Logical Properties of 'It Seems'

I am currently finishing a paper on the semantic and logical properties of 'seem'. As 'seem' is a subject-raising verb, we can treat 'it seems' as a sentential operator. This raises the question of how this operator behaves logically. Is it hyperintensional? Does it distribute over conjunction? Over disjunction? Over conditionals? Does it commute with negation?

I think it's fairly obvious that 'it seems' is hyperintensional. It seems to Lois Lane that Superman is not Clark Kent but it doesn't seem to her that Superman is not Superman. The other questions are harder.

Logically ‘it seems’ behaves in some respects like well-known quantifiers and sentential operators. Sentential operators like the knowledge operator and the universal quantifier distribute over conjunction but don’t distribute over disjunction. ‘It seems’ is similar in this respect. We can infer '(it seems that p) & (it seems that q)' from ‘it seems that (p & q)’ but we cannot infer '(it seems that p) or (it seems that q)' from ‘It seems that (p or q)'. It seems to me that it's raining or it isn't. But it doesn't follow that it seems to me that it is raining, or it seems to me that it isn't.

Like the knowledge operator, ‘it seems’ does not commute with negation. Your shirt may not seem like anything to me, in which case it’s not the case that it seems to me that your shirt is blue. So from ‘It’s not the case that it seems to that your shirt is blue’ it doesn’t follow that it seems to me that your shirt isn’t blue.

It’s a bit more difficult to determine whether ‘it seems’ distributes over indicative conditionals. It seems to me that if your mathematical proof [of Goldbach’s conjecture] is correct, then Goldbach’s conjecture is true (‘proof’ here is not used as a success term). Does it follow that if it seems to me that that your proof is correct, then it seems to me that Goldbach’s conjecture is true?

I am tempted to think that it doesn’t. It may seem to me that if your proof of Goldbach’s conjecture really is correct and confirmed to be so by trustworthy mathematicians, then Goldbach’s conjecture is true. But even if it seems to me that your proof is correct, then I could have different degrees of certainty concerning the propositions that your proof is correct and that Goldbach's conjecture is true. What makes me think this is that ‘it seems’, when not used perceptually, can be understood in terms of probabilistic belief, yet the meaning of ‘seem’ does not involve specific probabilities. So, even if we keep the meaning of ‘seem’ constant, the operant clauses in the final conditional could be made true by different probabilistic beliefs than the probabilistic belief that makes the premise true.