Outright
Belief

Ralph
Wedgwood†

Abstract

What is the relation between (a) “*full*” or “*outright*” belief and (b) the various *levels of confidence* that agents can have in the propositions that
concern them? This paper argues for a new answer to this question. Decision
theory implies that in making decisions, rational agents must treat certain
propositions as though they were completely certain; but on most forms of
decision theory, these propositions are not ones for which any finite agent
could have maximal justification – the agent will clearly have less
justification for these propositions than for elementary logical truths. Thus, every
adequate model of a finite rational agent’s belief-system must involve *two* set of credences – *theoretical* credences (the belief-states
that keep track of how much justification the agent has for the propositions in
question) and *practical* credences
(the belief-states on which the agent bases on her practical decisions). A full
or outright belief in *p* can be
defined as the state of being stably disposed to assign a *practical* credence of 1 to *p*,
for all *normal* practical purposes.
This definition allows for a kind of reconciliation between the pragmatist and
intellectualist approaches in epistemology.

*1.
**Outright belief and levels
of confidence*

Sometimes, we think of belief as a
phenomenon that comes in *degrees* –
that is, as consisting in the many different *levels of confidence* that an agent might have in various
propositions. Sometimes, we think of belief as a simple two-place relation that
holds between an agent and a proposition – that is, as what I shall here
call *outright belief*.

Both ways of thinking
of belief seem to be in evidence in everyday folk-psychological discourse. It
is perfectly natural to say, for example, you are considerably *more confident* that Dublin is the
capital of Ireland than that Dushanbe is the capital of Tajikistan. But it is
also natural to say that you simply *believe*
that Dublin is the capital of Ireland, without giving any further
qualifications about how confidently or strongly you believe it. In general, we
move back and forth easily between talking about what one simply* *believes, and talking about how
confident one is of various propositions.

This raises the
question: How are outright beliefs and levels of confidence related to each
other? Philosophers have defended many different answers to this question. A
few philosophers – such as Richard Jeffrey (1970) and Patrick Maher (1993,
152–5) – have taken an *eliminativist*
stance towards outright beliefs, claiming that strictly speaking, outright
beliefs do not really exist. A larger number of philosophers have attempted
some kind of *reductive* program. Thus,
many philosophers – including Scott Sturgeon (2008) and Richard Foley
(2009) among others – maintain that outright beliefs are reducible to
levels of confidence, while other philosophers – such as Mark Kaplan
(1999) and Brian Weatherson (2005) – claim that outright beliefs are
reducible to facts about the relevant agent’s preferences or utilities together
with facts about the agent’s levels of confidence. Others – most notably,
Gilbert Harman (1986) – maintain that levels of confidence are reducible
to facts about outright beliefs. Finally, a few recent philosophers –
especially, Jacob Ross and Mark Schroeder (2012) and Jonathan Weisberg (2011) –
maintain that neither of these two kinds of belief is reducible to the other.

In
this essay, I shall propose yet another answer to this question. This answer
has two main components. The first component is the thesis that there are in
fact *two different kinds* of levels of
confidence, which I shall call “practical credences” and “theoretical
credences” respectively. The second component is the thesis that outright
beliefs are reducible to facts about our *practical*
credences – but not to facts about our theoretical credences.

*2.
**Outright
belief and levels of confidence: Some distinguishing marks*

To begin our investigation of outright
beliefs and levels of confidence, I shall enumerate some of the distinguishing
marks of these two types of belief. I shall start by listing some of the
distinguishing marks of outright belief.

First, we typically
assume that a sincere *assertion* of a proposition *p* expresses an
outright belief in *p*. If you meet someone who asserts that the capital
of Uzbekistan is Samarkand, you will typically assume that if the speaker is
sincere, then the speaker has an outright belief that the capital of Uzbekistan
is Samarkand.[1]

Secondly, there seems
to be a fundamental connection between outright belief and *knowledge*. Whenever an agent knows
a proposition *p*, the agent must have an outright belief in *p*.
If you know that the capital of Uzbekistan is Tashkent and not Samarkand, then
you must have an outright belief that the capital of Uzbekistan is Tashkent and
not Samarkand.[2]

Finally, there is a
common way of using the term ‘believe’ – which we find in the
philosophical literature on knowledge, when it is claimed that everyone who
knows a proposition *p* must also
believe *p* – such that when it is
said simply that an agent “believes” a proposition *p*, what is meant is that have they have an outright belief in *p*.

I shall now list some
of the distinguishing marks of levels of confidence. Your levels of confidence
create an (at least partial)* ranking*
of the propositions that you have beliefs towards. For example, you might have
more confidence that 1 + 1 = 2 than that Dushanbe is the capital
of Tajikistan; so your levels of confidence rank the proposition that
1 + 1 = 2 above the proposition that Dushanbe is the capital of
Tajikistan.

We can illustrate this
point by imagining an (at least partially) ordered collection of boxes: your
levels of confidence in effect sort these propositions into these boxes. For
example, the proposition that 1 + 1 = 2 will presumably be sorted
into the box at the very top of this ordered collection; the proposition that
Dushanbe is the capital of Tajikistan is sorted into a box that comes somewhat
lower in this ordering; the proposition that Dushanbe is the capital of France
is sorted into a box that comes much lower down in this ordering; and the
proposition that 1 + 1 = 5 comes in the box at the very bottom of the
whole ordered collection, along with other obviously false and absurd
propositions.[3]

In fact, it seems that
our levels of confidence may do more than simply rank propositions into an
ordering of this sort. We can also make sense, not just of the thought that you
are *more* confident of *p* than of *q*, but also of the thought that you are *much* more confident of *p*_{1}
than of *q*_{1}, or that you
are only *slightly* more confident of *p*_{2} than of *q*_{2}. In effect, we can at
least sometimes compare *differences*
in levels of confidence. For example, we can ask: Is the degree to which you
are more confident of *p*_{1}
than of *q*_{1} *greater* or *smaller* than the degree to which you are more confident of *p*_{2} than of *q*_{2}?

This makes it plausible
that it may be possible to *measure* an
agent’s levels of confidence, by means of some kind of *credence function*, which assigns real numbers to the various
propositions that the agent has beliefs towards.[4]
However, even if the agent’s levels of confidence can be represented by means
of such a credence function, we should not assume that this credence function
will be a *probability* function. There
are at least two ways in which the agent’s credence function may fall short of
being a full-blown probability function.

First, the credence
functions that represent the agent’s levels of confidence may not be
probabilistically coherent. For example, there may be cases in which the agent
assigns different levels of confidence to two propositions *p* and *q*, even though in
fact *p* and *q* are logically equivalent. Later in this essay, I shall assume for
the sake of argument that a perfectly rational agent’s levels of confidence
would always be probabilistically coherent; but I shall not be able to defend
that assumption here. At all events, it is clear that not all actual agents are
perfectly rational; so we should not assume that the credence functions that
represent the confidence levels of actual agents must be probabilistically
coherent in this way.

Secondly, a probability
function is defined over an infinite propositional *algebra*. (For example, for every pair of propositions in this
algebra, the algebra must also contain their disjunction and their negations,
and so on.) I shall not assume that any real believer has a level of confidence
in every proposition in a complete propositional algebra of this sort. There
may be many “gaps” – many propositions towards which the believer is
totally “attitudeless”, and so has no level of confidence at all.

Many
philosophers hold that human believers like you and me will inevitably have a
set of levels of confidence that is “gappy” in this way. Indeed, given that the
core realization of our minds is found in the physical state of our brains, and
given the limited information density of matter, it may be that it would be *impossible* for us to have a belief of
some kind in every proposition in an infinite propositional algebra. I shall
not attempt to address this issue here. However, even if it is possible for
some human believers to have attitudes towards infinitely many propositions in
this way, this essay will focus exclusively on *finite agents* – that is, on agents who only have attitudes
towards finitely many propositions.

*3.
**The
functional role of these kinds of belief*

To give an adequate account of the
relationship between outright beliefs and levels of confidence, we need to get
beyond these distinguishing marks, and to investigate the essential properties
of these kinds of belief. I shall suppose here that one essential property of
each type of mental state is its *functional
role* – the role that the mental state is disposed to play in our
thinking and reasoning.[5]

There is fairly
widespread agreement among philosophers about the essential functional role of
outright belief. If you have an outright belief in *p*, you will simply *take p
for granted*, treating *p* as a starting point for further reasoning
(including both practical and theoretical reasoning). You have a picture of the
world that you rely on in all such further reasoning, and this proposition *p* is part of that picture of the world.

So far, this account of
the functional role of outright belief is largely metaphorical. It might take a
long time to give a complete account in less metaphorical terms. But it seems
plausible that this functional role involves at least the following two
elements – one of them concerning the role of outright beliefs in one’s
theoretical reasoning (that is, in the process of forming and revising one’s
beliefs and the like) and the other concerning the role of outright beliefs in
one’s practical reasoning (that is, in the process of forming and revising
one’s intentions about how to act).

On the theoretical
side, the essential functional role of outright beliefs involves a disposition
towards* deductive cogency*. Deductive
cogency itself has two components. First, it involves a kind of *multi-premise closure*: if you deduce a
conclusion from a certain set of premises, and (even after carrying out the
deduction) you have an outright belief in each of those premises, then to be
deductively cogent, you must have an outright belief in the conclusion as well.
(The distinctive feature of this kind of multi-premise closure is that it
allows for two different ways of responding to one’s having carried out such a
deduction – *either* by having an
outright belief in the conclusion *or*
by not having an outright belief in each of the premises. It is only if one has
an outright belief in each of the premises even *after* carrying out the deduction that this kind of closure requires
an outright belief in the conclusion.)

Secondly, deductive
cogency involves a kind of *overall
consistency*. Specifically, for you to be deductively cogent, the
propositions in which you have outright beliefs must form a logically
consistent set. It is part of the functional role of outright beliefs that they
are disposed to conform to the requirements of this sort of deductive cogency.[6]

On the practical side,
having an outright belief in *p*
involves what John Hawthorne (2003, 29) calls “using *p* as a premise in one’s practical reasoning”. As Robert Stalnaker
(1984, 15) puts it, “to believe [*p*]
is to be disposed to act in ways that would tend to satisfy one’s desires,
whatever they are, in a world in which *p*
(together with one’s other beliefs) were true.” I shall assume that this means
something like the following: if the set of propositions in which you have
outright beliefs is *P*, and together
with your values, *P* entails that the *optimal* options for you to choose
constitute a certain subset {*A*_{k}, … *A*_{m}} of the available options
{*A*_{1}, … *A*_{n}}, then you must be
disposed to choose an option that belongs to that subset {*A*_{k}, … *A*_{m}}.
So, in particular, if *P* together with
your values implies that *A* is the uniquely
best available option, then you must be disposed to choose *A*.

What is the functional
role of levels of confidence? For the purposes of this essay, a particular
answer to this question will simply be assumed, at least for the sake of
argument. This answer is based on an account of what it is for a believer’s
levels of confidence to be *rational*.
The central idea is that the essential functional role of levels of confidence
is that they are disposed to *approximate*
to conforming to the requirements of rationality that apply to them.

Broadly speaking, we
shall assume a version of the orthodox Bayesian account of what it is for an
agent’s levels of confidence to be rational – with the crucial amendment
that it is not assumed here that the believer must have a level of confidence
in *every* proposition in a whole
propositional algebra. This Bayesian account of rationality has two components:
one component is an account of *theoretical*
rationality; and the other component is an account of *practical* rationality. (It will turn out to be important for our
purposes to keep these two components clearly separated from each other.)

According to the
Bayesian account of theoretical rationality, rational levels of confidence can
be represented by a unique real-valued *credence
function*, and this function must meet the following two requirements:
first, must be *probabilistically coherent*;
and secondly, it must evolve in response to experience by means of some kind of
*conditionalization*.[7]
(According to this account of theoretical rationality, this credence function
may be partial or “gappy” – that is, it may not assign a credence to every
proposition in the whole propositional algebra. However, it is assumed that a
perfectly rational agent’s levels of confidence can be represented by a *unique* credence function: that is, for
every proposition *p* that the agent
has any attitude towards, all credence functions that can represent the agent’s
levels of confidence must assign the very same credence to *p*.[8])

According to the
Bayesian account of *practical*
rationality, rationality requires every agent’s levels of confidence to
constrain her *choices* and *intentions* in a certain distinctive way.
Let us say that whenever one makes a certain choice, or has a certain
intention, the object of one’s choice or intention is an “option”. Then
according to this conception of practical rationality, there is some sort of
“value” that options might exemplify such that the rational agent’s choices and
intentions will always be for options that maximize the *expectation* of this sort of value – where the relevant
“expectation” is defined in terms of these levels of confidence. There are
several tricky technical questions about how best to understand this sort of
“expectation”, but the key idea is this: there is an appropriate “partition” of
“states”, such that in each of these states each of the relevant options
exemplifies the relevant value to some precise degree; and if the agent’s
levels of confidence can be interpreted as assigning a *probability* to each of these states, an option’s “expected value”
can be identified with the probability-weighted sum of the option’s value in
each of these states, weighting the value that the option has in each of these
states by the relevant probability of the state.[9]

This account of the
functional roles of these types of belief seems to create insuperable
difficulties for the view that levels of confidence are reducible to outright beliefs.
How could we reconcile the view that levels of confidence are reducible to
outright beliefs with the idea that it is part of the essential functional role
of levels of confidence to approximate to these broadly Bayesian requirements
of rationality? It seems that the only way to achieve this reconciliation would
be by identifying the state of having level of confidence *n* in a proposition *p* with
an outright belief in the proposition that *p*
has a probability of *n*.

However,
this identification seems to imply that few if any human agents have *any* levels of confidence at all. The
very idea of precisely measuring the probability of propositions did not emerge
until the 17^{th} century.[10]
Brilliant agents such as Plato and Aristotle and the ancient Greek mathematicians
never had any beliefs whatsoever in any proposition of the form ‘Proposition *p* has a probability of *n*’. So, on this view, these agents never
had any levels of confidence at all. Since this result seems incredible, we
should conclude that if this account of the functional role of these types of
belief is correct, then levels of confidence simply cannot be reduced to
outright beliefs. Levels of confidence are either more fundamental than
outright beliefs, or the two kinds of belief are equally fundamental and
irreducible. At all events, from now on I shall simply set aside all attempts
to reduce levels of confidence to outright beliefs; I shall turn to evaluating
other theories of these types of belief instead.

*4.
**A
dilemma for non-trivial theories of outright belief*

In fact, there is a glaring problem with
the account of the functional role of outright belief that I gave in the
previous section. The functional role that this account ascribes to outright
belief in *p* is essentially identical
to the functional role of having the *maximum
possible* level of* *confidence in *p*. The functional role that this account
ascribes to outright belief is certainly different from the functional roles of
all levels of confidence that *fall short*
of maximum confidence; but it does not differ from the functional role of
maximum confidence.

On the theoretical side,
the functional role that my account ascribes to outright involves deductive
cogency: that is, it involves having an outright belief in every proposition
that you competently deduce from any set of propositions in each of which you
have an outright belief, and not having outright beliefs in all members of any inconsistent
set of propositions. But if your levels of confidence are probabilistically
coherent, then the attitude of having maximum confidence in a proposition will
also satisfy deductive cogency: you will have maximum confidence in every
proposition that you competently deduce from any set of propositions in which you
have maximum confidence; and the set of propositions in which you have maximum
confidence will also form a consistent set.

On the other hand, for
every level of confidence *n* that is lower
than the maximum possible, the attitude of having the level of confidence *n* in a proposition will not in general
satisfy deductive cogency. For example, suppose that your level of confidence
in each member of a certain set of propositions is 0.9. If the set has 10 or
more members, this set of propositions need not be logically consistent; and
your level of confidence in the *conjunction*
of any two or more members of this set may well be lower than 0.9.

On the practical side,
the functional role that my account ascribes to outright beliefs involves treating
the propositions in which you have outright beliefs as “premises” in your
practical reasoning – that is, in effect, basing your choices simply on your
outright beliefs, and not on any levels of confidence that you have in any
propositions that are incompatible with what you believe. This is obviously the
same as the practical side of the functional role of the maximum level of
confidence. Yet if your level of confidence in *p* is even slightly less than the maximum possible, there will be
perfectly normal situations in which you would base your choices on your level
of confidence in propositions that are incompatible with *p*. For example, consider a situation in which the available options
do not differ in value in any way if *p*
is true, but differ enormously in value if *p*
is false; in this case, you would base your choice on your level of confidence
in the *negation* of *p*, not just on your level of confidence
in *p*. So again, if your level of
confidence in *p* is even slightly less
than the maximum possible, your level of confidence in *p* will not have the functional role that my account ascribes to
outright belief.

In this way, my account
of the functional roles of these kinds of belief differentiates outright belief
from all levels of confidence that fall short of the maximum possible, but it does
not differentiate outright belief in *p*
from the attitude of having maximum confidence in *p*. In this way, my account of this functional role suggests that
having an outright belief in *p* is a
way of having full confidence in *p* –
in other words, a way of treating *p*
as certain.

This is not an
idiosyncratic feature of my account. The phrases which philosophers typically
use to describe the state of having an outright belief in a proposition *p* – phrases such as ‘taking *p* for granted’ or ‘treating *p* as though it were true’ – all
seem to imply that having an outright belief in *p* involves treating *p* as
though it were (for all intents and purposes) *certain* – that is, treating *p* as though one had maximum confidence in it.[11]

Yet we all seem to have
outright beliefs in propositions of which we are *not *maximally confident. For example, you can surely have an
outright belief both in ‘Dushanbe is the capital of Tajikistan’ and in ‘1 + 1 =
2’, even if you have *less* confidence in the former than in the latter.
But if you have *less* confidence in
‘Dushanbe is the capital of Tajikistan’ than in ‘1 + 1 = 2’, you cannot have *maximum*
confidence in the former.

Indeed, virtually all
philosophers who have explored the relationship between levels of confidence
and outright belief assume that it must be possible to have outright beliefs in
propositions of which one is not maximally confident. They have good reason for
assuming this. In everyday speech, ‘I believe that *p*’ seems to be a distinctly weaker claim that ‘I am certain that *p*’. Moreover, it seems that unless it is
possible to have an outright belief in propositions of which one is not
maximally confident, the notion of an outright belief will effectively be *trivialized*: it would simply be the
notion of the highest possible level of confidence, not a notion that plays any
distinctive and interesting role of its own in our thinking about agents.

However, non-trivial
theories of outright belief face a dilemma. On the one hand, theorists who give
reductive accounts of outright belief are forced to *deny* that the functional role of outright belief really is what it
seems to be. On the other hand, those who recognize both outright belief and
levels of confidence as equally fundamental and irreducible types of belief
seem committed to the conclusion that having non-trivial outright beliefs
involves a form of *incoherence* –
in effect, the incoherence of simultaneously treating the relevant propositions
as though they were both certain and not certain.

The best-known attempts
to reduce outright beliefs to levels of confidence all seem to be impaled on
the first horn of the dilemma. This is particularly clear with versions of the *threshold* view of philosophers such as
David Christensen (2004) and Scott Sturgeon (2008) – that is, the “Lockean”
view that to have an outright belief in a proposition *p* is to have a level of confidence in *p* that exceeds a certain threshold *r*. These views more or less explicitly imply that the functional
role of outright beliefs is not, strictly speaking, the same as the functional
role of maximum confidence.

Brian Weatherson (2005)
takes a slightly different approach, attempting to reduce outright beliefs to
facts about the agent’s preferences as well as her levels of confidence.
Weatherson’s account comes closer to implying that outright belief has the functional
role that it seems to have: his account implies that an agent who has an
outright belief in *p* has the same
preferences over options as she would if she had maximum confidence in *p*; and at least if certain (in my view,
rather questionable) assumptions about the agent’s preferences are true, the
agent’s outright beliefs will also conform to a certain closure principle.

It does not follow,
however, that Weatherson’s account vindicates the thesis that outright belief
really has this functional role. The functional role of a type of belief is a
matter of the *dispositions* that characterize
this type of belief, and the notion of a disposition is a causal or explanatory
notion. According to Weatherson’s account, the explanation of *why* an agent has the preferences that
she has does not appeal to the agent’s disposition to use the propositions that
she has outright beliefs in as “premises in her practical reasoning”. Instead,
this explanation appeals to the functional role of preferences and levels of
confidence, and to the fact that the agent has levels of confidence both in the
propositions that she has outright beliefs in and in other incompatible
propositions as well. Similarly, the explanation of why the agent’s outright
beliefs conform to the closure principle does not appeal simply to the
disposition to conform to the requirements of deductive cogency, but rather to
certain assumptions about the agent’s preferences, as well as to the functional
roles of preferences and levels of confidence.[12]
So Weatherson’s approach also fails to vindicate the thesis that the functional
role of outright belief is what it seems to be.

On the other hand, the
theorists who accept that outright belief has the same functional role as the
maximum level of confidence are impaled on the second horn of this
dilemma – they seem to be imputing a kind of *incoherence* to all agents who have outright beliefs in any
propositions of which they are not maximally confident. Indeed, the threat of
incoherence is obvious: by having an outright belief in a proposition *p* of which one is not maximally
confident, one is in a way simultaneously treating *p* as though it were certain and also treating *p* as though it were not certain.

It is this apparent
incoherence that lies behind the notorious *lottery
paradox* of Henry Kyburg (1961). Suppose that one carries out a
multi-premise deduction. Then treating the premises as though they were certain
would commit one to treating the conclusion as though it were also
certain – while treating the premises as though they were not certain
would in many cases permit one to have *less*
confidence in the conclusion than in *any*
of the premises.

In fact, however, the
incoherence that concerns us will arise even in cases of *single-premise* inference. According to my account of the functional
role of outright belief, if you deduce *q*_{1}
from *p*, and even after performing
this deduction, you still have an outright belief in *p*, you will be disposed to have an outright belief in *q*_{1} as well. But “deducing” *q*_{1} from *p* is effectively inferring *q*_{1}
from *p* in a way that is sensitive to
the fact that the conditional probability of *q*_{1} given *p* is
1. We should also consider how you will respond to inferring a proposition* q*_{2} from *p* if you have an outright belief in *p*, but the conditional probability of *q*_{2} given *p* is
slightly less than 1.

For example, suppose
that you have an outright belief in *p*,
and you infer *q*_{2} from *p*, while being sensitive to the fact
that the conditional probability of *q*_{2}
given *p* is 0.99. Moreover, suppose
that even though you have an outright belief in *p*, you do not have a maximal level of confidence in *p* – instead, your level of
confidence in *p* is just 0.9. Still,
we are assuming that having an outright belief in *p* involves treating *p* as
though it were (at least for all intents and purposes) certain. So, how can it
be coherent for you to continue to have a confidence level of just 0.891 in *q*_{2} when you are effectively
treating *p* as though it were certain,
and you are sensitive to the fact that the conditional probability of *q*_{2} given *p* is 0.99? Suppose that you simultaneously infer both *q*_{1} and *q*_{2} from *p*,
while having an outright belief in *p*,
and being sensitive to the facts that the conditional probability of *q*_{1} given *p* is 1, and that the conditional probability of *q*_{2} given *p* is 0.99. Surely you should now treat *q*_{2} as though your credence in *q*_{2} were (for all intents and purposes) 0.99, not as
though your credence were just 0.891. After all, you treat *q*_{1} as (for all intents and purposes) certain – just
because the conditional probability of *q*_{1}
given *p* is 1. It is surely incoherent
to treat *q*_{1} and *q*_{2} so differently in this
way. So, on this view, having an outright belief in a proposition in which one
has a non-maximal level of confidence seems to involve an unavoidable
incoherence.

This
seems to me to be a grave problem with this view. Nonetheless, we have seen
problems with all of the alternative approaches as well. So we seem to need a
new approach. Somehow this new approach must allow that it can be perfectly
coherent to have an outright belief in a proposition *p*, even if one is not maximally confident of *p*, and even if, in some crucial way, having an outright belief in *p* really does involve treating *p* as if it were (for all intents and
purposes) certain.

*5.
**The
solution to the dilemma*

To solve this problem, I propose, we
should recognize that in many cases, an agent’s belief-system can only be
adequately represented by *two* systems
of credences – call them the agent’s *theoretical
credences* and the agent’s *practical
credences*. Both of these two kinds of credences come in degrees. The core
distinction is this: theoretical credences represent the way in which the agent
registers, or keeps track of, the amount of justification that she has in
favour of the relevant propositions, while practical credences are the
credences on the basis of which the agent maintains and revises her intentions
about how to act.

How exactly could it be
that “theoretical credences represent the way in which the agent registers, or
keeps track of, the amount of justification that she has in favour of the
relevant propositions”? Presumably, if you rationally have a greater
theoretical credence in *p* than in *q*, that is the way in which you register
the fact that you have *more justification* for *p* than for *q* – or as many philosophers would put it, the fact that your *evidence*
gives more *support* to *p* than to *q*.

More precisely, I
suggest, your theoretical credences are disposed to approximate to the
requirements of *theoretical*
rationality listed in Section 3 above. These credences are not only disposed to
be probabilistically coherent; they are also disposed to change in response to
experience, and the changes dictated by experience are propagated throughout
the whole set of credences by means by some kind of conditionalization. These
are the only kinds of changes in one’s theoretical credences that are involved
in these theoretical credences’ essential functional role.

On the other hand, your
*practical* credences represent the
beliefs or doxastic states on the basis of which you maintain, form, and revise
your *intentions* for *action*. These practical credences are
disposed to approximate to the requirements of *practical* rationality listed in Section 3 above. That is, these
practical credences are disposed to constrain the agent’s intentions, in such a
way that those intentions tend to maximize the appropriate kind of expected
value, when the relevant kind of “expectation” is defined in terms of these
practical credences.

You can have one set of
practical credences for the purposes of making one decision, and a *different* set of practical credences for
the purposes of making another decision. For the purposes of making most
decisions, I treat it as though it were quite certain that Dublin is the
capital of Ireland. However, if I am offered a bet that pays $1 if Dublin is
the capital of Ireland and results in my being tortured horribly to death if
Dublin is not the capital of Ireland, I will probably base my decision on a
different set of practical credences, in which it is not treated as certain
that Dublin is the capital of Ireland. (It even seems possible for an agent who
is engaged in “multi-tasking” to make two decisions at the same time, basing
one decision on one set of practical credences, while basing the other decision
on a second set of practical credences.)

When one has a
practical credence of 1 in *p*, one is
in effect treating *p* as though it
were *practically certain*; when one
has a theoretical credence of 1 in p, one is in effect treating *p* as though it were *theoretically certain*. On the face of it, there need be no
incoherence in simultaneously treating a proposition *p* as though it were practically certain and also treating *p* as though it were not theoretically
certain. The two kinds of doxastic or credal attitudes towards *p* do not conflict with each other,
because they play fundamentally different roles: the state of being practically
certain of *p* plays the role of
guiding the way in which one maintains and revises one’s intentions about how
to act, while the state of being theoretically uncertain of *p* plays the role of registering the fact
that one does not have the highest possible degree of justification for *p*. In this way, we can resolve the
dilemma that confronted us in the previous section.

This distinction
between theoretical credences and practical credences can now be used to give
an account of the nature of outright belief: an outright belief in *p* is the state of being *stably disposed* to have a practical
credence of 1 in *p*, for at least all *normal* practical purposes.

It might be suggested
that to believe a proposition *p* is
simply to treat *p* as though it were
practically certain. But that suggestion would overlook the fact that believing
a proposition is an essentially *dispositional*
state, while “treating” a proposition *p*
in a certain way is an occurrent process of thinking or reasoning with *p*. This is why I propose instead that
having an outright belief is state of being stably *disposed* to treat *p* as
though it were practically certain.

Return to the example
that we have just considered – the case in which you are offered a bet
that will pay $1 if Dublin is the capital of Ireland, and will result in your
being tortured to death if Dublin is not the capital of Ireland. In this case,
it would be natural for you to say, “Look: I’m not going to take this bet. Even
though I definitely believe that Dublin is the capital of Ireland, I’m not so
fantastically confident of that to make it rational for me to take this insane
bet!”[13]

It seems to me that
this is the natural response for you to make in this case because for all *normal* practical purposes, your
practical credence in the proposition that Dublin is the capital of Ireland is
1; this is why you count as believing *p*.
But for the highly *abnormal* purpose
of deciding whether to take this bet, your practical credence is less than 1.

At
all events, this seems to me the best account of the state of outright belief:
it can explain why it seems that outright belief in *p* has the functional role of credence 1 – because outright
belief really does have the functional role of (being disposed to) have
(practical) credence 1 in *p*. At the
same time, it also makes it possible for an agent to have an outright belief in
propositions of which she is not (theoretically) certain, without lapsing into
any of the troubling forms of incoherence that we explored in the previous
section.

*6.
**Why
finite agents need both practical and theoretical credences*

Even if it is *possible* for some agents to have both practical and theoretical
credences, of the sort that I described in the previous section, why should we
think that any *actual* agents have two
systems of credences in this way? In this section, I shall argue that finite
agents – that is, agents whose belief systems are, as I put it above,
“gappy” – can be rational *only if*
they have both practical and theoretical credences. First, I shall argue that
to be rational, a finite agent must have practical credence of 1 in some
propositions for which she does not have the highest possible degree of
justification – that is, propositions for which she definitely has less
justification than for elementary logical truths and the like. Then I shall
argue that to be rational, a finite agent must also have a theoretical credence
of *less* than 1 in such propositions.
If these arguments are correct, then rationality requires all finite agents to
have both practical and theoretical credences.

My argument for the
first step hinges on the definition of the decision-theoretic notion of a
choice’s “expected value”. Within many forms of decision theory –
including both causal decision theory and the “benchmarks theory” that I have
defended elsewhere (Wedgwood 2011) – the notion of a choice’s “expected
value” is defined in terms of a *partition*
of *states of nature*.

A “state of nature” is
a state of affairs that has the following two features: first, it is utterly *beyond the agent’s control* which state
of nature actually obtains; and secondly, each state of nature is sufficiently *detailed* to determine exactly how good
each of the available acts would be if it were performed in that state of
nature. To fix ideas, I shall follow David Lewis (1981) in conceiving of these
states of nature as *conjunctions of
non-backtracking counterfactuals* – where each of these counterfactuals
has the form ‘If I did act *A _{n}*,
outcome

Admittedly, there is
one well-known approach to decision theory – the *evidential* decision theory (EDT) of Richard Jeffrey (1981) –
that does not need to appeal to a partition of states of nature, and can define
the notion of a choice’s “expected value” in terms of any partition whatsoever.
I shall assume here that EDT is false, and that the correct decision theory
(whatever exactly it is) will appeal to a partition of states of nature in this
way.[14]

The crucial point here
is that to say that a set of states of nature forms a “partition” is to say
that the relevant agent is *certain*
that *exactly one* of these states of
nature obtains. In other words, the relevant agent must be *certain* of the *disjunction*
of these states of nature, and she must also be certain of the *negation* of the *conjunction* of any pair of these states of nature.

Intuitively, however, it
seems clear that a disjunction of such states of nature will never be the kind
of proposition for which an agent whose system of beliefs is “gappy” has as much
justification as for the most elementary logical truths. In particular, the
disjunction of these states of nature will not itself be a logical truth –
not even an instance of the law of excluded middle ‘*p* ˅ ¬*p*’.
The *negation* of a state of nature is
not itself a state of nature: the negation of a conjunction of non-backtracking
counterfactuals is not itself a conjunction of non-backtracking counterfactuals
(it is effectively a disjunction of negations of non-backtracking
counterfactuals, which is quite another thing). So, even though the agent has
the highest possible degree of justification for the proposition that either
the state of nature or its negation obtains, this is not enough for having such
a high degree of justification for a disjunction of *states of nature*.

For example, consider a
simple case, where I am deciding whether to travel from London to Oxford by bus
or by train. I may just consider two states of nature: one state of nature* S*_{1}, in which the traffic on
the roads is light, and so taking the bus would result in my arriving in Oxford
after a journey of about an hour and a half, and a second state of nature *S*_{2}, in which the traffic on
the roads is heavy, and so taking the bus would take at least two hours. But I
certainly do not have as much justification for the disjunction of *S*_{1} and *S*_{2} as I have for the simplest logical truths: after all,
my evidence makes it quite clear that there is a non-zero chance that the bus
will crash on the motorway, with the result that I never reach Oxford at all.
Nonetheless, for the purposes of deciding whether or not to catch the bus or
the train, I do not consider the state of nature in which the bus crashes on
the motorway. Indeed, since my belief system is “gappy” in the way that I have
described, I may have no doxastic attitude whatsoever towards the proposition
that the bus will crash: I may be totally “attitudeless” towards that
proposition; and even if I do consider the possibility that the bus will crash,
there will inevitably be other possibilities that I do not consider – such
as that the bus will be hi-jacked by terrorists or the like.

Thus, if the agent is *ever* to be capable of guiding her
choices and intentions by this sort of expectation, the relevant partition will
have to consist in a set of states of nature such that the agent treats the
disjunction of these states of nature as *practically
certain*, even though she does not have the highest possible degree of
justification for this disjunction.

Arguably, a similar
issue arises at another point in the decision-making process. It is arguably
necessary for agents to treat it as practically certain that each of the
options that they are choosing between is *practically
available* – roughly in the sense that if they choose the option, they
will execute the choice and act accordingly. But the *availability* of an option is surely another proposition for which a
finite agent will never have the maximum degree of justification. (There is
always a remote possibility of suddenly dying, or being totally paralysed, in
the time lag between making and executing the choice.) So here too it seems
that rational agents will need to be practically certain of some proposition
for which they have less than the highest possible degree of justification.

As I shall now argue,
however, although a rational finite agent needs to be practically certain of
some propositions of this sort, she also needs *not* to be theoretically certain of them. That is, she needs to
register, or keep track of, the fact that she does not have the highest degree
of justification for these propositions.

Suppose that you start
by being practically certain of a proposition *p*, but then the practical stakes change – for example, because
you are offered a bet that pays $1 if *p*
is true, and results in your dying a horrible death if *p* is false. (Suppose that you know that the proposition *p* was chosen at random, so that being
offered this bet does not itself change the degree of justification that you
have for *p*.) If you are rational, you
will not accept the bet, because in making this decision, you will shift to a
new set of practical credences, which do not involve being practically certain
of *p*. But it seems that some kind of
beliefs must guide you in shifting from one set of practical credences to
another in this way – where these guiding beliefs must have been present *before* this shift in your practical
credences, and must include some kind of belief in the very proposition *p* that is at issue. It seems that these
guiding beliefs cannot be your pre-existing practical credences – since
those credences involve treating *p* as
if it were certain, and so cannot allow you to envisage the risks that would be
involved in believing *p* if *p* is false. Instead, these guiding
beliefs must include some non-maximal level of confidence in *p* itself.

Intuitively,
for this sort of shift to be rational, these guiding beliefs must be a set of
credences that keep track of the degree of justification that the agent has for
each of the propositions involved. In effect, they must be what I have called
theoretical credences. This seems to show that to be rational, finite agents
need theoretical credences as well as practical credences: for finite agents
like us, both practical and theoretical credences are indispensable.

*7.
**What
practical credences will a rational agent have?*

According to the argument that I have
just given, the agent’s theoretical credences – presumably including
credences about the practical stakes in the practical situation at hand –
can somehow guide a rational agent to shift from one set of practical credences
to another. To be confident that this argument is correct, we should give a
more detailed account of how exactly a rational agent will shift from one set
of practical credences to another in this way.

One suggestion that
might be considered is that the rational agent will have a set of practical
credences that *maximizes expected value*
(when the relevant “expectation” is defined in terms of the agent’s *theoretical* credences). But this
suggestion has a fatal problem. Practical credences that maximize expected value
in this way might include a practical credence of 1 in a proposition *p* such that the agent is *theoretically* almost* certain* that *p* is *false*.

If you treat a
proposition *p* as though it were
certain, for the purposes of making a given decision, even though in fact you
are theoretically almost certain that *p*
is false, then your attitude towards *p*
is surely not a state of belief at all. It seems rather to be the state that
Michael Bratman (1992) has called “acceptance”. For example, for the purposes
of planning your household budget, you might “accept” that the repairs that you
have to carry out to the roof of your house will cost at least $25,000 –
even though you are theoretically almost certain that the roof repairs will
cost less than that. Or to take another example, which is due to Keith Frankish
(2009, 86), a lawyer might “accept that their client is innocent for the
purposes of defending them” – even if the lawyer is theoretically almost
certain that their client is guilty.

In general, it seems
that no rational believer could *ever*
have a practical credence of 1 in a proposition *p* in which her theoretical credence is less than 0.5. This is a
point that our account of which practical credences the rational agent will
have must somehow explain.

In general, I propose, *any* kind of belief in a proposition *p* – including both theoretical and
practical credences in *p* – is
rationally required to be guided purely by certain distinctively *epistemic* values. Very roughly, being
guided by these distinctively epistemic values involves pursuing the goals of *believing p if p is true* and *not believing p if p is false*. Practical
considerations (such as the costs and benefits at stake in the agent’s
situation) only make a difference to the practical credence that the agent has
in *p* by making a difference to the
way in which these two goals – *believing
p if p is true* and *not believing p if
p is false* – are *balanced*
against each other.

For example, if the
goal of believing *p* when *p* is true is weighted relatively
heavily, while the goal of not believing *p*
if *p* is false is not weighted
particularly heavily, then this weighting seems to favour having practical
credences that are more *adventurous*
and *extreme* (that is, closer to 0 and
1) than one’s theoretical credences. On the other hand, if the goal of not
believing *p* if p is false is weighted
very heavily, and the goal of believing *p*
if *p* is true is not weighted
especially heavily, then this alternative weighting seems likely to favour
practical credences that are more *cautious*
and *non-committal* (that is, closer to
0.5) than one’s theoretical credences.[15]

We can articulate this
idea more precisely by introducing the idea of the “epistemic disvalue” of a
credence. A credence’s “epistemic disvalue” is a measure of the extent to which
it *falls short* of achieving these two
goals – by involving either (i) *insufficient*
credence in a *true* proposition or
(ii) *excessive* credence in a *false* proposition. In general, if *p* is true, then the *higher* one’s credence in *p*
is, the *lower* the epistemic disvalue
of one’s practical credence in *p*;
while if *p* is false, the *higher* one’s credence in *p*, the *greater* the epistemic disvalue of one’s practical credence in *p*. More specifically, let us assume that
the epistemic disvalue of one’s credence in *p*
is an increasing function of the *distance*
between that credence and *p*’s actual
truth-value.[16]

This notion of the
“epistemic disvalue” of an agent’s credence in a particular proposition *p* can be generalized to obtain a more
general notion of the epistemic disvalue of a whole *set* of credences. So long as we are only comparing different
possible sets of credences in the *same*
set of propositions, we may simply take the epistemic disvalue of each possible
set of credences to be the *sum* of the
epistemic disvalues of all the individual credences in that set.[17]

With this notion of the
“epistemic disvalue” of credences in hand, we can now make sense of a notion of
“*practically modulated *epistemic
disvalue” (or “*practical-epistemic
disvalue*” for short). Practical-epistemic disvalue always agrees with
epistemic disvalue in its *ranking* of
credences on the basis of their distance from the truth. So, for example, if *p* is true, it will always be better, in
terms of practical-epistemic value, to have a credence of 0.9 in *p* than to have a mere credence of 0.5 in
*p*. However, *how much* better
(in terms of practical-epistemic value) it is, when *p* is true, to have a
credence of 0.9 in *p* than to have a credence of 0.5 in *p* may
depend on what is at *stake* in the
relevant *practical situation*. In
effect, what is at stake in the practical situation may determine *which* increasing function of the
distance between the credence and the actual truth-value is the appropriate
measure of practical-epistemic disvalue in that situation.

So, for example, there
may be some situations in which, if *p*
is true, the degree to which having a practical credence of 0.9 in *p* is better (in terms of
practical-epistemic value) than merely having a practical credence of 0.5 in *p* is unusually great – whereas if *p* is false, the degree to which having a
practical credence of 0.9 in *p* is
worse than having a credence of 0.5 in *p*
is relatively small. Intuitively, these are the situations in which the
appropriate measure of practical-epistemic value favours having more
adventurous or extreme practical credences.

More precisely, we may
now propose the following account of which practical credences a rational agent
will have. According to this account, a rational agent’s practical credences
will always be a probabilistically coherent set of credences that *minimizes expected practical-epistemic
disvalue*, according to the measure of practical-epistemic value that is
appropriate for particular choice situation at hand – where the relevant
“expectation” is defined in terms of the agent’s *theoretical* credences.

This approach can
explain why the rational believer will never have a practical credence of 1 in
a proposition in which her theoretical credence is 0.5 or less, if it is true
that in *all* practical situations, the
appropriate measure of practical-epistemic disvalue is a *convex* increasing function of the distance between one’s credence
and the actual truth-value.[18]
Then the agent will never minimize expected practical-epistemic disvalue by
having a practical credence of 1 in a proposition in which her theoretical
credence is less than or equal to 0.5.

One famous measure of a
credence’s disvalue is the *Brier score* –
that is, the *square* of the distance
between one’s credence in the proposition and the proposition’s actual
truth-value. The Brier score has several well-known properties, which guarantee
that in any situation in which the Brier score is appropriate measure of
practical-epistemic disvalue, the practical credences that minimize expected
disvalue will be those that are *identical*
to the agent’s theoretical credences.[19]

However, there may be
other practical situations in which the appropriate measure of
practical-epistemic disvalue differs from the Brier score. For example, in some
practical situations, the appropriate measure might be the *cube* of the distance between one’s credence in the proposition and
the proposition’s actual truth-value. With this “cubic” measure, the practical
credences that minimize expected practical-epistemic disvalue will be *more cautious* and *non-committal* (that is, more clustered around intermediate
credences like 0.5) than one’s theoretical credences. So in any practical
situation in which the appropriate measure of practical-epistemic disvalue is
this “cubic” measure, a rational agent’s practical credences will be more
cautious and non-committal in this way.

In some other practical
situations, the appropriate measure of practical-epistemic disvalue might be
the distance between the credence and the actual truth-value raised to the
power of some real number *n* such that
1 < *n* < 2 –
for example, the appropriate measure might be the distance between the credence
and the actual truth-value raised to the power of 1.1. With this sort of
measure, the practical credences that minimize expected practical-epistemic
disvalue will be *more extreme* and *adventurous* (that is, closer to the
extremes of 0 and 1) than one’s theoretical credences.

According to the arguments of Section 6,
most practical situations will in fact be situations of this latter kind, in
which the appropriate measure of practical-epistemic disvalue is one of these
more “adventurous” measures. It is not clear that there are any situations in
which the appropriate measure of practical-epistemic value is one of the
“cautious” or “non-committal” measures that I have just described. Still,
perhaps we should allow for the possibility-in-principle of situations of this
sort. At all events, I propose the following two theses: first, there is an
appropriate measure of practical-epistemic value for each practical situation
that an agent is in; and secondly, a rational agent’s practical credences will
always be a probabilistically coherent set of credences that minimizes expected
practical-epistemic disvalue, according to the measure of practical-epistemic
value that is appropriate to the particular practical situation at hand –
where the relevant “expectation” is defined in terms of the agent’s theoretical
credences.

*8.
**Conclusion: Pragmatism vs.
intellectualism*

I shall conclude by noting the
way in which this approach allows for a kind of reconciliation of the rival pragmatist
and intellectualist approaches in epistemology.[20]

First, according to this approach, a form of intellectualism is true of
the theoretical credences. Which theoretical credences a rational believer will
have is sensitive only to the degree of justification that she has in favour of
the relevant propositions – that is, as most philosophers would put it, to
the degree to which these propositions are supported by the believer’s
evidence.

Secondly, a kind of
pragmatism is true of the practical credences: which practical credences the
rational agent will have *is* sensitive
to practical considerations – such as the needs, costs and benefits that
are at stake in the agent’s particular practical situation – since as I
have explained, these practical considerations may make a difference to the
measure of practical-epistemic value that is appropriate in that situation.

Finally, a *qualified* pragmatism is true of outright
beliefs: which propositions the rational believer has an outright belief in is
determined by the needs, costs, and benefits that are at stake in *normal* situations – though not
necessarily to the needs, costs, and benefits at stake in the particular
situation at hand.

This account of belief is undeniably complex and
elaborate. However, I hope that the arguments given here have made it plausible
that belief is itself a complex phenomenon, so that no less elaborate account
could do justice to this phenomenon itself.*

References

Bratman, M. E. 1992, “Practical Reasoning and Acceptance in a
Context”, Mind 101, pp. 1–15

Christensen, D. 2004, *Putting
Logic in its Place: Formal Constraints on Rational Belief*, Oxford: Oxford
University Press

Elga, A. 2010, “Subjective Probabilities should be Sharp”, Philosophers’
Imprint 10, no. 5, pp. 1–11 <www.philosophersimprint.org/010005/>

Foley, R. 2009, “Beliefs, Degrees of Belief, and the Lockean
Thesis”, in: F. Huber and C. Schmidt-Petri, eds., *Degrees of Belief* (Synthese Library Vol. 342), Berlin: Springer, pp. 37–47

Frankish, K. 2009, “Partial Belief and Flat-Out Belief”, in: F. Huber
and C. Schmidt-Petri, eds., *Degrees of
Belief* (Synthese
Library Vol. 342), Berlin: Springer, pp. 75–93

Hacking, I. 1975, *The
Emergence of Probability*, Cambridge: Cambridge University Press

Harman, G. 1986, *Change in View*,* *Cambridge, Massachusetts: MIT Press

Hawthorne, J. 2003, *Knowledge
and Lotteries*, Oxford: Oxford University Press

James, W. 1979, *The Will to
Believe and Other Essays in Popular Philosophy*, Cambridge, Massachusetts:
Harvard University Press

Jeffrey, R. C. 1970, “Dracula Meets Wolfman: Acceptance *vs*. Partial Belief”, in: M. Swain, ed., *Induction, Acceptance, and Rational Belief*,
Dordrecht: Reidel, pp. 157–85

Jeffrey, R. C. 1981, *The Logic
of Decision* (revised edition), Chicago: University of Chicago Press

Jeffrey, R. C. 2004, *Subjective
Probability: The Real Thing*, Cambridge: Cambridge University Press

Joyce, J. M. 1999, *Foundations
of Causal Decision Theory*, Cambridge: Cambridge University Press

Joyce, J. M. 2009, “Accuracy and Coherence: Prospects for an
Alethic Epistemology of Partial Belief”, in: F. Huber and C. Schmidt-Petri,
eds., *Degrees of Belief* (Synthese Library Vol. 342), Berlin: Springer, pp. 263–97

Kaplan, M. 1996, *Decision
Theory as Philosophy*, Cambridge: Cambridge University Press

Krantz, D. H., Luce,
R. D., Suppes, P., and Tversky, A. 1971, *Foundations of Measurement I: Additive and Polynomial Representations*,
London: Academic Press

Kyburg, H. 1961,
*Probability and the Logic of Rational Belief*, Middletown, Connecticut:
Wesleyan University Press

Levi, I. 1980, *The
Enterprise of Knowledge: An Essay on Knowledge, Credal Probability, and Chance*,
Cambridge, Massachusetts: MIT Press

Lewis, D. 1981, “Causal Decision Theory”, *Australasian Journal
of Philosophy* 59, pp. 5–30

Maher, P. 1993, *Betting on
Theories*, Cambridge: Cambridge University Press

Nozick,
R. 1981, *Philosophical Explanations*,
Cambridge, MA: Harvard University Press

Plantinga, A. 1993, *Warrant: The Current Debate*, Oxford:
Clarendon Press

Pollock, J. L., and Cruz,
J. 1999, *Contemporary Theories of
Knowledge*, Totowa, New Jersey: Rowman & Littlefield

Ross, J. and Schroeder,
M. 2012, “Belief, Credence, and Pragmatic Encroachment”, Philosophy and
Phenomenological Research, DOI: 10.1111/j.1933-1592.2011.00552.x

Stalnaker, R. 1984, *Inquiry*,
Cambridge, Massachusetts: MIT Press

Sturgeon, S. 2008, “Reason and the
Grain of Belief”, Noűs 42, pp. 139–65

Weatherson, B. 2005, “Can We Do Without Pragmatic Encroachment?”, Philosophical
Perspectives 19, pp. 417–43

Wedgwood, R. 2007, *The Nature of Normativity*, Oxford:
Clarendon Press

Wedgwood, R. 2011, “Gandalf’s
Solution to the Newcomb Problem”, Synthese Online First, DOI: 10.1007/s11229-011-9900-1

Weisberg, J. 2011, “Belief: Partial
and Full”, unpublished

† University of Southern California, School of Philosophy, 3709 Trousdale Parkway, Los Angeles, CA 90089-0451, USA, Email: wedgwood@usc.edu

[1]
Compare Maher (1993, 131): “It is … *part*
of the folk concept of belief that it is a mental state expressed by sincere,
intentional assertions.”

[2]
Most traditional epistemologists claim that knowledge implies belief: for
example, Nozick (1981, chap. 3) is typical in this respect. It seems clear that
when these epistemologists claim this, they mean that knowledge implies *outright* belief (not just some
more-or-less high level of confidence).

[3]
The approach that starts out with the idea that our levels of confidence
fundamentally consist in a *ranking* of
the relevant propositions is explored by Joyce (1999, 91, 138, 256).

[4]
For example, it would certainly be possible to measure your levels of
confidence by means of a real-valued credence function (unique up to at least
positive affine transformation) if they formed a *positive difference structure* of the sort that is defined by Krantz
et al. (1971, chap. 4). Moreover, even if your levels of confidence did not
form a complete difference structure, the facts about which differences in
levels of confidence are greater or smaller than others would impose numerous *constraints* on the credence functions
that could represent these levels of confidence, and so it would still be
possible to represent your levels of confidence by means of a *set* of real-valued credence functions.

[5]
I am not assuming any *reductive* form
of functionalism here: I am assuming only that it is a *necessary* feature of each type of mental state that it is disposed
to play a certain functional role. For more discussion of the various forms of
functionalism, see Wedgwood (2007, chap. 8).

[6] The overwhelming majority of philosophers who have written on this topic accept that the functional role of outright belief involves a disposition towards this sort of deductive cogency: see e.g. Kaplan (1999, 95–8) and Frankish (2009, 80). There are a few dissenters, however: see especially Christensen (2001, Chap. 4) and Sturgeon (2008, Section 5).

[7] For a statement of such a Bayesian conception of theoretical rationality, see e.g. Jeffrey (2004). This conception is undeniably controversial: for some critics, see e.g. Plantinga (1993) and Pollock and Cruz (1999); unfortunately I shall not be able to answer these criticisms here.

[8] For an argument that the perfectly rational agent’s levels of confidence must be “sharp” in this way, see Elga (2010).

[9] For one version of this account of practical rationality, see Wedgwood (2011).

[10] This history was famously studied by Hacking (1975).

[11]
This is left implicit in most philosophers’ accounts of outright belief. The
main exception is Levi (1980), who insists that to have an outright belief in a
proposition *p* is to have credence 1
in *p*.

[12]
In other words, even if Weatherson’s account implies that all agents who have
an outright belief in *p* have the same
preferences as they would have if they really treated *p* as certain, it does *not*
entail that they *reason* in the same
way as they would if they treated *p*
as certain. On this point, compare Ross and Schroeder (2012, Section 2).

[13] I owe this point to Jacob Ross (personal communication) – although the general idea seems to be implicit in Ross and Schroeder (2012).

[14] The argument for the conclusion that finite agents will need to rely on outright beliefs in their decision making that I give here is broadly similar to an argument that is given by Ross and Schroeder (2012, Section 1.3). However, their argument is at least slightly less precise than mine: one way in which their argument is less precise is that they do not make it clear that the argument has to assume that the correct theory of rational decision is not EDT, but a theory that appeals to “states of nature” or the like.

[15] This feature of the idea of “pursuing the truth” seems to have first been noticed by James (1979).

[16] In the terminology of Joyce (2009), the appropriate measure of epistemic disvalue should meet the conditions of truth-directedness, normality, and extensionality.

[17]
It would be a challenging matter to extend this notion of the “epistemic
disvalue” of sets of credences so that it can compare sets of credences in *different* sets of propositions; I cannot
address this issue here.

[18] For a discussion of the significance of convexity, see Joyce (2009, 280–4).

[19] For a discussion of the properties of the Brier score, see Joyce (2009, 290–93).

[20] For an illuminating discussion of the debate between pragmatism and intellectualism, see Weatherson (2005).

* The first draft of this paper was presented at a workshop at the University of Glasgow in December 2009, at the University of Konstanz in January 2010, and at Monash University in August 2010. I am grateful to all those audiences for their helpful comments. Eventually – perhaps particularly because of the comments of Toby Handfield at Monash – I realized that this first draft had to be drastically revised from beginning to end. In preparing the new and revised version, I particularly benefited from the comments of Jacob Ross, Bernhard Salow, Mark Schroeder, Timothy Williamson, and the editors of this volume. The first draft was written during my tenure of a Research Fellowship from the Leverhulme Trust, whom I should like to thank for their generous support.