I'm including some Kierkegaard in an Introduction to Philosophy course, where I concentrate on questions of knowledge. Kierkegaard is not obviously a reference for Epistemology for most people, but I believe he made an important contribution. My teaching is drawing on Concluding Unscientific Postscript to Philosophical Fragments and on Johannes Climacus or De omnibus dubitandum est, and that's what I will mostly refer to here.
Kierkegaard against Descartes and Epistemic Scepticism
Kierkegaard questions the method of doubt in philosophy. He contrasts Cartesian doubt with the 'wonder' with which the Ancient Greeks regarded the world. Descartes claims that philosophy begins with doubt. Kierkegaard's reply is that doubt in Descartes is only possible after philosophy has begun. Descartes' method of doubt casts doubt on previous philosophical positions. Wonder better describes a pre-philosophical attitude of curiosity and questioning with regard to the world. Descartes' claim, or implicit claim, that modern philosophy begins with doubt, leaves two unanswered problems: what was philosophy before doubt? From where did the method of doubt originate? In general the idea of philosophy as the appearance of pure doubt in the mind, leaves finite consciousness in a confrontation with the absoluteness of pure doubt. Consciousness cannot grasp such an abrupt intrusion of an external absolute. That is another reason why we need to begin with 'wonder'. With regard to general positions in Epistemology, Kierkegaard is against scepticism. It should follow that he rejects Foundationalist attempts to find pure foundations, beyond doubt, for Epistemology.
Kierkegaard against Hegel and Epistemic Coherentism or Internalism: Definitions
A lot of what Kierkegaard writes in many texts is directed against Hegel. Where Kierkegaard talks about Hegel, he is also referring to earlier Rationalism, particularly Spinoza. We can also take him to be referring to Kant and to the more recent phenomena of Coherentist and Internalist Epistemology; and Analytic Hegelianism. Coherentist Epistemology argues that the criterion for there being a state of knowledge, is that a set of beliefs cohere with each other. Internalism develops from this position, because it argues that there is knowledge where is agreement amongst inner beliefs. Donald Davidson's paper 'A Coherence Theory of Truth and Knowledge' is probably the best known argument for Coherentism in recent philosophy. Hegel could be taken as a forerunner of Coherentism, certainly the Preface and Introduction to the Phenomenology of Spirit are open to that interpretation. Hegel is not so clearly an Internalist, because he takes consciousness is general as what knows. The earlier Fichte (first and second editions of the Grundlage der Gesamten Wissenschaftslehere, mistranslated into English as Science of Knowledge) might be a better example of an Internalist in German ıdealist philosophy, a he takes the 'I' as a starting point for philosophy. We might also think of recent 'Analytic Hegelianism' as a kind of Coherentism (thinking particularly of John McDowell and Robert Brandom).
Kierkegaard against Hegel and Epistemic Coherentism or Internalism: The Critique
Kierkegaard asserts that objective knowledge must be a contradiction. What he is referring to here is knowledge as something outside subjectivity. Since it must be the subject that knows, what is know cannot completely stand outside subjectivity according to Kierkegaard. An object of knowledge is known by subjectivity and therefore cannot be absolutely objective. It mus tbe an object for a subject, though that should not be taken to mean that Kierkegaard is a Solipsist. He is an anti-Solipsist since he shows how the subject can have knowledge, if not absolutely objective knowledge. Coherentism is contradictory because we cannot enter the Coherentist circle of mutually supporting beliefs from outside. The beginning of Coherentist Epistemology cannot be epistemic since it does not already have a set of mutually supporting and connecting beliefs. There cannot be a entry into the Coherentist circles accept by weakening the criterion of Coherentism that there should be a complete set of strong mutually justşfying beliefs. Hegel was aware of this problem, as can be seen in his Prefaces and Introductions, where he announces the Preface of Introduction cannot be part of the system. He leaves open the question of why there is a Preface or Introduction. This will not do for Kierkegaard, it is the subject that knows. Hegel gets into the same problem as Descartes: knowledge is such an absolute it is not possible to understand how to enter it, and it is not possible to understand how one contingent consciousness can come into contact and union with it.
Kierkegaard's Epistemic Alternative: Realist and Subjectivist
Fichte. As was pointed out above, the early Fichte could be taken as a proto-Internalist. Indeed Fichtean has been critically examined by a major Analytic philosopher, RobertNozick as contributing to the internal understanding of the 'I'. I hope to return to Nozick and Fichte at a later date. Fichte was certainly very important for Kierkegaard, and Kierkegaard paid exhaustive attention to the internal reflections of consciousness. He also pays great attnetion to similar aspects in the work of other German ıdealists, as in theConcept of Anxiety, where he develops Kant's account of the inner traumas free will in Religion within the Bounds of Reason and Schelling's development of that account in the Essence of Human Freedom. Should we see Kierkegaard as an extension of Fichtean Internalism? Kierkegaard strongly criticises Fichte in his first book, his dissertation on The Concept of Irony. The context is irony in literary aesthetics. The starting point here, and in Kierkegaard's later work is an Ironic Subjectivism, in epistemic terms Internalism. However, this is not enough for Kierkegaard.
The Paradox. Like the method of doubt (and foundationalism), and Coherentism, Fictean Subjectivism runs into paradox. The paradox is good for Kierkegaard, it is the passion of paradox. The point though is to make a 'leap' beyond the paradox. It must be emphasised that the phrase 'leap of faith' is never used and that the leap is a twist in dialectical reasoning rather than an irrational unmotivated jump in to the beyond. There must be a dialectical move to surpass paradox, because the concepts must change. The paradox is never left behind, the existence of the paradox and the surpassing of the paradox belong together. Kierkegaard's Epistemology is Subjectivist. It is strongly Subjectivist, because it is based on a double reflection, in which reflective knowledge reflects on its belief that something is the case.
Time. However, Subjective consciousness escapes from the isolated moment of Subjectivity because that double reflection can only be grasped over time, in a moment known as the leap, as the reflection on the paradox that subjective knowledge is not knowledge of the objective. That movement in time establishes the self as existing over time as well as in moment to moment. The self is aware of something permanent in relation to itself, indepedent in relation to itself, but which is within it What is known is Real though Subjective. It does not disappear in a moment because it can be the object of double reflection, and subsequent indirect communication. It is only grasped through those movements. If what is known endures over time, it is Real and can be known to others, and we can communicate this even if only indirectly.