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:
- uS → uuS
- uS → kuS
- kS → ukS
The first parallels the kk-principle in standard epistemic logic. (2) and (3) I'm not sure about.
Then we have:
- (uS & (S → T)) → 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 uS → S)?