Suppose a u-operator for "it is understood that" and a k-operator for "it is known that"; let S stand for various sentences. Here are some possible rules for relating these:

  1. uS → uuS
  2. uS → kuS
  3. kS → ukS

The first parallels the kk-principle in standard epistemic logic. (2) and (3) I'm not sure about.

Then we have:

  1. (uS & (ST)) → uT

This is closure, and even if in dispute with respect to knowledge, it seems more plausible as a condition of understanding.

In fact, what I was thinking of here was a paraepistemic logic: draw up a list of factors that an "epistemic state" can involve, and then suppose an epistemic operator for each variation over the factors. Rather than debate whether "knowledge" is subject to closure, then, we stipulate that there are both closed and open epistemic states, and factive and non-factive ones, etc.

Now, it seems that there is no "understanding paradox" parallel to the knower paradox. What I mean is: "This sentence is not understood," doesn't seem to lead to a contradiction like, "This sentence is unknown," does. Note that the misunderstood sentence does have an intelligible erotetic counterpart: whereas, "Is this sentence known?" is not valid in erotetic logic, seeing as questions are not known as such, yet, "Is this sentence understood?" is valid.

Whence the asymmetry? Is this due to understanding not necessarily being factive (i.e. it is not necessarily true that uSS)?


Perhaps you could start by maintaining that uS means that I know the truth conditions of S. In general, understanding a proposition will involve more than this, but minimally, if I understand a proposition then I know its truth conditions. S may be true or may be false and I may not know which, but if I understand S then I at least know what it would be for S to be true (or false).

This would mean that uS is itself a kind of knowledge, though of course it is distinct from kS. So, if you are content with the transparency kk-principle, then uS → kuS holds good, since if I know the truth conditions of S then I know that I know them. Strictly speaking, you are not thereby committed to kk in general, only to its application to truth conditions.

You probably want to add kS → uS as an axiom, since if I know that S is true, then I must know its truth conditions.

kS → ukS is fairly uncontroversial on this position, since it says that if I know S, then I know the truth conditions of knowing S, which are themselves presumably nothing more than the truth conditions of S itself, together with whatever truth conditions you take to define your preferred account of knowledge.

uS → uuS then becomes the claim that if I know the truth conditions of S, then I know the truth conditions of knowing its truth conditions. This is no more controversial than kS → ukS. In fact, we might consider it to be an instance of it, since uS is a particular kind of knowledge.

I don't believe your axiom 4 is acceptable. We need a much stronger connection between S and T to ensure understanding of T. As it stands, S might be some false proposition whose truth conditions I know, and T any arbitrary proposition. At the very least I think we need (uS ∧ u(S ↔ T)) → uT.

As to the knowability (Fitch) paradox, I think you are correct about the factive nature of k. The derivation of the paradox depends on kP ⊢ P.

  • I would want to amend "know the truth conditions of" so that non-assertoric functions can be understood (e.g. I understand a question if and only if I know its answer conditions, or I understand an imperative if I know how to comply with it?). Jun 10 '21 at 14:34

Epistemic modal logic K has the advantage of reflexivity but with the very problematic closure property as you rightly mentioned only achievable by an idealized agent. While understanding modal logic has the opposite characteristic since understanding is a totally subjective psychological process which usually cannot be justified by others. But if you try to mix them up to try to get the best of each, it seems incompatible as correct justifiable understanding is a subtype of knowing while those self-claimed incorrect understanding is not JTB knowledge at all. So if you want to pursue your U-logic, you need to focus on those true understanding Ut-logic to have utility...

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