## 1. The Surprise Test Paradox



A teacher announces that there will be a surprise test next week. It
will be a surprise in that the students will not be able to know in
advance on which day the exam will be given. A student objects that
this is impossible: "The class meets on Monday, Wednesday, and
Friday. If the test is given on Friday, then on Thursday I would be
able to predict that the test is on Friday. It would not be a
surprise. Can the test be given on Wednesday? No, because on Tuesday I
would know that the test will not be on Friday (thanks to the previous
reasoning) and know that the test was not on Monday (thanks to
memory). Therefore, on Tuesday I could foresee that the test will be
on Wednesday. A test on Wednesday would not be a surprise. Could the
surprise test be on Monday? On Sunday, the previous two eliminations
would be available to me. Consequently, I would know that the test
must be on Monday. So a Monday test would also fail to be a surprise.
Therefore, it is impossible for there to be a surprise
test."



Can the teacher fulfill her announcement? We have an embarrassment of
riches. On the one hand, we have the student's elimination
argument. (For a formalization, see Holliday 2017.) On the other hand,
common sense says that surprise tests are possible even when we have
had advance warning that one will occur at some point. Either of the
answers would be decisive were it not for the credentials of the rival
answer. Thus we have a paradox. But a paradox of what kind?
'Surprise test' is being defined in terms of what can be
known. Specifically, a test is a surprise if and only if the student
cannot *know* beforehand which day the test will occur.
Therefore the riddle of the surprise test qualifies as an
*epistemic* paradox.



*Paradoxes* are more than edifying surprises. Here is an
edifying surprise that does not pose a paradox. Professor Statistics
announces she will give random quizzes: "Class meets every day
of the week. Each day I will open by rolling a die. When the roll
yields a six, I will immediately give a quiz." Today, Monday, a
six came up. So you are taking a quiz. The last question of her quiz
is: "Which of the subsequent days is most likely to be the day
of the next random test?" Most people answer that each of the
subsequent days has the same probability of being the next quiz. But
the correct answer is: *Tomorrow* (Tuesday).



Uncontroversial facts about probability reveal the mistake and
establish the correct answer. For the next test to be on Wednesday,
there would have to be a *conjunction* of two events: no test
on Tuesday (a 5/6 chance of that) *and* a test on Wednesday (a
1/6 chance). The probability for each subsequent day becomes less and
less. (It would be astounding if the next quiz day were a hundred days
from now!) The question is not whether a six will be rolled on any
given day, but when the *next* six will be rolled. Which day is
the next one depends partly on what happens meanwhile, as well as
depending partly on the roll of the die on that day.



This probability riddle is instructive and will be referenced
throughout this entry. But the existence of quick, decisive solution
shows that only a mild revision of our prior beliefs was needed. In
contrast, when our deep beliefs conflict, proposed amendments
reverberate unpredictably. Attempts to throw away a deep belief
boomerang. For the malfunctions that ensue from distancing ourselves
from the belief demonstrate its centrality. Often, the belief winds up
even more entrenched. "Problems worthy of attack prove their
worth by fighting back" (Hein 1966).



One sign of depth (or at least desperation) is that commentators begin
rejecting highly credible inference rules. The surprise test has been
claimed to invalidate the law of bivalence, the KK principle (if one
is in a position to know that \(p\), then one is also in
a position to know that one knows that \(p\)), and the
closure principle (if one knows \(p\) while also competently deducing q
from \(p\), one knows \(q\)) (Immerman 2017).



The surprise test paradox also has ties to issues that are not clearly
paradoxes - or to issues whose status as paradoxes is at least
contested. Consider the Monty Hall problem. There is a prize behind
exactly one of three doors. After you pick, Monty Hall will reveal
what is behind one of the doors that lacks the prize. He will then
give you an option to switch your choice to another closed door.
Should you switch to have the best chance to win the prize? When Marylin vos Savant answered yes in a 1990
*Parade m*agazine she was mistakenly scolded by many readers
-- including some scholars. The correct solution was provided by
the original poser of the puzzle decades earlier and was never
forgotten or effectively criticized. 



What makes the Monty Hall Problem interesting is that it is *not*a paradox. There has always been expert consensus on its
solution. Yet it has many of the psychological and sociological features
of a paradox. The Monty Hall Problem is merely a *cognitive
illusion*. Paradox status is also withheld by those who find only
*irony* in self-defeating predictions and only an
*embarrassment* in the "knowability paradox"
(discussed below). Calling a problem a 'paradox' tends to quarantine
it from the rest of our inquiries. Those who wish to rely on the
surprising result will therefore deny that there is any paradox. At
most, they concede there *was* a paradox. Dead paradoxes are
benign fertilizer for the tree of knowledge.



We can look forward to future philosophers drawing edifying historical
connections. The backward elimination argument underlying the surprise
test paradox can be discerned in German folktales dating back to 1756
(Sorensen 2003a, 267). Perhaps, medieval scholars explored these
slippery slopes. But let us turn to commentary to which we presently
have access.


### 1.1 Self-defeating prophecies and pragmatic paradoxes



In the twentieth century, the first published reaction to the surprise
text paradox was to endorse the student's elimination argument.
D. J. O'Connor (1948) regarded the teacher's announcement
as self-defeating. If the teacher had not announced that there would
be a surprise test, the teacher would have been able to give the
surprise test. The pedagogical moral of the paradox would then be that
if you want to give a surprise test do not announce your intention to
your students!



More precisely, O'Connor compared the teacher's
announcement to utterances such as 'I am not speaking
now'. Although these utterances are consistent, they
"could not conceivably be true in any circumstances"
(O'Connor 1948, 358). L. Jonathan Cohen (1950) agreed and
classified the announcement as a pragmatic paradox. He defined a
pragmatic paradox to be a statement that is falsified by its own
utterance. The teacher overlooked how the manner in which a statement
is disseminated can doom it to falsehood.



Cohen's classification is too monolithic. True, the
teacher's announcement does compromise one aspect of the
surprise: Students now know that there will be a test. But this
compromise is not itself enough to make the announcement
self-falsifying. The *existence* of a surprise test has been
revealed but that allows surviving uncertainty as to *which*
day the test will occur. The announcement of a forthcoming surprise
aims at changing uninformed ignorance into action-guiding awareness of
ignorance. A student who misses the announcement does not realize that
there is a test. If no one passes on the news about the surprise test,
the student with simple ignorance will be less prepared than
classmates who know they do not know the day of the test.



Announcements are made to serve different goals simultaneously.
Competition between accuracy and helpfulness makes it possible for an
announcement to be self-fulfilling by being self-defeating. Consider a
weatherman who warns 'The midnight tsunami will cause fatalities
along the shore'. Because of the warning, spectacle-seekers make
a special trip to witness the wave. Some drown. The weatherman's
announcement succeeds as a prediction by backfiring as a warning.


### 1.2 Predictive determinism



Instead of viewing self-defeating predictions as showing how the
teacher is refuted, some philosophers construe self-defeating
predictions as showing how the *student* is refuted. The
student's elimination argument embodies hypothetical predictions
about which day the teacher will give a test. Isn't the student
overlooking the teacher's ability and desire to thwart those
expectations? Some game theorists suggest that the teacher could
defeat this strategy by choosing the test date at random.



Students can certainly be kept uncertain if the teacher is willing to
be faithfully random. She will need to prepare a quiz each day. She
will need to brace for the possibility that she will give too many
quizzes or too few or have an unrepresentative distribution of
quizzes.



If the instructor finds these costs onerous, then she may be tempted
by an alternative: at the beginning of the week, randomly select a
single day. Keep the identity of that day secret. Since the student
will only know that the quiz is on some day or other, pupils will not
be able to predict the day of the quiz.



This plan is risky. If, through the chance process, the last day
happens to be selected, then abiding by the outcome means giving an
unsurprising test. For as in the original scenario, the student has
knowledge of the teacher's announcement and awareness of past
testless days. So the teacher must exclude random selection of the
last day. The student is astute. He will replicate this reasoning that
excludes a test on the last day. Can the teacher abide by the random
selection of the next to last day? Now the reasoning becomes all too
familiar.



Another critique of the student's replication of the
teacher's reasoning adapts a thought experiment from Michael
Scriven (1964). To refute predictive determinism (the thesis that all
events are foreseeable), Scriven conjures an agent
"Predictor" who has all the data, laws, and calculating
capacity needed to predict the choices of others. Scriven goes on to
imagine, "Avoider", whose dominant motivation is to avoid
prediction. Therefore, Predictor must conceal his prediction. The
catch is that Avoider has access to the same data, laws, and
calculating capacity as Predictor. Thus Avoider can duplicate
Predictor's reasoning. Consequently, the optimal predictor
cannot predict Avoider. Let the teacher be Avoider and the student be
Predictor. Avoider must win. Therefore, it is possible to give a
surprise test.



Scriven's original argument assumes that Predictor and Avoider
can simultaneously have all the needed data, laws, and calculating
capacity. David Lewis and Jane Richardson object:



> 
> ... the amount of calculation required to let the predictor
> finish his prediction depends on the amount of calculation done by the
> avoider, and the amount required to let the avoider finish duplicating
> the predictor's calculation depends on the amount done by the
> predictor. Scriven takes for granted that the requirement-functions
> are compatible: i.e., that there is some pair of amounts of
> calculation available to the predictor and the avoider such that each
> has enough to finish, given the amount the other has. (Lewis and
> Richardson 1966, 70-71)
> 



According to Lewis and Richardson, Scriven equivocates on 'Both
Predictor and Avoider have enough time to finish their
calculations'. Reading the sentence one way yields a truth:
against any given avoider, Predictor can finish and against any given
predictor, Avoider can finish. However, the compatibility premise
requires the false reading in which Predictor and Avoider can finish
against each other.



Idealizing the teacher and student along the lines of Avoider and
Predictor would fail to disarm the student's elimination
argument. We would have merely formulated a riddle that falsely
presupposes that the two types of agent are co-possible. It would be
like asking 'If Bill is smarter than anyone else and Hillary is
smarter than anyone else, which of the two is the
smartest?'.



Predictive determinism states that everything is foreseeable.
Metaphysical determinism states that there is only one way the future
could be given the way the past is. Simon Laplace used metaphysical
determinism as a premise for predictive determinism. He reasoned that
since every event has a cause, a complete description of any stage of
history combined with the laws of nature implies what happens at any
other stage of the universe. Scriven was only challenging predictive
determinism in his thought experiment. The next approach challenges
metaphysical determinism.


### 1.3 The Problem of Foreknowledge



Prior knowledge of an action seems incompatible with it being a free
action. If I know that you will finish reading this article tomorrow,
then you will finish tomorrow (because knowledge implies truth). But
that means you will finish the article even if you resolve not to.
After all, given that you will finish, nothing can stop you from
finishing. So if I know that you will finish reading this article
tomorrow, you are not free to do otherwise.



Maybe all of your reading is compulsory. If God exists, then He knows
everything. So the threat to freedom becomes total for the theist. The
problem of divine foreknowledge raises the possibility that theism
(rather than atheism) precludes free choice and thereby precludes our having any moral responsibility.



In response to the apparent conflict between freedom and
foreknowledge, medieval philosophers denied that future contingent
propositions have a truth-value. They took themselves to be extending
a solution Aristotle discusses in *De Interpretatione* to the
problem of logical fatalism. According to this truth-value gap
approach, 'You will finish this article tomorrow' is not
true *now*. The prediction will *become* true tomorrow.
A morally serious theist can agree with the *Rubaiyat of Omar
Khayyam*:



> 
> The Moving Finger writes; and, having writ,
>   
> 
> Moves on: nor all your Piety nor Wit
>   
> 
> Shall lure it back to cancel half a Line,
>   
> 
> Nor all your Tears wash out a Word of it.
> 



God's omniscience only requires that He knows every true
proposition. God will know 'You will finish this article
tomorrow' as soon it becomes true - but not before.



The teacher has freewill. Therefore, predictions about what he will do
are not true (prior to the examination). The metaphysician Paul Weiss
(1952) concludes that the student's argument falsely assumes he
knows that the announcement is true. The student can know that the
announcement is true *after* it becomes true - but not
before.



The logician W. V. O. Quine (1953) agrees with Weiss' conclusion
that the teacher's announcement of a surprise test fails to give
the student knowledge that there will be a surprise test. Yet Quine
abominates Weiss' reasoning. Weiss breeches the law of bivalence
(which states that every proposition has a truth-value, true or
false). Quine believes that the riddle of the surprise test should not
be answered by surrendering classical logic.


## 2. Intellectual suicide



Quine insists that the student's elimination argument is only a
*reductio ad absurdum* of the supposition that the student
*knows* that the announcement is true (rather than a
*reductio* of the announcement itself). He accepts this
epistemic *reductio* but rejects the metaphysical
*reductio*. Given the student's ignorance of the
announcement, Quine concludes that a test on any day would be
unforeseen. That is, Quine accepts that the student has no advance
knowledge of the time of the test, but he rejects that there is no
truth in advance as to when the test will be given.



Common sense suggests that the students are informed by the
announcement. The teacher is assuming that the announcement will
enlighten the students. She seems right to assume that the
announcement of this intention produces the same sort of knowledge as
her other declarations of intentions (about which topics will be
selected for lecture, the grading scale, and so on).



There are skeptical premises that could yield Quine's conclusion
that the students do not know the announcement is true. If no one can
know anything about the future, as alleged by David Hume's
problem of induction, then the student cannot know that the
teacher's announcement is true. (See the entry on
 the problem of induction.)
 But denying all knowledge of the future in order to deny the
student's knowledge of the teacher's announcement is
disproportionate and indiscriminate. Do not kill a fly with cannon --
unless it is killer fly and only a cannon will work!



In later writings, Quine evinces general reservations about the
concept of knowledge. One of his favorite objections is that
'know' is vague. If knowledge entails certainty, then too
little will count as known. Quine infers that we must equate knowledge
with firmly held true belief. Asking just how firm that belief must be
is akin to asking just how big something has to be to count as being
big. There is no answer to the question because 'big'
lacks the sort of boundary enjoyed by precise words.



> 
> There is no place in science for bigness, because of this lack of
> boundary; but there is a place for the relation of biggerness. Here we
> see the familiar and widely applicable rectification of vagueness:
> disclaim the vague positive and cleave to the precise comparative. But
> it is inapplicable to the verb 'know', even grammatically.
> Verbs have no comparative and superlative inflections ... . I
> think that for scientific or philosophical purposes the best we can do
> is give up the notion of knowledge as a bad job and make do rather
> with its separate ingredients. We can still speak of a belief as true,
> and of one belief as firmer or more certain, to the believer's
> mind, than another (1987, 109).
> 



Quine is alluding to Rudolf Carnap's (1950) generalization that
scientists replace qualitative terms (*tall*) with comparatives
(*taller than*) and then replace the comparatives with
quantitative terms (*being n millimeters in height*).



It is true that some borderline cases of a qualitative term are not
borderline cases for the corresponding comparative. But the reverse
holds as well. A tall man who stoops may stand less high than another
tall man who is not as lengthy but better postured. Both men are
clearly tall. It is unclear that 'The lengthier man is
taller'. Qualitative terms can be applied when a vague quota is
satisfied without the need to sort out the details. Only comparative
terms are bedeviled by tie-breaking issues.



Science is about what is the case rather than what ought to be case.
This seems to imply that science does not tell us what we ought to
believe. The traditional way to fill the normative gap is to delegate
issues of justification to epistemologists. However, Quine is
uncomfortable with delegating such authority to philosophers. He
prefers the thesis that psychology is enough to handle the issues
traditionally addressed by epistemologists (or at least the issues
still worth addressing in an Age of Science). This "naturalistic
epistemology" seems to imply that 'know' and
'justified' are antiquated terms - as empty as
'phlogiston' or 'soul'.



Those willing to abandon the concept of knowledge can dissolve the
surprise test paradox. But to epistemologists who find promise in less
drastic responses this is like using a suicide bomb to kill a fly.



Our suicide bomber may protest that the flies have been undercounted.
Epistemic eliminativism dissolves *all* epistemic paradoxes.
According to the eliminativist, epistemic paradoxes are symptoms of a
problem with the very concept of knowledge.



Notice that the eliminativist is more radical than the skeptic. The
skeptic thinks the concept of knowledge is coherent and definite in
its requirements. We just fall short of being knowers. The skeptic
treats 'No man is a knower' like 'No man is an
immortal'. There is nothing wrong with the concept of
immortality. Biology just winds up guaranteeing that every man falls
short of being immortal. Universal absence of knowledge would be
shocking. But the potential to shock us should not lead us to kill the
messenger (the skeptic) or declare unintelligible the vocabulary
comprising the message (specifically, the word
'know').



Unlike the messenger telling us 'No man is an immortal',
the skeptic has trouble *telling* us, 'There is no
knowledge'. According to Sextus Empiricus, assertion expresses
belief that one knows what is asserted (*Outlines of
Pyrrhonism*, I., 3, 226). He condemns the *assertion*
'There is no knowledge' (though not the proposition
expressed by the assertion) as *dogmatic* skepticism. Sextus
prefers agnosticism about knowledge rather than skepticism (considered
as "atheism" about knowledge). Yet it is just as
inconsistent to assert 'No one can know whether anything is
known'. For that conveys the belief that one knows that no one
can know whether anything is known.



The eliminativist has even more severe difficulties in stating his
position than the skeptic. Some eliminativists dismiss the threat of
self-defeat by drawing an analogy. Those who denied the existence of
souls were accused of undermining a necessary condition for asserting
anything. However, the soul theorist's account of what is needed
gives no reason to deny that a healthy brain suffices for mental
states.



If the eliminativist thinks that assertion only imposes the aim of
expressing a truth, then he can consistently assert that
'know' is a defective term. However, an epistemologist can
revive the charge of self-defeat by showing that assertion does indeed
require the speaker to attribute knowledge to himself. This
knowledge-based account of assertion has recently been supported by
work on our next paradox.


## 3. Lotteries and the Lottery Paradox



Lotteries pose a problem for the theory that a high probability for a
true belief suffices for knowledge. Given that there are a million
tickets and only one winner, the probability of 'This ticket is
a losing ticket' is very high. Yet we are reluctant to say this
makes the proposition known.



We overcome the inhibition after the winning ticket is announced. Now
the ticket is known to be a loser and tossed in the trash. But wait!
Testimony does not furnish *certainty*. Nor does perception or
recollection . When pressed, we admit there is a small chance that we
misperceived the drawing or that the newscaster misread the winning
number or that we are misremembering. While in this concessive mood,
we are apt to relinquish our claim to know. The skeptic syllogizes
from this surrender: *For any contingent proposition, there is a
lottery statement that is more probable and which is unknown. A known
proposition cannot be less probable than an unknown proposition. So no
contingent proposition is known* (Hawthorne 2004). That is too
much to give up! Yet the skeptic's statistics seem impeccable.



This skeptical paradox was noticed by Gilbert Harman (1968, 166). But
his views about the role of causation in inferential knowledge seemed
to solve the problem (DeRose 2017, chapter 5). The baby paradox was
dismissed as stillborn. Since the new arrival did not get the
customary baptism of attention, epistemologists did not notice that
the demise of the causal theory of knowledge meant new life for
Harman's lottery paradox.



The probability skeptic's ordinary suggestions about how we
might be mistaken contrast with the extraordinary possibilities
conjured by Rene Descartes' skeptic. The Cartesian
skeptic tries to undermine vast swaths of knowledge with a single
untestable counter-explanation of the evidence (such as the hypothesis
that you are dreaming or the hypothesis that an evil demon is
deceiving you). These comprehensive alternatives are designed to evade
any empirical refutation. The probabilistic skeptic, in contrast,
points to a plethora of pedestrian counter-explanations. Each is easy
to test: maybe you transposed the digits of a phone number, maybe the
ticket agent thought you wanted to fly to Moscow, Russia rather than
Moscow, Idaho, etc. You can check for errors, but any check itself has
a small chance of being wrong. So there is always something to check,
given that the issues cannot be ignored on grounds of
improbability.



You can check *any* of these possible errors but you cannot
check them *all*. You cannot discount these pedestrian
possibilities as science fiction. These are exactly the sorts of
possibilities we check when plans go awry. For instance, you think you
know that you have an appointment to meet a prospective employer for
lunch at noon. When she fails to show at the expected time, you
backpedal through your premises: Is your watch slow? Are you
remembering the right restaurant? Could there be another restaurant in
the city with the same name? Is she just detained? Could she have just
forgotten? Could there have been a miscommunication?



Probabilistic skepticism dates back to Arcesilaus who took over the
Academy two generations after Plato's death. This moderate kind
of skepticism, recounted by Cicero (*Academica* 2.74, 1.46)
from his days as a student at the Academy, allows for justified
belief. Many scientists feel they should only assign probabilities.
They dismiss the epistemologist's preoccupation with knowledge
as old-fashioned.



Despite the early start of the qualitative theory of probability, the
quantitative theory did not develop until Blaise Pascal's study
of gambling in the seventeenth century (Hacking 1975). Only in the
eighteenth century did it penetrate the insurance industry (even
though insurers realized that a fortune could be made by accurately
calculating risk). Only in the nineteenth century did probability make
a mark in physics. And only in the twentieth century do probabilists
make important advances over Arcesilaus.



Most of these philosophical advances are reactions to the use of
probability by scientists. In the twentieth century, editors of
science journals began to demand that the author's hypothesis
should be accepted only when it was sufficiently probable - as
measured by statistical tests. The threshold for acceptance was
acknowledged to be somewhat arbitrary. And it was also conceded that
the acceptance rule might vary with one's purposes. For
instance, we demand a higher probability when the cost of accepting a
false hypothesis is high.



In 1961 Henry Kyburg pointed out that this policy conflicted with a
principle of agglomeration: If you rationally believe \(p\)
and rationally believe \(q\) then you
rationally believe both \(p\) and \(q\).
Little pictures of the same scene should sum to a bigger picture of
the same scene. If rational belief can be based on an acceptance rule
that only requires a high probability, there will be rational belief
in a contradiction! To see why, suppose the acceptance rule permits
belief in any proposition that has a probability of at least .99.
Given a lottery with 100 tickets and exactly one winner, the
probability of 'Ticket \(n\) is a loser'
licenses belief. Symbolize propositions about ticket \(n\)
being a loser as \(p\_n\). Symbolize 'I
rationally believe' as \(B\). Belief in a
contradiction follows:


1. \(B{\sim}(p\_1 \amp p\_2 \amp \ldots \amp p\_{100})\),
  

by the probabilistic acceptance rule.
2. \(Bp\_1 \amp Bp\_2 \amp \ldots \amp Bp\_{100}\),
  

by the probabilistic acceptance rule.
3. \(B(p\_1 \amp p\_2 \amp \ldots \amp p\_{100})\),
  

from (2) and the principle that rational belief agglomerates.
4. \(B[(p\_1 \amp p\_2 \amp \ldots \amp p\_{100}) \amp{\sim}(p\_1 \amp
p\_2 \amp \ldots \amp p\_{100})]\),
  

from (1) and (3) by the principle that rational belief
agglomerates.



More informally, the acceptance rule implies this: each belief that a particular ticket will lose is probable enough to justify believing it. By repeated
applications of the agglomeration principle, conjoining all of these
justified beliefs together gives a justified belief. Finally,
conjoining that belief with the justified belief that one of tickets
is a winner gives the contradictory belief to the to the effect that
each will lose and one will win. Yet by agglomeration that too is
justified.



Since belief in an obvious contradiction is a paradigm example of
irrationality, Kyburg poses a dilemma: either reject agglomeration or
reject rules that license belief for a probability of less than one.
(Martin Smith 2016, 186-196) warns that even a probability of
one leads to joint inconsistency for a lottery that has infinitely
many tickets.) Kyburg rejects agglomeration. He promotes toleration of
*joint* inconsistency (having beliefs that cannot all be true
together) to avoid belief in contradictions. Reason forbids us from
believing a proposition that is necessarily false but permits us to
have a set of beliefs that necessarily contains a falsehood. Henry
Kyburg's choice was soon supported by the discovery of a
companion paradox.


## 4. Preface Paradox



In the preface of *Introduction to the Foundations of
Mathematics*, Raymond Wilder (1952, iv) apologizes for the errors
in the text. The 1982 reprint has three pages of errata that vindicate
Wilder's humility. D. C. Makinson (1965, 205) quotes Wilder's
1952 apology and extracts a paradox: Wilder rationally believes each
of the assertions in his book. But since Wilder regards himself as
fallible, he rationally believes the conjunction of all his assertions
is false. If the agglomeration principle holds, \((Bp \amp Bq)
\rightarrow B(p \amp q)\), Wilder would rationally believe the
conjunction of all assertions in his book and also rationally
disbelieve the same thing!



The preface paradox does not rely on a probabilistic acceptance rule.
The preface belief is organically generated in a qualitative fashion.
The author is merely reflecting on his humbling resemblance to other
authors who are fallible, his own past failing that he subsequently
discovered, his imperfection in fact checking, and so on.



At this juncture many philosophers join Kyburg in rejecting
agglomeration and conclude that it can be rational to have jointly
inconsistent beliefs. Kyburg's solution to the preface paradox
raises a methodological question about the nature of paradox. How can
paradoxes change our minds if joint inconsistency is permitted? A
paradox is commonly defined as a set of propositions that are
individually plausible but jointly inconsistent. The inconsistency is
the itch that directs us to scratch out a member of the set (or the
pain that leads us to withdraw from the stimulus). For instance, much
epistemology orbits an ancient riddle posed by the regress of
justification, namely, which of the following is false?


1. A belief can only be justified by another justified belief.
2. There are no circular chains of justification.
3. All justificatory chains have a finite length.
4. Some beliefs are justified.



Foundationalists reject (1). They take some propositions to be
self-evident or they permit beliefs to be justified by non-beliefs (such as perceptions or intuitions).
Coherentists reject (2). They tolerate some forms of circular
reasoning. For instance, Nelson Goodman (1965) has characterized the
method of reflective equilibrium as *virtuously* circular.
Charles Saunders Peirce (1933-35, 5.250) may have rejected (3).
The first clear rejector is Peter Klein (2007). For a book-length
defense, read Scott F. Aikin (2011). Infinitists believe that
infinitely long chains of justification are no more impossible than
infinitely long chains of causation. Finally, the epistemological
anarchist rejects (4). As Paul Feyerabend refrains in *Against
Method*, "Anything goes" (1988, vii, 5, 14, 19,
159).



Formulating a paradox as a set of individually plausible but jointly
inconsistent beliefs is a feat of data compression. But if joint
inconsistency is rationally tolerable, why do these philosophers
bother to offer solutions to paradoxes such as the regress of
justification? Why is it irrational to believe each of (1)-(4),
despite their joint inconsistency?



Kyburg might answer that there is a scale effect. Although the
sensation of joint inconsistency is tolerable when diffusely
distributed over a large body of propositions, the sensation becomes
an itch when the inconsistency localizes (Knight 2002). That is why
paradoxes are always represented as a *small* set of
propositions. A paradox is improved by reducing its membership --
as when a member of the set is exposed as superfluous to the
inconsistency. (Strictly speaking, a set can only change size in the
metaphorical way that a number grows or shrinks.)



If you know that your beliefs are jointly inconsistent but deny this
makes for a giant paradox, then you should reject R. M.
Sainsbury's definition of a paradox as "an apparently
unacceptable conclusion derived by apparently acceptable reasoning
from apparently acceptable premises" (1995, 1). Take the
negation of any of your beliefs as a conclusion and your remaining
beliefs as the premises. You should judge this jumble argument as
valid, and as having premises that you accept, and yet as having a
conclusion you reject (Sorensen 2003b, 104-110). If the
conclusion of this argument counts as a paradox, then the negation of
any of your beliefs counts as a paradox.



The resemblance between the preface paradox and the surprise test
paradox becomes more visible through an intermediate case. The preface
of Siddhartha Mukherjee's *The Emperor of All Maladies: A
Biography of Cancer* warns: "In cases where there was no
prior public knowledge, or when interviewees requested privacy, I have
used a false name, and deliberately confounded identities to make it
difficult to track." (2010, xiv) Those who refuse consent to be
lied to are free to close Doctor Mukherjee's chronicle. But
nearly all readers think the physician's trade-off between lies
and new information is acceptable. They rationally anticipate being
rationally misled. Nevertheless, these readers learn much about the
history of cancer. Similarly, students who are warned that they will
receive a surprise test rationally expect to be rationally misled
about the day of the test. The prospect of being misled does not lead
them to drop the course.



The preface paradox pressures Kyburg to extend his tolerance of joint
inconsistency to the acceptance of contradictions. For
Makinson's original specimen is a logician's regret at
affirming *contradictions* rather than false contingent
statements. Consider a logic student who is required to pick one
hundred truths from a mixed list of tautologies and contradictions
(Sorensen 2001, 156-158). Although the modest student believes
each of his answers, \(A\_1, A\_2, \ldots, A\_{100}\), he also believes
that at least of one these answers is false. This ensures he believes
a contradiction. If any of his answers is false, then the student
believes a contradiction (because the only falsehoods on the question
list are contradictions). If all of his test answers are true, then
the student believes the following contradiction: \({\sim}(A\_1 \amp
A\_2 \amp \ldots \amp A\_{100})\). After all, a conjunction of
tautologies is itself a tautology and the negation of any tautology is
a contradiction.



If paradoxes were always sets of propositions or arguments or
conclusions, then they would always be meaningful. But some paradoxes
are semantically flawed (Sorensen 2003b, 352) and some have answers
that are backed by a pseudo-argument employing a defective
"lemma" that lacks a truth-value. Kurt Grelling's
paradox, for instance, opens with a distinction between autological
and heterological words. An autological word describes itself, e.g.,
'polysyllabic' is polysllabic, 'English' is
English, 'noun' is a noun, etc. A heterological word does
not describe itself, e.g., 'monosyllabic' is not
monosyllabic, 'Chinese' is not Chinese, 'verb'
is not a verb, etc. Now for the riddle: Is 'heterological'
heterological or autological? If 'heterological' is
heterological, then since it describes itself, it is autological. But
if 'heterological' is autological, then since it is a word
that does not describe itself, it is heterological. The common
solution to this puzzle is that 'heterological', as
defined by Grelling, is not a well-defined predicate (Thomson 1962).
In other words, "Is 'heterological'
heterological?" is without meaning. There can be no predicate
that applies to all and only those predicates it does not apply to for
the same reason that there can be no barber who shaves all and only
those people who do not shave themselves.



The eliminativist, who thinks that 'know' or
'justified' is meaningless, will diagnose the epistemic
paradoxes as questions that only *appear* to be well-formed.
For instance, the eliminativist about justification would not accept
proposition (4) in the regress paradox: 'Some beliefs are
justified'. His point is not that no beliefs meet the high
standards for justification, as an anarchist might deny that any
ostensible authorities meet the high standards for legitimacy.
Instead, the eliminativist unromantically diagnoses
'justified' as a pathological term. Just as the astronomer
ignores 'Are there a zillion stars?' on the grounds that
'zillion' is not a genuine numeral, the eliminativist
ignores 'Are some beliefs justified?' on the grounds that
'justified' is not a genuine adjective.



In the twentieth century, suspicions about conceptual pathology were
strongest for the liar paradox: Is 'This sentence is
false' true? Philosophers who thought that there was something
deeply defective with the surprise test paradox assimilated it to the
liar paradox. Let us review the assimilation process.


## 5. Anti-expertise



In the surprise test paradox, the student's premises are
self-defeating. Any reason the student has for predicting a test date
or a non-test date is available to the teacher. Thus the teacher can
simulate the student's forecast and know what the student
expects.



The student's overall conclusion, that the test is impossible,
is also self-defeating. If the student believes his conclusion then he
will not expect the test. So if he receives a test, it will be a
surprise. The event will be all the more unexpected because the
student has deluded himself into thinking the test is impossible.



Just as someone's awareness of a prediction can affect the
likelihood of it being true, awareness of that sensitivity to his
awareness can also affect its truth. If each cycle of awareness is
self-defeating, then there is no stable resting place for a
conclusion.



Suppose a psychologist offers you a red box and a blue box (Skyrms
1982). The psychologist can predict which box you will choose with 90%
accuracy. He has put one dollar in the box he predicts you will choose
and ten dollars in the other box. Should you choose the red box or the
blue box? You cannot decide. For any choice becomes a reason to
reverse your decision.



Epistemic paradoxes affect decision theory because rational choices
are based on *beliefs* and desires. If the agent cannot form a
rational belief, it is difficult to interpret his behavior as a
*choice*. In decision theory, the whole point of attributing
beliefs and desires is to set up practical syllogisms that make sense
of actions as means to ends. Subtracting rationality from the agent
makes the framework useless. Given this commitment to charitable
interpretation, there is no possibility of you rationally choosing an
option that you believe to be inferior. So if you choose, you cannot
really believe you were operating as an anti-expert, that is, someone
whose opinions on a topic are reliably wrong (Egan and Elga 2005).



The medieval philosopher John Buridan (*Sophismata*, Sophism
13) gave a starkly minimal example of such instability:



(B)
You do not believe this sentence. 


If you believe (B) it is false. If you do not believe (B) it is true.
You are an anti-expert about (B); your opinion is reliably wrong. An
outsider who monitors your opinion can reckon whether (B) is true. But
you are not able to exploit your anti-expertise.



On the bright side, you are able to exploit the anti-expertise of
others. Four out of five anti-experts recommend against reading any
further!


### 5.1 The Knower Paradox



David Kaplan and Richard Montague (1960) think the announcement by the
teacher in our surprise exam example is equivalent to the
self-referential



(K-3)
 Either the test is on Monday but you do not know it before
Monday, or the test is on Wednesday but you do not know it before
Wednesday, or the test is on Friday but you do not know it before
Friday, or this announcement is known to be false. 


Kaplan and Montague note that the number of alternative test dates can
be increased indefinitely. Shockingly, they claim the number of
alternatives can be reduced to zero! The announcement is then
equivalent to



(K-0)
This sentence is known to be false. 


If (K-0) is true then it known to be false. Whatever is known to be
false, is false. Since no proposition can be both true and false, we
have proven that (K-0) is false. Given that proof produces knowledge,
(K-0) is known to be false. But wait! That is exactly what (K-0) says
- so (K-0) must be true.



The (K-0) argument bears a suspicious resemblance to the liar paradox.
Subsequent commentators sloppily switch the negation sign in the
formal presentations of the reasoning from \(K{\sim}p\) to
\({\sim}Kp\) (that is, from 'It is known that not-\(p\)',
to 'It is not the case that it is known
that \(p\)'). Ironically, this garbled
transmission results in a cleaner variation of the knower:



(K)
No one knows this very sentence. 


Is (K) true? On the one hand, if (K) is true, then what it says is
true, so no one knows it. On the other hand, that very reasoning seems
to be a proof of (K). Believing a proposition by seeing it to be
proved is a sufficient for knowledge of it, so someone must know (K).
But then (K) is false! Since no one can know a proposition that is
false, (K) is not known.



The skeptic could hope to solve (K-0) by denying that anything is
known. This remedy does not cure (K). If nothing is known then (K) is
true. Can the skeptic instead challenge the premise that proven is a
sufficient for knowing it? This solution would be particularly
embarrassing to the skeptic. The skeptic presents himself as a
stickler for proof. If it turns out that even proof will not sway him,
he bears a damning resemblance to the dogmatist he so frequently
chides.



But the skeptic should not lose his nerve. Proof does not always yield
knowledge. Consider a student who correctly guesses that a step in his
proof is valid. The student does not know the conclusion but did prove
the theorem. His instructor might have trouble getting the student to
understand why his answer constitutes a valid proof. The intransigence
may stem from the prover's intelligence rather than his
stupidity. L. E. J. Brouwer is best known in mathematics for his
brilliant fixed point theorem. But a doubtful reading of Immanuel
Kant's philosophy of mathematics led Brouwer retract his proof.
Brouwer also had philosophical doubts about the Axiom of Choice and the
Law of Excluded Middle. Brouwer persuaded a minority of mathematicians
and philosophers, known as intuitionists, to emulate his abstention
from non-constructive proofs. This led them to develop constructive
proofs of theorems that were earlier proved by less informative means.
Everybody agrees that there is more to learn from a proof of an
existential generalization that proceeds from a proved instance than
from an unspecific *reductio ad absurdum* of the corresponding
universal generalization. But this does not vindicate the
intuitionists's refusal to be persuaded by the *reductio ad
absurdum*. The intuitionist, even in the eyes of the skeptic, has
too high a standard of proof. An excessively high standard of proof
can prevent knowledge by proof. 



The logical myth that "You cannot prove a universal
negative" is itself a universal negative. So it implies its own
unprovability. This implication of unprovability is correct but only
because the principle is false. For instance, exhaustive inspection
proves the universal negative 'No adverbs appear in this
sentence'. A *reductio ad absurdum* proves the universal
negative 'There is no largest prime number'.



Trivially, *false* propositions cannot be proved true. Are
there any *true* propositions that cannot be proved true?



Yes, there are infinitely many. Kurt Godel's incompleteness
theorem demonstrated that any system that is strong enough to express
arithmetic is also strong enough to express a formal counterpart of
the self-referential proposition in the surprise test example
'This statement cannot be proved in this system'. If the
system cannot prove its "Godel sentence", then this
sentence is true. If the system can prove its Godel sentence, the
system is inconsistent. So either the system is incomplete or
inconsistent. (See the entry on
 Kurt Godel.)



Of course, this result concerns provability relative to a system. One
system can prove another system's Godel sentence. Kurt
Godel (1983, 271) thought that proof was not needed for knowledge
that arithmetic is consistent.



J. R. Lucas (1964) claims that this reveals human beings are not
machines. A computer is a concrete instantiation of a formal system.
Hence, its "knowledge" is restricted to what it can prove.
By Godel's theorem, the computer will be either
inconsistent or incomplete. However, any human being *could*
have consistent and complete knowledge of arithmetic. Therefore, necessarily, no
human being is a computer.



Critics of Lucas defend the parity between people and computers. They
think we have our own Godel sentences (Lewis 1999,
166-173). In this egalitarian spirit, G. C. Nerlich (1961)
models the student's beliefs in the surprise test example as a
logical system. The teacher's announcement is then a Godel
sentence about the student: There will be a test next week but you
will not be able to prove which day it will occur on the basis of this
announcement and memory of what has happened on previous exam days.
When the number of exam days equals zero the announcement is
equivalent to sentence K.



Several commentators on the surprise test paradox object that
interpreting surprise as unprovability changes the topic. Instead of
posing the surprise test paradox, it poses a variation of the liar
paradox. Other concepts can be blended with the liar. For instance,
mixing in alethic notions generates the possible liar: Is 'This
statement is possibly false' true? (Post 1970) (If it is false,
then it is false that it is possibly false. What cannot possibly be
false is necessarily true. But if it is necessarily true, then it
cannot be possibly false.) Since the semantic concept of validity
involves the notion of possibility, one can also derive validity liars
such as Pseudo-Scotus' paradox: 'Squares are squares,
therefore, this argument is invalid' (Read 1979). Suppose
Pseudo-Scotus' argument is valid. Since the premise is
necessarily true, the conclusion would be necessarily true. But the
conclusion contradicts the supposition that argument is valid.
Therefore, by reductio, the argument is necessarily invalid. Wait! The
argument can be invalid only if it is possible for the premise to be
true and the conclusion to be false. But we have already proved that
the conclusion of 'Squares are squares, therefore, this argument
is invalid' is necessarily true. There is no consistent judgment
of the argument's validity. A similar predicament follows from
'The test is on Friday but this prediction cannot be soundly
deduced from this announcement'.



One can mock up a complicated liar paradox that resembles the surprise
test paradox. But this complex variant of the liar is not an
*epistemic* paradox. For the paradoxes turn on the semantic
concept of truth rather than an epistemic concept.


### 5.2 The "Knowability Paradox"



Frederic Fitch (1963) reports that in 1945 he first learned of this
proof of unknowable truths from a referee report on a manuscript he
never published. Thanks to Joe Salerno's (2009) archival
research, we now know that referee was Alonzo Church.



Assume there is a true sentence of the form '\(p\)
but \(p\) is not known'.
Although this sentence is consistent, modest principles of epistemic
logic imply that sentences of this form are unknowable.




|  |  |  |
| --- | --- | --- |
| 1. | \(K(p \amp{\sim}Kp)\) | (Assumption) |
| 2. | \(Kp \amp K{\sim}Kp\) | 1, Knowledge distributes over conjunction |
| 3. | \({\sim}Kp\) | 2, Knowledge implies truth (from the second conjunct) |
| 4. | \(Kp \amp{\sim}Kp\) | 2, 3 by conjunction elimination of the first conjunct and then
conjunction introduction |
| 5. | \({\sim}K(p \amp{\sim}Kp)\) | 1, 4 Reductio ad absurdum |



Since all the assumptions are discharged, the conclusion is a
necessary truth. So it is a necessary truth that \(p \amp{\sim}Kp\) is
not known. In other words, \(p \amp{\sim}Kp\) is unknowable.



The cautious draw a conditional moral: If there are actual unknown
truths, there are unknowable truths. After all, some philosophers will
reject the antecedent because they believe there is an omniscient
being.



But secular idealists and logical positivists concede that there are
some actual unknown truths. How can they continue to believe that all
truths are knowable? Astonishingly, these eminent philosophers seem
refuted by the pinch of epistemic logic we have just seen. Also
injured are those who limit their claims of universal knowability to a
limited domain. For instance, Immanuel Kant (A223/B272) asserts that
all empirical propositions are knowable. This pocket of optimism would
be enough to ignite the contradiction (Stephenson 2015).



Timothy Williamson doubts that this casualty list is enough for the
result to qualify as a paradox:



> 
> The conclusion that there are unknowable truths is an affront to
> various philosophical theories, but not to common sense. If proponents
> (and opponents) of those theories long overlooked a simple
> counterexample, that is an embarrassment, not a paradox. (2000, 271)
> 



Those who believe that the Church-Fitch result is a genuine paradox
can respond to Williamson with paradoxes that accord with common sense
(and science -and religious orthodoxy). For instance, common
sense heartily agrees with the conclusion that something exists. But
it is surprising that this can be proved without empirical premises.
Since the quantifiers of standard logic (first order predicate logic
with identity) have existential import, the logician can deduce that
something exists from the principle that everything is identical to
itself. Most philosophers balk at this simple proof because they feel
that the existence of something cannot be proved by sheer logic. They
are not balking at the statement that is in accord in common sense
(that something exists). They are only balking at the statement that
it can be proved by sheer logic. Likewise, many philosophers who agree
that there are unknowables balk solely on the grounds that such a
profound result cannot be obtained from such limited means.


### 5.3 Moore's problem



Church's referee report was composed in 1945. The timing and
structure of his argument for unknowables suggests that Church may
have been inspired by G. E. Moore's (1942, 543) sentence:



(M)
I went to the pictures last Tuesday, but I don't believe that I did.



Moore's problem is to explain what is odd about declarative
utterances such as (M). This explanation needs to encompass both
readings of (M): '\(p \amp B{\sim}p\)' and '\(p
\amp{\sim}Bp\)'. (This scope ambiguity is exploited by a popular
joke: Rene Descartes sits in a bar, having a drink. The
bartender asks him if he would care for another. "I think
not," he says, and disappears. The joke is commonly criticized
as fallacious. But it is not given Descartes' belief that he is
essentially a thinking being.)



The common explanation of Moore's absurdity is that the speaker
has managed to contradict himself without uttering a contradiction. So
the sentence is odd because it is a counterexample to the
generalization that anyone who contradicts himself utters a
contradiction.



There is no problem with third person counterparts of (M). Anyone else
can say about Moore, with no paradox, 'G. E. Moore went to the
pictures last Tuesday but he does not believe it'. (M) can also
be embedded unparadoxically in conditionals: 'If I went to the
pictures last Tuesday but I do not believe it, then I am suffering
from a worrisome lapse of memory'. The past tense is fine:
'I went to the picture shows last Tuesday but I did not believe
it'. The future tense, 'I went to the picture shows last
Tuesday but I will not believe it', is a bit more of a stretch
(Bovens 1995). We tend to picture our future selves as better
informed. Later selves are, as it were, experts to whom earlier selves
should defer. When an earlier self foresees that his later self
believes \(p\), then the prediction is a reason to
believe \(p\). Bas van Fraassen (1984, 244) dubs this
"the principle of reflection": I ought to believe a
proposition given that I will believe it at some future time.



Robert Binkley (1968) anticipates van Fraassen by applying the
reflection principle to the surprise test paradox. The student can
foresee that he will not believe the announcement if no test is given
by Thursday. The *conjunction* of the history of testless days
and the announcement will imply the Moorean sentence:



(A')
 The test is on Friday but you do not believe it.


Since the less evident member of the conjunction is the announcement,
the student will choose not to believe the announcement. At the
beginning of the week, the student foresees that his future self may
not believe the announcement. So the student on Sunday will not
believe the announcement when it is first uttered.



Binkley illuminates this reasoning with doxastic logic ('doxa' is
Greek for belief). The inference rules for this logic of belief can be
understood as idealizing the student into an ideal reasoner. In
general terms, an ideal reasoner is someone who infers what he ought
and refrains from inferring any more than he ought. Since there is no
constraint on his premises, we may disagree with the ideal reasoner.
But if we agree with the ideal reasoner's premises, we appear
bound to agree with his conclusion. Binkley specifies some
requirements to give teeth to the student's status as an ideal
reasoner: the student is perfectly consistent, believes all the
logical consequences of his beliefs, and does not forget. Binkley
further assumes that the ideal reasoner is aware that he is an ideal
reasoner. According to Binkley, this ensures that if the ideal
reasoner believes p, then he believes that he will believe \(p\)
thereafter.



Binkley's account of the student's hypothetical epistemic
state on Thursday is compelling. But his argument for spreading the
incredulity from the future to the past is open to three
challenges.



The first objection is that it delivers the wrong result. The student
\(is\) informed by the teacher's announcement, so Binkley ought
not to use a model in which the announcement is as absurd as the
conjunction 'I went to the pictures last Tuesday but I do not
believe it'.



Second, the future mental state envisaged by Binkley is only
hypothetical: \(If\) no test is given by Thursday, the student will
find the announcement incredible. At the beginning of the week, the
student does not know (or believe) that the teacher will wait that
long. The principle of reflection 'Defer to the opinions of my future
self ' does not imply that I should defer to the opinions of my
*hypothetical* future self. For my hypothetical future self is
responding to propositions that need not be actually true.



Third, the principle of reflection may need more qualifications than
Binkley anticipates. Binkley realizes that an ordinary agent foresees
that he will forget details. That is why we write reminders for our
own benefit. An ordinary agent foresees periods of impaired judgment.
That is why we limit how much money we bring to the bar.



Binkley stipulates that the students do not forget. He needs to add
that the students know that they will not forget. For the mere threat
of a memory lapse sometimes suffices to undermine knowledge. Consider
Professor Anesthesiology's scheme for surprise tests: "A
surprise test will be given either Wednesday or Friday with the help
of an amnesia drug. If the test occurs on Wednesday, then the drug
will be administered five minutes after Wednesday's class. The
drug will instantly erase memory of the test and the students will
fill in the gap by confabulation." You have just completed
Wednesday's class and so temporarily know that the test will be
on Friday. Ten minutes after the class, you lose this knowledge. No
drug was administered and there is nothing wrong with your memory. You
are correctly remembering that no test was given on Wednesday.
However, you do not know your memory is accurate because you also know
that if the test was given Wednesday then you would have a
pseudo-memory indistinguishable from your present memory. Despite not
gaining any new evidence, you change your mind about the test
occurring on Wednesday and lose your knowledge that the test is on
Friday. (The change of belief is not crucial; you would still lack
fore*knowledge* of the test even if you dogmatically persisted
in believing that the test will be on Friday.)



If the students know that they will not forget and know there will be
no undermining by outside evidence, then we may be inclined to agree
with Binkley's summary that his idealized student never loses
the knowledge he accumulates. As we shall see, however, this overlooks
other ways in which rational agents may lose knowledge.


### 5.4 Blindspots



'I am a poet but I do not know it' expresses a proposition
I cannot know. But I can reach the proposition by other attitudes
such as hoping and wishing. A blindspot for a propositional attitude
is a consistent proposition that cannot be accessed by that
attitude. Blindspots are relative to the means of reaching the
proposition, the person making the attempt, and time at which he
tries. Although \(I\) cannot rationally believe 'Polar bears
have black skin but I believe they do not' *you* can
believe that I mistakenly believe polar bears do not have black
skin. The evidence that persuades you I am currently making that
mistake cannot persuade me that I am currently making that
mistake. This is an asymmetry imposed by rationality rather than
irrationality. Attributions of specific errors are personal blindspots
for the person who is alleged to have erred.



The anthropologist Gontran de Poncins begins his chapter on the arctic
missionary, Father Henry, with a prediction:



> 
> I am going to say to you that a human being can live without complaint
> in an ice-house built for seals at a temperature of fifty-five degrees
> below zero, and you are going to doubt my word. Yet what I say is
> true, for this was how Father Henry lived; .... (Poncins 1941
> [1988], 240])
> 



Gontran de Poncins' subsequent testimony might lead the reader
to believe someone can indeed be content to live in an ice-house. The
same testimony might lead another reader to believe that Poncins is
not telling the truth. But no reader ought to believe 'Someone
can be content to live in an ice house and everybody believes that is
not so'. That is a universal blindspot.



If Gontran believes a proposition that is a blindspot to his reader,
then he cannot furnish good grounds for his reader to share his
belief. This holds even if they are ideal reasoners. So one
implication of personal blindspots is that there can be disagreement
among ideal reasoners because they differ in their blindspots. 



This is relevant to the surprise test paradox. The students are the
surprisees. Since the announcement entails that the date of the
surprise test a blindspot for them, non-surprisees cannot persuade
them.



The same point holds for intra-personal disagreement over time.
Evidence that persuaded me on Sunday that 'This security code is
390524 but on Friday I will not believe it' should no longer
persuade me on Friday (given my belief that the day is Friday). For
that proposition is a blindspot to my Friday self.



Although each blindspot is inaccessible, a disjunction of blindspots
is normally not a blindspot. I can rationally believe that
'Either the number of stars is even and I do not believe it, or
the number of stars is odd and I do not believe it'. The
author's preface statement that there is some mistake in his
book is equivalent to a very long disjunction of blindspots. The
author is saying he either falsely believes his first statement or
falsely believes his second statement or ... or falsely believes
his last statement.



The teacher's announcement that there will be a surprise test is
equivalent to a disjunction of future mistakes: 'Either there
will be a test on Monday and the student will not believe it
beforehand or there will be a test Wednesday and the student will not
believe it beforehand or the test is on Friday and the student will
not believe it beforehand.'



The points made so far suggest a solution to the surprise test paradox
(Sorensen 1988, 328-343). As Binkley (1968) asserts, the test
would be a surprise even if the teacher waited until the last day. Yet
it can still be true that the teacher's announcement is
informative. At the beginning of the week, the students are justified
in believing the teacher's announcement that there will be a
surprise test. This announcement is equivalent to:



(A)
Either
 
i.
the test is on Monday and the student does not know it
before Monday, or
ii.
the test is on Wednesday and the student does not know
it before Wednesday, or
iii.
the test is on Friday and the student does not know it before
Friday.



Consider the student's predicament on Thursday (given that the
test has not been on Monday or Wednesday). If he knows that no test
has been given, he cannot also know that (A) is true. Because that
would imply


3. The test is on Friday and the student does not know it before
Friday.



Although (iii) is consistent and might be knowable by others, (iii)
cannot be known by the student before Friday. (iii) is a blindspot for
the students but not for, say, the teacher's colleagues. Hence,
the teacher can give a surprise test on Friday because that would
force the students to lose their knowledge of the original
announcement (A). Knowledge can be lost without forgetting
anything.



This solution makes who you are relevant to what you can know. In
addition to compromising the impersonality of knowledge, there will be
compromise on its temporal neutrality.



Since the surprise test paradox can also be formulated in terms of
rational belief, there will be parallel adjustments for what we ought
to believe. We are criticized for failures to believe the logical
consequences of what we believe and criticized for believing
propositions that conflict with each other. Anyone who meets these
ideals of completeness and consistency will be unable to believe a
range of consistent propositions that are accessible to other complete
and consistent thinkers. In particular, they will not be able to
believe propositions attributing specific errors to them, and
propositions that entail these off-limit propositions.



Some people wear T-shirts with *Question Authority!* written on
them. Questioning authority is generally regarded as a matter of
individual discretion. The surprise test paradox shows that it is
sometimes mandatory. The student is rationally required to doubt the
teacher's announcement even though the teacher has not given any
new evidence of being unreliable. For when only one day remains, the
announcement entails (iii), a statement is that is impossible for the student to know.
The student can foresee that this forced loss of knowledge opens an
opportunity for the teacher to give the surprise test. This
foreknowledge is available at the time the announcement*.*



This solution implies there can be disagreement amongst ideal
reasoners who agree on the same impersonal data. Consider the
colleagues of the teachers. They are not amongst those that the
teacher targets for surprise. Since 'surprise' here means
'surprise to the students', the teacher's colleagues
can consistently infer that the test will be on the last day from the
premise that it has not been given on any previous day. But these
colleagues are useless *to the students* as informants.


## 6. Dynamic Epistemic Paradoxes



The above anomalies (losing knowledge without forgetting, disagreement
amongst equally well-informed ideal reasoners, rationally changing
your mind without the acquisition of counter-evidence) would be more
tolerable if reinforced by separate lines of reasoning. The most
fertile source of this collateral support is in puzzles about updating
beliefs.



The natural strategy is to focus on the knower when he is stationary.
However, just as it is easier for an Eskimo to observe an arctic fox
when it moves, we often get a better understanding of the knower
dynamically, when he is in the process of gaining or losing
knowledge.


### 6.1 Meno's Paradox of Inquiry: A puzzle about gaining knowledge



When on trial for impiety, Socrates traced his inquisitiveness to the
Oracle at Delphi (*Apology* 21d in Cooper 1997). Prior to
beginning his mission of inquiry, Chaerephon asked the Oracle:
"Who is the wisest of men?" The Oracle answered "No
one is wiser than Socrates." This astounded Socrates because he
believed he knew nothing. Whereas a less pious philosopher might have
questioned the reliability of the Delphic Oracle, Socrates followed
the general practice of treating the Oracle as infallible. The only
cogitation appropriate to an infallible answer is interpretation.
Accordingly, Socrates resolved his puzzlement by inferring that his
wisdom lay in recognizing his own ignorance. While others may know
nothing, Socrates knows that he knows nothing.



Socrates continues to be praised for his insight. But his
"discovery" is a contradiction. If Socrates knows that he
knows nothing, then he knows something (the proposition that he knows
nothing) and yet does not know anything (because knowledge implies
truth).



Socrates could regain consistency by downgrading his meta-knowledge to
the status of a belief. If he believes he knows nothing, then he
naturally wishes to remedy his ignorance by asking about everything.
This rationale is accepted throughout the early dialogues. But when we
reach the *Meno*, one of his interlocutors has an epiphany.
After Meno receives the standard treatment from Socrates about the
nature of virtue, Meno discerns a conflict between Socratic ignorance
and Socratic inquiry (*Meno* 80d, in Cooper 1997). How would
Socrates recognize the correct answer even if Meno gave it?



The general structure of Meno's paradox is a dilemma: If you
know the answer to the question you are asking, then nothing can be
learned by asking. If you do not know the answer, then you cannot
recognize a correct answer even if it is given to you. Therefore, one
cannot learn anything by asking questions.



The natural solution to Meno's paradox is to characterize the
inquirer as only partially ignorant. He knows enough to recognize a
correct answer but not enough to answer on his own. For instance,
spelling dictionaries are useless to six year old children because
they seldom know more than the first letter of the word in question.
Ten year old children have enough partial knowledge of the
word's spelling to narrow the field of candidates. Spelling
dictionaries are also useless to those with full knowledge of spelling
and those with total ignorance of spelling. But most of us have an
intermediate amount of knowledge.



It is natural to analyze partial knowledge as knowledge of
conditionals. The ten year old child knows the spoken version of
'If the spelling dictionary spells the month after January as
F-e-b-r-u-a-r-y, then that spelling is correct'. Consulting the
spelling dictionary gives him knowledge of the antecedent of the
conditional.



Much of our learning from conditionals runs as smoothly as this
example suggests. Since we know the conditional, we are poised learn
the consequent merely by learning the antecedent (and by applying the
inference rule modus ponens: If \(P\) then \(Q\),
\(P\), therefore \(Q\)).
But the next section is devoted to some known
conditionals that are repudiated when we learn their antecedents.


### 6.2 Dogmatism paradox: A puzzle about losing knowledge



Saul Kripke's ruminations on the surprise test paradox led him
to a paradox about dogmatism. He lectured on both paradoxes at
Cambridge University to the Moral Sciences Club in 1972. (A descendent
of this lecture now appears as Kripke 2011.) Gilbert Harman transmitted
Kripke's new paradox as follows:



> 
> If I know that \(h\) is true, I know that any evidence
> against \(h\) is evidence against something that is
> true; I know that such evidence is misleading. But I should disregard
> evidence that I know is misleading. So, once I know that \(h\)
> is true, I am in a position to disregard any future
> evidence that seems to tell against \(h\). (1973, 148)
> 



Dogmatists accept this reasoning. For them, knowledge closes inquiry.
Any "evidence" that conflicts with what is known can be
dismissed as misleading evidence. Forewarned is forearmed.



This conservativeness crosses the line from confidence to
intransigence. To illustrate the excessive inflexibility, here is a
chain argument for the dogmatic conclusion that my reliable colleague
Doug has given me a misleading report (corrected from Sorensen
1988b):



(C\(\_1\))
My car is in the parking lot.
(C\(\_2\))
 If my car is in the parking lot and Doug provides evidence that
my car is not in the parking lot, then Doug's evidence is
misleading.
(C\(\_3\))
 If Doug reports he saw a car just like mine towed from the
parking lot, then his report is misleading evidence.
(C\(\_4\))
 Doug reports that a car just like mine was towed from the parking
lot.
(C\(\_5\))
 Doug's report is misleading evidence.


By hypothesis, I am justified in believing (C\(\_1)\). Premise
(C\(\_2)\) is a certainty because it is analytically true. The argument
from (C\(\_1)\) and (C\(\_2)\) to (C\(\_3)\) is valid. Therefore, my
degree of confidence in (C\(\_3)\) must equal my degree of confidence
in (C\(\_1)\). Since we are also assuming that I gain sufficient
justification for (C\(\_4)\), it seems to follow that I am justified in
believing (C\(\_5)\) by modus ponens. Similar arguments will lead me to
dismiss further evidence such as a phone call from the towing service
and my failure to see my car when I confidently stride over to the
parking lot.



Gilbert Harman diagnoses the paradox as follows:



> 
> The argument for paradox overlooks the way actually having evidence
> can make a difference. Since I now know [my car is in the parking
> lot], I now know that any evidence that appears to indicate something
> else is misleading. That does not warrant me in simply disregarding
> any further evidence, since getting that further evidence can change
> what I know. In particular, after I get such further evidence I may no
> longer know that it is misleading. For having the new evidence can
> make it true that I no longer know that new evidence is misleading.
> (1973, 149)
> 



In effect, Harman denies the hardiness of knowledge. The hardiness
principle states that one knows only if there is no evidence such that
if one knew about the evidence one would not be justified in believing
one's conclusion.



Harman's conclusion that new knowledge can undermine old
knowledge can be applied to the surprise test paradox: The students
lose knowledge of the test announcement even though they do not forget
the announcement or do anything else incompatible with their
credentials as ideal reasoners. A student on Thursday is better
informed about the outcomes of test days than he was on Sunday. He
knows the test was not on Monday and not on Wednesday. But he can only
predict that the test is on Friday if he continues to know the
announcement. The extra knowledge of the testless days undermines
knowledge of the announcement.



Most epistemologists accepted Harman's appeal to defeaters. Some
have tried to make it more precise with details about updating
indicative conditionals (Sorensen 1988b). This may vindicate and
generalize Harman's prediction that the future evidence will
change your mind about what is misleading evidence. Knowledge of such
conditionals is useless for a future *modus ponens.* The
dogmatist correctly says we know the conditional "If I know
that \(p\), then any evidence conflicting with \(p\) is
misleading evidence". Indeed, it is a tautology! But the
dogmatist fails to recognize this known tautology is useless
knowledge. Acquiring the misleading evidence will make me stop knowing
p. If an auditor foresees being presented with a biased list of facts,
he may utter the tautology to his assistant to convey another
proposition for which he has empirical support. That empirical
proposition need not be useless knowledge. When the predicted list is
presented, the forearmed auditor ignores the facts. But the basis is
not his *a priori* knowledge of the dogmatist's tautology.



Kripke notes that this solution will not stop the quick thinking
dogmatist who takes measures to prevent his acquisition of the
evidence that he now deems misleading (Kripke 2011, 43-44). A
second worry is that the dogmatist can still ignore weak evidence. If
I know a coin is fair then I know that if the first twenty tosses
comes out heads, then that is misleading evidence that the coin is not
fair. Such a run does not defeat my knowledge claim. (Substitute a
shorter run if you think it does defeat.) So Harman's solution does
not apply. Yet it is dogmatic to ignore this evidence. 



In addition to this problem of weak dogmatism, Rachel Fraser (2022)
adds a third problem of dogmatic bootstrapping. When Robert Millikan
and Harvey Fletcher measured elementary electric charge with tiny
charged droplets, they discounted some of the drops as misleadingly
wide of the plausible interval for the true value. Drops centrally located
within the interval were "beauties". Editing out the
outliers give Millikan a more precise measurement and a Nobel Prize in
1923. In 1978, the physicist Gerald Holton went through the notebooks
and was shocked by how much contrary data had gone unreported by
Millikan. Fraser thinks there is a vicious circularity in data purification. 



But the bootstrapping dogmatist will regard the circularity as virtuous. When the evidence is a mix of strong evidence and weak
counterevidence, the stronger body of evidence exposes the weaker body
as misleading evidence. Think of a jigsaw puzzle that has been polluted with stray pieces from another puzzle. When you manage to get a complete fit with a subset of the pieces, the remaining pieces are removed from view. Dimming the misleading evidence allows the leading evidence to shine more visibly. Therefore, we can indeed be more confident than
we were before dismissing the weak evidence. Millikan was being
responsible gatekeeper rather than a wishful thinker. Just as data
should control theory, theory should control data. The experimenter must strike a delicate balance between discounting too much contrary data and discounting too little. Proposed solutions to the dogmatism paradox have trouble sustaining this balance. 



I. J. Good (1967) demonstrated that gathering evidence maximizes
expected value *given that the cost of the evidence is
negligible*. Under this simplifying assumption, Good shows that
departures from the principle of total evidence is at least imprudent.
Given epistemic utilitarianism, this practical irrationality becomes
theoretical irrationality. Bob Beddor (2019) now adds the premise that
it is irrational to intend to do what one foresees to be irrational.
For instance, if you offered a million dollars to drink toxin tomorrow
that will make you ill for a day, you could profit from the offer
(Kavka 1983). But if the million would be earned immediately upon you
intending to drink the toxin, then you could not profit because you
know there would be no reason to follow through. By analogy, Beddor
concludes that it would be irrational to intend to avoid
counterevidence (Beddor 2019, 738). One is never entitled to discard
evidence, even after it has been foreseen as misleading evidence.



But if the cost of evidence is significant, the connection between
practical rationality and theoretical rationality *favors*
ignoring counterevidence. Judging that p embodies a resolution not to
inquire further into the question of whether p is true. Or so answers
the volitionist (Fraser 2022).


### 6.3 The Future of Epistemic Paradoxes



We cannot predict that any specific new epistemic paradox awaits
discovery. To see why, consider the prediction Jon Wynne-Tyson
attributes to Leonardo Da Vinci: "I have learned from an early
age to abjure the use of meat, and the time will come when men such as
I will look upon the murder of animals as they now look upon the
murder of men." (1985, 65) By predicting this progress, Leonardo
inadvertently reveals he *already believes* that the murder of
animals is the same as the murder of men. If you believe that a
proposition is true but will be first believed at a later time, then
you already believe it - and so are inconsistent. (The actual
truth is irrelevant.)



Specific regress can be anticipated. When I try to predict my first
acquisition of a specific truth, I pre-empt myself. When I try to
predict my first acquisition of a specific falsehood, there is no
pre-emption.



There would be no problem with predicting progress if Leonardo thinks
the moral progress lies in the moral preferability of the vegetarian
belief rather than the truth of the matter. One might admire
vegetarianism without accepting the correctness of vegetarianism. But
Leonardo is endorsing the correctness of the belief. This sentence
embodies a Moorean absurdity. It is like saying 'Leonardo took
twenty five years to complete *The Virgin on the Rocks* but I
will first believe so tomorrow'. (This absurdity will prompt
some to object that I have uncharitably interpreted Leonardo; he must
have intended to make an exception for himself and only be referring
to men of his kind.)



I cannot specifically anticipate the first acquisition of the true
belief that \(p\). For that prediction would show that
I already have the true belief that \(p\). The truth
cannot wait. The impatience of the truth imposes a limit on the
prediction of discoveries.