Defending Against the Trilemma

How Irrefutable Axioms Block the Trilemma's Attack on Foundationalism

We’re going to take a temporary break from economics, politics, and theology and return to philosophy.

In one of the first Contemplations on the Tree of Woe, I wrote about Münchhausen’s Trilemma. As a refresher, the Münchhausen trilemma is “a thought experiment to demonstrate the impossibility of proving any truth… If it is asked how any given proposition is known to be true, proof may be provided. Yet that same question can be asked of the proof, and any subsequent proof. The Münchhausen trilemma is that there are only three options when providing further proof in response to further questioning:

• A circular argument, in which proof of some proposition is supported only by that proposition;

• A regressive argument, in which each proof requires a further proof, ad infinitum; or

• A dogmatic argument, which rests on accepted precepts which are merely asserted rather than defended.

The Münchhausen trilemma establishes the basis for the three prevailing theories of justification used in contemporary analytic philosophy - coherentism, infinitism, and foundationalism. A theory of justification is, briefly, an explanation of why beliefs can be deemed true. It is to the foundationalist theory of justification we will turn.

To a foundationalist, it is not enough that a body of knowledge be supported merely by an integrated coherency. Foundationalism requires that justified beliefs be founded upon some secure ground. But upon what can belief be founded?

Since Aristotle, the answer to that question is that belief can be founded upon axioms. What is an axiom? Although Aristotle himself had a stricter standard, conventionally an axiom has been defined as a self-evident proposition.

That definition, however, begs the question: Self-evident to whom? Skeptics point to the fact that Euclid’s axioms, which were asserted to be self-evident, turned out to be merely arbitrary propositions. Mathematicians later proved non-Euclidean geometry was possible; physicists proved it was useful as a model for existent phenomena.

The skeptical assault on self-evident axioms proved devastating. By the 1950s, W.V.O Quine had persuaded analytic philosophers of epistemological holism, the view that “no individual statement can be confirmed or disconfirmed by an empirical test, but only a set of statements,” that “any belief networked to one's beliefs on all of reality, while auxiliary beliefs somewhere in the vast network are readily modified to protect desired beliefs.”

If Quine is right, then foundationalism is wrong. As philosopher Ronald E. Merrill explains,

The foundationalist program cannot be executed at all if axioms are regarded as arbitrary postulates with no inherent truth. It cannot be executed effectively if axioms are taken to be intuitively true, merely ‘obvious’ or ‘self-evident’. We require a stronger and stricter notion of what axioms are.

So what if we use an alternative definition of axiom? Instead of defining an axiom as a self-evident proposition, let us instead assert that an axiom be an inherently irrefutable proposition.

To refute a proposition is to prove it false or erroneous by logical argument or evidence. Now, some propositions may be contingently irrefutable. For instance, if I say “yesterday I had $5 in my pocket,” this proposition might be irrefutable if you have no evidence with which to verify it, but it is not inherently irrefutable. You might have x-ray video footage of me showing I had no money in my pocket, for instance.

For a proposition to be inherently irrefutable, it must be impossible by means of any logical argument or evidence by anyone to show the axiom to be false or erroneous. An argument built on inherently irrefutable propositions is unassailable. The skeptic cannot refute it.

Do irrefutable axioms exist? Yes! Examples of such axioms have been identified by, notably, Plato, Aristotle, Leibniz, Schopenhauer, Boole, Rand and most recently by Merrill in Axioms: The Eightfold Way

Rather than just list the axioms they have identified, let’s look at the process by which those axioms were derived. Which proposition are impossible to refute by any logical argument or evidence? Those propositions which underly all possible logical arguments and evidence.

Since the laws of logic underly all possible logical argument, the laws of logic are the best example of irrefutable axioms. One can deny the laws of logic, but one cannot refute the laws of logic.

• The Law of Identity: Whatever is, is.

• The Law of Non-Contradiction: Nothing can be and not be.

• The Law of the Excluded Middle: Everything must either be or not be.

Now, many contemporary theorists have argued that the Law of the Excluded Middle is not axiomatic, that it is merely a plausible but ultimately arbitrary assertion similar to Euclid’s postulates. Are they right? We should hope note, because rejecting the Law of the Excluded Middle inevitably leads to a form of postmodern relativism. Fortunately, the Law of the Excluded Middle is ontologically irrefutable. Multi-value logic has valuable and valid uses in cases of uncertainty and imperfect knowledge, but it ultimately relies on the Law of the Excluded Middle. As Merrill explains:

Attempts have been made to construct "non-Aristotelean" or so-called "multi-valued logics." But no such structure is truly assertable; to make an assertion is to claim that something is true rather than false. Note the absurdity of attempting to claim that it is true that a "three-valued" logic is valid, and therefore Aristotelean logic is invalid.

As a bulwark against the Trilemma, then, we can feel secure if we shield ourselves with the laws of logic. However, the laws of logic themselves only go so far. They do not address arguments relying on empirical evaluation of facts about the world, nor arguments relying on inductive logic, both of which are also susceptible to skeptical assault.

With regard to empirical evidence, I judge the following to be irrefutable axioms:

• The Axiom of Existence: Existence exists.

• The Axiom of Evidence: The evidence of the senses is not entirely unreliable evidence.

The Axiom of Existence has been thoroughly discussed by Rand, Merrill, and others working in the Neo-Aristotelian tradition. At its most basic, the Axiom of Existence simply states that no possible argument can refute the existence of existence, because in order to do so, the argument would have to exist.

The Axiom of Evidence is an axiom of my own formulation, although not my own creation. I first formulated the wording during a heated argument with Professors Scott Brewer and Robert Nozick at Harvard Law School.¹ The question had arisen: How can we know that our senses are reliable? After all, straws seem to bend in water; the same shade of gray can change in apparent hue based on nearby colors; hallucinations can confound our vision; and so on. My response was that all of the evidence for the unreliability of our senses itself arose from the senses. A true skeptic of sensory evidence could not even argue that the senses were totally unreliable because he’d have no evidence with which to do so. And even if he did have such evidence, he’d have no way to use it to refute a proposition, because that refutation could not be reliably made absent the senses.

Later, of course, I discovered that similar arguments had been made by the Academics and Peripatetics against the Atomists in ancient Greece. Having ideas that the ancient Greeks already had is more or less what it means to do philosophy, I guess.

In any case, there’s a popular meme where Donald Duck and Mickey Mouse argue about meaning. I’ve repurposed it for the Axiom of Evidence:

Of course, we still haven’t gotten very far. While it’s true that the proposition “the evidence of the senses is not entirely unreliable evidence” is irrefutable, the Axiom still leaves open the question of how much is reliable, and to what extent. That will be the topic of a future essay, where we will discuss the crossword puzzle theory of epistemology known as Foundherentism. Foundherentism, as we will see, is consistent with my theory of irrefutable axioms, and protected against the Trilemma.

1. Professor Robert Nozick taught Law and Philosophy, a graduate course I took in my third year at Harvard Law School. It was the last semester that he ever offered the course and he died not long thereafter at a prematurely young age. He was probably the most intelligent mind I encountered at Harvard. He was also the first person I ever met to discuss having been cancelled (because of his book Anarchy, State, and Utopia) and the stoic equanimity with which he handled it was inspirational to me years later when I enjoyed the same. Occasionally in life we interact with great minds. His was one. I am forever grateful to Professor Nozick for introducing me to a bigger world of ideas.