Where knowing stops: Agrippa's trilemma, counted

Vitaly Reznik

A child asks "why?". You answer. The child asks "why?" again, and again. Sooner or later every parent finds out that the game has only three endings. Either you keep answering forever. Or you come back to something you have already said. Or you stop: "Because I said so."

That is an old problem. About two thousand years ago the sceptic Agrippa listed five "modes" of doubt — disagreement, infinite regress, relativity, hypothesis and circularity — and three of them make up this one. (We know them through Sextus Empiricus, Outlines of Pyrrhonism, I.164–177.) In 1968 the philosopher Hans Albert put those three into one sentence and called it the Münchhausen trilemma, after the baron who pulled himself out of a swamp by his own hair: every attempt to justify a claim ends either in an infinite regress, or in a circle, or in a dogmatic stop. Albert concluded that no claim can be given a final, certain foundation, and that we should stop looking for one.

I accept the three endings. I dispute one hidden step: that they are equally bad. They are not, and the difference can be counted.

On this page some claims are marked proved: each is a theorem checked in Lean 4, a proof assistant in which a machine checks every step, and its name is given so that you can look it up. Others are marked my reading: what I take the theorems to mean for knowledge, which goes beyond them. And some are marked open: nobody, me included, has settled them.

One question

Take a chain of reasons and ask one question of it: in how many ways can the whole chain be true at once?

Here "because" is read strictly. "A, because B" means that A and B stand or fall together: they get the same truth value. "A, because not B" means they get opposite values. That is a model, not everything the word means in English; the everyday "because" also says why, and the model leaves that out on purpose. A reading of a chain is any assignment of "true" and "false" to its statements that keeps every "because" in it.

Now count the readings of each ending.

The dogma: "this is true, because it is true." Call it true: the "because" holds. Call it false: the "because" holds too. Two readings. The dogma settles nothing; it is a fork that looks like a stop. (Proved: dogma_two.)

The circle: "A, because B; B, because A." Both true works; both false works. Two readings — the same fork as the dogma, only longer. More generally, a loop in which every way back passes an even number of "not"s always reads twice. (Proved: circle_two, even_two.)

The bad circle: "A, because not A." Call it true, and it must be false; call it false, and it must be true. No readings at all. This is the shape of the liar, "this sentence is false". A loop with an odd number of "not"s on the way back has no reading. A chain with no readings is not a justification that failed; it is a paradox. (Proved: liar_zero, odd_zero.)

The regress: "A, because B; B, because C; ..." without end. If there are only finitely many statements to draw on, and each one's reasons are among them, this cannot happen: after as many steps as there are statements, some statement must come round again (the pigeonhole principle), and the chain ends where it repeats. (Proved: no_regress.) With infinitely many statements it can happen. The simplest such chain, "statement 1, because statement 2; statement 2, because statement 3; ...", reads twice: every link must agree, so either everything is true or everything is false. The same fork again. (Proved, for that chain: regress_two.)

The ground: a chain that ends at a statement which rests on nothing further — whose value is simply given, for instance by a calculation anyone can redo, or by an assumption laid down. Then the end has its value, every "because" passes it along, and there is exactly one reading. (Proved: forced_one.)

The converse holds too. In a finite system where each statement rests on one reason, a statement has exactly one reading if and only if its chain of reasons reaches a ground. (Proved: knowledge_iff_ground; the proof does not even need to assume that every statement is either true or false.)

So, counted:

ending	readings	what it is
dogma ("because it is so")	2	a fork
circle, even number of "not"s	2	the same fork
circle, odd number of "not"s (the liar)	0	a paradox
infinite regress	impossible in a finite system; the simplest infinite chain reads 2	the same fork
a chain that reaches a ground	1	a candidate for knowledge

The dogma and the circle do not fail because they stop or loop. They fail because they leave the answer open. A chain that reaches a ground does not. But that is not yet knowledge, and the reason is the weak spot of the count.

Acts and posits

An axiom — a statement simply laid down as true — also rests on nothing further. Written into a chain as a ground, it reads once. So the count alone cannot tell "I checked it" from "let us assume it". Albert would say: your ground is just my dogma with better manners.

The answer is to ask where a ground's value comes from. There are two kinds.

• An act: its value is computed from its own content. "7 is prime" is settled by dividing 7 by 2, 3, 4, 5 and 6: whoever does it gets the same answer. In the model, a chain that reaches an act has a reading whatever values you give to the assumptions, and all those readings agree. (Proved: act_absolute, act_reads.)
• A posit: its value is set by the starting assumptions — an axiom, a hypothesis, a fact about the world I take as given. Each choice of assumptions gives a chain that reaches a posit exactly one reading, but two choices can give it opposite values. (Proved: posit_relative, posit_each_once.)

To be honest about these theorems: they follow almost directly from the definitions. In the model a posit is a value you choose, and an act is a value computed from numbers. The real work is outside them: deciding which kind a real statement is. And the theorems say nothing about beliefs in general — only that no choice of values for the posits changes an act.

So the word I want needs both conditions. Knowledge, in the sense of this essay: one reading, and the same one whatever the starting assumptions. A chain that reaches an act has it. A chain that reaches a posit has one reading per choice of assumptions — and a rival choice is always possible.

My reading: the axiom of choice, a standard axiom of set theory, is a posit. Gödel (1938) and Cohen (1963) showed that, if set theory without it is consistent, then adding it and adding its negation are both consistent. That independence is known; calling the axiom a posit for that reason is my criterion.

Count it yourself

Take a sheet of paper.

1. "The meeting is at noon, because the email says so; the email says so, because the meeting is at noon."
2. "This statement is not true."
3. "7 is prime, because no number from 2 to 6 divides it."
4. "The meeting is at noon, because the email says so; the email says what the calendar says; the calendar says noon, because I put it there myself this morning."

How many readings does each have — and is it knowledge?

Answers. 1 — two: a circle, so a fork. 2 — none: the liar, a paradox. 3 — one, and it is knowledge: the chain ends at a division anyone can redo. 4 — one, but only given that my memory of this morning is right. "I put it there myself" is a posit: the memory sets its value, and a rival is thinkable. One reading, on credit.

Where Agrippa is right

My reading. On one influential interpretation (Michael Frede's), Sextus aims the modes at dogma in his own sense — agreeing to claims about what is not evident, about how things are by nature — and not at ordinary living: the sceptic eats, works and follows the customs of his country (Sextus, Outlines, I.13 and I.21–24; Frede, "The Sceptic's Beliefs", 1979). Others, notably Myles Burnyeat and Jonathan Barnes, read the modes as reaching ordinary beliefs as well, and Sextus applies them to the things we perceive. I take Frede's side here, and on it the line falls like this:

• For acts, the trilemma does not bite. An act rests on nothing, reads once, and no choice of assumptions changes it.
• For things, Agrippa is right. By my criterion, whatever we lay down rather than compute has a rival: Euclid's geometry and Lobachevsky's, set theory with the axiom of choice and without it, a story with centaurs and one without. Agrippa's modes of disagreement and relativity describe exactly that. His three endings are not a failure of reason; they show up wherever we choose our starting points instead of working them out.

(The sceptic's own "living" is wider than my acts: customs and crafts are, by my definition, posits, which he follows without agreeing to them.)

What about ordinary knowledge of the world — "it is raining"? On this line it is a posit: the world sets its value, and the opposite is thinkable. I call it knowledge on credit. The phrase comes from ZTL, the logic this site is about, which labels a "true" that rests on something unchecked ON CREDIT. One can say that looking out of the window is an act of measurement; then the credit moves to the instrument, and the instrument's reliability is again an assumption. Open: whether observation can ever be an act in the strict sense.

Objections

"Who chose the assumptions?" This was Albert's natural next question. For posits it is fair, and the answer is: someone did, and someone else could have chosen differently. For acts it has no grip, because the value does not depend on the choice at all.

"Your way of counting is a choice too." There is another common convention, used in logic programming, under which "p, because p" counts as false unless something else supports it. Then the dogma reads once. I do not adopt it, because that "false unless supported" is a default the chain itself does not supply — exactly the kind of chosen stop under examination. Whoever thinks the default is neutral will count differently, and should say so.

"An infinite regress can be fine." Peter Klein defends infinitism: a belief can be justified by an endless chain of reasons, none of them final. The count does not refute him; it admits the endless chain and counts it. It reads twice, like the dogma; whether that is a defect is the debate.

"Does a real act give a value at all?" In the spirit of Wilfrid Sellars, who argued that nothing is simply given to the mind: in the model an act is a computation on given numbers, and whether any act of a real person is like that is not something a theorem can show. Open.

"Is the ground knowledge, or something below knowledge?" For Wittgenstein, at the bottom of our justifications lies not a known proposition but our acting (On Certainty, §204). Whether to call the ground "knowledge" is a decision about words, not a theorem.

Several reasons at once

Real reasons combine: "A, because B and C", "A, because B or not C". With these the count sorts statements rather than endings, and one surprise appears. "0 = 0, or else because I say so" gets the same value in every reading, though the dogma inside it still reads twice: a true act outweighs it. (Proved: act_or_dogma.) What is proved in general is the sufficient direction: if a statement's reasons bottom out in acts alone, its value is the same under every choice of assumptions. (Proved: knows_of_good.) It is not the whole boundary: "p or not p" has one value whatever p is, with no act involved. Where exactly knowledge ends in such combinations is open.

What is old and what is new

The sorting is old. Saul Kripke's theory of truth (1975) separates grounded sentences, the ungrounded but harmless truth-teller ("this sentence is true"), and the paradoxical liar; Haim Gaifman (1992) gives the truth-teller two self-supporting values and the liar none. In graph theory the same structure has been studied as kernels of a directed graph, after a translation of the "not"s (Walicki and Dyrkolbotn 2022). Splitting the third ending into a ground and an unjustified stop has also been proposed, without the count, in a self-published paper by Day and Athos (2026), one of whose authors is an AI persona; their reliable stop needs support from outside the chain, mine is fixed inside the model.

What I claim as new is narrower: applying the count to Agrippa's trilemma, the line between acts and posits with its consequence that Agrippa is right about things, and the machine-checked proofs. The search for prior work was about thirty English-language queries run by AI sessions in October 2026, and the closest candidate, a 2009 paper by Michał Walicki, is not yet read. Not finding something is not the same as its not existing.

How to check

The theorems are in the open repository inventor1975/ZTL, in four Lean files: lean/Agrippa.lean (the count, knowledge_iff_ground, acts and posits), lean/AgrippaNet.lean (several reasons at once), lean/AgrippaZ.lean (the same count in ZTL's terms) and lean/ZDescent.lean (when assumptions add nothing below them) — 98 theorems with their helper lemmas. They use no axioms beyond Lean's core, and the repository's automatic checks verify that for every theorem on every change.

The full paper, with the model stated exactly, the literature and the limits only touched on here: Agrippa's Trilemma, Counted: Acts, Posits and the Formal Register, doi:10.5281/zenodo.23647180.

Back to the child

So when the child asks "why?" once more, there are more than three ways to stop. You can say "because I worked it out", and point to something anyone can redo: that answer holds whoever asks. Or you can say "because I chose to start there", and then the honest follow-up is "and here is what changes if you start somewhere else."