Posts on: Epistemology
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?
Standard decision theory studies one-shot decisions, where an agent faces a single choice. Real decision problems, one might think, are more complex. To find the way out of a maze, or to win a game of chess, the agent needs to make a series of choices, each dependent on the others. Dynamic decision theory (aka sequential decision theory) studies such problems.
There are two ways to model a dynamic decision problem. On one approach, the agent realizes some utility at each stage of the problem. Think of the chess example. A chess player may get a large amount of utility at the point when she wins the game, but she plausibly also prefers some plays to others, even if they both lead to victory. Perhaps she enjoys a novel situation in move 23, or having surprised her opponent in move 38. We can model this by assuming that the agent receives some utility for each stage of the game. The total utility of a play is the sum of the utilities of its stages.
A common worry about mathematical platonism is how we could know about an independent realm of mathematical facts. The same kind of worry arises for moral realism: if there are irreducible moral facts, how could we have access to them?
Benacerraf (1973) put the problem in terms of causation. Knowledge of maths, he suggested, would require some kind of causal connection between the mathematical facts and our mathematical beliefs, but modern platonists typically don't believe in such a connection.
Humean accounts of physical laws seem to have an advantage when it comes to explaining our epistemic access to the laws: if the laws are nothing over and above the Humean mosaic, it's no big mystery how observing the mosaic can provide information about the laws. If, by contrast, the laws are non-Humean whatnots, it's unclear how we could get from observations of the mosaic to knowledge of the laws. This line of thought is developed, for example, in Earman and Roberts (2005). Chen (2023) (as well as Chen (2024)) argues that it rests on a mistake. Eddy suggests that Primitivists about physical laws have no more trouble explaining our epistemic access than friends of the Best-System Analysis.
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).
I want to say something about a passage in Christensen (2023) that echoes a longer discussion in Christensen (2007).
Here's a familiar kind of scenario from the debate about higher-order evidence.
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:
Covid finally caught me, so I fell behind with everything. Let's try get back to the blogging schedule. This time, I want to recommend DiPaolo (2019). It's a great paper that emphasizes the difference between ideal ("primary") and non-ideal ("secondary") norms in epistemology.
The central idea is that epistemically fallible agents are subject to different norms than infallible agents. An ideal rational agent would, for example, never make a mistake when dividing a restaurant bill. For them, double-checking the result is a waste of time. They shouldn't do it. We non-ideal folk, by contrast, should sometimes double-check the result. As the example illustrates, the "secondary" norms for non-ideal agents aren't just softer versions of the "primary" norms for ideal agents. They can be entirely different.
Often there are many reasons for and against a certain act or belief. How do these reasons combine to an overall reason? Nair (2021) tries to give an answer.
Nair's starting point is a little more specific. Nair intuits that there are cases in which two equally strong reasons combine to a reason that is twice as strong as the individual reasons. In other cases, however, the combined reason is just as strong as the individual reasons, or even weaker.
To make sense this, we need to explain (1) how strengths of reason can be represented numerically, and (2) under what conditions the strengths of different reasons add up.
Isaacs and Russell (2023) proposes a new way of thinking about evidence and updating.
The standard Bayesian picture of updating assumes that an agent has some ("prior") credence function Cr and then receive some (total) new evidence E. The agent then needs to update Cr in light of E, perhaps by conditionalizing on E. There is no room, in this picture, for doubts about E. The evidence is taken on board with absolute certainty.
The standard picture thereby assumes that the agent's cognitive system is perfectly sensitive to a certain aspect of the world: if E is true, the agent is certain to update on E; if E is false, the agent is certain to not update on E.
Internalism about justification is often supported by intuitions about cases. Srinivasan (2020) argues that these intuitions can't be trusted, because there are analogous cases in which they go in the opposite direction. I'll explain why I'm not convinced.
I should say that I'm not sure what this debate is about. Are we talking about some pre-theoretic folk concept of justification? Or about a concept that plays some important theoretical role? Srinivasan acknowledges (in footnote 10) that there might not be a single, precise folk concept of justification. I agree. To clarify her topic, she says that she is interested in the kind of justification that is a precondition for knowledge. This doesn't really help me. I think that 'knowledge' is context-dependent, and that it sometimes means no more than 'true belief'. There is no interesting justification condition that is present in every case of knowledge.
Some people – important people, like Richard Jeffrey or Brian Skyrms – seem to believe that Laplace and de Finetti have solved the problem of induction, assuming nothing more than probabilism. I don't think that's true.
I'll try to explain what the alleged solution is, and why I'm not convinced. I'll pick Skyrms as my adversary, mainly because I've just read Skyrms and Diaconis's Ten Great Ideas about Chance, in which Skyrms presents the alleged solution in a somewhat accessible form.
There's a striking tension in Lewis's philosophy. His epistemology and philosophy of mind, on the one hand, leave no room for (non-trivial) a priori knowledge or a priori inquiry. Yet for most of his career, Lewis was engaged in just this kind of inquiry, wondering about the nature of causation, the ontology of sets, the extent of logical space, the existence of universals, and other non-contingent matters. My paper "The problem of metaphysical omniscience" explores some options for resolving the tension. The paper has just come out in a volume, Perspectives on the Philosophy of David K. Lewis, edited by Helen Beebee and A.R.J. Fisher.
A popular idea in recent (formal) epistemology is that an externalist conception of evidence is somehow useful, or even required, to block the threat of skepticism. (See, for example, Das (2019), Das (2022), and Lasonen-Aarnio (2015). The trend was started by Williamson (2000).)
Greaves (2013) describes a case in which adopting a single false belief would (supposedly) be rewarded by many true beliefs.
Emily is taking a walk through the Garden of Epistemic Imps. A child plays on the grass in front of her. In a nearby summerhouse are n further children, each of whom may or may not come out to play in a minute. They are able to read Emily's mind, and their algorithm for deciding whether to play outdoors is as follows. If she forms degree of belief 0 that there is now a child before her, they will come out to play. If she forms degree of belief 1 that there is a child before her, they will roll a fair die, and come out to play iff the outcome is an even number. […]
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?
Let Ep mean that your evidence entails p. Let an externalist scenario be a scenario in which either Ep holds without EEp or ¬Ep holds without E¬Ep.
It is sometimes assumed, for example in Gallow (2021) and Isaacs and Russell (2022), that any externalist scenario is a scenario in which you have evidence that you don't rationally respond to your evidence. On the face of it, this seems puzzling. Why should there be a connection between evidential externalism and evidence of irrationality? But the assumption actually makes sense.
I've read around a bit in the literature on higher-order evidence. Two different ideas seem to go with this label. One concerns the possibility of inadequately responding to one's evidence. The other concerns the possibility of having imperfect information about one's evidence. I have a similar reaction to both issues. I haven't seen it in the papers I've looked at. Pointers very welcome.
I'll begin with the first issue.
Let's assume that a rational agent proportions her beliefs to her evidence. This can be hard. For example, it's often hard to properly evaluate statistical data. Suppose you have evaluated the data, reached the correct conclusion, but now receive misleading evidence that you've made a mistake. How should you react?
Some (e.g. Christensen (2010)) say you should reduce your confidence in the conclusion you've reached. Others (e.g. Tal (2021)) say you should remain steadfast and not reduce your confidence.
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):
In the previous post I argued that rational priors must favour some possibilities over others, and that this is a problem for Richard Pettigrew's model of Jamesian permissivism. It also points towards an alternative model that might be worth exploring.
I claim that, in the absence of unusual evidence, a rational agent should be confident that observed patterns continue in the unobserved part of the world, that witnesses tell the truth, that rain experiences indicate rain, and so on. In short, they should give low credence to various skeptical scenarios. How low? Arguably, our epistemic norms don't fix a unique and precise answer.
Pettigrew (2021) defends a type of permissivism about rational credence inspired by James (1897), on which different rational priors reflect different attitudes towards epistemic risk. I'll summarise the main ideas and raise some worries.
(There is, of course, much more in the book than what I will summarise, including many interesting technical results and some insightful responses to anti-permissivist arguments.)
Last week I gave a talk in which I claimed (as an aside) that if you update your credences by conditionalising on a true proposition then your credences never become more inaccurate. That seemed obviously true to me. Today I tried to quickly prove it. I couldn't. Instead I found that the claim is false, at least on popular measures of accuracy.
The problem is that conditionalising on a true proposition typically increases the probability of true propositions as well as false propositions. If we measure the inaccuracy of a credence function by adding up an inaccuracy score for each proposition, the net effect is sensitive to how exactly that score is computed.
I've been teaching a course on classical epistemology this term, so I've thought
a little about knowledge.
A common judgement in the literature seems to be that knowledge is incompatible
with a certain kind of luck -- the kind of luck we find in Gettier cases. This
is then cashed out in terms of safety: for a belief to constitute knowledge it
must be true in all nearby possible worlds.
While I share the initial judgement, the development in terms of safety doesn't
look plausible to me. It has the wrong kind of structure.
Teaching for this semester is finally over.
Last week I gave a talk in Umea at a workshop on singular thought. I was pleased
to be invited because I don't really understand singular thought. Giving
a talk, I hoped, would force me to have a closer look at the literature. But then I was
too busy teaching.
People seem to mean different things by 'singular thought'. The target of my
talk was the view that one can usefully understand the representational content
of beliefs and other intentional states as attributing properties to
individuals, without any intervening modes of presentation. This view is often
associated with a certain interpretation of attitude reports: whenever we can
truly say `S believes (or knows etc.) that A is F', where A is a name, then
supposedly the subject S stands in an interesting relation of belief (or
knowledge etc.) to a proposition directly involving the bearer of that name.
Last week, I gave a talk in Manchester at a
(very nice) workshop on "David Lewis and His Place in the History of Analytic
Philosophy". My talk was on "Lewis's empiricism". I've now written it up as a
paper, since it got too long for a blog post.
The paper is really about hyperintensional epistemology. The question is how we
can make sense of the kind of metaphysical enquiry Lewis was engaged in if we
accept his models of knowledge and belief, which leave no room for substantive
investigations into non-contingent matters.
On the modal analysis of belief, 'S believes that p' is true iff p is
true at all possible worlds compatible with S's belief state. So
'believes' is a necessity modal. One might expect there to be a dual
possibility modal, a verb V such that 'S Vs that p' is true iff p is
true at some worlds compatible with S's belief state. But there
doesn't seem to be any such verb in English (or German). Why not?
What do we use if we want to say that something is compatible with
someone's beliefs? Suppose at some worlds compatible with Betty's
belief state, it is currently snowing. We could express this by "Betty
does not believe that it is not snowing". But (for some reason) that's
really hard to parse.
Dutch Book arguments are often used to justify various epistemic
norms – in particular, that credences should obey the
probability axioms and that they should evolve by
condionalization. Roughly speaking, the argument is that if someone
were to violate these norms, then they would be prepared to accept
bets which amount to a guaranteed loss, and that seems
irrational.
But it's hard to spell out how exactly the argument is meant to go. In
fact, I'm not aware of any satisfactory statement. Here's my
attempt.
My paper "Imaginary
Foundations" has been accepted at Ergo (after rejections from
Phil Review, Mind, Phil Studies, PPR, Nous, AJP, and Phil
Imprint). The paper has been in the making since 2005, and I'm quite
fond of it.
The question I address is simple: how should we model the impact of
perceptual experience on rational belief? That is, consider a
particular type of experience – individuated either by its
phenomenology (what it's like to have the experience) or by its
physical features (excitation of receptor cells, or whatever). How
should an agent's beliefs change in response to this type of
experience?
According to the Principle of Indifference, alternative
propositions that are similar in a certain respect should be given
equal prior probability. The tricky part is to explain what should
count as similarity here.
Van Fraassen's cube factory nicely illustrates the problem. A
factory produces cubes with side lengths between 0 and 2 cm, and
consequently with volumes between 0 and 8 cm^3. Given this
information, what is the probability that the next cube that will be
produced has a side length between 0 and 1 cm? Is it 1/2, because the
interval from 0 to 1 is half of the interval from 0 to 2? Or is it
1/8, because a side length of 1 cm means a volume of 1 cm^3, which is
1/8 of the range from 0 to 8?
According to a popular picture, some beliefs are justified by "seemings": under
certain conditions, if it seems to you that P, then you are justified
to believe that P, without the assistance of other beliefs. So
seemings provide a kind of foundation for belief, albeit a fallible
kind of foundation.
But most of our beliefs are not justified by seemings (or by
beliefs which are justified by seemings, etc.). I once learned that
Luanda is the capital of Angola and I've retained this belief for many
years, although I rarely think about Angola and thus rarely experience
any relevant seemings that could justify the belief.
You observe a process that generates two kinds of outcomes, 'heads'
and 'tails'. The outcomes appear in seemingly random order, with
roughly the same amount of heads as tails. These observations support
a probabilistic model of the process, according to which the
probability of heads and of tails on each trial is 1/2, independently
of the other outcomes.
How observations about frequencies confirm or disconfirm
probabilistic models is well understood in Bayesian epistemology. The
central assumption that does most of the work is the Principal
Principle, which states that if a model assigns (objective)
probability x to some outcomes, then conditional on the model, the
outcomes have (subjective) probability x. It follows that models that
assign higher probability to the observed outcomes receive a greater
boost of subjective probability than models that assign lower
probability to the outcomes.
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.
A lot of what I do in philosophy is develop models: models of
rational choice, of belief update, of semantics, of communication,
etc. Such models are supposed to shed light on real-world phenomena,
but the connection between model and reality is not completely
straightforward.
For example, consider decision theory as a descriptive model of
real people's choices. It may seem straightforward what this model
predicts and therefore how it can be tested: it predicts that people
always maximize expected utility. But what are the probabilities and
utilities that define expected utility? It is no part of standard
decision theory that an agent's probabilities and utilities conform in
a certain way to their publicly stated goals and opinions. Assuming
such a link is one way of connecting the decision-theoretic model with
real agents and their choices, but it is not the only (and in my view
not the most fruitful) way. A similar question arises for the agent's
options. Decision theory simply assumes that a range of "acts" are
available to the agent. But what should count as an act in a
real-world situation: a type of overt behaviour, or a type of
intention? And what makes an act available? Decision theory doesn't
answer these questions.
There has been a lively debate in recent years about the
relationship between graded belief and ungraded belief. The debate
presupposes something we should regard with suspicion: that there is
such a thing as ungraded belief.
Compare earthquakes. I'm not an expert on earthquakes, but I know
that they vary in strength. How exactly to measure an earthquake's
strength is to some extent a matter of convention: we could have used
a non-logarithmic scale; we could have counted duration as an aspect
of strength, and so on. So when we say that an earthquake has
magnitude 6.4, we characterize a central aspect of an earthquake's
strength by locating it on a conventional scale.
In discussions of the raven paradox,
it is generally assumed that the (relevant) information gathered from an
observation of a black raven can be regimented into a statement of the
form Ra & Ba ('a is a raven and a is
black'). This is in line with what a lot of "anti-individualist" or
"externalist" philosophers say about the information we acquire
through experience: when we see a black raven, they claim, what we
learn is not a descriptive or general proposition to the effect that
whatever object satisfies such-and-such conditions is a black raven,
but rather a "singular" proposition about a particular object --
we learn that this very object is black and a raven. It seems
to me that this singularist doctrine makes it hard to account for many
aspects of confirmation.
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?
Next, undermining. Suppose we are testing a model H according to
which the probability that a certain type of coin toss results in
heads is 1/2. On some accounts of physical probability, including
frequency accounts and "best system" accounts, the truth of H is
incompatible with the hypothesis that all tosses of the relevant type
in fact result in heads. So we get a counterexample to simple
formulations of the Principal Principle: on the assumption that H is
true, we know that the outcomes can't be all-heads, even though H
assigns positive probability to all-heads. In such a case, we say that
all-heads is undermining for H.
Suppose we are testing statistical models of some physical process
-- a certain type of coin toss, say. One of the models in question
holds that the probability of heads on each toss is 1/2; another holds
that the probability is 1/4. We set up a long run of trials and
observe about 50 percent heads. One would hope that this confirms the
model according to which the probability of heads is 1/2 over the
alternative.
(Subjective) Bayesian confirmation theory says that some evidence E
supports some hypothesis H for some agent to the extent that the
agent's rational credence C in the hypothesis is increased by the
evidence, so that C(H/E) > C(H). We can now verify that observation of
500 heads strongly confirms that the coin is fair, as follows.
Time-slice epistemology is the idea that epistemic norms are
history-independent: whether an agent at a time satisfies an epistemic
norm is always determined by the agent's state at that time,
irrespective of the agent's earlier states.
One motivation for time-slice epistemology is a kind of
internalism, the intuition that agents should not be epistemically
constrained by things that are not "accessible" at the relevant
time. Plausibly, an agent's earlier beliefs are not always accessible
in the relevant sense. If yesterday you learned that yew berries are
poisonous but since then forgot that piece of information, it seems
odd to demand that your current beliefs and actions should
nevertheless be constrained by the lost information.
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).)
Imagine the universe has a centre that regularly produces new stars
which then drift away at a constant speed. This has been going on
forever, so there are infinitely many stars. We can label them by age,
or equivalently by their distance from the centre: star 1 is the
youngest, then comes star 2, then star 3, and so on, without end. The
stars in turn produce planets at regular intervals. So the older a
star, the more planets surround it. Today, something happened to one
(and only one) of the planets. Let's say it exploded. Given all this,
what is your credence that the unfortunate planet belonged to the
first 100 stars? What about the second 100? It would be odd to think
that the event is more likely to have happened at one of the first 100
stars than at one of the next 100, since the latter have far
more planets. Similarly if we compare the first 1000 stars with the
next 1000, or the first million with the next million, and so on. But
there is no countably additive (real-valued) probability measure that
satisfies this constraint.
Two initially plausible claims:
- Sometimes, a possible chance function conditionalized on a proposition A yields another possible chance function.
- Any rational prior credence function Cr conditional on the hypothesis Ch=f
that f is the (actual, present) chance function should coincide with
f; i.e., Cr(A / Ch=f) = f(A) for all A (provided that Cr(Ch=f)>0).
Claim 1 is a supported by the popular idea that chances evolve by
conditionalizing on history, so that the chance at time t2 equals the
chance at t1 conditional on the history of events between t1 and
t2. Claim 2 is a weak form of the Principal Principle and often taken
to be a defining feature of chance.
You can't predict the stock market by looking at tea leaves. If an
episode of looking at tea leaves makes you believe that the stock
market will soon collapse, then -- assuming your previous beliefs did
not support the collapse hypothesis, nor the hypothesis that tea
leaves predict the stock market -- your new belief is unjustified and
irrational. So there are epistemic norms for how one's opinions may
change through perceptual experience.
Such norms are easily accounted for in the traditional Bayesian
picture where each perceptual experience is associated with an
evidence proposition E on which any rational agent should condition
when they have the experience. But what if perceptual experiences
don't confer absolute certainty on anything? Jeffrey pointed out that
if there is a partition of propositions { E_i } = E_1,...,E_n such
that (1) an experience changes their probabilities to some values {
p_i } = p_1,...,p_n, and (2) the experience does not affect the
probabilities conditional on any member of the partition, then the new
probability assigned to any proposition A is the weighted average of
the old probability conditional on the members of the partition,
weighted by the new probability of that partition. This rule is often
called "Jeffrey conditioning" and sometimes "generalised
conditioning", but unlike standard conditioning it isn't a dynamical
rule at all: it is a simple consequence of the probability
calculus. To get genuine epistemic norms on the dynamics of belief
through perceptual experience, Jeffrey's rule must be supplemented
with a story about how a given experience, perhaps together with an
agent's previous belief state, may fix the partition { E_i } and
values { p_i } that determine a Jeffrey update. This is the "input
problem" for Jeffrey conditioning.
Suppose a rational agent makes an observation, which changes the
subjective probability she assigns to a hypothesis H. In this case,
the new probability of H is usually sensitive to both the observation
and the prior probability. Can we factor our the prior probability to
get a measure of how the experience bears on the probability of H,
independently of the prior probability?
A common answer, going back to Alan Turing and I.J.Good, is to use
Bayes factors. The Bayes factor B(H) for H is the ratio
(P'(H)/P'(not-H))/(P(H)/P(not-H)) of new odds on H to old odds. Thus
the new odds on H are the old odds multiplied by the Bayes factor. For
example, if the prior credence in H was 0.25 and the posterior is 0.5,
then the odds on H changed from 1:3 to 1:1, and so the Bayes factor of
the update is 3. The same Bayes factor would characterise an update
from probability 0.01 to about 0.03 (odds 1:99 to 1:33) or from 0.9 to
about 0.96 (odds 9:1 to 27:1).
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.
Luc Bovens and Wlodek Rabinowicz (2010
and 2011)
present the following puzzle:
Three people are each given a hat to put on in the
dark. The hats' colours, either black or white, has been decided by
three independent tosses of a fair coin. Then the light goes on and
everyone can see the hats of the two others, but not their own. All of
this is common knowledge in the group.
Let's call the three players X, Y and Z. There are eight possible
distributions of hat colours, each with probability 1/8:
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.
Alice is randomly selected from her population to be tested for a
rare genetic disorder that affects about one in 10,000 people. The
test is accurate 99 percent of the time, both among subjects that have
the disorder and among subjects that don't. Alice's test comes back
positive.
Call the information in the previous paragraph E, and suppose it's
all you know about the situation. How confident are you that Alice has
the disorder?
Letting our subjective probabilities be guided by the stated
frequencies, we can use Bayes' Theorem to figure out that P(disorder |
positive) = P(positive | disorder) * P(disorder) / (P(positive |
disorder) * P(disorder) + P(positive | ~disorder) * P(~disorder)) =
0.99 * 0.0001 / (0.99 * 0.0001 + 0.01 * 0.9999) = 0.0098. Assume then
that your degree of belief is about 0.01.
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.
Compare the following two ways of responding to the weather report's
"probability of rain" announcement.
Good: Upon hearing that the probability of rain is x,
you come to believe to degree x that it will rain.
Bad: Upon hearing that the probability of rain is x, you
become certain that it will rain if x > 0.5, otherwise certain that
it won't rain.
The Bad process seems bad, not just because it may lead to bad
decisions. It seems epistemically bad to respond to a "70%
probability of rain" announcement by becoming absolutely certain that
it will rain. The resulting attitude would be unjustified and irrational.
Apropos Williamson. The following question came up last year when
we discussed The Philosophy of Philosophy in Canberra. I
thought it had a sensible answer that we just couldn't figure out, but
then Dorothy Edgington raised the same question at the recent
phloxshop workshop in Berlin, and even though there were quite a few
Williamsonians present, there was no agreement on what the answer is,
and the proposals didn't sound very convincing.
The question is simply how, on Williamson's account, we can have
knowledge of substantial metaphysical necessities, e.g. of the fact
that gold necessarily has atomic number 79. Williamson explains that
when we counterfactually imagine gold having atomic number 78 (knowing
that it has number 79), we will "generate a contradiction", because we
hold "such constitutive facts [as atomic number] fixed" (p.164). But
the distinction between constitutive and not-constitutive facts can
hardly be analysed as the distinction between whatever we happen to
hold fixed and the rest, given Williamson's commitment to strong
mind-independence of metaphysical modality. So what justifies our
holding fixed the atomic number?
Alvin Goldman has just been giving this year's summer school here
in Cologne. When he put forward his view that what distinguishes good
ways of belief formation from other ways is their truth-conduciveness, I
found myself disagreeing and claiming that there is no general principle that
distinguishes the good ways from others. This is somewhat surprising
given that I've often claimed in recent times that the only epistemic
criterion for evaluating belief-formation is truth-conduciveness. Here
is how I think the two claims can go together.
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?
First, a quick reminder of history. David Lewis once proposed a principle (the 'Principal Principle') linking rational credence and objective chance. It says (or rather, entails) that your rational credence in
any proposition A, on the assumption that the objective chance of A is x, should also be x, no matter what (further) evidence E you have:
OP: P(A | ch(A)=x & E) = x.
This principle, the 'Old Principle', is widely taken to suffer from two defects. First,
suppose your evidence E includes ~A. Then probability theory
ensures that P(A | ch(A)=x & E) = 0, irrespective of x. Lewis
responded by restricting OP to cases where E is 'admissible'. He suggested that a
(true) proposition is admissible iff it is entailed by the history of the world up to now
together with the laws of nature.
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.
We Bayesians are sometimes bugged about ultimate priors: what
probability function would suit a rational agent before the
incorporation of any evidence? The question matters not because anyone
cares about what someone should believe if they popped into existence
in a state of ideal rationality and complete empirical ignorance. It
matters because the answer also determines what conclusions rational
agents should draw from their evidence at any later point in their
life. Take the total evidence you have had up to now. Given this
evidence, is it more likely that Obama won the 2008 election or that
McCain won it? There are distributions of priors on which your
evidence is a strong indicator that McCain won. Nevertheless, this
doesn't seem like it's a rational conclusion to draw. So there must be
something wrong with those priors.
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.
Okay. Here are some thoughts on a talk Frank Jackson gave last week on Williamson on thought experiments.
The question is what Gettier discovered in his famous article. According to Frank, he revealed a fact about our concept 'knowledge': that it is not the same as our concept of justified true belief. According to Williamson, Gettier has revealed a fact about knowledge itself: that it is not justified true belief. A discovery merely about our concepts, Williamson says, "would show little of philosophical interest"; it would be "of significance primarily to theorists of concepts, not to epistemologists". For "the primary concern of epistemology is with the nature of knowledge, not with the nature of our concept of knowledge". (All of these are from p.206 of The Philosophy of Philosophy.) Frank disagrees. He thinks that results about the key concepts of a discipline are quite important to that discipline.
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!
Mostly, when we don't believe something, we don't know it either. But arguably not always. The timid student thinks she's merely guessing, while in fact she knows. She knows, but she lacks the confidence required for belief. It would be nice to have an analysis of knowledge that allowed for such cases, but also explained why they are rare.
Lewis's analysis tries to do that. On Lewis's account, you know p iff your evidence rules out any relevant situation where ~p. Among the rules for what counts as 'relevant', the 'rule of belief' tells us that any possibility with non-negligible subjective probability counts as relevant. Now suppose you don't believe p. Then you give non-negligible probability to ~p situations. So you know p only if your evidence rules out all those ~p situations. Moreover, your present evidence 'rules out' a situation iff you have different evidence in that situation than you actually have. So if you have knowledge without belief, you must assign positive probability to situations where you have different evidence than you actually have. On a suitable understanding of evidence, those cases will be rare, because we are normally confident that we have the evidence that we have.
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.
A coin is to be tossed. Expert A tells you that it will land heads with probability 0.9; expert B says the probability is 0.1. What should you make of that?
Answer: if you trust expert A to degree a and expert B to degree b and have no other relevant information, your new credence in heads should be a*0.9 + b*0.1. So if you give equal trust to both of them, your credence in heads should be 0.5. You should be neither confident that the coin will land heads, nor that it will land tails. -- Obviously, you shouldn't take the objective chance of heads to be 0.5, contradicting both experts. Your credence of 0.5 is compatible with being certain that the chance is either 0.1 or 0.9. Credences are not opinions about objective chances.
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.
Here is Lewis's 1996 analysis of knowledge:
S knows proposition P iff P holds in every possibility left uneliminated by S's evidence. ("Elusive Knowledge", p.422 in Papers)
By evidence, Lewis explains, he means perceptual experiences and memories; a possibility W counts as eliminated iff the subject does not have the same evidence in W: "When perceptual experience E (or memory) eliminates a possibility W [...], W is a possibility in which the subject is not having experience E" (424). It follows that everyone trivially knows what perceptual experiences they have: In every possibility W in which I have experience E, I obviously have experience E.
Following up on Weng-Hong (1, 2, 3), here are a few thoughts on thresholds for belief.
If beliefs come in different degrees or strength, what do we mean when we say not that Fred believes that P with strength x, but simply that Fred believes that P? Perhaps we mean that Fred believes that P with sufficient strength, where context may help determining what counts as sufficient. However, on this account, the following principles should be obviously invalid (both descriptively and normatively):
My officemate Jens-Christian, my flatmate Weng Hong and his officemate Aidan have started a blog on bunnies probabilitiy, possibility and rationality. There's already a couple of good posts by Weng Hong.
We had a little chat about the normativity of rationality today. Unlike with moral norms, I cannot imagine people who vastly disgree with me on the norms of rationality and who actually act upon their different norms. Can you imagine people who usually infer "~P" from "P and Q", update their beliefs by counter-conditionalizing P'(H) = 1-P(H|E), and always try to minimize their expected utility? I can't.
By contrast, I find it easy to imagine people who value torturing innocent people and do so. This indicates that so-called norms of rationality are to a large part not real norms at all, but conceptual necessities. So is the "ought" or "must" in "if you believe P and Q, you must/ought to also believe P" like the "must" in "if it is true that P and Q, then it must also be true that P"? I think it's more like the "ought" in "if you go 'File' -> 'Save', the program ought to save the current document". Software can be buggy and fail to do what it's supposed to do according to its design specification. It is inconceivable that a word processor generally doesn't do any of the things that characterize a word processor. But it is conceivable that it fails occasionally and under specific conditions. (Then perhaps what Dutch books arguments try to show is that if you don't obey the probability axioms, you do something -- viz. give different evaluations to the same states of affairs -- which, if you did the same thing on a large scale, would rob you of your status as an agent with beliefs and desires.)
Lots of interesting stuff came up at the Summer School and the GAP and the A Priori workshop. Here's just two quick notes on something Jason Stanley mentioned in his talk on "Knowledge and Certainty".
Jason argued that knowledge does not entail certainty. He pointed out that in Unger's arguments to the opposite conclusion, "know" is always emphasized, as in:
In July, I tried to show that Williamson's argument against luminosity fails
for states that satisfy a certain infallibility condition. I now think that (for basically the same reason) Williamson's argument fails for any state whatsoever, including knowing something and being such that it's raining outside. (The latter of course isn't luminous, but this is not established by
Williamson's argument.)
Tim Williamson argues that no interesting conditions are such that if they obtain, then one is in a position to know that they obtain. I'll try to show that his argument fails for all conditions for which one can only non-inferentially believe that they obtain if they really do obtain. It seems to me that many interesting conditions -- probably including feeling cold and knowing that one feels cold -- are of this kind. I haven't checked the secondary literature, so what I'm going to say is probably old. Anyway, here goes.
For the "Philosophische Club" at the university of Bielefeld, I've made a short paper out of that entry on perceptual content. The proposal is still that the information we acquire through perception is the information that we have just those perceptual experiences. But more needs to be said about what that amounts to: if "having just those experiences" means having experiences with this fundamental phenomenal charater, the proposal is incompatible with physicalism; if it means having just this brain state, the proposal is false. So I end up defending a kind of analytical functionalism even about demonstratives like "this experience". The main argument has something to do with skeptical scenarios. I won't repeat it here, as the paper itself is short enough.
Looking out of the window, I come to believe that it's snowing
outside. I don't just add this single belief to my stock of beliefs; I
conditionalize on something. On what?
It doesn't seem to be the proposition that the scene before my eyes
contains the very features that caused my perception. Arguably, what
caused my perception is H2O falling from the sky. If that was what I
conditionalize on, I would take my present experience as
evidence that snow is made of H2O, rather than XYZ. But I don't.
For some reason, I find Moore's refutation of idealism ("here is a hand; therefore there is an external world") much more convincing than his refutation of skepticism ("I know that here is a hand; therefore I know that I am not a brain in a vat".) Why is that?
In both cases, Moore's argument would not convince his opponent who would obviously reject Moore's premise. So that's not the difference. I think the difference also isn't that skepticism is a philosophically stronger position than idealism. Rather, it seems to me that the premise against idealism is much more certain than the anti-skeptical premise. That here is a hand (or at least that there are hands) is about as certain as non-logical truths get, that I know that here is a hand is not. If I were to compile a list of Moorean facts -- of facts that are at least as certain as any philosophical argument against them --, I would include all kinds of facts about material objects, other people, experiences, mathematics and modality, but knowledge claims probably wouldn't make the list.
What does it take for something to be a perfectly reliable indicator
of something else?
I'm not really familiar with discussions of reliability in epistemology, and I'd be grateful for pointers. Anyway, here is my own suggestion.
First, we need a mapping from (possible) states of the indicator to
the indicated facts (or states or propositions).
Let's say that the indicator displays that p, for short: I(p), if its state is mapped to p by that mapping. The mapping may
be any old function (but the 'states' may not be any old Cambridge
states): there is a good sense in which a clock that consistently runs 8 minutes fast is reliable; the tricky bit is only to read what it says, to figure out
the mapping. This is the sense of "reliable" I'm interested in.
I once believed that in non-contingent matters, knowledge is true,
justified belief. I guess my reasoning went like this:
How do we come to know, say, metaphysical truths? Not by direct
insight, usually. Nor by simple reflection on meanings, sometimes.
Rather, we evaluate arguments for and against the available
options, and we opt for the least costly position. If that's how
we arrive at a metaphysical belief, the belief is clearly justified
-- we have arguments to back it up. But it may not be knowledge:
it may still be false. Metaphysical arguments are hardly ever
conclusive. But suppose we're lucky and our belief is true. Then it's knowledge: what more could we ask for? Surely not any causal connection to the non-contingent matters.
But now that Antimeta has asked for Gettier cases in mathematics, it seems
to me that there are perfectly clear examples (I've posted a comment over there, but it seems to have gone lost):
Some people intuit that
- the subject in a Gettier case has knowledge;
- Saul Kripke has his parents essentially;
- "Necessarily, P and Q" entails "Necessarily, P";
- whenever all Fs are Gs and all Gs are Fs, the set of Fs equals the
set of Gs;
- the liar sentence is both true and not true;
- the conditional probability P(A|B) is the probability of the
conditional "if B then A";
- it is rational to open only one box in Newcomb's problem;
- switching the door makes no difference in the
Monty Hall problem;
- propositions are not classes;
- people are not swarms of little particles;
- a closed box containing a duck weighs less when the duck inside
the box flies;
- spacetime is Euclidean;
- there is a God constantly interfering with our world.
They are wrong. All that is false.
Conservatism as a methodological principle says that we should prefer new theories that resemble our old theories. (I don't mean the principle that a new theory should be at least as good as its predecessors, nor the principle that it should explain the success and failures of its predecessors. Very non-conservative theories can do that.)
What is the status of conservatism? Is it a primitive rule telling us that even if we know that some revisionary theory is as good as a conservative one -- that both explain roughly the same data, make roughly the same predictions, are equally simple, etc. --, we should prefer the conservative theory? (An otherwise good theory according to which there are no birds, but only bird-halluzinations, say, just seems incredible, in particular if a more credible alternative is available.) In this case, conservatism would resemble the simplicity principle that tells us to always prefer the simpler of otherwise equal theories.
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?
Eliezer Yudkowsky, in his Intuitive Explanation of Bayesian Reasoning, argues that it is irrational to justify the belief that if a biological war will break out it won't wipe out humanity by pointing out that one is an optimist:
p(you are currently an optimist | biological war occurs within ten years and wipes out humanity) =
p(you are currently an optimist | biological war occurs within ten years and does not wipe out humanity)
I'm always worried when a philosopher claims that it's a virtue of his theory that it rules out certain kinds of scepticism, or when a philosopher criticizes another philosopher (say, a contextualist) for not doing so.
I suppose it would be a good thing if newspapers always told the truth. But what would you say if I offered you a theory on which it is ruled out a priori that something false could be written in a newspaper? That wouldn't be a point in favour of my theory. For it seems intuitively obvious that something false could be written in a newspaper. A theory isn't good just because it entails something which, if true, would be good.
Let P be a proposition of which you neither believe that it's true nor that it's false, say Goldbach's Conjecture. Since you know that you don't believe P (otherwise you couldn't have chosen it), your conditional subjective probability for [P and I don't believe P] given P should be close to 1. However, if you were to learn that P, your subjective probability for [P and I don't believe P] shouldn't be close to 1, but close to 0. So is this a case were you shouldn't conditionalize?
Jonathan Schaffer argues (in Analysis 2001) that Relevant Alternatives Theories of knowledge (RATs) such as Lewis's fail because of Missed Clues cases:
Professor A is testing a student, S, on ornithology. Professor A shows S a goldfinch and asks, 'Goldfinch or canary?' Professor A thought this would be an easy first question: goldfinches have black wings while canaries have yellow wings. S sees that the wings are black (this is the clue) but S does not appreciate that black wings indicate a goldfinch (S misses the clue). So S answers, 'I don't know'.
We want to say that S doesn't know that the bird is a goldfinch. Yet it seems that S's evidence rules out all relevant alternatives. For situations with goldfinch-perceptions but no goldfinches are skeptical scenarios and usually regarded as irrelevant.
This is still a bit vague, but anyway.
As I remarked in the first part of this little series, from an implementation perspective, it is not surprising that applying one's beliefs and desires to a given task requires processing. Consider a 'sentences in boxes' implementation of belief-desire psychology: I have certain sentence-like items stored in my belief module, and other such items in my desire module. When I face a decision, I run a query on these modules. Suppose the question is whether I should take an umbrella with me. The decision procedure may then somehow find the sentences "It is raining" and "If I take an umbrella, I don't get wet" (or rather, their Mentalese translations) in the belief box and "I don't get wet" in the desire box. From these it somehow infers the answer, that I should take the umbrella.
Why not simply use a notion of content on which belief isn't closed under strict implication? Then it will be much easier to say that reasoning always delivers new content.
There is no shortage of fine-grained notions of content. We could use English sentences, or classes of intensionally isomorphic sentences, or bundles of tuples of objects and properties ('singular propositions') together with modes of presentation, or whatever. The tricky part is to say what determines whether a subject has a belief or desire with such a content.
As Robbie Williams remarked in the comments, perhaps what we do when we reason is putting parts of our fragmented belief space together. However, I doubt that this will do as a general solution.
First, at least in the context of an interpretationist account of content, it doesn't suffice for fragmentation that the relevant beliefs are somehow stored in different parts of the brain. Rather, if my beliefs are fragmented, say, into a compartment in which I believe P and one in which I don't, this must show up in my behaviour, more or less as follows: 1) In some contexts, the best explanation of some of my actions involves the assumption that I take the world to be P; but also 2) in some contexts, the best explanation of some of my actions involves the assumption that I don't take the world to be P; Moreover, 3) the discrepancy can't be explained as a change of belief.
One might suggest that in fact resoning, like (factual) learning, always means acquiring new information. After all, it is possible to acquire new information by learning that P even if what one previously knew already entailed P. In this case the new information can't be P, but it can be something else. To use Robert Stalnaker's favourite example. when you learn that all ophtalmologists are eye-doctors, the possibilities you can thereby exclude are not possibilities where some ophtamologists aren't eye-doctors -- there are no such possibilites. Rather, they are possibilities where "ophtalmologist" means something different. You've acquired information about language. Perhaps what you learn when you learn that the square root of 1156 is 34 is similarly something about language, in this case about mathematical expressions. That explains why we can't replace synonymous expressions in the content attribution: Just as it would be wrong to say you've learned that all eye-doctors are eye-doctors, so here it would be wrong to say you've learned that the square root of 34*34 is 34.
What do we do when we draw inferences? We don't acquire new information, at least not if the reasoning is deductively valid. Rather, we try to find new representations of old information. The point of that is perhaps that we can only make our actions depend on representations of information, not directly on the information itself, and some forms of representation lend themselves more easily to guide certain actions than others.
The problem is familiar in programming: to accomplish a given task it is often crucial to find a data structure that makes the relevant properties of the stored data easily accessible. In principle, every data set could be represented as a huge number, but in pratice it helps a lot to represent it in terms of arrays or strings or objects with suitable properties.
This argument is not deductively valid:
The best available theory says p;
Therefore, p.
For even the best available theory can be false. It's not even clear that the premiss makes the conclusion very probable. So is it fallacious to argue for a claim by pointing out that it is entailed by the best available theory? No. The argument may be valid in another sense: in the sense that it is irrational to accept the premiss but reject the conclusion. For if you accept that the best available theory says p, rejecting p means to knowingly reject the best available theory -- and that may well be irrational. It's always irrational to knowingly reject the best available theory in favour of another, worse, theory. The only rational alternative is agnosticism. But if the best theory is sufficiently good and much ahead of its rivals then agnosticism too is irrational. That's because rationality demands that you increase your credence in a proposition in the light of good reasons.
Brian Weatherson:
We know that positive conceivability is a good inductive guide to possibility. And we know negative conceivability is a good inductive guide to possibility.
What kind of induction is this? What we do know is that sometimes what seems conceivable on first sight later turns out to be incoherent (and thus inconceivable in the technical sense introduced by Dave Chalmers and deployed by Brian). We also know that this doesn't happen very often, and that it happens mainly when we consider rather complicated stories or hypotheses. So we have good inductive reason to assume that there is no hidden contradiction in, say, the hypothesis that there could be an apple in a basket. But this only supports the claim that prima facie conceivability is a good inductive guide to ideal conceivability.
If a statement p is impossible, then empirical information and a priori reasoning usually suffice to establish its impossibility. So if despite carrying out the relevant empirical investigations and a priori reasonings no impossibility shows up, this is a good reason to believe that p is possible. One might be tempted to say that our knowledge of possibility is always based on such a failure to detect the respective impossibility. This is what Bob Hale calls an asymmetric approach to modal epistemology. (See his "Knowledge of Possibility and of Necessity", Proceedings, 2003.)
Sometimes people say that for logical reasons there can be no examples of unknown or unknowable truths. The logical reason is this: to know that p is an unknown truth requires knowing that p is true, which contradicts the requirement of p being unknown.
Before I give examples of unknown and unknowable truths let me give examples of philosophers who died more than 100 years ago: Hume, Leibniz, Kant, and the philosopher first born in the 16th century. One might have thought that it is impossible for physical reasons to give such examples. After all, a philosopher who died more than 100 years ago just isn't there any more, so he can't be given as an example. But not so. In order to give an example of a dead philosopher it suffices to name or describe one; it is not necessary to dig him out.
Brute necessity is hard to accept, much harder than brute possibility. If someone claims that necessarily there are no purple cows, I expect an explanation. Perhaps he knows what kind of DNA is essential for cowhood and also that this kind of DNA can never produce purple beings, and he also believes that the laws of nature are necessary. This would make his claim understandable. But suppose he had no such explanation. Suppose in fact that we all know that only a minor mutation would be required to produce purple cows, a mutation perfectly compossible with the laws of nature. And still he claims that there could not be any purple cows. This would seem bizarre.
John Hawthorne has some
nice arguments for the view that knowledge is closed under known
implication. I don't know much about knowledge, but it seems to me that
there is a good reason to believe that at least justification -- and hence
presumably also justified true believe -- is not so closed. The reason
is this:
E is some evidence, H and S are alternative and incompatible hypotheses.
(Obvious examples are skeptical scenarios, like E = visual evidence of
a zebra, H = there is a zebra, S = there is a mule disguised as a zebra.) E
strongly supports H: It raises its probability of truth from about 0.3 to
about 0.9. And H implies Not-S. Yet E does not raise the probability of
Not-S. On the contrary, it raises the probability of S.
Let "S(p)" abbreviate "p is strongly supported by the availble evidence".
The picture shows that
S(p) and S(p -> q) does not imply S(q);
S(p & q) does not imply S(p); (let p=-S, q=H)
S(p) does not imply S(p v q); (let p=H, q=-S).
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:
It is often said, correctly I think, that there are contingent but a priori
sentences, e.g. "water is the dominant liquid on earth". Are these
sentences analytic or synthetic? That is, what puts you in a position to
know these sentences? Does understanding suffice, or do you have to invoke
some other a priori means, like Gödelian insight? To me this seems
wildly and unnecessarily mysterious. Of course understanding suffices, at
least in ordinary cases. So there are contingent but analytic sentences. I
wonder why this is hardly ever said. Does anyone really believe that those
statements are synthetic a priori?
There are many ways to update a belief system. For example, 1) believe every proposition that comes to your mind; 2) believe everything that makes you feel good; 3) believe everything Reverend Moon says. In "A Priority as an Evaluative Notion", Hartry Field argues that there is no fact of the matter as to which way is best.
In one sense, this is trivial. Of course the normative question which way you should choose does not have a purely factual answer. Which way you should choose depends on what you want from your belief system.
