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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?

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    The simplest answer is : we have to start somewhere... – Mauro ALLEGRANZA Jul 2 '16 at 18:52
  • you do not "accept" the axioms; the scientific community of experts works according to theories, more or less fundamental, based on axioms. Those "shared" theories are learned and studied during our "scientific training". Sometimes, new theories (with new axioms) are invented/discovered and some old ones are rejected/modified/ "absorbed" into new ones. But for sure, never in a mathematics curricula someone ask you (or me) if we "agree" on e.g. the Peano axioms. – Mauro ALLEGRANZA Jul 2 '16 at 18:55
  • As Wittgenstein put it, "there has to be a way to grasp a rule which is not an interpretation", for any interpretation of a rule would in its turn have to be interpreted. You are dealing with a similar regress: there is a way to grasp that you know something without performing an "act of acceptance", just as there is a way to ride a bicycle without being able to explain how. Knowledge-that is rooted in knowledge-how about which "How?" can no longer be asked philosophy.stackexchange.com/questions/34384/… – Conifold Jul 2 '16 at 21:22
  • @Conifold Can you make that an answer please? I'll hit the green check if you do. – APCoding Jul 3 '16 at 16:25
  • As a very informal answer, you "accept something as true" the moment you cease to consider the possibility that that thing might be false. (This only works well because of the law of the excluded middle, but most arguments I have seen regarding logic do use that law). – Cort Ammon Aug 31 '16 at 20:41
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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.

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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

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