I recently tried to use the trilemma to better understand the limitations of full semantics of second order logic. I have now the opinion that the simplicity of the trilemma is treacherous, and it is not really clear what it says exactly.
I will now try to explain how the axiomatic argument caused me some confusion, and why I think that the regressive argument is treacherous, unclear and needs further elaboration.
The axiomatic argument is unclear about the situation where additional means for knowledge are postulated as axiomatic arguments. In my case I wondered about transfinite induction, but I think the problem is easier explained by one of Shakespeare's characters: The ghost of his death father appears to Hamlet and tells him that the new king is a murderer. Now even if Hamlet, by an act of faith, accepts that listening to the ghost is a way to acquire new certain knowledge, the knowledge itself will not be entirely based on the axiomatic argument, because also the fact that the ghost claimed it is important. However, Hamlet is only willing to believe the ghost, because he has other indications that the claims of the ghost may be true. So Hamlet also accepts the knowledge itself by an act of faith, and hence this example (and also my own example) is unable to disprove the axiomatic argument.
The real weakness of the trilemma is the regressive argument. Every argument that can ever be muttered is finite. But this implies that the intuitive interpretation of "ad infinitum" or "infinite regress" to mean a series of proofs that goes on forever cannot be the correct interpretation. A more reasonable interpretation is that of an unfinished argument. But it should be an unfinished argument where we have the option to hear further parts, but we don't know whether the argument will lead to an axiomatic foundation, or a circular argument, or stay a regressive argument. Another issue is that an actual argument is not a linearly ordered series of propositions, but a tree of propositions (or a directed graph, if we want to take circular arguments seriously). Some of the propositions in this tree might be proved by a circular argument, some by an axiomatic argument, and for some we can't decide yet how or whether they will be proved.
The regressive argument fails to distinguish different possible (valid and invalid) means of proof, because it doesn't investigate the consequences of "we can't decide yet where they will lead". If we can decide that an argument will never lead anywhere (for example because the propositions become less plausible instead of more plausible as the argument goes on), then it should no longer count as a regressive argument. Or at least it should be distinguished from a "proper" regressive argument. There might be more different cases hidden behind the regressive argument, but what has been said is already enough to clarify why I feel that the regressive argument needs further elaboration.