Munchausen’s trilemma and the impossibility of certain truth

One day the Baron Munchausen found himself stuck in a mire together with his horse. The situation was dire, but he managed to save himself (and his horse!) by pulling his hair up until he was lifted out of the mud.

Obviously, Munchausen’s feat is impossible, as it violates the law of gravity. So it is fitting that it gives the name to the most compelling demonstration of the impossibility of another impossibility that human beings have been after for quite some time: certain knowledge.

One of the earliest demonstrations that certainty isn’t something that human beings can reasonably aspire to was given by the ancient skeptics. Julia Annas (who, incidentally, will be one of the speakers of the forthcoming STOICON event in New York City, on 15 October) presents the argument as articulated by Sextus Empiricus (in her translation of Outlines of Scepticism):

“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.”

The modern version of the argument relies on three alternative paths to certain knowledge, all judged to be dead ends (hence Munchausen’s tri-lemma, also known as Agrippa’s trilemma, from the Greek skeptic to whom Diogenes Laertius attributes the original formulation). If someone states something to be certainly true, we are well within our rights to ask him how does he know that. To which there can be only three classes of answers:

1. A circular argument, where at some point the theory and the alleged proof support each other, however indirectly.

2. An argument from regression, in which the proof relies on a more basic proof, which in turn relies on an even more basic one, and so on, in an infinite regress.

3. An axiomatic argument, where the proof stems from a (hopefully) small number of axioms or assumptions which, however, are not themselves subjected to proof.

It is self-evident why none of the above options are good enough, if one’s objective is to arrive at certainty. And I should immediately add that these are the only three modes available not just in the case of deductive logic (which means most of mathematics), but also in the case of inductive inference (which means the rest of math and all of scientific as well as common knowledge — see Hume’s problem of induction).

There are, of course, different ways of biting the bullet, and they correspond to some of the major schools of epistemology. Say you find the first option (circularity) as the most palatable — or the least distasteful — one. Then you are a coherentist about knowledge, arguing for something like Quine’s web of belief approach. If you’d rather go for infinite regression you are, quite appropriately, an infinitist (which, as far as I know, is not a popular position among epistemologists). But if your taste agrees more with the idea of unproven axioms, then you are a foundationalist, someone who thinks of knowledge as built, metaphorically, like an edifice, on foundations (which, however, cannot be further questioned).

If none of the above does it for you, then you can go more radical. One way to do so is to be a fallibilist, that is someone who accepts that human knowledge cannot achieve certainty, but that we can still discard notions because they have been shown to be false (see Popper’s falsifiability criterion).

Karl Popper, who wrote about Munchausen’s trilemma in his The Logic of Scientific Discovery (a book that I’m re-appreciating the more I am sent to it by way of other readings) opted for a mixed approach: he thought that a judicious combination of dogmatism (i.e., assuming certain axioms), regress, and perceptual experience is the best we can do, even though it falls short of the chimera of certainty.

It has to be noted that Munchausen’s trilemma does not imply that we cannot make objective statements about the world, nor that we are condemned to hopeless epistemic relativism. The first danger is avoided once we realize that — given certain assumptions about whatever problem we are focusing on — we can say things that are objectively true. Just think, for instance, of the game of chess. Its rules (i.e., axioms) are entirely arbitrary, invented by human beings out of whole cloth. But once accepted, chess problems do admit of objectively true solutions (as well as of a large number of objectively false ones). This ought to clearly show that arbitrariness is not equivalent to lack of objectivity.

The second danger, relativism, is pre-empted by the fact that some solutions to whatever problem do work (whatever the criterion for “working” is) better than others. It is true that engineers have to make certain assumptions about the laws of nature, as well as accept the properties of the materials they use as raw facts. But it is equally true that bridges built in a certain way stay up and function properly, while bridges built in other ways have a nasty tendency to collapse.

So, it looks like the quest for certainty, which has plagued both philosophy and science since the time of Plato, is doomed to failure. But are we certain of this? If so, then doesn’t that certainty itself undermine our very contention that there can be no certainty to begin with? Nice try, but no, because we do not actually have a proof that there can be no certainty. Munchausen’s trilemma is a reasonable conclusion arrived at by logical reasoning. But logic itself has to make certain assumptions in order to work, so there…