In epistemic logic, axiom 4 says that if I know p, then I know that I know p. What is the philosophical value of such an axiom?

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    See Epistemological Principles in Epistemic Logic; it is a quite obvious principle: "Kφ → KKφ states that what is known is known to be known." But we may use logic where this principle is not valid: logic related to frames where the relation R on worlds is not transitive. Oct 21 at 9:36
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    This is called the KK thesis, and it reflects the old Cartesian idea of "luminosity of knowledge" - while one can be uncertain about external things they are supposed to have unlimited access to their own mind, and hence cannot know unknowingly, see IEP. The thesis is also linked to not having knowledge by luck, see SEP, and standing ready to justify what is known, Wikipedia. Any of those motivations give strong grounds for rejecting KK.
    – Conifold
    Oct 21 at 10:53
  • It's a tautology and tautologues are useful as they are always true. There's an alternative which goes Pa => PPa. That is, if I can prove a, then I can prove that I can prove a. Technically, this is useful in that you can prove Godel's theorems on incompleteness. If you want details you should ask on Math Overflow. Oct 21 at 14:42

Axiom 4 is a philosophical axiom in epistemic logic. Philosophers, in general, have a very high interest in knowledge and the relationship between knowledge and belief. The nature of epistemic possibilities involving knowledge is important to many philosophers, so much so that it has its own name: the paradox of knowability. Axiom 4 is what some philosophers might call a "knowledge operator".

In logic, knowledge operators are similar to the modal-logic operators used in modal-logic epistemic logics. This axiom is a result of the studies done by philosophers attempting to understand natural language and what can be derived from its use. The philosophical value of this axiom is that it can shed light on concepts in epistemology.

A very famous philosophical problem is the paradox of knowability. This "paradox" occurs when one realizes that many things are not known, even though they are possible to know. For example, I do not know whether or not there is another planet like Earth orbiting the sun. Why? Many people would say that this planet does not exist, but that is simply an assumption. However, there is no proof that it cannot exist. Thus, our intuition about what exists might be wrong when applied to all possibilities.

A coincidence in the number of axioms and postulates in epistemic logic with those in modal-logic epistemic logics is of note. In modal-logic, there are 4 axioms and 3 postulates. In epistemic logic, there are 4 axioms and 3 postulates. This implies that the study of knowledge has a lot more to do with modal-logic than it first appears.

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