Posts on: Self Location
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'm moderately confident that I don't live in a computer simulation.
My reasoning goes like this.
A priori, simulation scenarios are less probable than
non-simulation scenarios.
My evidence is more likely in non-simulation scenarios than in
simulation scenarios.
So: It is highly improbable, given my evidence, that I'm in a
simulation scenario.
By a "simulation scenario", I mean a scenario in which a subject's
experiences of themselves and their environment are generated by a
computer program that simulates an ordinary (non-simulated) subject and
their environment.
I assume that it is a priori possible for a computer program to
generate experiences (and a "subject") by simulating an ordinary subject
with experiences. I'm not 100% sure this is true. (If not, premise 1 can
be strengthened: simulation scenarios have probability 0.) But it seems
plausible, especially if we're liberal about what qualifies as a
computer program and as a simulation.
Sensory information is centred. Right now, for example, my visual
system conveys to me that there's a red wall about 1 metre
ahead (among much else); it does not convey that Wolfgang
Schwarz is about 1 metre away from a red wall on 22 January 2026 at
12:04 UTC.
We can quibble over what exactly is part of the sensory information.
We can also quibble over what "sensory information" is even meant to be.
But it should be uncontroversial that we gain information from our
senses. My point is that, on any plausible way of spelling this out, the
information we receive is centred: it doesn't have parameters that fix a
unique location in space and time. If I were unsure about what time it
is or who I am, looking at the wall in front of me wouldn't help. The
underlying reason, of course, is that photoreceptors are insensitive to
differences in spatiotemporal location: they don't produce different
outputs depending on where or when they are activated by photons.
The standard dynamic norm of Bayesianism,
conditionalization, is clearly inadequate if credences are
defined over self-locating propositions. How should it be adjusted?
This question was popular at around 2005-2015. Chris Meacham and I
came up with the same answer, which we published in (Meacham 2010),
(Schwarz
2012), and (Schwarz 2015). I showed that the
replacement norm that we proposed has all the traditional virtues of
conditionalization. For example, (under the usual idealized conditions)
following the norm uniquely maximizes expected accuracy, and an agent is
invulnerable to diachronic Dutch books iff they follow the norm.
In 2009, at the ANU, Mike Titelbaum organized a small workshop on the Sleeping Beauty problem. I gave a talk in which I argued that the answer to the problem depends on whether we accept genuinely diachronic norms on rational belief: if yes, halfing is the most plausible answer; if no, we get thirding. A successor of this talk is now forthcoming in Noûs. Here's a PDF. In this post, I want to discuss a surprisingly hard question Kenny Easwaran raised in the Q&A after my talk:
How confident should Beauty be on Wednesday that the coin has landed heads?
I occasionally teach the doomsday argument in my philosophy classes, with the hope of raising some general questions about self-locating priors. Unfortunately, the usual formulations of the argument are problematic in so many ways that it's hard to get to these questions.
Let's look at Nick Bostrom's version of the argument, as presented for example in Bostrom (2008).
Wilhelm (2021) and Lando (2022) argue that the Sleeping Beauty problem reveals a flaw in standard accounts of credence and chance. The alleged flaw is that these accounts can't explain how attitudes towards centred propositions are constrained by information about chance.
I assume you remember the Sleeping Beauty problem. (If not, look it up: it's fun.) Wilhelm makes the following assumptions about Beauty's beliefs on Monday morning.
First, Beauty can't be sure that it is Monday:
There are two paths to Shangri La. One goes by the sea, the other by the mountains. You are on the mountain path and about to enter Shangri La. You can choose how your belief state will change as you enter through the gate, in response to whatever evidence you may receive. At the moment, you are (rationally) confident that you have travelled by the mountains. You know that you will not receive any surprising new evidence as you step through the gate. You want to maximize the expected accuracy of your future belief state – at least with respect to the path you took. How should you plan to change your credence in the hypothesis that you have travelled by the mountains?
If a certain hypothesis entails that N percent of all observers in the universe have a certain property, how likely is it that we have that property – conditional on the hypothesis, and assuming we have no other relevant information?
Answer: It depends on what else the hypothesis says. If, for example, the hypothesis says that 90 percent of all observers have three eyes, and also that we ourselves have two eyes, then the probability that we have three eyes conditional on the hypothesis is zero.
This effect is easy to miss because many hypotheses that appear to be just about the universe as a whole secretly contain special information about us. Consider the following passage from Carroll (2010), cited in Arntzenius and Dorr (2017):
Imagine you and I are walking down a long path. You are ahead,
but we can communicate on the phone. If you say, "there are strawberries here" and I trust you, I should not come to believe that there
are strawberries where I am, but that there are strawberries wherever
you are. If I also know that you are 2 km ahead, I should come to
believe that there are strawberries 2 km down the path. But what's the
general rule for deferring to somebody with self-locating beliefs?
What makes the Sleeping Beauty problem non-trivial is Beauty's
potential memory loss on Monday night. In my view, this means that
Sleeping Beauty should be modeled as a case of potential epistemic
fission: if the coin lands tails, any update Beauty makes to her
beliefs in the transition from Sunday to Monday will also fix her
beliefs on Tuesday, and so the Sunday state effectively has two
epistemic successors, one on Monday one on Tuesday. All accounts of
epistemic fission that I'm aware of then entail halfing.
It is widely agreed that conditionalization is not an adequate norm
for the dynamics of self-locating beliefs. There is no agreement on
what the right norms should look like. Many hold that there are no
dynamic norms on self-locating beliefs at all. On that view, an
agent's self-locating beliefs at any time are determined on the basis
of the agent's evidence at that time, irrespective of the earlier
self-locating belief. I want to talk about an alternative approach
that assumes a non-trivial dynamics for self-locating beliefs. The
rough idea is that as time goes by, a belief that it is Sunday should
somehow turn into a belief that it is Monday.
Let's look at the third type of case in which credences can come apart from known chances. Consider the following variation of the Sleeping Beauty problem (a.k.a. "The Absentminded
Driver"):
Before Sleeping Beauty awakens on Monday, a coin is
tossed. If the coin lands tails, Beauty's memories of Monday will be
erased the following night, and the coin will be tossed again on
Tuesday. If the Monday toss lands heads, no memory erasure or further
tosses take place. Beauty is aware of all these facts.
When Beauty awakens on Monday morning and learns that today's toss
has landed tails (alternatively: that the Monday toss has landed
tails), how should that affect her credence in the hypothesis that the
coin is fair?
Fred has bought a duplication machine at a discount from a series
in which 50 percent of all machines are broken. If Fred's machine
works, it will turn Fred into two identical copies of himself, one
emerging on the left, the other on the right. If Fred's machine is
broken, he will emerge unchanged and unduplicated either on the left
or on the right, but he can't predict where. Fred enters his machine,
briefly loses consciousness and then finds himself emerge on the
left. In fact, his machine is broken and no duplication event has
occurred, but Fred's experiences do not reveal this to him.
An evil scientist might have built a brain in vat that has all the
experiences you currently have. On the basis of your experiences, you
cannot rule out being that brain in a vat. But you can rule out
being that scientist. In fact, being that scientist is
not a skeptical scenario at all. For example, if the scientist in question
suspects that she is a scientist building a brain in a vat, then that
would not constitute a skeptical attitude.
Given some evidence E and some proposition P, we can ask to what
extent E supports P, and thus to what extent an agent should believe P
if their only relevant evidence is E. The question may not always have
a precise answer, but there are both intuitive and theoretical reasons
to assume that the question is meaningful – that there is a kind
of (imprecise) "evidential probability" conferred by evidence on
propositions. That's why it makes sense to say, for example, that one
should proportion one's beliefs to one's evidence.
In 2008, I wrote a post on Stalnaker on self-location,
in which I attributed a certain position to Stalnaker and raised some
objections. But the position isn't actually Stalnaker's. (It might be
closer to Chisholm's). So here is another attempt at figuring out
Stalnaker's view. (I'm mostly drawing on chapter 3 of Our Knowledge
of the internal world (2008), chapter 5 of Context (2014),
and a forthcoming paper called "Modeling a perspective on the world"
(2015).)
Plausible moral theories should be agent-relative. They should
permit us to care more about close friends than about distant
strangers. They can prohibit killing ten innocent people even in
circumstances where eleven innocent people would otherwise be killed
by somebody else. They might say that it would be right for Alice to
dance with Bob, but wrong for Bob to dance with Alice.
But how should we think about agent-relative values? It may seem
that the state of affairs in which Alice dances with Bob is either
right or not right. How could it be right relative to Alice but wrong
relative to Bob? Or consider a case where I can prevent you from
killing eleven by killing ten myself. If it is wrong that you kill the
eleven, then surely I have a moral reason to see to it that you don't
kill the eleven, just as I have a moral reason to see to it that I
don't kill the ten. Moreover, presumably it is worse if you kill
eleven than if I kill ten. So shouldn't my reason to prevent you from
killing the eleven outweigh my reason to not kill the ten?
Dilip Ninan has also argued on a number of occasions that attitude
contents cannot in general be modelled by sets of qualitative centred
worlds; see especially his "Counterfactual
attitudes and multi-centered worlds" (2012). The argument is
based on an alleged problem for the centred-worlds account applied to what he
calls "counterfactual attitudes", the prime example being imagination.
Since the problem concerns the analysis of attitudes de re,
we first have to briefly review what the centred-worlds account might
say about this. Consider a de re belief report "x believes that y is
F". Whether this is true depends on what x believes about y, but if
belief contents are qualitative, we cannot simply check whether y is F
in x's belief worlds. We first have to locate y in these
qualitative scenarios. A standard idea, going back to Quine, Kaplan
and Lewis, is that the belief report is true iff there is some
"acquaintance relation" Q such that (i) x is Q-related uniquely to y
and (ii) in x's belief worlds, the individual at the centre is
Q-related to an individual that is F. For example, if Ralph sees
Ortcutt sneaking around the waterfront, and believes that the guy
sneaking around the waterfront is a spy, then Ralph believes de re of
Ortcutt that he is a spy.
If we want to model rational degrees of belief as probabilities,
the objects of belief should form a Boolean algebra. Let's call the
elements of this algebra propositions and its atoms (or
ultrafilters) worlds. Every proposition can be represented as a
set of worlds. But what are these worlds? For many applications, they
can't be qualitative possibilities about the universe as a whole, since
this would not allow us to model de se beliefs. A popular
response is to identify the worlds with triples of a possible universe,
a time and an individual. I prefer to say that they are maximally
specific properties, or ways a thing might be. David Chalmers (in
discussion, and in various papers, e.g. here and there) objects that
these accounts are not fine-grained enough, as revealed by David
Austin's "two tubes" scenario. Let's see.
If beliefs are modeled by a probability distribution over centered
worlds, belief update cannot work simply by conditionalisation. How
then does it work? The most popular answer in philosophy goes as
follows.
Let P an agent's credence function at time t1, P' the credence function
at t2, and E the evidence received at t2. Since E is a centered
proposition, it can be true at multiple points within a world.
Suppose, however, that the agent assigns probability 0 to worlds at
which E is true more than once. Then to compute P', first
conditionalise P on the uncentered fragment of E -- i.e. the strongest
uncentered proposition entailed by E. This rules out all worlds at
which E is true nowhere. Second, move the center of each remaining
world to the (unique) point at which E is true.
A lot has been written in the last 10 years or so on updating
self-locating beliefs, mostly in the context of the Sleeping Beauty
problem. One thing almost all of these papers have in common is that
they quote Lewis's remark in "Attitudes de dicto and de se" (1979,
p.534), where he says:
it is interesting to ask what happens to decision theory
if we take all attitudes as de se. Answer: very little. We replace the
space of worlds by the space of centered worlds, or by the space of
all inhabitants of worlds. All else is just as before.
This is supposed to imply that Lewis took standard
conditionalisation to be the correct update rule for self-locating
belief.
Suppose tonight you will fission into two persons. One of your
successors will wake up Mars and one on Venus. There are then two
possibilities for how things might be for you tomorrow: you
might wake up on Mars, and you might wake up on Venus. These are
distinct centered possibilities that do not correspond to distinct
uncentered possibilties. There is just one possibility for the
world, but two possibilities for you. Indeed, the two possibilities
are two actualities: you will wake up on Mars, and you will
wake up on Venus. It is tempting to go further and say that there are also two
possibilities for you now. I want to discuss three quite
different reasons for making this move.
One of the grave threats to the development of mankind in general,
and philosophy in particular, is the assumption that the objects of
propositional attitudes can be expressed by that-clauses. The
assumption is often smuggled in via a definition, e.g. when propositions
are defined as things that are 1) objects of attitudes and 2)
expressed by that-clauses. No effort is made to show that anything
satisfies both (1) and (2) -- let alone that the things that satisfy (1)
coincide with the things that satisfy (2).
Rational credence should match the expectation of objective
chance. Here I will have a brief look at what happens
to this connection between credence and chance on the assumption that
credence is centered and chance is not.
1. Fixing the time. Both credences and chances evolve over time. When a
coin is tossed twice, the chance of two heads may initially be 1/4;
after the first toss has come up heads, it is 1/2. So when your
beliefs should match the assumed chance, it can only match the chance
you assume to obtain at some particular time. At what time?
In the last entry, I have suggested that
EEP) P_2(A) = P_1(+A|+E)
is a sensible rule for updating self-locating beliefs. Here, E is the
total evidence received at time 2 (the time of P_2), and '+' denotes a
function that shifts the evaluation index of propositions, much like
'in 5 minutes': '+A' is true at a centered world w iff A is true at
the next point from w where new information is received. (EEP) therefore
says that upon learning E, your new credence in any proposition A
should equal your previous conditional credence that A will obtain at the next
point when information comes in, given that this
information is E.
I've been participating in a couple of workshops here at ANU lately,
and I thought I'd share some notes. First, we had a little Sleeping Beauty workshop where Terry Horgan
and Mike Titlebaum defended thirding, and me halfing. Unfortunately, I
think we didn't quite get to the heart of our disagreement. Each of us
said their own thing, without saying enough about what's wrong with
the reasoning of the other sides. So I'll do that here. I start with
Terry's account.
Here are some notes on Stalnaker's account of self-locating beliefs,
in chapter 3 of Our Knowledge of the Internal World. I find the
discussion there slightly intransparent, so I'll start with a
presentation of what I take to be Stalnaker's account, but in my own
words. This will lead to a few objections further down.
We start with extreme haecceitism. Every material object and every
moment in time has, in addition to its normal, qualitative properties
also a non-qualitative property, its 'haecceity', that distinguishes it
from everything else. My haecceity belongs to me with metaphysical
necessity, and could not belong to anyone else. Moreover, it is my only
(non-trivial) essential property. (This is the 'extreme' part in extreme
haecceitism.) In this world, I am a human being, but in other worlds, I
am a cockatoo, or a poached egg. My haecceity is freely combinable with
any qualitative property.
Continuing the topic of the last post, suppose I'm certain that no-one
else in the history of the universe ever had (or will have) exactly the experiences
that I have now. Then I can 'translate' any centered proposition into
an uncentered propositions in such a way that the translation is certain to preserve
truth-values. For instance, "it is raining" gets translated into
"it is raining at all times and places where someone has such-and-such
experiences". In this case, one might think, purely centered information can never
affect my uncentered beliefs. For purely centered information only distinguishes
between multiple centers within a single world; but if no world has multiple
possible centers, then there is nothing to learn from such information.
(This line of reasoning is related to what Mike Titelbaum says
in his forthcoming paper "The Relevance of Self-Locating Beliefs", though I
don't think Mike would endorse the argument I present here.)
Darks clouds are gathering. Soon it will be raining. When it does, I will
believe that it is raining. I do not yet believe that it
is raining even though I do believe that my well-informed future self
will believe that it is raining. I thereby violate the 'Principle of
Reflection'. Once we allow for centered propositions that change their truth-value between times
and places, Reflection, like its close cousin Conditioning, become very implausible
norms of rationality.
Hey there. I've been a bit busy moving house, sitting in the garden, watching the falling leaves, etc. I've also thought some more about the absentminded driver. Here's something odd: on a certain interpretation of this case, we get a an unstable decision problem that remains interestingly unstable even when mixing (randomization) is allowed.
Some background. A decision problem is unstable if a decision to do one thing
inevitably makes another thing preferable. In a classic example, Death, who is very good at predicting people's whereabouts, has predicted where you will be tomorrow and awaits you there. Should you stay where you are (in Damascus) or flee to Aleppo?
A curious aspect of the Sleeping Beauty debate is the role of Dutch Books. At first sight, it looks as if Dutch Book considerations support thirding (see e.g. Hitchcock 2004). However, as Halpern 2006 shows, Beauty can also be Dutch Booked if she is a thirder. Some have argued that these arguments might fail because in Sleeping Beauty type cases, credences and betting odds can come apart (see e.g. Bradley and Leitgeb 2006). I disagree. Instead, I will argue that her vulnerability to Dutch Books doesn't show that Beauty is irrational -- at least not if she is a halfer.
Suppose beliefs locate us in centered logical space: to believe something is to rule out not only ways a universe might be, but ways things might be for an individual at a time. Then there will be two kinds of rational belief change: we can learn something new about our present situation, and we can change our situation and adjust our beliefs to this change. The rule for changes of the first kind is conditionalization. The rule for changes of the second kind doesn't have an official name yet, as far as I know. (In the AGM/KM framework, it is called "update", but we Bayesians often use "update" for conditioning.) In practice, the two rules always go hand in hand: you never learn something new without changing your situation, and you hardly ever change your situation without learning anything new.
In this paper, I try to spell out the two rules, and their combination: Believing in afterlife: conditionalization in a changing world (PDF).
I'm a bit unhappy with some parts of the story, and I should probably say more about alternative accounts in the literature, and why I don't like them. So hopefully there will be an update soon. In the meantime, comments are as always very welcome!
This is a follow-up to the previous post on Shangri La. As before, the story is that a fair coin decides which path you take to Shangri La: on heads, you travel by the Mountains, on tails, by the Sea. If you arrive at Shangri La via the Sea, the guardians will replace your Sea memories with Mountain memories.
In the other post, I said that if you actually traveled by the Mountains, you should remain confident that you traveled by the Mountains, even though you would have ended up with the same evidence had you traveled by the Sea.
(This is more or less the talk I gave at the "Epistemology at the Beach" workshop last Sunday.)
"A wise man proportions his belief to the evidence", says Hume. But to what evidence? Should you proportion your belief to the evidence you have right now, or does it matter what evidence you had before? Frank Arntzenius ("Some problems for conditionalization and reflection", JoP, 2003) tells a story that illustrates the difference:
...there is an ancient law about entry into Shangri La:
you are only allowed to enter, if, once you have entered, you no
longer know by what path you entered. Together with the guardians you
have devised a plan that satisfies this law. There are two paths to
Shangri La, the Path by the Mountains, and the Path by the Sea. A fair
coin will be tosssed by the guardians to determine which path you
will take: if heads you go by the Mountains, if tails you go by the
Sea. If you go by the Mountains, nothing strange will happen: while
traveling you will see the glorious Mountains, and even after you
enter Shangri La you will for ever retain your memories of that
Magnificent Journey. If you go by the Sea, you will revel in the
Beauty of the Misty Ocean. But just as you enter Shangri La, your
memory of this Beauteous Journey will be erased and replaced by a
memory of the Journey by the Mountains.
What about this much simpler argument for halfing:
As usual, Sleeping Beauty wakes up on Monday, knowing that she will have an indistinguishable waking experience on Tuesday iff a certain fair coin has landed tails. Thirders say her credence in the coin landing heads should be 1/3; halfer say it should be 1/2.
Now suppose before falling asleep each day, Beauty manages to write down her present credence in heads on a small piece of paper. Since that credence was 1/2 on Sunday evening, she now (on Monday) finds a note saying "1/2".
I've thought a bit about belief update recently. One thing I noticed is that it is often assumed in the literature (usually without argument) that if you know that there are two situations in your world that are evidentially indistinguishable from your current situation, then you should give them roughly the same credence. Although I agree with some of the applications, the principle in general strikes me as very implausible. Here is a somewhat roundabout counter-example that has a few other interesting features as well.
Does the semantic value of expressions in a language sometimes depend on other things than their utterance context? That depends on what is meant by "semantic value", but for the most part, I think not.
It can appear otherwise if one identifies the content of an utterance with the main proposition it conveys to competent hearers.
Alice, Bob and Carol are searching for honey. Alice sees a bee hive on a tree near Bob and wants to inform both Bob and Carol about this. That is, she wants Bob to acquire the self-locating belief that there is a bee hive on the tree near him, and she wants Carol to acquire the belief that there is a bee hive on the tree over there near Bob. She achieves both goals simultaneously by pointing at the relevant tree and saying, "there's a bee hive on the tree over there".
Since Alice conveys two different (centered) propositions to Bob and Carol with her sentence, one might conclude that her sentence expresses two different contents, one relative to Bob's context of assessment and one relative to Carol's. Content, then, is relative to both an utterance context and an assessment context. However, it is quite implausible that Alice's utterance really has these two propositions as its literal semantic value. Instead, what she expressed was just the proposition that there is a bee hive on the tree she is pointing at, and Bob and Carol figured out the centered propositions they were meant to learn from this information.
I've been assigned some boring administrative work, but that's finished now, I hope. Here are some rough thoughts on indifference and Adam Elga's Dr. Evil paper (PDF).
There are many possible individuals whose mental state is subjectively indistinguishable from my current mental state insofar as they all share my current phenomenal experiences and my (real or quasi-) memories. Some of them inhabit worlds that are exactly as I believe the actual world is, and are located in that world exactly where I believe I am located in the actual world. Others occupy very different places in very different worlds: they are brains in vats or inhabitants of gruesome counterinductive worlds. How should I distribute my credence among all these possibilities?
Another nice
problem from Brian Weatherson's weblog: Farrington is 50% confident
that it's after 4:30, and 50% confident that a certain coin
landed tails. Now he comes to know that iff the coin landed tails, some
researchers create a brain-in-a-vat duplicate of himself at exactly 4:30
today. What are the probabilities he should assign to the 5 open
possibilities: