wo's weblog

Coded communication

Fri, 26 Jun 2026 14:42:55 +0000

Imagine a community of people who pass encrypted messages to one another, without knowing what they mean. Agent X has encrypted a message and handed it to messenger A, who passes it to messenger B, who passes it to agent Y, who has the codebook to decrypt the message. When A utters the message to B, she has no idea what it says; neither does B.

Intuitively, the meaning or content of A's utterance is the content of the decrypted message. That's why A and B don't know what the utterance means.

This might become less obvious if the code has the grammatical form of an ordinary sentence. Suppose it is "Heynya is noor." It's easy to imagine that messengers like A and B start to embed these codes in other sentences. A might ask "Have I already told you that Heynya is noor?" B might ask, "Do you know if Heynya is noor?", and A might answer, "I do". They might say, "'Heynya is noor' is true iff Heynya is noor". If they learn to say such things, someone might claim that they do know what the codes mean. They know the relevant T-theory, for example.

To see that they still don't really know what the codes mean, we have to look beyond their verbal behaviour. Suppose the decrypted meaning of "Heynya is noor" is that the Ashka'ri tribe is planning an attack. If B learned that the Ashka'ri tribe was planning an attack, he would take precautions. His "belief that Heynya is noor" has no such effects.

We might assign a different kind of meaning to the codes, a meaning that is genuinely known to the messengers. What B learns when A utters "Heynya is noor" is that whatever the codebook assigns to "Heynya is noor" is the case. This, we might say, is the actual meaning of the coded sentence among the population of messengers. (Here I assume that the messengers have reason to believe that the coded messages are generally accurate.)

We might call this known type of meaning the primary intension of a code; the secondary intension is given by the codebook: it is the code's "original" meaning, the one that isn't genuinely known among the messengers.

At this stage, primary intensions are fairly uninformative. We can change this by giving the messengers partial knowledge of the codebook. More simply, let's assume that the code uses some words in their ordinary meaning: it only encrypts singular terms. The message passed by A to B might now say "Heynya is planning an attack", where "Heynya" is the only encrypted part, and this is known among the messengers.

As before, B doesn't know the original, secondary meaning of the message. But now he acquires useful information. For one, he learns that someone is planning an attack, which may be sufficient to take precautions. He also learns that whoever composed the original message knows that this party is planning an attack. And he learns that whatever earlier messages said about "Heynya" applies to the party planning an attack. For example, if an earlier message went "Heynya is powerful", he can now infer that a powerful party is planning an attack.

In this setting, the primary intension of "Heynya is planning an attack" is the proposition that whoever the codebook assigns to "Heynya" is planning an attack. It has become much more informative.

A grammar that maps every message to its primary intension can play an important role in a systematic theory of the messengers' practice. For example, we might want a grammar to deliver meanings that fill the blank in: when someone hears an utterance of S, they tend to act in a way that would be reasonable on the assumption that —. Primary intensions can fill this blank, secondary intensions can't.

On the other hand, the primary intensions probably won't fit how the messengers talk about their own language. If we ask B what he learned from A, he might well answer "that Heynya is planning an attack", not "that whoever the codebook assigns to 'Heynya' is planning an attack". So the primary intensions may not match inner-language judgements about "what is said".

Also, the primary intensions are still uniform for all singular terms. All singular terms have the same kind of primary intension: "whatever the codebook assigns to '…'". There's no variation. So a grammar doesn't really need a separate entry for each singular term. We can instead have a kind of meta-rule that fills in all these entries at once.

In fact, we can reconstruct the "primary grammar", the grammar that maps expressions to their primary intensions, from the "secondary grammar" that maps expressions to their secondary intensions. Suppose, as before, that the codebook maps "Heynya" to the Ashka'ri tribe. So the secondary intension of "Heynya" rigidly picks out the Ashka'ri, and the secondary intension of "Heynya is planning an attack" is the proposition that the Ashka'ri tribe is planning an attack. What's the primary intension? Well, it replaces the rigid secondary intension of "Heynya" by its non-rigid primary intension, and we know what that is, since it's essentially the same for all singular terms.

So while the primary intensions can play a useful role, it is understandable that people studying the language of the messenger population would focus on the secondary grammar.

In some respects, English is a messenger language. Technical terms like 'boson' are widely used to mean "whatever the experts say they mean"; the experts have the codebook. Proper names like "Gödel" are widely used to mean "whoever stands at the origin of our use of this name"; the causal chain is the codebook.

I think this explains why people studying English tend to focus on the secondary grammar, identifying the meaning of "Gödel" with Gödel and ignoring primary intensions.

Having learned to think about meaning from Frege and Lewis, I always found this puzzling. Why, I thought, should we focus on a notion of meaning that makes meanings opaque to competent speakers and hearers? Shouldn't we rather identify meanings with something that is actually, transparently conveyed from speaker to hearer?

Well, I still think we should, but the reason is more subtle than I used to think. English only works approximately like the messenger language. "Jack the Ripper" is a proper name, but its primary intension isn't just "whoever stands at the origin of our use of this name". "Hesperus" and "Phosphorus" have substantively different primary intensions. "Boson" and "elm" certainly do. The primary intensions of names and kind terms in English aren't uniform, and they aren't determined by their secondary intensions. So something gets lost when we focus on secondary intensions.

Lewis 1969 on the probability of conditionals

Thu, 18 Jun 2026 11:02:08 +0000

I finally got around to adding the papers from "Janssen-Lauret and Macbride 2023 to the search corpus at "https://www.david-lewis.org. It's a wonderful collection with lots of treasures. I want to comment on an intriguing passage on pp.71f., from an abandoned 1969 textbook project on confirmation theory.

First, some context. At this point in the manuscript, Lewis has introduced \(\mathcal{M}\) as a probability measure on the propositions expressible in a language \(\mathcal{L}\) with classical boolean connectives; \(\mathcal{C}\) is the associated conditional probability measure, defined by the ratio formula. Lewis notes that conditional probabilities are often read as "the probability of C if A", which suggests that \(\mathcal{C}(C/A)\) might equal \(\mathcal{M}(C\textit{ if }A)\), where '\(C\textit{ if }A\)' is the material conditional. But that's obviously false. Lewis continues:

Lewis here suggests that one can identify conditional probabilities with probabilities of conditionals – a hypothesis that he famously refuted in "Lewis 1976.

Lewis's proposal has two parts.

The first is a slight revision to the material analysis of conditionals: '\(C\textit{ if }A\)' has the truth-conditions of the material conditional, but also carries a "presupposition" that \(A\) is true.

We then define, in the second part, a new probability measure \(\mathfrak{M}\) on the sentences of \(\mathcal{L}\), which conditionalizes on the presupposition of the sentences to which it is applied. If \(A\) and \(C\) carry no presupposition, it follows that \(\mathfrak{M}(C\textit{ if }A) = \mathcal{C}(C/A)\).

"Nolan 2026, 246f. comments on this passage:

Lewis's proposal here fails, so far as I can see, for the same reasons so many other accounts of the probability of the conditional in terms of conditional probability fail, especially when embedded conditionals are considered. (See Lewis's "Lewis 1976 and "Lewis 1986 and "Hájek 1989 for some among many of the technical problems.)

I disagree. Lewis's proposal is incomplete. But it doesn't fall prey to triviality results, for essentially the same reason for which the trivalent approach to conditionals, going back to "de Finetti 1936, avoids triviality.

On the trivalent approach, '\(C\textit{ if }A\)' is true if \(A\) and \(C\) are both true, false if \(A\) is true and \(C\) false, and "undefined" if \(A\) is false. One can then define a sentential probability operator \(P^*\) that satisfies the usual axioms for bivalent sentences (i.e., for sentences that are never undefined) and that evaluates non-bivalent sentences by conditionalizing on definedness, so that \(P^*(A)\) is the probability that \(A\) is true conditional on \(A\) having a truth-value. For bivalent \(A\) and \(C\), we then have \(P^*(C\textit{ if }A) = P^*(C/A)\).

The third truth-value here plays essentially the same role as the presupposition in Lewis's account. Its effect is that conditionals are associated with two regions in logical space, one of which serves to restrict the probability operator \(P^*\), while the other divides the worlds in the restricted set into those where the conditional is true and those where it is false.

On the trivalent approach, we have further work to do. We have to decide how negation, conjunction, and disjunction work if one of the arguments has the "undefined" truth-value. We also have to decide how to interpret conditionals with undefined antecedents or consequents, and whether entailment should be understood in terms of preservation of truth or in terms of preservation of non-falsity.

Analogous questions arise for the presuppositional approach. Here, we have to decide how the presuppositions of complex sentences are determined, and whether entailment should be understood in terms of simple truth-preservation or in terms of truth-preservation when the presuppositions of the conclusion are satisfied ("Strawson entailment").

Lewis says nothing about these issues. That's why his account is incomplete. For example, Lewis doesn't tell us if a nested conditional \(A \Rightarrow (B \Rightarrow C)\) presupposes just \(A\) or if the presupposition of the embedded conditional projects to the whole sentence, so that \(A \Rightarrow (B \Rightarrow C)\) presupposes \(A \land B\). The latter option seems better. It predicts that \(\mathfrak{M}(A \Rightarrow (B \Rightarrow C)) = \mathcal{C}(C/A \land B)\).

Note that, in this case, we usually don't have \(\mathfrak{M}(A \Rightarrow(B \Rightarrow C)) = \mathcal{C}(B \Rightarrow C \;/\; A)\), since \(\mathcal{C}(B \Rightarrow C \;/\;A) = \mathcal{C}(B \supset C \;/\; A)\), and this usually won't equal \(\mathcal{C}(C/A \land B)\). So the last sentence in the displayed passage isn't quite right: the equality \(\mathfrak{M}(C\textit{ if }A) = \mathcal{C}(C/A)\) only holds if \(C\) doesn't itself carry a nontrivial presupposition.

The problem with the equality \(\mathfrak{M}(C\textit{ if }A) = \mathcal{C}(C/A)\) is that the conditional probability operator \(\mathcal{C}\) is insensitive to presuppositions. In a more comprehensive account, we'd want to introduce a presupposition-sensitive conditional probability operator \(\mathfrak{C}\) that relates to \(\mathcal{C}\) in the way \(\mathfrak{M}\) relates to \(\mathcal{M}\). For presupposition-free \(C\), we'll have \(\mathfrak{C}(C/A) = \mathcal{C}(C/A)\). But if \(C\) carries presupposition \(P\), we'll want to conditionalize on that presupposition, so that \(\mathfrak{C}(C/A) = \mathcal{C}(C\;/\;A\land P)\).

So the right equality is \(\mathfrak{M}(C\textit{ if }A) = \mathfrak{C}(C/A)\). This holds even if \(C\) is itself a conditional.

Compared to the trivalent approach, Lewis's presuppositional approach seems to have some advantages.

Remember that the trivalent approach has to decide how conjunction and disjunction etc. behave when one of the arguments has "undefined" truth-value. It's natural to think (with de Finetti) that \(A \land B\) is true only if \(A\) and \(B\) are both true. But then a "partitioning sentence" of the form '\((C\textit{ if }A) \land (D\textit{ if }\neg A)\)' could never be true: one of the conjuncts must have a false antecedent, which renders that conjunct undefined. But many such sentences are surely true.

To avoid this problem, we'd have to say that \(A \land B\) is true not only if \(A\) and \(B\) are both true, but also if one of \(A\) and \(B\) is true and the other is undefined. ("Égré, Rossi, and Sprenger 2021 tentatively endorse this response.) But then conjoining a conditional with a tautology turns the conditional into a material conditional and we predict that \(P^*(\top \land (C\text{ if }A)) = P^*(A \supset C)\), which seems wrong.

So the trivalent account seems forced to make false predictions either about partitioning sentences or about conditionals conjoined with tautologies.

Lewis's presuppositional account has more flexibility. The presupposition of a conjunction \(A \land B\) doesn't have to be determined pointwise, by somehow combining the values of \(A\) and \(B\) at each world to determine whether that world satisfies the presupposition of \(A \land B\). There's no principled reason why we couldn't say that when \(A\) and \(B\) have incompatible presuppositions then these presuppositions cancel each other out in \(A \land B\), while \(\top \land A\) inherits all the presuppositions of \(A\).

Of course, we'd like to see the general rules for how presuppositions project. The familiar rules from "Heim 1982 don't deliver the desired results. But it's clear anyway that what Lewis here calls "presuppositions" are not presuppositions of the ordinary kind: when I say 'C if A' I'm not taking for granted that A is true. (It would be bizarre to respond with "hey wait a minute! I didn't know that A.")

I haven't explained why I don't think Lewis's proposal is vulnerable to the triviality arguments. The reason is that the operators \(\mathfrak{M}\) and \(\mathfrak{C}\) in the equality \(\mathfrak{M}(C\textit{ if }A) = \mathfrak{C}(C/A)\) don't conform to the rules of probability, which the triviality arguments assume.

For example, Lewis's 1976 argument assumes that \(P((C\textit{ if }A) \land C) = P(C\text{ if }A \;/\; C) P(C)\). This is licensed by standard probability theory. But \(\mathfrak{C}(C\textit{ if }A \;/\;C) = \mathfrak{C}(A \supset C\;/\;C \land A) = 1\), and there's no plausible account of how presuppositions project out of conjunction on which we'll have \(\mathfrak{M}((C\textit{ if }A) \land C) = \mathfrak{M}(C)\). Indeed, the conditional probability operator \(\mathfrak{C}\) doesn't even satisfy the ratio formula.

In this regard, the presuppositional approach is again on a par with the trivalent approach, on which the "probability" operator \(P^*\) that figures in the equality \(P^*(C\textit{ if }A) = P^*(C/A)\) also violates the rules of probability, in essentially the same way, which blocks all triviality arguments – as explained in "Lassiter 2020.

In his published work, Lewis never mentioned his presuppositional 1969 account. Why did he abandon it?

I suspect that three reasons played a role. One, he realised how hard it is to fill in the missing parts: to specify how the postulated "presuppositions" project out of conjunction, disjunction, etc. Two, he realised that \(\mathfrak{M}\) doesn't behave like a standard probability measure, so that it's misleading to call \(\mathfrak{M}(C\textit{ if }A)\) the "probability" of \(C\textit{ if }A\). Three, he realised that more needs to be said about the (unusual) sense in which conditionals "presuppose" their antecedent.

These are all good worries. But they equally apply to the trivalent approach, which is very much alive today. So the abandoned 1969 proposal might still be worth exploring.

de Finetti, Bruno. 1936. “La Logique de La Probabilité.” In Actes Du Congrès International de Philosophie Scientifique, 4:1–9. Paris: Hermann Editeurs.

Égré, Paul, Lorenzo Rossi, and Jan Sprenger. 2021. “De Finettian Logics of Indicative Conditionals Part I: Trivalent Semantics and Validity.” Journal of Philosophical Logic 50 (2): 187–213. "https://doi.org/10.1007/s10992-020-09549-6.

Hájek, Alan. 1989. “Probabilities of Conditionals: Revisited.” Journal of Philosophical Logic 18 (4): 423–28. "https://www.jstor.org/stable/30226421.

Heim, Irene. 1982. “The Semantics of Definite and Indefinite Noun Phrases.” PhD thesis, University of Massachusetts Amherst.

Janssen-Lauret, Frederique, and Fraser Macbride, eds. 2023. Philosophical Manuscripts. Oxford, New York: Oxford University Press.

Lassiter, Daniel. 2020. “What We Can Learn from How Trivalent Conditionals Avoid Triviality.” Inquiry 63 (9-10): 1087–1114. "https://doi.org/10.1080/0020174X.2019.1698457.

Lewis, David. 1976. “Probabilities of Conditionals and Conditional Probabilities.” The Philosophical Review 85: 297–315.

Lewis, David. 1986. “Probabilities of Conditionals and Conditional Probabilities II.” The Philosophical Review 95: 581–89.

Nolan, Daniel. 2026. “Philosophical Manuscripts, by David Lewis, Edited by Frederique Janssen-Lauret and Fraser MacBride.” Mind 135 (1): 244–50. "https://doi.org/10.1093/mind/fzae064.

Time travel and sortal-relative predication

Mon, 08 Jun 2026 13:49:59 +0000

Here's an attractive picture. All there really is, at a fundamental level, are fields in spacetime (or something like that). The world as we know it, with its rocks and chairs and cats and people, somehow emerges from this basis: all truths about rocks and chairs etc. are made true by truths about fields in spacetime. But how? To explain this, it would help if we could locate the familiar objects – rocks and chairs etc. – in the physical description of reality. With the help of classical mereology, which is "plausibly analytic, this seems possible: ordinary objects can be identified with aggregates of spacetime points. They are regions in spacetime. With this, we can explain how simple facts involving ordinary objects can emerge. For example, what makes it true that my chair has steel legs is that its region has a certain kind of subregion with high-amplitude excitations of quark and electron fields in a certain arrangement.

On this picture, objects are sortal-independent. If a lump of clay is created in the form of a statue and later destroyed in an explosion, the statue and the lump of clay are the same object. They are the same region in spacetime, with the same field values.

To be sure, we sometimes draw a distinction between the statue and the lump. The lump could survive smashing, the statue couldn't. But perhaps these modal predications require a special analysis. Counterpart theory provides the standard solution (as in "Lewis 1986, 253–58). "Fine 2003 claims that the statue and the lump also differ in non-modal properties: the statue is Romanesque (say), the lump isn't. I'm not sure I agree, but suppose it's true. I can certainly imagine that we learn to speak this way. What would this show?

Surely it wouldn't follow that there is more in fundamental reality than fields in spacetime. It would only follow that the simple analysis of ordinary predications doesn't work: we couldn't say that an ordinary "a is F" statement is true because "a" designates a certain physically specifiable object, "F" expresses a physically specifiable property, and the object in question has that property. Instead, we might say that ordinary singular terms are associated with a referent and a sort. The referent is a spacetime region; the sort isn't really an object, but an index that affects the interpretation of predicates: "a is F" is true because the region picked out by "a" has the physical property expressed by "F-qua-G", where G is the sort (compare "Loets 2021). If even that fails, we might have to go for a syncategorematic analysis of the kind we probably need for statements like "The oil price is rising". ("The oil price" hardly picks out a region of spacetime.)

Anyway, we might hope that there is at least a wide range of ordinary "a is F" predications whose truth or falsity can be explained by assuming that "a" picks out a certain spacetime region and "F" a physical property of spacetime regions.

But now suppose we live in a universe in which time travel is possible. And suppose some ordinary object (a rock, a chair, a person, or even an electron) travels back in time, but only a short stretch, so that it arrives at a time at which it already existed. Concretely, let's say Tim travels back in time in order to warn his younger self of the perils of time travel. He then exists twice at the relevant time t, once in an older form, once in a younger form. What shall we say about Tim at t?

If young Tim weighs 70 kg at t and old Tim 80 kg, we don't want to say that Tim weighs 150 kg at t. But if "Tim" picks out the relevant spacetime region, why doesn't this region have a mass of 150 kg at t?

Compare Tim's bikini, which happens to be lying on the floor at t. (Don't ask.) The bikini consists of a top part and a bottom part. Its total mass is the sum of the masses of the two parts. So when we consider the bikini as a spacetime region R, and we ask about its mass at t, we have to add up all the mass in R at t. (Very loosely speaking.) Why isn't this true for Tim?

The problem obviously generalizes beyond mass. We could ask about Tim's shape, his volume, his velocity, his temperature, his electric charge. In each case, the intuitive answer doesn't aggregate the older and younger Tim. Not even if they stand very closely together.

It's not entirely clear what we want to say about Tim at t. I guess we want to say that he weighs both 70 kg and 80 kg. That's OK. We can give an explanation of these predications. Roughly: "Tim weighs 70 kg at t" is true because "Tim" picks out a spacetime region that divides into parts which stand in a certain relation – the "R-relation" of "Lewis 1976 – to each other, and one of these parts is entirely located at t and has a mass of 70 kg. (See "Wasserman 2018, sec. 9.2.)

But here's the problem. This analysis of "Tim weighs 70 kg at t" seems to crucially rely on the fact that we think of Tim as a person. If Tim is simply a region in spacetime, we should be able to think of him in other ways as well. Suppose we think of him as "the region with such-and-such boundaries", or as "the mereological fusion of Tim before the time-travel and Tim after the time-travel". If we ask about the shape or mass (etc.) of these objects, we get a different answer. We get the bikini-style answer on which the mass at t is 150 kg.

So even predications involving mass, shape, volume, temperature, and velocity are sortal-relative.

Fine, Kit. 2003. “The Non-Identity of a Material Thing and Its Matter.” Mind 112: 195–234.

Lewis, David. 1976. “Survival and Identity.” In The Identities of Persons, edited by Amélie Oksenberg Rorty, 17–40. Berkeley: University of California Press. "https://doi.org/10.1525/9780520353060-002.

Lewis, David. 1986. On the Plurality of Worlds. Malden (Mass.): Blackwell.

Loets, Annina J. 2021. “Qua Qualification.” Philosopher’s Imprint 21 (27).

Wasserman, Ryan. 2018. Paradoxes of Time Travel. Oxford, New York: Oxford University Press.

Teaching mathematical logic

Thu, 30 Apr 2026 15:38:00 +0000

I taught two courses this year that I haven't taught before. One of them was our 4th-year undergraduate course on mathematical logic, "Logic, Computability, and Incompleteness". As usual, I ended up writing my own textbook. "Here it is as PDF and "here as HTML.

Why yet another textbook? Two reasons mainly. One is that many existing textbooks are addressed at maths students. This shows up not only in the examples and illustrations, but also in the fact that comparatively little time is spent motivating, explaining, and discussing definitions, proof ideas, or results. I wanted more of that.

Second, and relatedly, I wanted to cover more than what's covered in most existing textbooks. Many of my students have never heard of compactness, ZFC, ordinals, or the arithmetical hierarchy. But these are interesting and important topics. I obviously couldn't cover all of them in depth, but I think it's useful to have at least heard of them.

The course was reasonably well received, especially given that it was a first run and scheduled at 9 am. Nonetheless, I feel ambivalent about it. The topic is fascinating and rewarding. But it's also, in many ways, settled. The central results were established in the 1930s, almost a century ago. There are hardly any accessible open questions. This is in strong contrast to, for example, the decision theory course that I often teach for 4th-year students. That course also has a formal character, but it constantly touches on issues that are under debate: almost every week I can tell my students a few questions that would make good PhD projects. (And postgraduate research is naturally on many minds in the final undergrad year.) I couldn't do this in the logic course. But I might try again.

The tyranny of the objective

Fri, 20 Feb 2026 16:39:33 +0000

A widely held view in philosophy is that ordinary information and ordinary belief are concerned with "objective" propositions whose truth-value doesn't vary between perspectives or locations within a world.

Some hold that all genuine content is objective, and that the appearance of counterexamples is an illusion that can somehow be explained away. (See, e.g., "Stalnaker 1981, "Magidor 2015, or "Cappelen and Dever 2013.) Even those who accept that there is genuinely perspectival or self-locating information tend to treat it as a special case that requires special rules for integration with ordinary, non-perspectival information. (See, e.g., "Bostrom 2002, "Meacham 2008, "Moss 2012, "Titelbaum 2013, "Builes 2020, or "Isaacs, Hawthorne, and Russell 2022).

I think of this as the tyranny of the objective. (Compare "Chalmers 1998.)

In my view, all ordinary belief, and all ordinary information, is perspectival. Our senses tell us how things are here and now, around us. By scientific experiments and observations, we can find out more about our solar system or about the biology of organisms on our planet. When we learn such facts, we may also learn objective facts: by coming to know that it is raining, I also come to know that it is raining somewhere in the history of the world. But this objective belief is unusual and derivative. Ordinary confirmation is always confirmation of perspectival hypotheses by perspectival evidence.

"Lewis 1979 explained how this can be modelled formally. We simply need to replace the uncentred worlds of traditional confirmation theory with centred worlds.

The clearest sign that something is amiss with the objectivist mainstream is that it can't account for elementary facts about reasoning with perspectival information. As "Isaacs, Hawthorne, and Russell 2022, 252 put it: "All the precise theories we know of face very serious objections." I agree.

Of course, dropping the objectivist starting point doesn't automatically solve the difficult puzzles discussed in "Bostrom 2002 or "Isaacs, Hawthorne, and Russell 2022. But I think it sets the ground for a solution, and explains where certain arguments go wrong.

To see what I mean, let's have a closer look at "Isaacs, Hawthorne, and Russell 2022.

The paper explores whether our evidence supports the hypothesis that there are many universes. In the main part of the paper, the authors (henceforth, IRH) assume that centred credences are derived from uncentred priors by special rules. In Appendix B, IRH consider the possibility of starting with centred priors, but they argue that this doesn't affect the conclusion that our evidence supports the multiverse hypothesis.

Concretely, IRH prove two theorems. The theorems are complicated, but a simple example illustrates the key moves.

Let H1 be the hypothesis that there is exactly one universe, and H2 the hypothesis that there are two universes. Assume that each universe has a fixed chance p of being inhabited. Assume that p < 0.5. For simplicity, let's assume that an inhabited universe contains exactly one centre from which it is observed. The evidence that is received at such a centre is "local" insofar as it doesn't reveal anything about what might be the case in other universes. But it reveals (among other things) that this universe is inhabited.

Does such evidence support H2 over H1?

To answer this question, we need some assumptions about the (centred) priors.

Let Pr be a rational prior credence function. Let I=1 be the hypothesis that there is exactly one inhabited universe. Since the chance of any universe being inhabited is p, we might expect that

(1)Pr(I=1 | H1) = p and

(2)Pr(I=1 | H2) = 2p(1-p).

For (1), the idea is that if there's just one universe, and any universe has a fixed chance p of being inhabited, then the probability of this one universe being inhabited is p.

For (2), we assume that there are two universes, U1 and U2. Each has an independent chance p of being inhabited. There are two ways for there to be exactly one inhabited universe: U1 is inhabited and U2 isn't, or U2 is inhabited and U1 isn't. Each scenario has probability p(1-p). So the total probability of I=1 is 2p(1-p).

Now let E be our evidence. Plausibly,

(3)Pr(E | H1 ∧ I=1) = Pr(E | H2 ∧ I=1).

The idea here is that our evidence E is not made any more or less probable by the presence of a second, uninhabited universe.

Since p < 0.5, it follows (by a little maths) that Pr(E | H2) > Pr(E | H1). And so E supports H2 over H1.

IRH's "Theorem 3" generalizes this result.

The point I want to make is that this line of reasoning is highly dubious if we take centred priors seriously.

Let's return to the priors. We have to make a choice: Should the prior Pr assign positive probability only to inhabited points, or can it also assign positive probability to uninhabited points?

This is a somewhat arcane theoretical question, since any evidence will immediately rule out uninhabited points anyway.

Suppose we decide that Pr assigns positive probability only to inhabited points. Then (1) is false. Given that there is exactly one universe, the prior probability that this universe is inhabited must be 1, not p. (2) is also false. In general, on this approach we can't assume that the prior probabilities align with the chances.

We can hold on to (1) and (2) only if we allow uninhabited points to have positive prior probability. But if we do that, we should give up (3).

To see why, let be a two-universe world in which E is true at the first universe and the second universe is uninhabited. Let be a one-universe world in which E is true. If Pr assigns positive probability to both locations in then Pr(E | ) is less than 1, while Pr(E | ) is 1. So Pr(E | H2 ∧ I=1) < Pr(E | H1 ∧ I=1).

There is no plausible view on which (1)-(3) are all true. So we don't get "Theorem 3". Nor do we get the stronger "Theorem 4".

IRH assume that the prior Pr assigns positive probability only to inhabited points, except in worlds that are entirely uninhabited: here, the prior assigns positive probability to a "dummy" centre. This is formally consistent and makes it possible to accept (1)-(3), but it is an entirely implausible account of rational priors.

Bostrom, Nick. 2002. Anthropic Bias: Observation Selection Effects in Science and Philosophy. New York: Routledge.

Builes, David. 2020. “Time-Slice Rationality and Self-Locating Belief.” Philosophical Studies 177 (10): 3033–49. "https://doi.org/10.1007/s11098-019-01358-1.

Cappelen, Herman, and Josh Dever. 2013. The Inessential Indexical. Oxford: Oxford University Press.

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