Per your own links, the answer is a definitive "no".
Here is a crucial selection from your link on Reliabilist Epistemology:
A reliable inferential process, for example, confers justification on an output belief only if the input beliefs (premises) are themselves justified. How could their justifiedness have arisen? Presumably, by having been caused by earlier applications of reliable processes. This chain must ultimately terminate in reliable processes that themselves have no doxastic inputs. Perceptual inputs are a good candidate for such processes.
The writer is admitting that one can, and should, ask for "justification" for the process used, which throws the entire Reliabilist project back into the trilemma. This author then argues for foundationalism -- that there are epistically unquestionable beliefs, in this case arguing for "perception" as such a belief. Perception, of course, can be wrong, and it is not actually raw data, it is the integrated judgement of our unconscious processes, so perception is not even a credible candidate for such a role. That can only be "experiences" (which can still be wrong, but are at least not constructed artifacts). So this "solution" involves endorsing unjustified premises, following the first horn of the Trilemma, and does so on false pretenses of claiming those premises are unquestionable despite their documented fallibility.
Additionally one cannot get to "reliable processes" from experience alone (or perceptions) -- one must also invoke some reasoning methodology, which is NOT experiential nor perceptual. This is the valid part of Sellars' attack on the "myth of the given". To actually go with a 1st horn solution to the Trilemma, Reliabilists need to have unquestionable reasoning processes to operate to produce the higher tier processes, as well as unquestionable data they can operate on. Yet the article admits that none of these processes are unquestionable:
Justification is conferred on a belief by the truth-ratio (reliability) of the process that generates it. Just how high a truth-ratio a process must have to confer justification is left vague, just as the justification concept itself is vague. The truth-ratio need not be 1.0, but the threshold must surely be greater (presumably quite a bit greater) than .50.
So -- Goldman's reliabilism fails the Munchausen Trilemma, by falsely claiming to credibly satisfy the first leg.
Plantinga's Proper Functionalism has different but related problems. While Goldman has given up on the "true" part of "justified true knowledge", Plantinga continues to hold by "true" as a criteria for justification, which is why he has so much trouble with Gettier problems. The reason Gettier problems are "problems" is because indirect realism is the case for us -- we do not know directly what is true, we only infer it, and our inferences can be wrong (or for Gettier, right for the wrong reason). This situation makes it impossible to have "justified TRUE knowledge", rather than just justified knowledge.
Plantinga's attempts to get to "true" by a linguistic transform from belief, know and justify to an alternate term set that includes warrant, functioning cognitive design, and a favorable condition. However, the section 1d details how Gettier problems continue to plague his new set of terms as well. The summary of that section shows that Plantinga himself admits he has not been able to show that his approach is valid:
Plantinga maintains that the needed resolution condition may be stated as follows:
(RC) A belief B produced by an exercise of cognitive powers has warrant sufficient for knowledge only if MBE (the mini-environment with respect to B and E) is favorable for E.
This, of course, raises the question of just what it is for a mini-environment to be “favorable.” Plantinga has, in the past, offered various proposals for what favorableness consists in that he has subsequently admitted to be unsatisfactory.
While I have focussed so far on how Plantinga is confounded by his attempts to justify "true" in his formulation, he also faces the problem of justifying his methodology. Here is one area where his formulation runs afoul of the Trilemma:
The notion of a design plan at issue here is whatever notion is presupposed by talk of proper function for biological systems (as when a physician determines that a human heart is functioning the way it is supposed to on account of its pumping at 70 beats per minute). Plantinga himself gives a theistic account of this notion, but other proper functionalists, such as Ruth Millikan (1984) and Peter Graham (2012), have offered naturalistic, evolutionary accounts.
His approach depends upon a "design plan", which requires a design process and or a designer, and the design has to be:
- Properly functioning,
- well designed
- truth directed
- and operating in the right environment
None of these, nor the existence of a designer or design process, are unquestionable givens. All need to be themselves justified. And each justification itself can be challenged, and will need further justification. Plantinga's "solution" is entirely enmeshed in the Trilemma.
So no, neither of these "externalist" approaches to justification are able to solve the Trilemma.
Three asides: 1) Reliabilists are spotty in the abandonment of "truth" as part of justification. One of the examples cited of the two birders, assumed that we have access to a "Truth check":
Two bird-watchers, a novice and an expert, are together in the woods when a pink-spotted flycatcher alights on a branch. Both form a belief that it’s a pink-spotted flycatcher. The expert is immediately justified in believing this but the novice isn’t; the latter just jumps to this conclusion out of excitement.
The text assumes three knowledge states, the novice, the expert, and the "true" state that we the readers are to presume while considering the other two. But there are only two witnesses, and two knowledge states available to us, or the birders, the novice and the expert opinions. And BOTH could be mistaken!
2) Reliabilism is implicitly accepting reasoning as empirically based -- reasoning processes are justified by their empirical reliability
3) I was struck by the identification of actual scientific open mindedness as a "problem" for reliabilism. In science, all hypotheses and theories are tentative, and can be overturned by contrary evidence. And reliabilism embraces this process of evidence-based thinking. Yet when a thought problem suggested that one could have evidence for reliable clairvoyance, and should therefore accept the reliability of clairvoyance justifications, this was seen as a critical flaw:
The principle example here is due to Laurence BonJour (1980). His strongest example describes a subject, Norman, who has a perfectly reliable clairvoyance faculty, but no evidence or reasons for or against the general possibility of a clairvoyant power or for or against his possessing one. One day Norman’s clairvoyance faculty produces in him a belief that the President is in New York City, but with no accompanying perception-like experience, just the belief. Intuitively, says BonJour, he isn’t justified in holding this belief; but reliabilism implies that he is.
Per Popperian SCIENCE he is justified!!!!
As a final note, while the Trilemma refutes rationalism, and neither of these authors successfully address it, one can PRAGMATICALLY live with the Trilemma, and still hold "as good as rational" beliefs. Her is a link to an answer I gave on another Trilemma question: Is the Münchhausen trilemma really a trilemma?