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The Varieties of Uncertainty

Timeline, logical, semantic, and metaphysical credence

This chapter is a draft — it is readable but still changing.

Eliezer Yudkowsky has compressed a deep claim about probability into a single sentence: the only true randomness, the only irreducible uncertainty, is the kind that “results from standing in more than one place and being unable to tell who you are” — indexical uncertainty. I think he is right, and rightness of this kind is rare enough in the foundations of probability to be worth dwelling on. But the claim is also easy to over-read. If indexical uncertainty is the only irreducible uncertainty, it is tempting to conclude that it is the only thing a credence can legitimately be about — that every honest probability is secretly a self-location probability, and everything else is confusion. That conclusion is wrong, and seeing exactly why it is wrong yields something useful: a complete map of the kinds of uncertainty a rational agent can carry, and a demonstration that the Bayesian machinery runs coherently on every one of them.

Standing in More Than One Place

Start with what Yudkowsky gets right. Indexical uncertainty arises whenever an observer’s evidence is consistent with more than one location, identity, or vantage point — whenever there are several places you could be standing and nothing in your experience settles which. The classic cases are self-location puzzles: Sleeping Beauty waking without knowing whether it is Monday or Tuesday; the simulation hypothesis, on which your experiences are consistent with being flesh or being software; the observer outside Schrödinger’s box, about to open it.

In the Quantum Branching Universe (QBU) — the Everettian picture whose primer, and whose twin terms of art Measure and Credence, I set out in Measure and Credence — this is not one exotic species of uncertainty among many. It is what quantum uncertainty is. When a measurement branches the world, every outcome occurs, weighted by Measure, and there is a successor of you in each branch. Before you look, your evidence is consistent with all of them. Your uncertainty about “what the result will be” is exactly uncertainty about which timeline you are on — and that is an indexical question. The two framings are one phenomenon described twice:

This identification earns its keep. It dissolves the old mystery of quantum “randomness”: nothing in the physics is random — the universal wavefunction evolves deterministically — and the irreducible uncertainty Yudkowsky points at is not in the world but in the self-locating predicament of an observer who is about to stand in more than one place at once. It is also the native habitat of empirical Credence generally. Empirical knowledge, as I argued in what knowledge is, is knowledge of which branch you are in, and Bayesian updating on observation is the trimming of the timelines consistent with your evidence. Whether it rains tomorrow, whether you carry the genetic variant, what the die will show: all timeline uncertainty, all indexical.

How the numbers should actually be assigned in self-location puzzles — why you are not a random sample from the observers, and not a random branch from the tree — is a hard enough question to need two chapters of its own, You’re Not a Random Sample and You’re Not a Random Branch. Here I need only the taxonomy point: timeline uncertainty is the central case of Credence, the one that empirical evidence bears on and Measure objectively underwrites.

Central — but not exhaustive. Consider three questions a rational agent can be genuinely, quantifiably unsure about, none of which is a question about which branch it occupies.

Logical Uncertainty

What is your credence that the trillionth digit of π is a 7? Something near one tenth, presumably — and that number is doing real work: you would take one side of a bet at twenty-to-one and the other side at five-to-one, and you would be right to.

But this uncertainty has nothing to do with timelines. The trillionth digit of π is the same in every branch of the multiverse; no observation, no measurement, no self-location will vary it. Mathematics is the one domain where standing in more than one place changes nothing. Your uncertainty here is a fact about you, not about the world: you lack the computation. A being with enough compute would have no credence to assign — it would just know. Logical uncertainty is the epistemic shadow of bounded resources, and it covers everything from unresolved digits to open conjectures: your credence that Goldbach’s conjecture is true is a probability about a fact that was never in suspense.

Semantic Uncertainty

What is your credence that baldness begins at fewer than 500 hairs?

Here the uncertainty lives in the word, not the world. Nothing about scalps is hidden from you; what is unsettled is what “baldness” means, precisely — where a vague predicate’s boundary falls, how a definitional dispute will resolve, which way an ambiguous term will be sharpened. This is uncertainty about the conditions of a statement rather than about its subject matter. Conditionalism locates it exactly: a truth claim is shorthand for a conditional, and when the conditions are left underspecified — the territory of when statements fail — an agent can still hold a credence about how they will be filled in. Semantic credence is what it is rational to carry while the conditions are in flux: your credence about how a court will construe “vehicle,” how a community will end up using “planet,” whether a borderline case will fall inside a category once the category is forced to commit.

Metaphysical Uncertainty

What is your credence that consciousness requires a biological substrate? That there are moral facts? That the branching picture of quantum mechanics is itself the right one, rather than some rival interpretation?

These are not questions about which timeline you occupy — the answers, whatever they are, hold across all of them. Nor are they mere computation deficits or definitional fuzz. They are uncertainty about the fundamental furniture of reality, about which explanatory framework is true. And credence applies here too. My own confidence in the QBU is high but it is a credence — I hold the theory the way a rational agent holds any deep theory, with a probability short of one. This is the position I defend in defense of Bayes: explanatory theories contain no probabilities, but credences about theories are perfectly coherent, and metaphysical uncertainty is exactly such credence at maximum depth.

So the map has four regions:

  1. Timeline (indexical) uncertainty — which branch am I on? The empirical core, objectively underwritten by Measure.
  2. Logical uncertainty — what do my premises entail? Bounded computation facing unbounded mathematics.
  3. Semantic uncertainty — what do the words mean? Unsettled conditions on vague or ambiguous statements.
  4. Metaphysical uncertainty — what is reality fundamentally like? Credence about frameworks themselves.

Yudkowsky’s claim survives intact, properly scoped: timeline uncertainty is the only irreducible variety, the only one that would persist for an agent with unlimited computation, perfected language, and the true theory in hand. Even that ideal agent, standing at a quantum branchpoint, cannot know which successor it is about to be. The other three varieties are reducible in principle — compute more, define sharper, discover the truth — but no agent that actually exists has done the reducing, and Credence is the honest bookkeeping of agents that actually exist.

Betting on Mathematics

The taxonomy raises a sharp objection, and answering it is the payoff of this chapter.

Timeline credence has something objective to answer to: Measure. When I say the coin has a 50 percent chance of heads, there is a physical branch weight my credence is tracking, and that is what makes the probability calculus more than etiquette. But the other three varieties have no objective probability underneath. There is no Measure over the digits of π; the trillionth digit does not occur in 10 percent of branches, it is what it is in all of them. So why should logical credences obey the probability axioms at all? What disciplines the numbers, when there is no fact of chance for them to be right about? A critic can put it harshly: your “credence” about π is just a feeling with a decimal point.

The answer comes from Logical Induction, a formal framework due to Garrabrant, Benson-Tilsen, Critch, Soares, and Taylor. Its mechanism is a market. Picture the unresolved statements of mathematics as tradeable contracts — a share of “the trillionth digit of π is 7” pays out a dollar if true, nothing if false — and picture a population of traders, each an algorithm implementing some computable betting strategy: one exploits digit statistics, another hunts partial proofs, another arbitrages related theorems against each other. The market price of each contract is the system’s credence in that statement. As logical evidence accumulates — a proof lands, a computation terminates, a heuristic pans out — traders profit or bleed, prices move, and the credences update.

The central theorem is that this market cannot be systematically exploited: no polynomial-time trading strategy can pump unbounded profit out of its prices. That is Dutch-book resistance, generalized — and it is the answer to the critic. The reason credences must obey the probability laws was never, at bottom, that objective chances exist. It is that incoherent credences are exploitable: an agent whose betting odds violate the axioms is an agent someone can construct a guaranteed win against. Coherence is a discipline on the agent, not a mirror of chance in the world. Logical Induction proves the discipline is achievable where no chance exists — its credences satisfy the probability axioms in the limit, converge toward the truth as evidence arrives, and answer to nothing but the demand that no bookie beat them.

And notice what updating looks like inside the mechanism: a proof is evidence, a computation is an observation, a partial result shifts the posterior. Bayesian practice survives unchanged across all four regions of the map. What varies is only the target — Measure in the timeline case, and in the other three cases nothing at all except the standing requirement of coherence. Bayes never needed an objective probability under every credence. It needed agents who cannot be turned into money pumps.

A Meta-Credence Check

Is the four-fold classification exhaustive? I have reasoned through each category and I cannot rule out a subtler variety the map fails to mark. So the honest answer is itself a credence: around 95 percent that the list is complete, responsibly short of certainty.

And the residual 5 percent is the chapter in miniature. It is not timeline uncertainty — no branch of the multiverse contains a fifth category that other branches lack. It is logical uncertainty about what my own distinctions entail, semantic uncertainty about where the category boundaries fall, and metaphysical uncertainty about whether reality has joints my taxonomy misses. The map classifies even the doubt about the map. That is what a framework in good working order looks like: not certainty, but uncertainty that knows its own varieties.