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It claims there are three options of which none of them are satisfying.

Circular argument doesn't prove anything because it's just when the premise is the same as the conclusion.

x ∵ x

Infinite Regress isn't clear on why we'd need to do this let alone even possible to implement so it also has no proofs.

x ∵ ... (never ending chain)

Foundationalism is different in that it's actually useful. Mathematics is it's epitome. Math has tons of proofs. Math isn't only cohesive but also adhesive. What more do we want?

y ∵ x; where x is presumably true

As for it's criticism: Why start at claim x?

Because presumably, it's either true (or hopefully true via self-evidence or empirical research) or it doesn't matter because sometimes we only want to see where the premise takes us.

Maybe an interesting take on my question is: how do we know it's a trilemma?

The trilemma is about justification of a given proposition. Any justification, so the story goes, takes ultimately one of these forms if faced with skepticism.

Therefore, the third option is about people who answer to the question "But how do you know that x really is true" dogmatically, e.g. with "Because it is", "Because I say so", etc.

Ultimately, the answers "It is self-evident" and "Empirical research shows it" are, if taken as an end-point of justification, arbitrary since the skeptical objections "Why do you assume self-evidence here, I do not follow" and "What are the standards of this research, why should this be a particularly good standard for the truth of the proposition" are still valid.

Sure, we accept that for statements about the world, empirical research is probably the gold standard and brings us as close to the truth as we might get, but that's not the setting in which the trilemma is supposed to work.

It is about forms of justification, and ending the chain of justification at an arbitrary point is dogmatic, no matter how well-justified the point itself might be. If I claim that this justification was unquestionable and ultimately justifies the whole chain as true, I am dogmatic.

Circular argument

We know it's a trilemma because the argument is founded on logic and proofs, and all proofs will end in either circular logic, infinite regression, or a foundational assumption.

Infinite regress

You can always break a proof into parts. Those parts get simpler and simpler. Keep breaking them up long enough, and all parts will eventually become circular logic, infinite regression, or foundational assumptions.

Foundational assumptions

We know it's a trilemma.

All kidding aside, we find this pattern is generally true. Logical arguments end up in one of these buckets. Sometimes you can change which bucket it ends up in (such as calculus, which permits us to replace some infinite regression arguments like Zeno's paradox with foundational assumptions about how limits behave), but they end up in one of these buckets in the end.

The main value of the trilemma is to provoke thought. Its to make you think about what must come of rational thought. If it does so, then it has done its job. Proving it "right" however, is more difficult. The whole point of the trilemma is that people who believe in it are not satisfied with the result of any attempt at a proof.

To prove it, you would have to select what concept of "proof" you wish to explore and what tools you are willing to consider in the proof thereof. Those are rather personal, which means there's not a one size fits all answer, besides the satirical version I wrote above.

I would argue that you have a clear opinion as to what proofs you find appealing. You appear to find the foundational ones appealing while the others feel like "not an argument." Consider, however, the infamous Turtles argument:

The following anecdote is told of William James. [...] After a lecture on
cosmology and the structure of the solar system, James was accosted by
a little old lady.

"Your theory that the sun is the centre of the solar system, and the
earth is a ball which rotates around it has a very convincing ring to
it, Mr. James, but it's wrong. I've got a better theory," said the
little old lady.

"And what is that, madam?" inquired James politely.

"That we live on a crust of earth which is on the back of a giant
turtle."

Not wishing to demolish this absurd little theory by bringing to bear
the masses of scientific evidence he had at his command, James decided
to gently dissuade his opponent by making her see some of the
inadequacies of her position.

"If your theory is correct, madam," he asked, "what does this turtle
stand on?"

"You're a very clever man, Mr. James, and that's a very good
question," replied the little old lady, "but I have an answer to it.
And it's this: The first turtle stands on the back of a second, far
larger, turtle, who stands directly under him."

"But what does this second turtle stand on?" persisted James
patiently.

To this, the little old lady crowed triumphantly,

"It's no use, Mr. James—it's turtles all the way down."

Now you don't have to agree with the opinions in this story, but you must admit that there something worth calling a "proof" here, and it ends in infinite regress. As a thought experiment, how would you go about responding to this? Would you be inclined to tell the little old lady that it is not valid to use infinite regression in proofs?

Consider how we model numbers with Peano arithmetic. We always have an axiom of induction. We may call that simply a foundational axiom, but if we look into why we think it is a valid axiom, it starts to look an aweful lot like an infinite regression argument. We just tucked it away inside a foundational axiom so that we didn't have to worry about it polluting the rest of the proofs. So, perhaps, in a way, such fundamentals of arithemtic simply prove that infinite regress is a valid method of proof in some carefully constructed circumstances!

You have misunderstood the point of the Munchausen Trilemma. It plays a key role in the process of philopshy showing that none of our beliefs are justified knowledge, per the standards of "reasoning".

Most people hold that they have knowledge and beliefs based on justified reasons, and that beliefs SHOULD be justified, and knowledge isn't knowledge unless it is justified somehow. The term for this is the Principle of Sufficient Reason: https://plato.stanford.edu/entries/sufficient-reason/

In its first usage, by Spinoza and Leibniz, the PSR basically called for a proof, before one could satisfy it.

Another key term in the history of knowledge is Justified True Belief -- the criteria that knowledge is only knowledge if one has both justifications, AND it is true. https://plato.stanford.edu/entries/knowledge-analysis/

However, most empiricists since Locke have been indirect realists -- holding that the world is only inferred, not known directly. And if we cannot know the world directly, then one cannot EVER know of a belief about the world is TRUE or not, and JTB is unachievable. https://www.iep.utm.edu/perc-obj/#H2

In practice today, most "reasoning" people follow a softened version of the PSR -- where "sufficient" reasons are just "supporting justifications are stronger than refutations".

However, one can apply the Munchausen trilemma to all "supporting justifications" to challenge what THEIR support is. And the answer, in every case, because we cannot complete an infinite series, will lead to and UNJUSTIFIED supposition, or a circular argument.

If beliefs need to be justified to be reasonably held, then all those justifications, to be held reasonably, must themselves be justified. But they are not and cannot be. The Munchausen Trilemma refutes all claims to be "reasonable" or hold "justified beliefs" based on our current standards of knowledge or reasoning.

The response among many philosophers has been to embrace larger networks of supporting assumptions, this is the coherence reply to this dilemma. https://www.iep.utm.edu/coherent/ The reasoning is that while a simple circle might be a fallacy, a complex web of justifications is not. The name of the Munchausen Trilemma, with critiques the circularity of coherentism, is a ridicule of this claim. Baron von Munchausen cannot pull himself out of the mud, or pull his horse out of the mud, but by pulling on his hair, then lifting himself this way, he was able indirectly to lift his horse through the stirrups!

If no beliefs are justified by our current form of rationality/reasoning, then philosophy and knowledge must both become non-rationalist. The response to this has varied between Europe and the US. In Europe, it has been an embrace of radical relativity -- IE postmodernism. In the US it has typically lead to pragmatism, where one can justify beliefs based on the pragmatic criteria of their utility/effectiveness.