One way to answer the Münchhausen Trilemma is an axiomatic approach. This approach ends the regressive arguments' "how do you know?" questions by simply saying "I accept this is true". But how do you know that you accept that this is true? If you say you accept that you accept that it is true, how do you know that's true?
Phrased another way: How do you know that you accept something as true?
How do you know that you accept something as true?
Same way you know you accept anything: you are sincere in your acceptance. For example, this is the same way you are sincere when you say "I feel glad" and you sincerely mean it.*
To paraphrase Ayer: "Axiom is true simply because we do not allow it to be otherwise" (from Language, Truth and Logic, ch.4 "The A Priori" p. 41)
*see section 3-6
You don't know that an axiom is true, you just accept it as if it was true.
People will take the axiom, and other axioms, and logical rules, and make deductions. You then look at the deductions. If the deductions result in contradictions then your axioms are not well chosen.
For example, in mathematics there is the axiom "for any two real numbers a and b, exactly one of the statements a < b, a = b, or a > b is true". If we replaced this with "for any two real numbers a and b, exactly two of the statements a < b, a = b, or a > b are true", then we would eventually end up deductions producing contradictions.
Again in mathematics, we know that there are statements that can neither be proven to be correct nor can they be proven to be wrong. If you have such a statement S, you can take as an axiom. We get a slightly different mathematics. We could have taken (not S) as an axiom, and again get a different mathematics. So in this case, any one out of two contradicting statements can be taken as an axiom.
Definition of axiom
1: a statement accepted as true as the basis for argument or inference.
2: an established rule or principle or a self-evident truth.
Axioms can be construed as either weaker than self-evidence or exactly the same thing as self-evidence, definitions vary. When axioms are construed as the same thing as self-evidence:
In epistemology, the Münchhausen trilemma is a thought experiment used to demonstrate the impossibility of proving any truth, even in the fields of logic and mathematics.
The dogmatic argument, of the Münchhausen trilemma which rests on accepted precepts which are merely asserted rather than defended meets one of the two definitions of "axiom".
Self-evidence In epistemology (theory of knowledge), a self-evident proposition is a proposition that is known to be true by understanding its meaning without proof...
Expressions of language that are self-evidently true are understood to be necessarily true entirely on the basis of their meaning. We can know that this sentence is comprised of words and not comprised of empty boxes of Crackerjacks entirely on the basis of the meaning of its words.
This example refutes the claim of the Münchhausen trilemma and establishes the foundational basis of analytic knowledge: Semantic tautologies which are a semantically interconnected set of truisms.
