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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... 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. Jul 2 '16 at 18:55
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    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
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    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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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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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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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.

enter image description here

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    Since a semantically inert truism cannot serve as a foundation of other knowledge, I do not see how this answers the question.
    – Philip Klöcking
    May 2 at 19:17
  • "the basic conclusion of the Münchhausen trilemma it that there are no expressions of language that can that can be relied upon as definitely true" - that's not true. It just says that if there is no circular or infinite chain of justifications for a given proposition, then either it is an axiomatic/dogmatic proposition itself, or its justification is ultimately based on an axiomatic/dogmatic proposition, ie. accepted as true without further justification. Have you even read the link you provided?
    – Philip Klöcking
    May 3 at 19:39
  • @PhilipKlöcking "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." That the following sentence is true is completely proved entirely on the basis of the meaning of its words: "This sentence is comprised of words".
    – polcott
    May 3 at 19:53
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    Saying that the truth lied in the semantic meaning of its words (plus syntax, obviously) is not a proof, it is an assertion, which is exactly what the dogmatic solution is about. It does not matter whether this assertion is self-evidently true. It's about whether it can be proven to be true by any form not mentioned.
    – Philip Klöcking
    May 3 at 20:01
  • @PhilipKlöcking So in other words you are saying that there does not exist any totally reliable way to know for sure that this sentence is not entirely comprised of empty boxes of Crackerjacks.
    – polcott
    May 3 at 20:03

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