## 1. How Far Can Consistency and Coherence Take Us? Categorical Self-Doubting Belief



It might seem that consistency and coherence are not strong enough to
tell us what the relation should be between the first order and the
second order in cases of epistemic self-doubt, in the same way that
they don't seem sufficient for explaining what is wrong with
Moore-paradoxical statements (see entry on
 epistemic paradoxes).
 In the latter I assert either "*p* and I don't
believe *p*", or "*p* and I believe
not-*p*". There is a lack of fit between my belief and my
belief about my belief in either case, but the beliefs I hold
simultaneously are not inconsistent in content. What I say of myself
would be consistent and quite sensible if said about me by someone
else, thus: "*p*, but she doesn't believe
*p*". Similarly, there is nothing inconsistent in the claim
"There is a cat in the distance and she is severely
near-sighted", though there does seem to be a problem with the
first-person claim "There is a cat in the distance and I am
severely near-sighted" if my assertion about the cat is made on
the basis of vision and I give no cue that I mean the second clause as
a qualification of the first. My confidence about the cat should have
been tempered by my awareness of the limitation of my vision. If there
are general principles of rationality that govern self-doubt of our
faculties or expertise it seems that they will have to go beyond
consistency among beliefs.



However, consistency and coherence do impose constraints on what a
subject may believe about the reliability of her beliefs if combined
with an assumption that the subject knows what her beliefs are. (This
is also found for Moore's paradox, and is used in
Shoemaker's approach to that problem; Shoemaker 1994.) One way
of formulating an extreme case of believing that one's epistemic
system doesn't function well is as attributing to oneself what
Sorensen calls anti-expertise (Sorensen 1988: 392f.). In the simplest
type of case, *S* is an *anti-expert* about *p* if
and only if




Anti-expertise (A)
  

Either *S* believes *p* and *p* is false or *S*
does not believe *p* and *p* is true. 




Sorensen pointed out that if *S* is consistent and knows
perfectly what her beliefs are, then she cannot believe she is an
anti-expert. For if *S* believes *p* then, by perfect
self-knowledge, she believes that she believes *p*, but her
beliefs that *p* and that she believes *p* are together
inconsistent with both of the disjuncts of A. Similarly for the case
where *S* does not believe *p*. This phenomenon generalizes
from outright belief to degrees of belief, and from perfect knowledge
of one's beliefs to decent but imperfect knowledge of them (Egan
& Elga 2005: 84ff.).



Believing you are an anti-expert is not compatible with coherence and
decent knowledge of your own beliefs. Denying that knowledge of our
own beliefs is a requirement of rationality would not be helpful,
since usefully doubting that one's beliefs are soundly formed
would seem to require a good idea of what they are. Egan and Elga
favor the view that your response to this fact about anti-expertise
should be to maintain coherence and decent self-knowledge of belief,
and refrain from believing that you are an anti-expert. However, one
can imagine examples where the evidence that you are incompetent is so
overwhelming that one might think you should believe you are an anti-expert even if
it makes you incoherent (Conee 1987; Sorensen 1987, 1988; Richter
1990; Christensen 2011).



The problem with self-attributing unreliability coherently
doesn't go away if the degree of unreliability is more modest.
Consider the following property:




I'm not perfect (INP)
  

\(P((P(q) \gt .99 \amp -q )\textrm{ or } (P(q) \lt .01 \amp q)) \gt .05\) 





This says that you are at least 5% confident that you're highly confident
of *q* though it's false or lack confidence in *q*
though it's true. It is a softened version of anti-expertise,
and you can't coherently fulfill it, have \(P(q) >.99\), and
have perfect knowledge of your beliefs. For in that case \(P(P(q) >
.99) = 1\), which means INP can only be true if \(P(-q) > .05\).
But \(P(-q) > .05\) implies \(P(q) < .95\) which contradicts
\(P(q) > .99\). The point survives if you have imperfect but good
knowledge of what your beliefs are. The self-doubt expressed through
INP is quite modest, but is no more consistent than attributing to
oneself anti-expertise, and this will be so for any value of
INP's right hand side that is not equal to \(P(-q)\). 



Egan and Elga think that the significance of evidence of
anti-reliability is taken into account by seeing it as obligating a
subject to revise her first-order belief. However, their view implies
that to be rational such a revision must be done without attributing
anti-expertise to oneself. One can revise whenever one wants of
course, but any revision should have a reason or motivation. If one
does not give any credence at all to the possibility one is an
anti-expert, then what is one's reason to revise one's
first-order belief? There would seem to be no other way to take on
board and acknowledge the evidence of your anti-expertise than to give
some credence to its possibility. Egan and Elga say that the belief
the evidence should lead a subject to is that she has been an
anti-expert and that should lead her to revise (Egan & Elga 2005:
86). But if she avoids
incoherence by attributing anti-expertise only to a previous self,
then that belief can't be what leads her to revise her current
view. If she does not attribute anti-expertise to her current self
then she doesn't give her current self any reason to revise.



The same problem can be seen with Egan and Elga's treatment of
cases of self-attributing less extreme unreliability (such as INP)
that they regard as unproblematic. Consider a person with growing
evidence that his memory isn't what it once was. What effect
should this have on his beliefs about the names of students? They
compare what happens to his confidence that a given student is named
"Sarah" when he hears counterevidence--overhearing
someone calling her "Kate" for example--in the case
where he has and the case where he hasn't taken the evidence of
his decline in memory into account. Via a Bayesian calculation they
conclude that when he hasn't taken into account the evidence
about his memory, the counterevidence to his particular belief that
the student is named Sarah does reduce his belief that she is Sarah,
but it does so much less than it would have if he had taken the
evidence about his memory into account.



But this analysis represents taking into account the evidence about
one's memory only implicitly, as an effect that that evidence
already had on one's prior probability that the student is
Sarah. That effect is the difference between a .99 and a .90 prior
probability, or degree of belief. The distinction that is then derived
between the effects that counterevidence can have on the self-doubter
and the non-self-doubter is just the familiar point that
counterevidence will have a greater effect the lower one's
initial probability. 



This doesn't tell us how to assimilate news about one's
decline in reliability to the first-order belief, but only how to
treat other evidence about the first-order matter once one has done
so. The question was supposed to be how the evidence about memory
should affect our beliefs, and to answer that requires saying how and
why that evidence about his memory should make our subject have a .90
rather than .99 initial confidence that the student was Sarah. Surely
one must attribute reduced reliability to oneself if one is to have
any reason to revise one's first-order belief that the student
was Sarah on the basis of evidence of diminished reliability. Even a
raw feel of redness of a flower must become an ascription of red to
the flower in order for one's experience of it to affect other
beliefs such as that it would or wouldn't be an appropriate
gift. Even evidence suggesting a small amount of unreliability, as
with INP above, presents us with a trilemma: either we incoherently
attribute unreliability to ourselves but revise and have a
justification for doing so, or we coherently fail to attribute
unreliability and revise without justification for doing so, or we
remain coherent by failing to attribute unreliability and don't
revise, ignoring evidence of our unreliability. It seems that it is
not possible for a rational subject to acknowledge evidence of her own
unreliability and update her first-order belief on the basis of
it.



This approach using consistency (or coherence) plus self-knowledge of
belief gives a way of representing what a state of self-doubt is. It
sensibly implies that it is irrational to stay in such a state but it
also implies that it is irrational to be in it in the first place,
making it unclear how self-doubt could be a reason to revise. The
approach identifies a kind of matching that rationality demands: give
no more credence to the possibility that one is a bad judge of
*q* than one gives to not-*q*. However, it leaves other
questions unanswered. If the rational subject finds herself doubting
her judgment, should she defer to her first-order evidence, or her
second-order evidence about the reliability of her first-order
judgment? What are the rules by which she should decide, and how can
they be justified?


## 2. Conditional Principles


### 2.1 Synchronic Reflection and Self-Respect



We might do better at understanding the relations rationality requires
between your beliefs and your beliefs about them by adding to the
requirements of consistency and coherence a bridge principle between
the two orders expressed via conditional (subjective) probability.
Conditional probabilities say what your degree of belief in one
proposition is (should be) given another proposition, here the
relevant propositions being a first-order proposition *q*, and
the proposition that one has degree of belief *x* in *q*,
respectively. A first pass at how to represent a situation where my
beliefs at the two levels don't match comes from its apparent
conflict with the synchronic instance of the Reflection Principle (van
Fraassen 1984).




Reflection
  

\(P\_0 (q\mid P\_1 (q) = x) = x \) 




Reflection says that my current self's degree of belief in
*q* given that my future self will believe it to degree *x*
should be *x*. It is implied by the fact that her degrees of
belief are represented as probabilities that my future self is
coherent, but that alone does not rule out the possibility that her
judgment is compromised in some other way--as for example when
Ulysses anticipated that he would be entranced by the Sirens--and
the principle can be questioned for such cases (Sobel 1987;
Christensen 1991; van Fraassen 1995). However, the self-doubt we are
imagining is one that the subject has about her current beliefs, and
the synchronic version of Reflection




Synchronic Reflection (SR)
  

\(P\_{0}(q\mid P\_{0}(q)=x) = x \) 




which says that my degree of belief in *q* now given that I now
believe *q* to degree *x* should be *x*, seems less
open to question. Christensen (2007b) also calls this principle
Self-Respect (SR). This is not the tautology that if I believe
*q* to degree *x* then I believe *q* to degree
*x*, for in a logically equivalent form the principle is




Synchronic Reflection/Self-Respect (SR)
  

\([P\_{0}(q \amp P\_{0}(q)=x)\mid P\_{0}(P\_{0}(q)=x)] = x \) 




which does not follow from either deductive logic or the probability
axioms alone. But SR has been widely endorsed as unobjectionable, and
according to some even undeniable, as a requirement of rationality
(van Fraassen 1984: 248; Vickers 2000: 160; Koons 1992:
23--Skyrms 1980 sees
as useful a version of it he calls Miller's Principle, though he
also shows it is subject to counterexamples).



While I can sensibly imagine my future self to be epistemically
compromised, unworthy of my deference, violating SR would require
regarding my current self as epistemically compromised, as having a
degree of belief that should be other than it is. This appears to be
something that doubt of my own judgment would call for, in which case
whether self-doubt can be rational depends on whether SR is a
requirement of rationality.



SR can be defended as a rational ideal by Dutch Strategy arguments,
though not by the strongest kind of Dutch Book argument (Sobel 1987;
Christensen 1991, 2007b: 328-330, 2010b; Briggs 2010--Roush
2016 argues it can't be defended as a requirement by a
Dutch book argument at all). It has been argued to be questionable if
not false on grounds that it conflicts with the *Epistemic
Impartiality Principle* that says we should not in general take
the mere fact that we have a belief as a reason to have that belief
any more so than we do with the mere fact that others have that belief
(Christensen 2000: 363-4; Evnine 2008: 139-143; Roush
 2016).[1]



Nevertheless the probabilist--one who thinks that rationality
requires probabilistic coherence--will have a difficult time
resisting SR since, analogously to what we saw above with
anti-expertise, SR follows from coherence if it is supplemented with
the further assumption that the subject has perfect knowledge of her
own beliefs. Still, this does little to explain intuitively why SR
should be binding; even someone who has perfect knowledge that he has
a belief should be able to sensibly wonder whether it is a belief he
ought to have. That perfect knowledge of our beliefs is a requirement
of rationality can be doubted in a variety of ways (Williamson 2000;
Christensen 2007b: 327-328; Roush 2016). However, as
above so for the discussion here, denying that rationality requires
knowledge of our own beliefs does not overcome the problem.
Self-correction that will be of any use requires some degree of
accuracy about one's beliefs, and even if a subject
doesn't have perfect self-knowledge coherence still makes
reflective demands; Christensen (2007b: 332) has noted that the closer
a coherent subject comes to perfect knowledge of his beliefs the more
nearly he will satisfy SR.



SR has something to recommend it, but it seems to be a rule a
self-doubter will violate. Consider our underslept doctor. It looks as
if once it is pointed out to her how long it has been since she slept,
she should regard her current confidence in *q*, her diagnosis of
embolism, as higher than it ought to be. I.e., she would instantiate a
principle we could call Refraction:




Refraction
  

\(P\_{0}(q\mid P\_{0}(q)=x) < x \) 




Apparently, her degree of belief that it's an embolism given
that she has degree of belief *x* that it's an embolism
should be less than *x*, contradicting SR. Or imagine that the
person who tells me a hallucinogenic drug has been slipped into my
coffee is a trusted friend who is neither in the habit of joking nor
currently smirking. I seem to have an obligation to
regard some of my current degrees of belief as higher than they should
be.



Refraction is a way of representing a state of self-doubt, one in
which I don't regard the degree of belief I (think I) have as
the right one to have. But despite the fact that it doesn't have
the subject attributing unreliability to herself categorically as we
had in the last section, Refraction is not compatible with the
combination of coherence and knowledge of one's beliefs, since
the latter two together imply SR. In this representation of what
self-doubt is, it is not rational according to the probabilistic
standard.



One might defend this verdict by saying that the exception proves the
rule: if I really think that my degree of belief in *q* should be
different than it is, say because I realize I am severely underslept,
then surely I should change it accordingly until I come to a credence
I do approve of, at which point I will satisfy SR. However even if it
is ideal to be in the state of self-respect that SR describes, it
seems wrong to say that a state of disapproving of one's
first-order belief when faced with evidence of one's impaired
judgment is irrational. In such a case it would seem to be irrational
*not* to be in a state of self-doubt. Moreover, it is unclear
how a revision from a state violating SR to one in conformity with SR
can be rational. According to many probabilists the rational way to
revise beliefs is via conditionalization, where one's new
degrees of belief come from what one's previous function said
they should be given the new belief that prompts the revision (see
entries on
 interpretations of probability
 and
 Bayes' Theorem).
 That is, all change of belief is determined by the conditional
probabilities of the function one is changing from. Thus change of
belief on the basis of a belief about what my belief is will depend on
the value of \(P\_i(q\mid P\_i(q) = x)\). If the value of \(P\_i(q\mid
P\_i(q)=x)\) isn't already *x* then a conditionalization
using that conditional probability will not necessarily make
\(P\_f(q\mid P\_f(q)=x) = x\), as required by SR, and it is hard to see
how it could. In this approach, similarly to the previous, the whole
cycle of epistemic self-doubt and resolution appears to be unavailable
to a probabilistically rational subject.



If we represent epistemic self-doubt as a violation of Synchronic
Reflection (Self-Respect), then it is not rational for a coherent
person who knows what her beliefs are. This is a general rule giving
the same verdict in all cases, that the orders must match, and it
gives the form of that matching in terms of a conditional probability.
The second-order is in the driving seat since the condition in
SR's conditional probability that determines a value for the
first-order proposition *q* is itself a statement of probability,
but SR can't lead to change of first-order belief unless one
does not know what one's belief in *q* is. As in the
approach via categorical statements above, this simple approach via
conditional probability does not represent the cycle of self-doubt and
resolution as available to a rational subject.


### 2.2 What Would the Maximally Rational Subject Do?



Another way of representing self-doubt using conditional probability
follows the intuitive thought that I should tailor my first-order
degree of belief to the confidence I think the maximally rational
subject would have if she were in my situation (Christensen 2010b:
121). This would be a sensible explanation of the authority of higher
order beliefs, of why taking them into account would be justified. It
gives half of an answer to the question when the first-order should
and shouldn't defer to the second order by identifying a class
of second-order statements to which the first order must always defer.
However it pushes back the question of which individual statements
those are to the question of which probability function is the
maximally rational.



A conditional principle that would capture the idea of deferring to
the view of an ideal agent who was in one's shoes is:



\[
Cr(q\mid P\_{M}(q) = x) = x
\]


(Christensen 2010b) which says that one's credence in *q*
given that the maximally rational subject in one's situation has
credence *x* in *q*, should be *x*. The maximally
rational subject obeys the probability axioms and possibly has further
rationality properties that one might not possess oneself, though she
is assumed to be in your situation, having no more evidence than you
do. If one obeys the probability axioms oneself, then this principle
becomes:




RatRef
  

\(P(q\mid P\_{M}(q) = x) = x \) 




This says that your credence in *q* should be whatever you take
the maximally rational credence to be for your situation, an idea that
seems hard to argue with. It is a variant of a principle used by Haim
Gaifman (1988) to construct a theory of higher-order probability.
There the role of \(P\_{M}\) was given to what has become known as an
*expert function* corresponding, in his use, to the
probabilities of a subject who has maximal knowledge.



RatRef gives a sensible account of cases like the underslept doctor.
It would say the reason that the realization she is sleep-deprived
should make her less confident of her diagnosis is that a maximally
rational person in her situation would have a lower confidence.
Moreover this provides us with a way of representing the state of
self-doubt coherently, even with perfect knowledge of what one's
belief states are. One can have degree of belief *y* in *q*,
and one can even believe that one has degree of belief *y*, that
is, believe that \(P(q) = y\), consistently with believing that the
maximally rational agent has degree of belief *x*, i.e.,
\(P\_{M}(q) =x\), because these are two different probability
functions.



Because self-doubt is not defined as a violation of the conditional
probability RatRef, as it was with SR, we can also see a revision that
takes you from self-doubt to having your confidence match that of the
maximally rational subject as rational according to
conditionalization. You may have degree of belief *y* in
*q*, discover that the maximally rational subject has degree of
belief *x*, and because you have the conditional probability
RatRef put yourself in line with that ideal subject. Note that it is
not necessary to have an explicit belief about what your own degree of
belief in *q* is for this revision to occur or to be
rational.



RatRef has problems that are easiest to see by considering a
generalization of it:




Rational Reflection (RR)
  

\(P(q\mid P' \textrm{ is ideal}) = P'(q) \) 




Rational Reflection (Elga 2013) maintains the idea that my degree of
belief in *q* should be in line with that I think the maximally
rational subject would have in my situation, but also highlights the
fact that my determining what this value is depends on my identifying
which probability function is the maximally rational one to have. I
can be coherent while being uncertain about that, and there are cases
where that seems like the most rational option. This by itself is not
a problem because RR is consistent with using an expected value for
the ideal subject, a weighted average of the values for *q* of
the subjects I think might be the maximally rational subject. But it
is not only I who may be uncertain about who the maximally rational
subject is. Arguably the maximally rational subject herself may be
uncertain that she is--after all, this is a contingent fact, and
one might think that anyone's confidence in it should depend on
empirical evidence (Elga 2013).



The possibility of this combination of things leads to a problem for
RR, for if the subject who is actually the maximally rational one is
unsure that she is, then if she follows RR she won't fully trust
her own first-order verdict on *q* but will as I do correct it to
a weighted average of the verdicts of those subjects she thinks might
be the maximally rational one. In this case my degree of belief in
*q* *given* that she is the maximally rational subject
should not be her degree of belief in *q*. It should be the one
she would have in case she were certain that she were the maximally
rational subject:




New Rational Reflection (NRR)
  

\(P(q\mid P' \textrm{ is ideal}) = P'(q\mid P' \textrm{ is ideal}) \)





This principle (Elga 2013) also faces problems, which are developed
below through the approach to self-doubt via higher-order
evidence.



The approach asking what the maximally rational subject would do gives
a motivation for the idea that second-order evidence has authority
with respect to first-order beliefs, and like the SR approach puts the
second order in the driver's seat by having a statement of
probability in the condition of the conditional probability. This may
appear to give unconditional authority to second-order evidence, but
second-order evidence won't change the subject's
first-order verdict if she believes the latter is already what the
maximally rational subject would think. The approach represents
self-doubt as a coherent state that one can also coherently revise via
conditionalization. It identifies a state of matching between the
orders--matching one's confidence to one's best guess
of the maximally rational subject's confidence. This gives a
general rule, and demands that same matching for all cases, though
gives no explicit guidance about how to determine which is the
maximally rational subject or degree of belief.


## 3. Higher-Order Evidence



Questions about the rationality (or reasonableness or justifiedness)
and import of epistemic self-doubt can be developed as questions about
whether and how to respect evidence about one's evidence.
Higher-order evidence is evidence about what evidence one possesses or
what conclusions one's evidence supports (see entry on
 evidence).
 This issue about the upshot of higher-order evidence does not in the
first instance depend on whether we take such evidence as necessary
for justification of first-order beliefs. The question is how our
beliefs should relate to our beliefs about our beliefs when we do
happen to have evidence about our evidence, as we often do (Feldman
2005; Christensen 2010a; Kelly 2005, 2010).



Self-doubt is a special case of responding to higher-order evidence.
Not all evidence about our evidence arises from self-doubt because not
all such evidence is about oneself, as we will see below. Also,
representing self-doubting situations as responding to evidence about
my evidence takes information about my capacities to be important just
insofar as it provides evidence that I have either incorrectly
identified my evidence or incorrectly evaluated the support relation
between my evidence and my conclusion. For example, in the case of the
doctor above who receives evidence that she is severely underslept,
the reason she should reconsider her diagnosis is because this is
evidence that she might be wrong either in reading the lab tests or in
thinking that the evidence of lab tests and symptoms supports her
diagnosis. By contrast the fourth approach below via calibration does
not see the implications of self-doubt as necessarily proceeding via
evidence about our evidence or evidential support.



Like the approach via the maximally rational agent, the evidential
approach has the virtue of identifying a justification for responding
to the second-order beliefs that self-doubt brings. Their authority
comes from the facts that they are evidence relevant to whether one
has good evidence for one's first-order belief, and that one
should respect one's evidence. This raises the hope that what we
already know about evidence can help settle when negative second-order
evidence should override a first order belief and when not. Many
authors have thought that in either kind of case rationality demands
that the two orders eventually match, in some sense, but we will see
below that more recent thinking about evidence has led some to defend
the rationality of having the first- and second-order beliefs in
tension, for some cases. Another virtue of the evidential approach is
that merely knowing what your beliefs are doesn't automatically
imply that a state of self-doubt is inconsistent or incoherent as it
did in first two approaches above, via categorical beliefs and
conditional principles. There is no obvious contradiction in believing
both *q* and that one's evidence doesn't support
*q*, even if one also has a correct belief that one believes
*q*, so what may be irrational about the state must be based on
further considerations.



The higher-order evidence approach can be usefully developed through
the example of hypoxia, a condition of impaired judgment that is
caused by lack of sufficient oxygen, and that is rarely recognized by
the sufferer at its initial onset. Hypoxia is a risk at altitudes of
10,000 feet and higher (Christensen 2010b: 126-127). Suppose you
are a pilot who does a recalculation while flying, to conclude that
you have more than enough fuel to get to an airport fifty miles
further than the one in your initial plan. Suppose you then glance at
the altimeter to see that you're at 10,500 feet and remember the
phenomenon of hypoxia and its insidious onset. You now have evidence
that you might have hypoxia and therefore might have misidentified the
support relations between your evidence and your conclusion. Are you
now justified in believing that you can get to the more distant
airport? Are you justified in believing that your evidence supports
that claim?



Letting *F* be the proposition that you have sufficient fuel to
get to the more distant airport, the following four answers are
possible:


1. You are justified in believing *F*, but no longer justified
in believing that your (1st-order) evidence supports
*F*.
2. You are justified in believing *F* and justified in
believing that your (1st-order) evidence supports
*F*.
3. You are not justified in believing *F*, and not justified in
believing that your (1st-order) evidence supports
*F*.
4. You are not justified in believing *F*, but you are
justified in believing that your (1st-order) evidence
supports *F*.



4) doesn't seem plausible; even if you can't actually
bring yourself to believe *F*, being justified in believing your
evidence supports *F* prima facie justifies you in believing
*F*.



However none of the other answers seem to be entirely adequate either.
It may seem, as in 1, that you could still be justified in believing
*F*--in case your calculation was actually right--but
no longer have strong enough reason to believe that the calculation
was right. However, this would also mean that you could justifiably
believe "*F*, but my overall evidence doesn't support
*F*". Feldman (2005: 110-111) argues that it is
impossible for this belief to be both true and reasonable since the
second conjunct undermines the reasonableness of the first conjunct
(cf. Bergmann 2005: 243; Gibbons 2006: 32; Adler 2002). And if you
were aware of having this belief then you would be believing something
that you know is unreasonable if true. You would be, in the view of
Feldman and others, disrespecting the evidence. The state in which you
believe "F and my evidence does not support F" is a case
of "level-splitting", also called *epistemic
akrasia*, because you believe you ought not to have a particular
belief state but you have it anyway.



The second reply--you are justified in believing *F* and
justified in believing that your evidence supports
*F*--might seem reasonable in some cases, for example if the
evidence about one's evidence comes in the form of skeptical
philosophical arguments, which one may think are too
*recherche* to command revisions in our everyday
beliefs. But this attitude hardly seems acceptable in general since it
would mean never giving ground on a first-order belief when presented
with evidence that you may be wrong about what your evidence implies.
When flying airplanes this kind of rigidity could even be hazardous.
However, Feldman counts the second reply as a possible way of
respecting the evidence; it might be fitting not only when faced with
radical skeptical arguments but also in cases where one's
initial view of what the first-order evidence supports is actually
correct.



The third reply, that after noting the altimeter evidence one
isn't justified in believing that one's evidence supports
*F* and also isn't justified in believing *F* has the
virtue of caution but also the consequence that the altimeter evidence
deprives you of justification for believing *F* even if you do
not suffer from hypoxia, which Feldman takes to be problematic.
However, this response, unlike the first answer, respects the
higher-order evidence; the altimeter evidence gives you some reason to
believe that you might suffer from hypoxia, which gives you some
reason to believe your evidence does not support *F*. The
misfortune of being deprived of your knowledge even if you don't
actually have hypoxia is an instance of the familiar misfortune of
misleading evidence in general. However, as we will see shortly,
misleading higher-order self-doubting evidence is distinct from other
higher-order evidence, and some recent authors have been led by this
to the view that option 1 above--akrasia--can be more
rational than option 3 in some cases.



Notably, in both of the replies that Feldman counts as possible ways
of respecting the evidence, 2) and 3), the first-order and
higher-order attitudes match; one either is justified in believing
*F* and justified in believing one's evidence supports
*F*, or isn't justified in believing *F* and also
isn't justified in believing one's evidence supports
*F*. On getting evidence suggesting one's evidence does not
support one's conclusion, one should either maintain that it
does so support and maintain the first-order belief--be
"steadfast"--or grant that it might not support
one's first-order belief and give up the latter--be
"conciliatory". If one thinks that which of these
attitudes is the right response varies with the case, then the
"total evidence" view will be attractive. On this view
whether the first order should concede to the second depends on the
relative strength of the evidence at each level. (Kelly 2010)



In the conciliatory cases, self-doubting higher-order evidence acts as
a defeater of justification for belief, which raises the question of
its similarities to and differences from other defeaters. In John
Pollock's (1989) terminology, some defeaters of justification
for a conclusion are rebutters, that is, are simply evidence against
the conclusion, while other defeaters are undercutters; they undermine
the relation between the evidence and the conclusion. (These are also
referred to as Type I and Type II defeaters.) The pilot we imagined
would be getting a rebutting defeater of her justification for
believing that she had enough fuel for an extra 50 miles if she looked
out her window and saw fuel leaking out of her tank. However, if the
altimeter reading is a defeater, then as evidence about whether she
drew the right conclusion from her evidence it is definitely of the
undercutting type.



All undercutters are evidence that has implications about the relation
between evidence and conclusion, and to that extent are higher-order
evidence. But the higher-order evidence that leads to self-doubt is
distinct from other undercutting-type evidence. In the classic Type II
defeater case one's justification for believing a cloth is red
is that it looks red and then one then learns that the cloth is
illuminated with red light. This evidence undermines your
justification for believing that the cloth's looking red is
sufficient evidence that it is red, by giving information about a
feature of the lighting that gives an alternative explanation of the
cloth's looking red. This is higher-order evidence because it is
evidence about the cause of your evidence, and thereby evidence about
the support relation between it and the conclusion, but higher-order
evidence in the cases of the doctor and the pilot are not about how
the evidence was caused, and not directly about how matters in the
world relevant to one's conclusion are related to each other.




Self-doubting defeaters are about agents and they are in addition
agent-specific (Christensen 2010a: 202). They are based on information
about you, the person who came to the conclusion about that support
relation, and have direct negative implications only for your
conclusion. In the case of the cloth, anyone with the same evidence
would have their justification undercut by the evidence of the red
light. The evidence that the doctor is underslept, however, would not
affect the justification possessed by some other doctor who had
reasoned from the same evidence to the same conclusion using the same
background knowledge. The evidence that the pilot is at risk of
hypoxia would not be a reason for a person on the ground, who had
reasoned from the same instrument readings to the same conclusion, to
give up the belief that the plane had enough fuel for fifty more
miles.



Christensen argues that the agent-specificity of self-doubting
higher-order evidence requires the subject to "bracket"
her first-order evidence in a way that other defeating evidence does
not. He thinks this means that in no longer using the evidence to draw
the conclusion, she will not be able to give her evidence its due
(Christensen 2010a: 194-196). In contrast, in the case of the
red light and other cases not involving self-doubt, once the redness
of the light is added to the evidence, discounting the appearance of
the cloth doesn't count as failing to respect that evidence
because one is justified in believing it is no longer due respect as
evidence of redness. However, arguably, the difference is not that the
self-doubter must fail to give the evidence its due. In the
higher-order self-doubt cases we have seen the undercutting evidence
does not give the subject reason to believe the evidential relation
she supposed was there is not there. It gives reason to think that she
doesn't know whether the evidential relation is there, even if
it is. If it isn't then in bracketing her first-order evidence
she isn't failing to give it its due; it isn't due any
respect. Because self-doubting defeating evidence concerns the
subject's knowledge of the evidential support relation and not
the relation itself it appears weaker than typical defeating evidence.
However it is potentially more corrosive because it doesn't give
the means to settle whether the evidential relation she endorsed is
there and so whether the first-order evidence deserves respect.



If the pilot gives up the belief in *F* and she had been right
about the evidential relation, then she will have been the victim of a
misleading defeater. Misleading defeaters present well-known
difficulties for a theory of justification based on the idea of defeat
because Type II defeaters may be subject to further defeaters
indefinitely. For example, if one had learned that the light
illuminating the cloth was red via testimony, the defeat of
one's justification for believing the cloth was red would be
defeated by good evidence that one's source was a pathological
liar. If we say that justified belief requires that there be no
defeaters then that leads us to disqualify any case where a misleading
defeater exists, and a subject will lose justification she might have
had, even if the misleading defeaters are distant facts the
she isn't aware of. But if we refine the view to say that
only defeaters for which no defeater exists will undermine
justification then a subject will count as justified even if she
ignores evidence that looks like a defeater for all she knows, because
of the existence of a defeater defeater she doesn't know about.
 In general we will face the question how many and which of the
existing defeater defeaters matter to whether we have a justified
belief (Harman 1973; Lycan 1977).



If despite being at an altitude of 10,500 feet our pilot did do the
calculation correctly, then her first-order evidence deserved her
belief and her evidence from her altitude and the phenomenon of
hypoxia was a misleading defeater. It was good reason to worry that
her blood oxygen was low, but it might not have been low, and it would
be possible in principle for her to get further evidence that would
support this view, such as from the reading of a finger pulse
oximeter. Misleading defeaters are not new, but few would be tempted
to say in the case where one gets evidence that the light is red that
it would be rational for the subject to believe both "my
evidence doesn't support the claim that the cloth is red"
and "the cloth is red". However, for self-doubting type II
defeaters several authors have claimed that such level-splitting can
be rational.



For example, Williamson (2011) has argued that it is possible for the
evidential probability of a proposition to be quite high, while it is
also highly probable *that* the evidential probability is low.
For instance one's evidence about oneself might indicate that
one has made a mistake evaluating one's evidence, a kind of
mistake that would lead one to believe an unsupported conclusion,
*F*. One evaluates the evidential probability of *F* as high
because of one's view of its evidence, but thinks *F* might
well be true without one's belief in it being knowledge. 



Another way of arguing that it can be rationally required to respond
to a real support relation--that is, for the pilot, to believe
*F*--even when one has evidence it might not exist, and so,
should also believe one's evidence does not (or might not)
support that belief, is with the thought that a rational norm does not
cease to apply just because a subject has evidence she hasn't
followed it (Weatherson 2008, 2010 (Other Internet Resources); Coates
2012). This rationale would not sanction akrasia for one who learned
the light was red, because the point is restricted to cases where the
defeating evidence concerns the subject; we saw above that that makes
the defeating evidence weaker, and it is weaker in just the right way
to support this approach.



Another way of arguing that akrasia can be rational is to take the
existence of a support relation as sufficient for justification of
belief in a proposition whether the subject has correct beliefs about
that support relation or not (Wedgwood 2011). This is motivated by
externalism about justification (see entry on
 internalist and externalist conceptions of epistemic justification),
 which might be more plausible for justifications subject to
self-doubting higher order evidence because it is weaker than other
undercutting evidence. In a different tack, it has been argued that a
general rule that takes negative self-doubting higher-order evidence
to always exert some defeating force on first-order beliefs will be
very hard to come by. Because the subject is being asked to behave
rationally in the face of evidence that she has not behaved
rationally, she is subject to norms that give contradictory advice,
and fully general rules for adjudicating between such rules are
subject to paradoxes (Lasonen-Aarnio 2014).



Possibly the only thing harder than defending a fully general rule
requiring the first-order and second-order to match is accepting the
intuitive consequences of level-splitting or akrasia. In this
situation one believes a certain belief state is (or might be)
irrational but persists in it anyway. Horowitz (2014) has defended the
Non-Akrasia Constraint (also sometimes referred to as an Enkratic
Principle) which forbids being highly confident in both
"*q*" and "my evidence does not support
*q*", in part by arguing that allowing akrasia
delivers highly counter-intuitive follow-on consequences in paradigm
cases of higher-order evidence. For example, if our pilot maintains
confidence that *F*, she has enough fuel, how should she explain
how she got to a belief in *F* that she thinks is true when she
also thinks her evidence does not support *F*? It would seem that
she can only tell herself that she must have gotten lucky.



She could further tell herself that if the reason she persisted in
believing *F* despite the altimeter reading was that she did in
fact have low blood oxygen, then it was really lucky she had that
hypoxia! Otherwise in evaluating her total evidence correctly she
would have come to a false belief that not-*F*. Reasoning so, the
pilot would be using her confidence in *F* as a reason to believe
the altimeter reading was a misleading defeater, which does not seem
to be a good way of finding that out. Moreover, if she did this
argument a number of times she could use the track record so formed to
bootstrap her way to judging herself reliable after all (Christensen
2007a,b; White 2009; Horowitz 2014--for general discussion of
what is wrong with bootstrapping, see Vogel 2000 and Cohen 2002).
Akrasia also sanctions correspondingly odd betting behavior.



To the extent that New Rational Reflection (of the previous section)
calls for a match between first-order and second-order beliefs it
counts as a Non-Akrasia principle. However, this particular matching
requirement is subject to several problems about evidence that are
brought out by Lasonen-Aarnio (2015). It requires substantive
assumptions about evidence and updating our beliefs that are not
obvious, and does not appear to respect the internalism about
rationality that apparently motivates it, namely that one's
opinions about what states it is rational to be in match what states
one is actually in. Moreover it does not seem that New Rational
Reflection could embody the attractive idea that in general a subject
may rationally always be uncertain whether she is rational, i.e., even
the ideal agent can doubt that she is the ideal agent--an idea
that RatRef failed to conform with and that led to the formulation of
this new principle. This is because New Rational Reflection must
assume that some things, such as conditionalization, cannot be doubted
to be rational, i.e., to be what the ideal agent would do. It is not
clear that we should have expected that everything can be doubted at
once (Vickers 2000; Roush et al. 2012), but this is an ongoing area of
research (Sliwa & Horowitz 2015).



Another problem some have seen with any version of Rational Reflection
is that it ultimately doesn't allow the subject to remain unsure
what degree of belief it is rational for her to have. It forces her to
match her first-order degree of belief to a specific value, namely, a
weighted average of the degrees of belief that she thinks it might be
rational to have. It collapses her uncertainty about what is rational
to a certainty about the average of the possibilities and forces her
to embrace that precise value. This doesn't allow a kind of
mismatch or akrasia that might be the right way to respond to some
higher-order evidence, where one is confident that *q* and also
thinks it likely that the evidence supports a lower confidence than
one has but is unsure what that lower confidence should be. Maybe one
should not defer to the higher-order evidence in this case, because
one is not sure what its verdict is (Sliwa & Horowitz 2015). See
the higher-order calibration approach below for a way of representing
this uncertainty that can give a justification for thinking that
matching to averages is rational.



The evidential approach locates the authority that second-order
information about our judgment has over us in the idea that it is
evidence and we should respect our evidence. A state of self-doubt on
this view is confidence both that *q* and that one's
evidence may well not support *q*. This state does not make the
subject inconsistent, but is a state of level-splitting or akrasia.
Matching on this view is constituted by agreement between one's
level of confidence in *q* and how far one thinks one's
evidence supports *q*, and doesn't allow akrasia, but
taking an evidential approach does not by itself settle whether
rationality requires matching, or under what circumstances first- or
second-order evidence should determine one's first order
confidence. General rules about how to adjudicate in self-doubting
cases between the claims of the two orders of evidence may be hard to
get due to paradoxes, and to the need in every instance to hold some
features of rationality as undoubtable in order to initiate and
resolve one's doubt.


## 4. Calibration and Higher-Order Objective Probability



Another approach to self-doubt explains the authority that
second-order evidence sometimes has over one's first-order
beliefs by means of the idea that such evidence provides information
about the relation of one's first-order beliefs to the way the
world is which one is obligated to take into account. That is,
evidence like the altitude reading, sleep-deprivation, and empirical
studies of the unreliability of eyewitness testimony, provides
information about whether your beliefs are reliable indicators of the
truth. We take the reading of a thermometer no more seriously than we
regard the instrument as reliable. Our beliefs can be viewed as
readings of the world and treated the same way (Roush 2009; White
2009; Sliwa & Horowitz 2015).


### 4.1 Guess Calibration



One way of formulating a constraint that says we should be no more
confident than we are reliable is by requiring *guess
calibration* (GC):




If I draw the conclusion that *q* on the basis of evidence
*e*, my credence in *q* should equal my prior expected
reliability with respect to *q*. (White 2009; Sliwa &
Horowitz 2015)




Your expected reliability with respect to *q* is understood as
the probability--chance or propensity--that your guess of
*q* is true. You may not be sure what your reliability is so you
will use an expected value, a weighted average of the values you think
are possible, and this should be a prior probability, evaluated
independently of your current belief that *p*.



Self-doubt on this picture would be a state in which you've
drawn conclusion *q*, say because your confidence in *q*
exceeded a given threshold, but also have reason to believe your
reliability with respect to *q* is not as high as that
confidence, and such a state would be a violation of GC. Whether this
self-doubting state can be coherent when a subject knows her own
beliefs depends very much on how the reliability aspect is formulated.
If chances and propensities that your guesses are true are determined
by frequencies of ordered pairs of guesses of *q* and truth or
falsity of *q* then self-doubt will make one incoherent here in
the way that representing oneself as an anti-expert did above, because
coherence and expected reliability will be logically
 equivalent.[2]
 In any case, GC requires matching between the orders in all cases, and
tells us that the match is between your confidence and your
reliability.



GC makes sense of the intuition in some cases of self-doubt that the
subject should drop her confidence. The pilot on looking at the
altimeter reading should cease to be so sure she has enough gas for
fifty more miles because it gives her reason to think she is in a
state where her calculations will not reliably turn out truths.
Similarly, the doctor realizing she is severely underslept acquires
reason to think she is in a state where her way of coming to beliefs
does not reliably lead to true conclusions. The main dissatisfaction
with GC has been that it apparently cedes all authority to
second-order evidence. In fact in GC's formulation the
confidence you should finally have in *q* does not depend on how
far the evidence *e* supports *q* or how far you think
*e* supports *q*, but only on what you think is your
propensity for or frequency of getting it right about *q*,
whether you used evidence or not.



There may be cases where second-order evidence is worrying enough that
one's first order conclusion should be mistrusted entirely, even
if it was in fact soundly made--maybe the pilot and doctor are
such cases since the stakes are high. But as we saw above it
doesn't seem right across the board to take first-order evidence
to count for nothing when second-order evidence is around. Here we can
see that by supposing that two people, Anton and Ana, reason to
different conclusions, *q* and not-*q*, on the basis of the
same evidence, Anton evaluating the evidence correctly, Ana not.
Suppose both are given the same undermining evidence, say that people
in their conditions only get it right 60% of the time. According to
GC, rationality requires both of them to become 60% confident in their
conclusions. Anton, who reasoned correctly from the evidence, is no
more rational than Ana, and has no right to a higher confidence in his
conclusion *q* than Ana who reasoned badly has for her conclusion
not-*q* (Sliwa & Horowitz 2015).


### 4.2 Evidential Calibration



It seems wrong that second-order evidence should always swamp the
first-order verdict completely, so the calibration idea has been
re-formulated so as to incorporate dependence on first-order evidence
explicitly, in the *evidential calibration* constraint
(EC):




When one's evidence favors *q* over not-*q*,
one's credence in *q* should equal the [prior] expected
reliability of one's educated guess that *q*. (Sliwa &
Horowitz 2015)




Your educated guess corresponds to the answer that you have the
highest credence in. Reliability of such a guess is defined as the
probability that you would assign the highest credence to the true
answer if you had to choose, and as above this probability is
understood as your propensity to guess correctly. What is used in EC,
as in GC, is an expected rather than actual reliability so it is
weighted by how likely you think each possible reliability level is.
The difference between GC and EC is that in the latter the calibration
requirement depends explicitly on which conclusion the first-order
evidence actually supports. On this principle, Anton, who reasoned
correctly with the first-order evidence, is rational to be .6
confident in *q* rather than not-*q* because *q* is the
conclusion the first-order evidence actually supports. The
contribution of the second-order evidence is to reduce his confidence
in that conclusion from a high value to .6.



According to Sliwa and Horowitz EC implies that Ana is not rational to
have a .6 confidence in not-*q* because not-*q* is not the
conclusion the evidence actually favors. It would be rational for her
to have .6 confidence that *q*, the same as Anton. This claim
highlights ambiguities in the phrase "one's educated guess
that *q*". Expected reliability is the probability that you
would assign the highest credence to the true answer, and the
undermining evidence both Anton and Ana were given said that in the
conditions they were in they had a 60% chance of their guess being the
right
 answer.[3]
 If so, then neither of them have enough information to know the
expected reliability of a guess *that *q**. If the
probabilities it appeared they were given, for one's guess about
*q* whatever it is, are to be usable, then the phrase in EC would
need to be interpreted "one's educated guess that *q*
or not-*q*".



The fact that Ana did not actually guess that *q* makes for a
difficulty on either interpretation of the higher-order evidence and
EC. Suppose the 60%-probability evidence they were given was indeed
only about guesses that *q*, and that "educated guess that
*q*" in EC refers only to guesses that *q*. If the
word "one's" in the phrase "one's
educated guess that *q*" refers narrowly to the individual
EC is being applied to, then EC does not imply anything for Ana, since
she did not guess that *q*. If "one's" refers
broadly to anyone who guesses that *q* in the conditions Anton
and Ana were in, then it does follow that what is rational for Ana is
to believe *q* with 60% confidence. However, whether they are
given general evidence about guesses that *q* or not-*q*, or
separate statistics on the success of *q*-guesses and of
not-*q* guesses, that higher-order evidence would have given Ana
no means to correct herself. Because she got it wrong in the first
step by incorrectly concluding the first-order evidence supported
not-*q* she will also lack the means to correct herself, that is,
to know whether she should be 60% confident in *q* or in
not-*q*. What EC says it is rational for her to do in the
situation is not something she has the ability to do. It might be
possible to avoid these difficulties in a reformulation, but they are
consequences of the move in EC to add deference to the actual
first-order evidential support relation.



EC rules out many cases of bootstrapping that level-splitting views
allow. For example, a bootstrapping doctor with evidence that he is
unreliable assembles a sterling track record of success in his
first-order decisions by judging the correctness of his conclusions by
his confidence in those conclusions. He thinks the evidence of his
unreliability that he started out with has now been outweighed, so he
concludes that he is reliable after all. EC doesn't allow this
to be rational because it doesn't allow him to assemble the
track record in the first place, since he is obligated in every
instance to take into account the expected (un)reliability that he has
evidence for. It is unclear, however, that EC similarly rules out
bootstrapping for a subject who begins with no evidence at all about
her reliability.



The EC reformulation of GC takes a different view of whether
rationality requires us to get it right about the first-order support
relation or merely to get it right by our own lights, but this
question is of course not specific to the topic of the relation of
first-order and second-order evidence. For example, in a probabilistic
account evidential support relations are dictated completely by
conditional probabilities. In a subjective Bayesian version of this
picture rationality requires one to have the confidence dictated by
the subjective conditional probabilities that follow from one's
confidences in other propositions. In an objective version rationality
would obligate one to have a confidence that is in line with the
objective conditional probabilities. There are other ways of making
out subjective vs. objective views of the relevant evidential support
relations, and whether we should favor one or the other depends on
more general considerations that could provide independent reason to
favor one or the other view in the current debate about order
relations.



Though this distinction is not specific to the current context it
appears to have played a role in some authors' intuitions about
level-splitting above. For example, when Weatherson and Coates say
that the subject ought to believe what the first-order evidence
actually supports because a norm does not cease to apply just because
one has evidence one hasn't followed it, they assume that the
relevant norm and evidential support relation are objective.
Wedgwood's appeal to externalism about justification also takes
its bearings from what the first-order evidence actually supports
rather than what to one's own point of view it seems to support.
A challenge for these approaches that achieve some additional
authority for first order evidence over the second order by requiring
deference to the actual evidential relation at the first order is to
explain why this is an obligation at the first order but a subject
need only take into account the *expected* reliability at the
second order.


### 4.3 Calibration in Higher-Order Probability



Another approach that sees rationality constraints between the two
orders as based on taking evidence of one's expected reliability
into account derives the constraints top-down from general, widely
held, subjective Bayesian assumptions about evidential support, and
explicit representation of second-order reliability claims in
higher-order objective probability (Roush 2009). Like approach 2 above
it uses subjective conditional probability to express the match that
is required between the two orders, but it avoids the consequence we
saw in most of those approaches--and in the categorical approach
and the other calibration approaches just discussed--that a state
of self-doubt combined with knowledge of one's beliefs is
incoherent. Unlike the first two calibration approaches it gives an
explanation why calibration is part of rationality; it does this by
deriving the constraint from another widely accepted assumption, the
Principal Principle.



We can write a description of the relation of the subject's
belief in *q* to the way the world is--her
reliability--as an objective conditional probability:




Calibration Curve
  

\(\PR(q\mid P(q)=x) = y \) 




The objective probability of *q* given that the subject believes
*q* to degree *x* is *y*. This is a curve, a function
that allows reliability *y* to vary with the independent variable
of confidence, *x*, with different variables used in order to
allow for the possibility that the subject's degree of belief
tends not to match the objective probability, and that the level and
direction of mismatch can vary with the level of confidence. The curve
is specific to proposition *q* and to the subject whose
probability function is *P*. A subject is calibrated on *q*,
on this definition, if his calibration curve is the line \(x =
 y\).[4]



Calibration curves are widely studied by empirical psychologists who
find that human beings' reliability tends on average to vary
systematically and uniformly with confidence, with for example high
confidence tending to overconfidence, as in eyewitness testimony.
Despite the averages found when subjects take tests in controlled
settings, the curves also vary with sub-group, individual traits,
professional skills, and particular circumstances. All manner of
higher-order evidence about a subject's belief-forming
processes, methods, circumstances, track-record, and competences are
relevant to estimating this function. In real life no one could get
enough evidence in one lifetime to warrant certainty about an
individual's calibration curve for *q* in a set of
circumstances, but if one is a Bayesian one can form a confidence
about what a person's calibration curve is, or what value it has
for some argument *x*, that is proportional to the strength of
one's evidence about this, and one can have such a confidence
about one's own calibration curve.



On this approach epistemic self-doubt is a state where one is
confident and more or less correct that one believes *q* to
degree *x*, that is, \(P(q) = x\), but also has an uncomfortably
high level of confidence, say \([?] .5\), that one is unreliable
about *q* at that confidence. That is, one has confidence \([?]
.5\) that the objective probability of *q* when one has
*x*-level of confidence in *q* is different from *x*,
which we would write \(P(\PR(q\mid P(q)=x) \ne x) [?] .5\). Let us
say that the different value is *y*, so \(P(\PR(q\mid P(q)=x) =
y) [?] .5\), \(y \ne x\). Whether or not the reason for this
unreliability is that one tends to mistake evidential support
relations and whether one does or does not think a given evidential
support relation obtains, make no general difference to this
evaluation which is simply about whether one tends to get things right
when doing the sort of thing one did in coming to be confident in
*q* to level
 *x*;[5]
 it is about the relation between one's confidence and the way
things are.



On this view, a state of self-doubt involves a combination of states
of the following sort:

 

\[
\begin{align\*}
P(q) & = x\\
P(P(q) = x) & = .99 \qquad\textrm{(high)}\\
P(\PR(q\mid P(q) = x) = y) & [?] .5, \qquad y \ne x\\
\end{align\*}
\]


You actually believe *q* to degree *x*, you are confident
(say at .99) that you so believe, and you have an uncomfortably high
level of confidence that you are not calibrated for *q* at
*x*, that the objective probability of *q* when you are
*x* confident of *q* is *y*. This state escapes
incoherence for two reasons. One is that one's confidence either
about one's degree of belief or one's reliability is not
1, and unlike some conditional probability formulations of self-doubt
above, the slightest uncertainty is enough to make it coherent to
attribute a large discrepancy between your believed confidence and
your believed reliability.



This is made possible by the second factor, that (un)reliability is
expressed here as an objective conditional probability, and coherence
alone does not dictate how subjective and objective probabilities must
relate. This is analogous to the reason that the approach via the
maximally rational subject above was able to represent a state of
self-doubt as coherent, namely, that in evaluating my own *P* I
compare it to a different probability function. However, in this case
the second function is not an expert function that declares
unconditionally what value the maximally rational subject's
value for *q* would be, but a calibration function, a conditional
probability that tells one what objective probability is indicated by
one's subjective probability. One difference between the two
approaches is that there are obvious ways to investigate calibration
curves empirically, whereas it would be hard to recruit enough
maximally rational subjects for a statistically significant study so
we tend to be left appealing to intuitions about what seems
rational.



Once the defeating information about the relation of a subject's
credences to the world is expressed in objective probability it can be
represented explicitly as a consideration the subject takes on board
in assessing the quality of the degree of belief she takes herself to
have in *q* and resolving the question what her degree of belief
should be, thus:

 

\[
P(q\mid P(q)=x \amp \PR(q\mid P(q)=x)=y) = ? 
\]


This asks for the degree of belief the subject should have in *q*
on condition that she actually has degree of belief *x* in
*q* and the objective probability of *q* given that she has
degree of belief *x* in *q* is *y*. This expression is
the left-hand side of Self-Respect/Synchronic Reflection with a
further conjunct added to its condition. SR doesn't specify what
to do when there is another conjunct and so is not suited to
explicitly represent the question of self-doubt, which means that the
self-doubting examples above are not counterexamples to it (Roush
2009). However, some in the past have endorsed variants on an
unrestricted version of SR (Koons 1992; Gaifman 1988) where the value
of this expression is *x* regardless of what other conjunct might
be present:




Unrestricted Self-Respect
 (USR)[6]
  

\(P(q\mid P(q)=x \amp r) = x\), for *r* any proposition 




Dutch book arguments that might give support to SR do not do the same
for USR, leaving us with a need to find other ways of evaluating it
when r is the statement of a calibration curve.



It is not incoherent but it is baldly counterintuitive to suppose that
the subject should have degree of belief *x* when she believes
that her so believing is an indicator that the objective probability
of *q* is not *x*, and a principled argument can also be
made to this effect (Roush 2009). Unpacking the condition \(P(q)=x
\amp \PR(q\mid P(q)=x) = y\), it seems to say that my credence is
*x* and when my credence is *x* the objective probability is
*y*, inviting us to discharge and infer that the objective
probability is *y*. If
 so,[7]
 then the expression would reduce to:

 

\[
P(q\mid \PR(q)=y) = ?
\]


which is the left-hand side of a generalization of the Principal
Principle (see entry on
 David Lewis)




Principal Principle
 (PP)[8]
  

\(P(q\mid Ch(q)=y) = y\) 




from chance to any type of objective probability. PP says that your
credences in propositions should conform to what you take to be their
chances of being true, and, admissibility issues notwithstanding, it
is hard to deny that there exists a domain in which the Principal
Principle is compelling, and surely one where the generalization to
any type of objective probability is too. If so then the answer to the
question what the subject's credence in *q* ought to be in
light of her consideration of information about her reliability
is:




Cal
  

\(P(q\mid (P(q)=x \amp \PR(q\mid P(q)=x)=y)) = y \) 




Cal says that your credence in *q* given that your credence in
*q* is *x* and the objective probability of *q* given
that your credence in *q* is *x* is *y*, should be
*y*.



Cal is a synchronic constraint, but if we revise our credences by
conditionalization then it implies a diachronic constraint:




Re-Cal
  

\(P\_{n+1}(q) = P\_{n}(q\mid (P\_{n}(q)=x \amp \PR(q\mid P\_{n}(q)=x)=y))
= y \) 




This calibration approach tells the subject how to respond to
information about her cognitive impairment in every case. It uses the
information about herself to correct her belief about the world.
Intuitively it is a graded generalization of the thought that if you
knew of someone (or yourself) that he invariably had false beliefs,
then you could gain a true belief by negating everything he said.



Cal and Re-Cal give an explicit characterization of self-doubt and
justification of a unique and determinate response to it on the basis
of deeper principles that are compelling independently of the current
context. Cal follows from only two assumptions, first that
probabilistic coherence is a requirement of rationality, and second
that rationality requires one's credences to align with what
according to one's evidence are the objective probabilities.
Re-Cal comes from further assuming that updating our beliefs should
occur by conditionalization.



Although self-doubt under the current definition of it is not an
incoherent state, Cal implies that rationality always requires a
resolution of the doubt that brings matching between the two levels,
and tells us that the matching consists in the alignment of subjective
and perceived objective probabilities. High confidences in
"*q*", "I have confidence *x* in
*q*", and "the objective probability of *q* when
I have confidence *x* in *q* is low" are not
incoherent, but they do violate the Principal Principle. Re-Cal tells
us how to get back in line with PP.



Though Re-Cal has us conditioning on second-order evidence, the
adjustment it recommends depends on both first- and second-order
evidence and does not always favor one level or the other. How much
authority the second-order claim about the reliability/calibration
curve has depends very much on the quality of one's evidence
about it. This can be seen by imagining being uncertain about, e.g.,
one's calibration curve, i.e., \(P(\PR(q\mid P(q) = x) = y) <
1\), and doing a Jeffrey conditionalization version of Re-Cal (Roush
2017, Other Internet Resources). But even in case one has perfect
knowledge of one's calibration curve, the role of the
first-order evidence in determining one's first-order belief is
ineliminable. The verdict, the level of confidence, that the first
order gave you for *q* is the index for determining which point
on the calibration curve is relevant to potentially correcting your
degree of belief. To understand why this is far from trivial, recall
that the curve can in principle and does often in fact have different
magnitudes and directions of distortion at different confidences. The
verdict's dependence on the first-order evidential support
relation is different from that of EC in another way, since it uses
not the objective support relation at the first order but the
consequences of the subject's take on it. Thus, Ana above would
not be left not knowing how to make herself rational.



The fact that the update proceeds by conditionalization means that all
of the kinds of evaluation of evidence that conditionalization imposes
come along with that. Misleading self-doubting defeaters troubled some
authors above and led them to level-splitting views, but they are
handled by Re-Cal as conditionalization always handles them.
Self-doubting defeaters are processed at face value as relevant to the
calibration curve in proportion to their quality as evidence.
Convergence theorems tell us that if the world isn't
systematically deceptive then the misleading defeaters will be washed
out, that is, defeated by some other evidence, in the long run. In
some cases that will happen only long after we're all dead, but
if one views that as inadequate then that is a dissatisfaction with
subjective Bayesianism, and is not specific to its usage here.



The approach to epistemic self-doubt in terms of higher-order
probability allows the state of self-doubt to be rational (coherent),
and to be rationally resolved. Cal expresses a requirement of matching
between the two orders in all cases, though it does not imply that
*attributing* a mismatch to oneself is incoherent. Neither
order is always dominant; both orders always make a contribution to
determining the resolution at the first order of conflicts between
orders, and their relative contribution depends on the quality of the
evidence at each order. Cal and Re-Cal explain why one should revise
in light of higher-order evidence, when one should, by reference only
to probabilistic coherence, the Principal Principle, and
conditionalization. Cal and Re-Cal are general and make available all
of the resources of the Bayesian framework for analysis of
higher-order evidence. A further notable fact about the framework is
that Re-Cal allows for cases where news about one's reliability
should increase one's confidence, which would be appropriate for
example in cases, easy to imagine, where one acquired evidence that
one was systematically *under*confident. Thus, it is possible
for second-order evidence to make it rational to be not only steadfast
or conciliatory, but even emboldened.