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Epistemic Paradoxes
First published Wed Jun 21, 2006; substantive revision Fri Dec 30, 2011
Epistemic paradoxes are riddles that turn on the concept of knowledge (episteme is Greek for
knowledge). Typically, there are conflicting, well-credentialed answers to these questions (or
pseudo-questions). Thus the riddle immediately informs us of an inconsistency. In the long run,the riddle goads and guides us into correcting at least one deep error if not directly about
knowledge, then about its kindred concepts such as justification, rational belief, and evidence.
Such corrections are of interest to epistemologists. Historians can date the origin of
epistemology by the appearance of skeptics. As manifest in Plato's dialogues featuring
Socrates, epistemic paradoxes have been discussed for twenty five hundred years. Given their
hardiness, some of these riddles about knowledge will be discussed for the next twenty five
hundred years.
1. The Surprise Test Paradox
1.1 Self-defeating prophecies and pragmatic paradoxes
1.2 Predictive determinism
1.3 The Problem of Foreknowledge
2. Intellectual suicide
3. Lotteries and the Lottery Paradox
4. Preface Paradox
5. Anti-expertise
5.1 The Knower Paradox
5.2 The Knowability Paradox
5.3 Moore's problem
5.4 Blindspots
6. Dynamic Epistemic Paradoxes
6.1 Meno's Paradox of Inquiry: A puzzle about gaining knowledge
6.2 Dogmatism paradox: A puzzle about losing knowledge
6.3 The Future of Epistemic Paradoxes
Bibliography
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Other Internet Resources
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1. The Surprise Test Paradox
A teacher announces that there will be a surprise test next week. 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.
The riddle is: Can the teacher fulfill his announcement? We have an embarrassment of riches.
On the one hand, we have the student's elimination argument. 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. 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
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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 very surprising 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 riddle is instructive. 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. Problems worthy of attack prove their worth by
fighting back Paul Erdos.
The solution to a complex epistemic paradox relies on solutions (or partial solutions) to more
fundamental epistemic paradoxes. For instance, many approach the surprise test as a nested
sequence of puzzles: Inside the enigma of the surprise test is the preface paradox; inside the
preface paradox is Moore's paradox. In addition to this depth-wise connection, there are
lateral connections to other epistemic paradoxes such as the knower paradox and the problem
of foreknowledge.
There are also ties to issues that are not clearly paradoxes or to issues whose status as
paradoxes is at least contested. Some philosophers find only irony in pragmatic paradoxes,
only cognitive illusion in the lottery paradox, only an embarrassment in the knowability
paradox. Calling a problem a paradox tends to quarantine it from the rest of our inquiries.
Those who wish to dis-inhibit us will therefore deny that there is any paradox and admonish us
for not making use of all our evidence.
The surprise test paradox has yet more oblique connections to some paradoxes that are not
epistemic, such as the liar paradox and Pseudo-Scotus' paradoxes of validity. They will be
mentioned in passing, chiefly to set boundaries.
We can look forward to future philosophers drawing surprising 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 me turn to commentary to which we presently have
access.
1.1 Self-defeating prophecies and pragmatic paradoxes
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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 sentences such as I
remember nothing at all and I am not speaking now. Although these sentences 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 there is 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 intelligence 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.
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As Professor Statistics taught us, students can 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.
Unfortunately, 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 he 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, 7071)
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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 defeat
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 knowledgeimplies 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 insinuates that
theism precludes morality.
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. 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.
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The teacher has freewill. Therefore, predictions about what he will do are not true (prior to the
examination). Accordingly, 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.
W. V. 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
W. V. 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 reductio. Given the student's ignorance of the
announcement, Quine concludes that a test on any day would be unforeseen.
Common sense suggests that the students are informed by the announcement. The teacher is
assuming that the announcement will enlighten the students. He seems right to assume that
the announcement of this intention produces the same sort of knowledge as his other
declarations of intentions (about which topics will be selected for lecture, the grading scale,how long he will be absent for minor surgery, and so on).
There are extreme, philosophical 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 suggested by David Hume's problem of induction, then the student cannot know
that the teacher's announcement is true. But denying all knowledge of the future in order to
deny the student's knowledge is like using a cannon to kill a fly.
In later writings, Quine evinces general reservations about the concept of knowledge. One of
his pet objections is that know is vague. If knowledge entails absolute certainty, then too little
will count as known. Quine infers that we must equate knowledge with firmly held true belief.
Asking just how firm the belief must be is like 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.
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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 Rudolph 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 big man who stoops may stand
less high than another big man who is not as lengthy. Both men are clearly big. It is unclearthat The lengthier man is bigger. 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 worthaddressing 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, 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, epistemicparadoxes 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 fine. 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.
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Unlike the believer in No man is an immortal, the skeptic has trouble asserting There is no
knowledge. For assertion expresses the belief that one knows. That is why Sextus Empiricus
(Outlines of Pyrrhonism, I., 3, 226) condemns the assertion There is no knowledge as
dogmatic skepticism. Sextus often seems to prefer agnosticism about knowledge rather than
skepticism (considered as atheism about knowledge). Yet it also seems 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.
Agnostics overestimate how easy it is to identify what cannot be known. To know, one need
only find a single proof. To know that there is no way to know, one must prove the negative
generalization that there is no proof. After all, inability to imagine a proof is commonly due to
a failure of ingenuity rather than the non-existence of a proof. In addition to being a more
general proposition, a proof of unknowability requires epistemological premises about what
constitutes proof. Consequently, meta-proof is even more demanding than proof.
The agnostic might be tempted to avoid presumptuousness by converting to meta-
agnosticism. But this retreats in the wrong direction. Meta-meta-proof even more
demanding than meta-proof. Meta-meta-proof need both the epistemological premises about
what constitutes proof that meta-proof needs and, in addition, meta-meta-proof needs
epistemological premises about what constitutes meta-proof.
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 used to be accused of undermining a necessary condition for asserting
anything. However, the soul theorist's account of what is needed to make an assertion begs
the question against those who believe that a healthy brain is enough 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 supportedby a paradox that originated among philosophers of science rather than philosophers of
language.
3. Lotteries and the Lottery Paradox
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Lotteries pose a problem for the theory that we can assert whatever we think is true. Given
that there are a million tickets and only one winner, the probability of This ticket is a losing
ticket is very high. If our aim were merely to utter truths, we should be willing to assert the
proposition. Yet we are reluctant.
What is missing? Speakers will assert the proposition after seeing the result of the lottery
drawing or hearing about the winning ticket from a newscaster or remembering what the
winning ticket was. This suggests that asserters represent themselves as knowing. This in turn
suggests that there is a rule, or norm, governing the practice of making assertions that requires
us to assert only something we know. This knowledge norm explains why the hearer can
appropriately ask How do you know? (Williamson 2000, 249255). Perception, testimony,
and memory are reliable processes that furnish answers to this challenge.
Do these processes furnish certainty? 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 conciliatory mood, we are apt to relinquish our claim to know.
The skeptic generalizes from this surrender (Hawthorne 2004). 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.
Notice that the probability skeptic's mild suggestions about how we might be mistaken are not
the extraordinary possibilities invoked by Rene Descartes' skeptic. The Cartesian skeptic tries
to undermine vast swaths of knowledge with a single 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 pedestrian counter-explanations
that are easy to verify: 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 something goes wrong. For instance, you think you know that you have an
appointment to meet a prospective employer for lunch at noon. When he fails to show at the
expected time, you begin a forced march backwards through your premises: Is your watch
slow? Are you remembering the right restaurant? Could there be another restaurant in the city
with same name? Is he just detained? Could he have just forgotten? Could there have been a
miscommunication?
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Probabilistic skepticism dates back to Arcilaus 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 are
attracted to probabilism and 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 (despite the fortune to be
made by accurately calculating risk that should have been obvious to those in the business of
insuring against 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 Arcelius.
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 acknowledged 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 doxastic logic
(the logic of belief). Logicians thought that rational belief agglomerates: If you rationallybelieve p and rationally believe q then you rationally believe both p and q. Little pictures
should sum to a big picture. These logicians also required that rational belief be consistent. But
if rational belief can be based on an acceptance rule that only requires a high probability, there
will be rational belief in a contradiction! 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 pn. Symbolize I rationally believe as B. Belief in a contradiction
follows:
1.B~(p1 & p2 & & p100),
by the probabilistic acceptance rule.
2.Bp1 & Bp2 & & Bp100,
by the probabilistic acceptance rule.
3.B(p1 & p2 & & p100),
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from (2) and the principle that rational belief agglomerates.
4.B[(p1 & p2 & & p100) & ~(p1 & p2 & & p100)],
from (1) and (3) by the principle that rational belief agglomerates.
Since belief in an obvious contradiction is a paradigm example of irrationality, Kyburg poses a
dilemma: either reject agglomeration or reject probabilistic acceptance rules. Kyburg chooses
to reject 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 D. C. Makinson's (1965) preface paradox, an author rationally believes each of the assertionsin his book. But since the author regards himself as fallible, he rationally believes the
conjunction of all his assertions is false. If the agglomeration principle holds, (Bp & Bq) B(p
& q), the author must both rationally believe and disbelieve the conjunction of all the
assertions in his book!
The preface paradox does not rely on a probabilistic acceptance rule. The preface belief is
generated in a qualitative fashion. The author is merely reflecting on his similarity to other
authors who are fallible, his own past failing that he subsequently discovered, his imperfection
in fact checking, and so on.
At this point 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. Paradoxes force us to change our minds in a highly structured way. Forinstance, much epistemology responds to a 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.
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3.All justificatory chains have a finite length.
4.Some beliefs are justified.
Foundationalists reject (1). They take some propositions to be self-evident. 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 Peirce (193335, 5.250) rejected (3), an approach later refined by Peter Klein (2007) and most recently
defended at book-length by 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).
Very elegant! But if joint inconsistency is rationally tolerable, why do these philosophers
bother to offer solutions? Why is it not rational to believe each of (1)(4), in spite of their joint
inconsistency?
Kyburg might answer that there is a scale effect. Although the dull pressure of joint
inconsistency is tolerable when diffusely distributed over a large set of propositions, the pain
of contradiction becomes unbearable as the set gets smaller (Knight 2002). And indeed,
paradoxes are always represented as a small set of propositions.
If you know that your beliefs are jointly inconsistent, then you should reject R. M. Sainsbury's
definition of a paradox as an apparently unacceptable conclusion derived by apparentlyacceptable 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, 104110). If the conclusion of this argument counts
as a paradox, then by a similar argument for 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 contains a warning: 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. 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
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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 (Sorensen 2001, 156158). Consider a logic student who isrequired to pick one hundred truths from a mixed list of tautologies and contradictions.
Although the modest student believes each of his answers, A1, A2, , A100, 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: ~(A1 & A2 & & A100). 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 genuine 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 would not be the anarchist theme that ostensible authorities fail
to meet a minimal standard of legitimacy. The eliminativist unromantically diagnoses justifiedas a pathological term; like heterological, declarative sentences that apply the word fail to
express a proposition. 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.
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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 is expecting.
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. You cannot rationally choose an option that you believe to be inferior. So if you make a
choice, then you cannot really believe that you were doing so 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.
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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.
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 tests can be increased indefinitely.
Shockingly, they claim the number of alternatives can be reduced to zero! The announcementis 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 stinks of the liar paradox. Subsequent commentators sloppily switch the
negation sign in the formal presentations of the reasoning from K~p to ~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). Proving a proposition is sufficientfor 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
proving a proposition is sufficient for knowing it? This solution would be particularly
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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 looks more like the dogmatist he so frequently chides.
But the skeptic should not lose his nerve. A student taking a logic examination can be surprised
that he soundly deduced a theorem. The student did not know the conclusion because itseemed implausible and he was only guessing that a key inference rule was valid. His instructor
might have trouble getting the student to understand why his answer constitutes a valid proof
(rather than merely a desperate bid for partial credit).
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. 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 Gdel'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 Gdel sentence, then thissentence is true. If the system can prove its Gdel sentence, the system is inconsistent. So
either the system is incomplete or inconsistent. (See the entry on Kurt Gdel.)
Of course, this result concerns provability relative to a system. One system can prove another
system's Gdel sentence. Kurt Gdel (1983, 271) thought that mathematical intuition gave him
knowledge that arithmetic is consistent. Human knowledge is not restricted to what human
beings can prove.
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 Gdel's theorem, the computer will be either inconsistent or incomplete. However,
Lucas draws an invidious comparison: a human being with a full command of arithmetic can be
consistent (even if he is actually inconsistent due to inattention or wishful thinking).
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Other philosophers defend the parity between people and computers. They think we have our
own Gdel sentences (Lewis 1999, 166173). 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 Gdel 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 i t 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). If Pseudo-Scotus' argument is valid then, since its premise is true, its conclusion is
true which means it is invalid. If Pseudo-Scotus' argument is invalid, it is possible for the
premise to be true and conclusion false. But if an argument is invalid, it is necessarily invalid. 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 isconsistent, modest principles of epistemic logic imply that sentences of this form are
unknowable.
1. K(p & ~Kp) (Assumption)
2. Kp & K~Kp 1, Knowledge distributes over conjunction
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3. ~Kp 2, Knowledge implies truth (from the second conjunct)
4. Kp & ~Kp 2, 3 by conjunction elimination of the first conjunct and then conjunction
introduction
5. ~K(p & ~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 & ~Kp is not known. In other words, p & ~Kp is unknowable.
The cautious will 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 many idealists and virtually all logical positivists and other secular verificationists concede
that there are some actual unknown truths while also maintaining that all truths are knowable.
Astonishingly, they seem refuted by this pinch of epistemic logic.
Timothy Williamson doubts such astonishment 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)
The polemical intent of denying that the result is paradox is to remove an inhibition.
Williamson does not want us to quarantine the theorem with such suspicious characters as the
liar paradox.
Those who believe that the Church-Fitch result is a paradox can respond to Williamson with
examples of paradoxes that accord with common sense. For instance, common sense heartily
agrees with conclusion that something exists. But it is surprising that this can be provedwithout 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. Likewise,
many philosophers balk at the proof of unknowables because they feel that such a profound
result cannot be obtained from such limited means.
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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 by inspired 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 & B~p and p & ~Bp. (This scope
ambiguity is behind my favorite joke about Rene Descartes: Descartes is sitting in a bar, having
a drink. The bartender asks him if he would like another. I think not, he says, and
disappears.)
The common explanation of Moore's absurdity is that the speaker has managed to contradicthimself 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 in third person counterparts of (M). Anyone else can say about me, with
no paradox, Camels have three eye lids but Roy Sorensen does not believe it. (M) can also be
embedded unparadoxically in conditionals: If those membranes are eye lids, then camels have
three eye lids but I do not believe it. The past tense is fine: Camels have three eye lids but I
did not believe it. The future tense, Camels have three eye lids 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 weaker element of the conjunction is the announcement, the student will not 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.
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Binkley fortifies this reasoning with doxastic logic. The principle of 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 anymore 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 Canada extends to the North Pole 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. A principle thattells me to 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 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.
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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 foreknowledge 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
A blindspot is a consistent but inaccessible proposition. Blindspots are relative to the means of
reaching the proposition, the person making the attempt, and time he makes the attempt.
Although I cannot know the blindspot There is intelligent extra-terrestrial life but no one
knows it, I can suspect it. Although I cannot rationally believe Polar bears have black skin but I
do not believe it you can. This means there can be disagreement between ideal reasoners
(even under strong idealizations such as Binkley's). 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 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
doubt that Poncins is telling the truth. But no reader ought to believe Someone can be
content to live in an ice house and I doubt it.
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 blindspots is that there can be disagreement among ideal reasoners because
they differ in their blindspots.
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This is relevant to the surprise test paradox. The students are the surprisees. Since the date of
the surprise test is 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 My new locker combination is 183614 but on Friday I will not believe itshould 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 untilthe 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
(iii) The test is on Friday and the student does not know it before Friday.
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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 evidence of being unreliable.
Indeed, the student can foresee that their change of mind opens a new opportunity for
surprise.
Another consequence is that 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 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.
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.
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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 whenwe reach the Meno, one his interlocuters 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
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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. Knowledge
of the conditional is conditional knowledge (that is, conditional upon learning the antecedent
and 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 reliablecolleague Doug has given me a misleading report (corrected from Sorensen 1988b):
(C1) My car is in the parking lot.
(C2) 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.
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(C3) If Doug reports he saw a car just like mine towed from the parking lot, then his report is
misleading evidence.
(C4) Doug reports that a car just like mine was towed from the parking lot.
(C5) Doug's report is misleading evidence.
By hypothesis, I am justified in believing (C1). Premise (C2) is a certainty because it is
analytically true. The argument from (C1) and (C2) to (C3) is valid. Therefore, my degree of
confidence in (C3) must equal my degree of confidence in (C1). Since we are also assuming that
I gain sufficient justification for (C4), it seems to follow that I am justified in believing (C5) 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. New knowledge cannot undermine old knowledge.
Harman disagrees.
Harman's belief that new knowledge can undermine old knowledge may be relevant to the
surprise test paradox. Perhaps 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
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announcement. Perhaps the extra knowledge of the testless days undermines knowledge of
the announcement.
6.3 The Future of Epistemic Paradoxes
We cannot coherently 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 is showing that he already believes that the murder of
animals is the same as the murder of men.
There would be no problem 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.
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