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The Münchhausen trilemma is a term used in epistemology to stress the purported impossibility to prove any truth even in the fields of logic and mathematics. The name Münchhausen-Trilemma was coined by the German philosopher Hans Albert in 1968 in reference to a Trilemma of "dogmatism vs. infinite regress vs. psychologism" used by Karl Popper; it is a reference to the problem of "bootstrapping", after the story of Baron Münchhausen, pulling himself and the horse on which he was sitting out of a mire by his own hair.
It is also known as Agrippa's trilemma, after a similar argument by Sextus Empiricus, which was attributed to Agrippa the Skeptic by Diogenes Laertius; Sextus' argument, however, consists of five (not three) "modes". Popper in his original 1935 publication mentions neither Sextus nor Agrippa, but attributes his trilemma to Jakob Fries.
If we ask of any knowledge: "How do I know that it's true?", we may provide proof; yet that same question can be asked of the proof, and any subsequent proof. The Münchhausen trilemma is that we have only three options when providing proof in this situation:
- The circular argument, in which theory and proof support each other (i.e. we repeat ourselves at some point)
- The regressive argument, in which each proof requires a further proof, ad infinitum (i.e. we just keep giving proofs, presumably forever)
- The axiomatic argument, which rests on accepted precepts (i.e. we reach some bedrock assumption or certainty)
The first two methods of reasoning are fundamentally weak, and because the Greek skeptics advocated deep questioning of all accepted values, they refused to accept proofs of the third sort. The trilemma, then, is the decision among the three equally unsatisfying options.
In contemporary epistemology, advocates of coherentism are supposed to be accepting the "circular" horn of the trilemma; foundationalists are relying on the axiomatic argument. Views that accept the infinite regress are called infinitism. Advocates of fallibilism, though, point out that while it is indeed correct that a theory cannot be proven true, it can be proven false (test method) or it can be deemed unnecessary (Occam's razor). Thus, conjectural theories can be held as long as they have not been refuted. This position is taken for granted in the natural sciences.
Agrippa and the Greek skeptics
The following tropes for Greek skepticism are given by Sextus Empiricus, in his Outlines of Pyrrhonism. According to Sextus, they are attributed only "to the more recent skeptics" and by Diogenes Laertius we attribute them to Agrippa. The tropes are:
- Dissent – the uncertainty of the rules of common life, and of the opinions of philosophers
- Progress ad infinitum – All proof requires some further proof, and so on to infinity.
- Relation – All things are changed as their relations become changed, or, as we look upon them from different points of view.
- Assumption – The truth asserted is merely a hypothesis.
- Circularity – The truth asserted involves a vicious circle (see regress argument, known in scholasticism as diallelus).
 According to the mode deriving from dispute, we find that undecidable dissension about the matter proposed has come about both in ordinary life and among philosophers. Because of this we are not able to choose or to rule out anything, and we end up with suspension of judgment.  In the mode deriving from infinite regress, we say that what is brought forward as a source of conviction for the matter proposed itself needs another such source, which itself needs another, and so ad infinitum, so that we have no point from which to begin to establish anything, and suspension of judgment follows.  In the mode deriving from relativity, as we said above, the existing object appears to be such-and-such relative to the subject judging and to the things observed together with it, but we suspend judgment on what it is like in its nature.  We have the mode from hypothesis when the Dogmatists, being thrown back ad infinitum, begin from something which they do not establish but claim to assume simply and without proof in virtue of a concession.  The reciprocal mode occurs when what ought to be confirmatory of the object under investigation needs to be made convincing by the object under investigation; then, being unable to take either in order to establish the other, we suspend judgment about both.
With reference to these five tropes, the first and third are a short summary of the ten original grounds of doubt which were the basis of the earlier scepticism. The three additional ones show a progress in the sceptical system, and a transition from the common objections derived from the fallibility of sense and opinion, to more abstract and metaphysical grounds of doubt.
According to Victor Brochard, "the five tropes can be regarded as the most radical and most precise formulation of skepticism that has ever been given. In a sense, they are still irresistible today."
- supported by infinite regress
- based on perceptual experience (psychologism)
The first two possibilities are rejected by Fries as unsatisfactory, requiring us to adopt the third option. Karl Popper argued that a way to avoid the trilemma was to use an intermediate approach incorporating some dogmatism, some infinite regress, and some perceptual experience. 
The argument proposed by Hans Albert runs as follows: All of the only three ("tri"-lemma) possible attempts to get a certain justification must fail:
- All justifications in pursuit of 'certain' knowledge have also to justify the means of their justification and doing so they have to justify anew the means of their justification. Therefore, there can be no end. We are faced with the hopeless situation of 'infinite regression'.
- One can justify with a circular argument, but this sacrifices its validity.
- One can stop at self-evidence or common sense or fundamental principles or speaking ex cathedra or at any other evidence, but in doing so, the intention to install 'certain' justification is abandoned.
An English translation of a quote from the original German text by Albert is as follows:
Here, one has a mere choice between:
- An infinite regression, which appears because of the necessity to go ever further back, but is not practically feasible and does not, therefore, provide a certain foundation.
- A logical circle in the deduction, which is caused by the fact that one, in the need to found, falls back on statements which had already appeared before as requiring a foundation, and which circle does not lead to any certain foundation either.
- A break of searching at a certain point, which indeed appears principally feasible, but would mean a random suspension of the principle of sufficient reason.
Albert stressed repeatedly that there is no limitation of the Münchhausen trilemma to deductive conclusions. The verdict concerns also inductive, causal, transcendental, and all otherwise structured justifications. They all will be in vain.
Therefore, certain justification is impossible to attain. Once having given up the classical idea of certain knowledge, one can stop the process of justification where one wants to stop, presupposed one is ready to start critical thinking at this point always anew if necessary.
This trilemma rounds off the classical problem of justification in the theory of knowledge.
The failure of proving exactly any truth as expressed by the Münchhausen trilemma does not have to lead to dismissal of objectivity, as with relativism. One example of an alternative is the fallibilism of Karl Popper and Hans Albert, accepting that certainty is impossible, but that it is best to get as close as we can to truth, while remembering our uncertainty.
In Albert's view, the impossibility to prove any certain truth is not in itself a certain truth. After all, one needs to assume some basic rules of logical inference to derive his result, and in doing so must either abandon the pursuit of "certain" justification, as above, or attempt to justify these rules, etc. He suggests that it has to be taken as true as long as nobody has come forward with a truth which is scrupulously justified as a certain truth. Several philosophers defied Albert's challenge; his responses to such criticisms can be found in his long addendum to his Treatise on Critical Reason and later articles.
- Critical rationalism
- Duhem-Quine thesis
- Gödel's incompleteness theorems
- Tarski's undefinability theorem
- What the Tortoise Said to Achilles
- Dogmatismus - unendlicher Regreß - Psychologismus Albert, Traktat über kritische Vernunft, 1968, p. 11, cited after Westermann, Argumentationen und Begründungen in der Ethik und Rechtslehre, 1977, p. 15.
- Robert Nola, "Conceptual and Non-Conceptual Content", in : ^Karl Popper: A Centenary Assessment vol 2, 2006, p. 158.
- Diogenes Laërtius, ix.
- Sextus Empiricus, Pyrrhōneioi hypotypōseis i., from Annas, J., Outlines of Scepticism Cambridge University Press. (2000).
- Brochard, V., The Greek Skeptics.
- J. F. Fries, Neue oder anthropologische Kritik der Vernunft (1828 to 1831).
- Karl Popper, "The Logic of Scientific Discovery", p. 87
- Albert, H., Traktat über kritische Vernunft, p. 15 (Tübingen: J.C.B. Mohr, 1991).
- Hans Albert, Treatise on Critical Reason, Princeton University Press, 1985, chap. I, sect. 2.
- For Hans Albert's scientific articles see List of Publications in Hans Albert @ opensociety.de