To begin with, let's briefly recall how thinking works in Aquinas, whose epistemology is probably the scholastic one most familiar to any readers coming here. For Aquinas it goes, roughly speaking, like this: first the senses receive sense impressions. From these sense impressions the mind forms phantasms, reproductions or representations of the sense-data in the imaginative power. Out of the phantasms the intellect abstracts universal intelligible content, forming intelligible species, which are always and necessarily universal. Understanding consists in the apprehension of these intelligible species. When we want to understand something about individuals, what we do is take our understanding of the universal intelligible content and apply it back, in a kind of reflex action, upon the individual phantasms, which always and necessarily represent singulars, understanding that this universal is the intelligible content of this singular. (For Aquinas anything not universal in the singular material object is matter, which is intrinsically unintelligible.)
For Scotus things are different. Besides what is abstracted as a universal there is an irreducible singularity to a form, a haecceitas or thisness which makes the thing not only "such-and-such" a thing but also "this" thing. Since the principle of individuation is formal, not material, it is also intelligible, which means that, contrary to Aquinas, it is not the case that all intellection is of universals.
In addition to this Scotus recognizes a kind of cognition which Aquinas either does not or else doesn't flesh out very fully. In addition to abstractive cognition there is also a kind of cognition he calls intuitive, which is related to abstractive roughly as vision is to imagination, and indeed Scotus calls intuitive cognition a kind of intellectual "vision". Abstractive cognition abstracts from actual existence, and its object can be considered regardless of whether it is real or imaginary or hypothetical or past or distant; intuitive cognition is the kind of knowledge we have of something as existing and present to us in its actual existence. Key texts in which Scotus considers this distinction are found in Ordinatio Book II Dist. 3 P.2 Q.2, and in Questions on the Metaphysics Book VII Q.15. Here's a snippet from the latter:
There is a double intellection; one quidditative which abstracts from existence; the other, which is called “vision,” is of the existent as existent. The first, although it is generally of the universal, can be primarily of the singular . . . For the singular of itself is not determined to existence, because is abstracts from it just as the universal does. The second intellection is of both together, i.e. of the singular insofar as it is existing. And in this way . . . it does not include some accident but only existence, which does not pertain to the individual’s formula, neither insofar as it is a quiddity, nor insofar as the singular participates in this quiddity.
In addition to the familiar kind of cognition in which the intellect abstracts a universal from the phantasm, here Scotus argues that there is another way for our minds to encounter things, a direct and immediate “act of simple awareness in which some object is grasped . . . as present and existing here and now.” (The quote is from Allan Wolter.) The fact that we apprehend our own mental acts and memories has been cited as evidence that our minds grasp singulars not only in their common natures but also qua singulars, even if not in their very singularity. This distinction between these two kinds of cognitions makes this possible, although it may require some elucidation.
I cannot apprehend my thoughts or my memories (or take mental cognizance of any of my experiences) only as thoughts or memories in general, but only as my thought which I am thinking now, or my experience which was given to me then and there. Grasping my thought in this way is different from grasping the absolute nature of thought; knowing my thoughts involves knowing of their actual existence, even if in this particular thought of mine which I consider I can find nothing to distinguish it from every other thought. Thinking about the nature of thought in general (as when I think about logic) and considering the particular train of thought I am engaged in now are examples of “a double intellection; one quidditative which abstracts from existence; the other, which is called ‘vision,’ is of the existent as existent.” An analogy can be drawn between these two cognitions and the two faculties of sense and imagination; both sense and imagination involve the presentation of images to the awareness but imagination can take place without the presence or existence of its object: it is “abstractive”; while sensation only happens when an existent object is present to act upon the sense-faculty to directly produce the sense act: it is “intuitive.” Now just as the imagination presents singular images abstracted from existence, so the intellect in the type of thinking Aristotelians are most familiar with considers concepts whose objects may either exist or not. These abstractive concepts, Scotus points out, are usually but not always universals: I can think about the rage of Achilles as well as the serenity of Socrates, even if the one probably never existed and the other did, for both Achilles and his rage are intelligible as individuals whether they ever actually were or not. And when I think of Achilles and Socrates I don’t primarily think “man” and then add some determining difference to distinguish them, but first I think of the individuals, under whom “Greek” and “man” and “ancient” are included (though only one of whom perhaps also includes “wise”).
On the other hand, just as sensations only occur when the sense-object is existing, present and acting on the sense-organ, so Scotus argues that there must be an analogous intellectual activity which is so moved. One reason to think this is because of an argument Scotus advanced at the beginning of the Question, that the intellect as the superior power must know whatever the inferior sense power does. Superior power or not, however, according to the accepted maxim whatever is in the intellect (at least as regards material objects) was first in the senses, and here the intellect shares some of the defects of the sense. The intellect, like the sense powers, perceives the singular as singular without apprehending the singularity precisely speaking. Socrates and Achilles can both then be understood as individuals having the same quiddity, and as either existing or not existing, “for the singular of itself is not determined to existence, because it abstracts from it just as the universal does.” Only Socrates, however (presuming Achilles to be a non-existing fictional character), can be both seen and understood as “this singular insofar as it is existing.” This intellectual grasp however is only sufficient for the intellect to know the existence of something as a “this”; for the knowledge of quiddities, either of the essential nature or the quiddities of the accidents, pertains to abstraction, and the sense is not moved by the singularity which contains the quiddities per se.
(By the way, . Much of the difficulty for other philosophers on this issue and on the intelligibility of singulars in general came from the traditional Aristotelian axiom that while sense was concerned with particulars, the understanding pertained to universals. Some non-Christian philosophers took this to mean that the intellect was as unconcerned with particulars as the senses were with universals, to the extent that even God does not know individuals, a doctrine which seems to contain traces of the platonic tendency against which Scotus has been fighting from the beginning. While none of Scotus’ Christian opponents could follow the implications of the axiom so interpreted to such extremes, they still had difficulties in grappling with it. In the passage just cited Scotus shows that the axiom can be interpreted to mean that the intellect can do something the senses cannot, without meaning that sense has a domain of its own from which the intellect is excluded, thus eliminating the difficulty. See Sebastian Day, Intuitive Cognition: A Key to the Later Scholastics, 59-60.)
In men, intuitive intellectual cognition, since it never acts without the sense in considering material things, is subject to the limitations of sense in a way that an intellect not hampered by such restrictions would not be. Scotus writes:
The intellect immediately receptive of the action of the object can be moved by singularity; not however that which is receptive through the mediacy of a natural action. Only the first is [true of] the angelic intellect which sees immediately the material singular. The second is [true of] our intellect, on which nature acts only through the mediacy of something begotten in the sense, which can be called a material natural action, with respect to that which is intelligible, operative on the intellect.
God, on the other hand, has no need for universal concepts. This is because God knows everything he knows by an immediate intuitive cognition. Not having to rely on intermediary senses and images to think with, he has no need to abstract anything from them. Our minds, however, are limited in what they can grasp intuitively and must rely on abstraction for the rest, which as it turns out is not wholly adequate.
Scotus' theory of how we think about sensible things seems to go something like this.. The sense faculty apprehends the substance as a “simul totum,” as a unified conglomeration of attributes, colors and noises and shapes and smells, etc., and alongside this activity the intellect has an intuitive grasp of the fact that the sense is perceiving this “simul totum,” this existing acting something. From the phantasm of the attribute-conglomeration the intellect removes accidents one by one until it grasps the nature underlying them: my mind understands that this short white bald Greek-speaking something is a man, to whom the accidents short and white as so forth belong. “And thus,” Scotus continues, “the terminus a quo and the terminus ad quem of the reflection is something confused and in the middle it is distinct.” I take it that the terminus a quo is the confused conglomeration in the phantasm. The nature stands in the middle and is understood distinctly because it is the primary principle of assimilable action in the substance. But the mind can penetrate no further before becoming entangled in the accidents. Socrates acts on my mind to the extent that I understand not only the common nature humanity but also the fact that this something acting on me (known by intuitive cognition) is this man (known by abstractive cognition). I can therefore name him not only by species but as an individual. But when I attempt to go on and grasp what makes Socrates himself and no one else, all I have to fall back on are his characteristics of being short, bald, ugly, wise—-none of which are unable to be true of other men, either singly or all together. Even though Socrateity does not form an accidental unity with humanity, I only understand him in a quasi-accidental way, as the (an) individual man with these accidental, non-unique attributes. Thus the terminus ad quem remains confused as well. I know that containing the humanity and supporting all the accidents of Socrates is Socrateity, but I do not ever reach a concept of Socrateity which actually picks him out of all other possible men. “And without such a concept we never conceive the singular distinctly.”
Besides the way we encounter sensible objects, however, Scotus' account of intuitive cognition has widespread implications ranging from how we know our own acts to the nature of the beatific vision.