserve a patient reading. But, in the interest of time, I will quote from the 15th section of St. Thomas' Commentary on I Corinthians, and then proceed to the first claim indicated in Oppy's quotation:
"For it is clear that the soul is naturally united to the body and is departed from it contrary to its nature and per accidens. Hence the soul devoid of its body is imperfect, as long as it is without the body. … In another way, because it is clear that man naturally desires his own salvation; but the soul, since it is part of man’s body, is not an entire man, and my soul is not I; hence, although the soul obtains salvation in another life, nevertheless, not I or any man. Furthermore, since man naturally desires salvation even of the body, a natural desire would be frustrated."
(1) What are our brain states but certain kinds of states of ourselves in action? Our neural structure takes on the synaptic weights, thresholds etc., that it does precisely by being informed by the numerous intelligible species we receive via sensation. As the old Scholastic saw has it, "Nothing is in the intellect but by way of the senses." Once it is "in us," however, the object's formal structure imposes itself on our proper matter. Thus the same form exists objectively under two (or more, depending on how many observers) material substances. Note that the same form does not exist in two supposita (that is, more or less, two 'individuals') by virtue of the same material construction (or, 'shape'). The tree we see in our lawn does not become perceived as a tree by making a little tree shape in our retina or occipital visual center. Rather, the same form is present in a way that 'fits' for the various kinds of matter involved. In the case of a tree, it will inform cellulose, chlorophyll, etc., into an actual tree, but in the case of our perception of the tree, it inform our synaptic response so that we can neurally 'possess' the form of a tree in a way that corresponds to our proper matter (i.e., flesh and blood). The same form can, of course, be present under a different matter, such as the pixels on your screen, or the electronic binary code that generates the computer image.
I am tickled to see unBeguiled did craft a pretty handsome restatement of this principle by saying, "An external phenomenon interacts with our sense organs which in turn causes our brain to produce a model of that phenomenon." But it's a one-hooray party, since not much later unBeguiled goes on to mangle the whole benefit of the Aristhomistic epistemological schema. Objecting to my entire usage of "form," as well as, fatally, my claim that a form "objectively exists" under the matter of both the percept and the perceiever, he complains:
I don't know why you are using the word 'form' here. First you used 'thing's formal structure', which I changed to 'phenomenon', so I will substitute again.
"Thus the same phenomenon exists under two material substances."
False. When I look at my dog, the model of the dog produced in my brain is not the same phenomenon as the dog.
In this light we can see why his restatement of the Scholastic wisdom was, alas, only skin-deep. For whereas the Scholastics invoked that principle in order to account for our objective, intimate immersion in and knowledge of the world, unBeguiled invocation of perceptual "modeling" sets the ship right into a Cartesian vortex of skepticism. By replacing an object's intelligible structure (i.e., form) with phenomenon, he has changed the whole nature of the game. Obviously, what happens in my brain versus what triggers it happening are distinct phenomena. But this is trivial: for an Aristhomist they have to be distinct, otherwise percepts exist wholly and ideally in our brains. For an Aristhomist, there must be a phenomenological difference between our perception of a thing and the thing's act of being (i.e., existence), otherwise we would either be perceiving our our perceptions (and so on ad infinitum) or there would never be any thing that we perceive. This is why I used the word "objectively;" it is not a trivial term. It conveys the idea, again, that we are not merely perceiving our own mental "representations" in some inner "theater" of consciousness (as Descartes, Locke, Berkeley, Hume, Kant, and Hegel all claimed, albeit each in his own way), but that we are actually, objectively informed with the exact same intelligible content that makes an X an X. A dog's proper matter (i.e., its body), the wood comprising a statue of a dog, the binary signals comprising a digitized dog, and our neural matter all have sufficiently similar potency (i.e., materiality) that they can all be informed by the same actuality (i.e., form). The stakes could not be higher: if a dog's actual form is not capable of "inhabiting" our perceptual potency, we have no way of really perceiving any dog, or any thing else besides. Hence, unless it is crucially qualified, unBeguiled's (perhaps more "scientific" sounding) use of "model" and "phenomenon" spells the doom of objective science as surely as Descartes did with his rationalistic skepticism (cf., to name only a few sources, E. A. Burtt's The Metaphysical Foundations of Modern Physical Science, chapter VI of The New Story of Science by Robert Augros & George Stanciu, chapter II of Wolfgang Smith's Cosmos & Transcendence, chapters I & II of his The Quantum Enigma, as well as his essay "The Plague of Scientistic Belief" [available from me upon request], for all, or most of, the gory details). Clearly, the "worldview-level" meta-problem that unBeguiled––and nearly every modern naturalist and materialist––has with form is no mere academic dispute. At stake is our very place in the world as natural knowers.
If this were not enough, however, unBeguiled also quibbles with the facticity of Scholastic adage by saying, "I think you are trying to say that all conscious brain states result from sensory input. But that's false. Dreams, hallucinations, illusions, neurosurgeons poking your brain etc." Nonetheless, the adage stands, because it only refers to the general contents of the mind, not the specific mental events that characterize, say, dreams and hallucinations. We are dependent, at a basic level, on the world, God's Creation, for our own inner world. Our cognition, our entire life as knowers and explorers, is constantly informed by our immersion in the Good Creation––we as creatures getting to know and love our fellow creature, the Universe. We know ourselves as knowers, as conscious rational beings, only by knowing Nature and our natural place in her nourishing womb. Her myriad natures inform our own nature, and our natures, in response, know nature in a way she cannot achieve without us: theoretically, formally, scientifically. "In knowing himself," says Étienne Gilson in From Aristotle to Darwin and Back Again (University of Notre Dame Press, 1984, p. 4):
"man knows nature in a unique way, because in this unique case the nature that he knows, he is. In and through the knowledge which man has of himself nature knows herself directly; she becomes conscious of herself in him, self-conscious one might say, and there i strictly nothing else that man can hope to know in this way."
Moreover, even the specific mental events do have a sensory basis, insofar as dreams occur when (at least) one module of our brains (say, our limbic system and hormone modulation) responds to the activity of some (one or more) other brain module, say, our short term memory. Likewise, hallucinations have to have some kind of sensory content, otherwise we would not be hallucinating about anything. Even in the most austere sensory settings, a hallucinating mind will begin "feeding on itself," such as when our proprioception seems to swell or our breathing and pulse become a roaring sea, or even when memories––as perceived brain events––trigger hallucinations. None of this means that our knowledge is strictly limited to what we immediately perceive, since the whole point of the adage is that what comes to us via the sense takes on an abstract life of its own, as it were, in the intellect. If we were purely sensory beings, we would be utterly dependent on stimuli for our mental content. But insofar as we are endowed with an intellect, we can abstract the intelligible species from sensible objects and imaginatively (and rationally) manipulate those abstract contents to extrapolate about our course of action, consequences, etc.
And you don't have to take my word, or Aristotle's or St. Thomas' word for it; you can read the 600-page book (Basic Books, 1999) by George Lakoff and Mark Johnson, Philosophy in the Flesh, to get the same fundamental claim based on the cutting edge of cognitive psychology and neuroscience. Consider:
"One might imagine a spiritual tradition [hey, let's call it Thomistic, just for kicks!] in which such a Soul is fundamentally embodied––shaped in important ways by the body, located forever as part of the body, and dependent for its ongoing existence on the body" (p. 563).
"Our corporeality is part of the corporeality of the world" (p. 565).
"The environment is not 'other' to us. It is not a collection of things that we encounter. Rather, it is part of our being" (p. 566).
I endorse their book, first, because it takes nature seriously, which is a sorely neglected preambula philosophiae in our age, and, second, because they strongly (and with a nice tip of the hat to Aristotle) deny the modern "sensory gap" that has plagued science since at least Descartes, if not Ockham. I must suspend my approbation, however, as soon as Lakoff and Johnson use (or, rather, sacrifice) their model of embodied cognition to a biological Kantianism, to wit,
"Conceptual categories of spatial relations [among other concepts] are created as a result of the structure of of our brains plus our experience of our bodies and how they function in space and how things are named in our language. … We take our spatial relations for granted because they work for us. But it is mistaken to think they are just objectively given features of the external world" (p. 575).
As I've stressed many times before, the brain must not be reified apart from its actual function in the larger wholeness of sentient organisms. Its proper function is but a function of its ordered role in the larger structure of our selves. Seeing as we are rational sentient animals, brain states can be the means by which we, as whole agents, effect rational decisions. But to say the brain states just are our rational contents is to make a mess of mental content and to impose a kind of idealism on the brain. The mental is unextended and, in many cases, highly abstract, whereas our brain states are, of course, extended and concrete.