Jonathan L

Jonathan L. Kvanvig

Serial Number: P035

THE EPISTEMIC PARADOXES

The four primary epistemic paradoxes are the lottery, preface, knowability, and surprise examination paradoxes. The lottery paradox begins by imagining a fair lottery with a thousand tickets in it. Each ticket is so unlikely to win that we are justified in believing that it will lose. So we can infer that no ticket will win. Yet we know that some ticket will win.

In the preface paradox, authors are justified in believing everything in their books. Some preface their book by claiming that, given human frailty, they are sure that errors remain, errors for which they take complete responsibility. But then they justifiably believe both that everything in the book is true, and that something in it is false.

The knowability paradox results from accepting that some truths are not known, and that any truth is knowable. Since the first claim is a truth, it must be knowable. From these claims it follows that it is possible that there is some particular truth that is known to be true and known not to be true.

The final paradox concerns an announcement of a surprise quiz next week. A Friday test will not be a surprise, yet, if the test cannot be on Friday, it cannot be on Thursday either. For if it has not been given by Wednesday night, and it cannot be a surprise on Friday, it will not be a surprise on Thursday. Similar reasoning rules out all other days of the week as well; hence, no surprise quiz can occur next week. On Wednesday, the teacher gives a quiz, and the students are taken completely by surprise.

1 Lottery and Preface Paradoxes

2 Knowability Paradox

3 The Surprise Examination Paradox

1. Lottery and Preface Paradoxes

The lottery paradox, first developed in Kyburg (1961), and the preface paradox, originally formulated in Makinson (1965), have a similar structure, though some (e.g., Pollock (1986)) hold that they have different solutions. Each hinges on a conflict between a rule of acceptance, a condition on the transfer of warrant, and an axiom about warrant.

Rule of Acceptance: There is some threshold short of certainty where acceptance of a claim is warranted or justified.

Transfer Condition: A set of warranted claims is closed under deduction. That is, a set of warranted claims includes all the deductive consequences of that set.

Warrant Axiom: It is not possible to be warranted in believing p and, for the same time and the same individual, be warranted in believing ~p.

One standard approach is to find some fault with the Transfer Condition. Denials of the transfer condition sometimes result from explicit consideration of these paradoxes, and other times from more general considerations within the theory of knowledge. For example, Kyburg addresses the lottery paradox explicitly and holds that it relies on the faulty conjunction principle, the principle according to which a person is warranted in believing a conjunction p&q if that person is warranted in believing p and warranted in believing q. In this way, outright contradictions are thought to be avoidable in the set of warranted beliefs for a person at a time, while still allowing that the set can be inconsistent, i.e., be such as to deductively imply a contradiction. Other epistemologists (e.g., Nozick (1981)) develop general theories of knowledge to answer the Gettier problem (SEE Gettier Problem; Deductive Closure Principle) in which the transfer condition is abandoned.

To accept such a solution requires abandoning a coherence theory of justification, for a minimal condition for coherence is logical consistency. If this approach is the best available for solving these paradoxes, the paradoxes will have taught us that coherence theories of justification must be abandoned.

A point of importance concerning those who abandon the transfer condition explicitly to solve the paradoxes is that abandoning the conjunction principle alone will not do the trick. For the paradoxes can be generated without employing any conjunction principle. In the lottery paradox, a person can infer from individual premises that each particular ticket will lose, together with the knowledge of exactly how many tickets there are in the lottery, that no ticket will win. This claim can be deduced apart from the conjunction principle, and yet it contradicts the knowledge that some ticket will win, thereby violating the warrant axiom according to which contradictory warranted claims are impossible.

Moreover, a fully satisfying solution to these paradoxes that denies the deductive closure condition would need to offer an alternative principle in its place. For it is obvious that some of our beliefs are warranted precisely because we deduce them from other things we know.

A different approach is to question the rule of acceptance. There are two approaches that could be taken here. The first is simply to succumb to the paradox, granting that no acceptance is ever warranted when a claim is less than certain. This solution, however, seems unduly restrictive. For the small chance of illusion, hallucination, and other types of misperception indicate that our ordinary perceptual beliefs are less than certain.

Alternatively, one may be suspicious of the idea of acceptance or belief, on Bayesian grounds. Bayesian epistemology arises within the context of an application of a subjective interpretation of probability theory to epistemological issues (SEE Probability Theory and Epistemology), and begins by noting that we believe some things more strongly than others (we believe that 2+2=4, and that there is no life on Mars, but we are much more confident of the first than the second). So, perhaps, the concept of belief in ordinary language is a coarse-grained way of talking about certain mental states that strictly speaking only come in degrees. Once such a viewpoint is accepted, two lines emerge as to how to treat the ordinary concept of belief. First, one might hold that there is no such thing as belief simpliciter, but rather there are only degrees of belief, and hence that the Acceptance Condition is ill-formed since it is formulated using the concept of belief simpliciter. Another alternative is to think that there is some way of understanding the ordinary concept of belief in terms of degrees of belief, using the lottery and preface paradoxes to show that such an definition cannot proceed in terms of some threshold of degree of belief above which the ordinary concept of belief applies (Kaplan 1981). The task for this kind of Bayesian view is to give an account of the ordinary concept of belief which, in conjunction with the Transfer Condition, does not violate the Warrant Axiom. In either case, the Bayesian approach gives some reason to be suspicious of the Acceptance Condition, and thereby suggests that the paradoxes might best be dealt with by abandoning or altering that condition.

2. Knowability Paradox

The knowability paradox derives from work by Fitch (1963) on value concepts. The paradox depends on two claims:

(1) Everything is knowable, i.e., for all propositions p, if p is true, then it is possible that it is known (by someone at some time) that p is true.

(2) Some things are not known, i.e., there exists some proposition p, such that p is true, and it is not known that p is true.

We then substitute an instance of (2)--a claim of the form "q is true and it is not known that q is true"--in as the value for p in the first claim. Since we are committed to the truth of this instance of (2), the antecedent of (1) is true when this instance of (2) is the value for p in (1). We thus deduce that it is possible that a certain conjunction is known, namely that q is true and that it is not known that q is true. Since to know a conjunction is to know that each of its conjuncts is true, we can infer that it is possible that it is known that q is true and that it is known that it is not known that q is true. However, since knowledge implies truth, the latter piece of knowledge implies that it is not known that q is true. So we thereby deduce that it is possible that it is both known and not known that q is true.

The crucial features of this argument are the distribution principle, i.e., that knowledge of a conjunction implies knowledge of each of the conjuncts, and the factive character of knowledge, i.e., that knowing p implies p. The remainder of the argument relies on no principles of inference beyond those of first-order quantification theory with modal operators. So the paradox has much broader implications than merely those concerning knowledge. Fitch, for example, used the paradox to argue that verificationism, the thesis that all truths are knowable, entails a very silly form of verificationism according to which all truths are known. The broader implications of the paradox, almost completely unaddressed in the literature, concern versions of anti-realism according to which truth is an epistemic notion, (e.g., truth is empirical confirmation in the long run, or verification by an ideal scientific community, or what is warrantedly assertible, etc.). Such anti-realisms fall within the scope of the paradox, for these epistemic notions distribute over conjunction every bit as much as knowledge does, and their theory of truth makes these epistemic notions factives. So, the paradox can be formulated in terms of the claims that some truths are not justified or verified in this special anti-realist way, and that all truths are epistemic in the favored anti-realist way.

The paradox could be resolved in a way that retains both claims (1) and (2) above if knowledge fails to distribute over conjunction. The paradox requires moving from knowledge of a conjunction to knowledge of each of the conjuncts, i.e., from K(A&B) to K(A) & K(B). Some theories of knowledge deny such a principle (e.g., Nozick (1981)), but these denials are not supported by independent argument but are rather only implications of the general theory. Because of this feature, the denial can easily appear to be a defect of the theory rather than a virtue. In any case, denying the principle requires a further explanation of the difference between cases where such distribution is acceptable and those where it is not, for it is clear that one can come to know, e.g., that the George owns a car by learning that George owns a car and a truck.

3. The Surprise Examination Paradox

The Surprise Examination Paradox first appeared in print in D.J. O'Connor (1948). The paradox originated earlier when a Swedish mathematician, Lennart Ekbom, discussed at Ostermalms College a difficulty he had noticed with an announcement by the Swedish Broadcasting Company during World War II. The announcement said that a civil defense exercise was to be held during a particular week. In order to ensure preparedness, no one was to know in advance which particular day of the week the exercise would be conducted. Ekbom noticed that the unexpectedness of the exercise was problematic, which forms the core of the Surprise Examination Paradox. This paradox appears in many guises and under many names, including among others the Prediction and Hangman paradoxes. All have essentially the same form as that represented by the surprise examination, on which we focus here.

Skeptical approaches to this paradox deny that what the announcement of a surprise examination warrants any beliefs about the future. Thus, Quine (1953) maintains that even a surprise examination announced only one day in advance would not be paradoxical, for such an examination would be a surprise as long as one could not know in advance that the examination would be given tomorrow.

Non-skeptical approaches to the paradox grant that one can know on the basis of the announcement that an examination will be offered. For such approaches, an examination given the last day of the week would thereby not be a surprise, contrary to the Quinean approach above. Such approaches, in order to resolve the paradox, must find something wrong with the announcement itself (Shaw (1958); Kaplan and Montague (1960)) or with the inferences by the students (Meltzer (1964)), for the paradox ends by having an examination occur that surprises the students. A standard approach following the first line is to show that the teacher's statement is self-referentially incoherent, as in the Liar paradox where "This statement is false" is incoherent. For example, the announcement might mean or imply "There will be a surprise examination next week, and there being no surprise examination is deducible from this entire statement."

Such approaches have been rejected for violating a coherence requirement on the announcement (Bosch (1972)), according to which it is plainly obvious that the announcement is coherent and hence that any adequate resolution of the paradox must locate an error in the students' reasoning.

More recent discussion has focused on different varieties of the paradox, and whether formal approaches to the paradox suffice to resolve it. In spite of being originally castigated as "rather frivolous" (O'Connor (1948)), this paradox has attracted by far the most attention of the epistemic paradoxes (near one hundred articles have been written on it, compared to only a handful for the others). Moreover, hardly anything uncontroversial is to be found regarding the paradox, including how to interpret the announcement, how to resolve the paradox, whether there is one or several paradoxes involved, whether the paradox is simply a variant of other well-known paradoxes, and what conditions a proper resolution of the paradox must satisfy. Some would hold that it is the deepest of the paradoxes; in any case, the attention it has received shows that it is far from frivolous.

Bibliography

Binkley, R. (1968) `The Surprise Examination in Modal Logic', Journal of Philosophy 65: 127-36. (Develops a skeptical approach to the paradox interpreted as involving modal operators.)

*Bosch, J. (1972) `The Examination Paradox and Formal Prediction', Logique et Analyse 15: 505-25. (Argues that the paradox must find a flaw in the students' reasoning rather than in the announcement itself.)

Edgington, D. (1985) `The paradox of Knowability', Mind 95: 557-68. (Attempts to develop a variation of verificationism immune to the knowability paradox.)

*Fitch, F. (1963) `A Logical Analysis of Some Value Concepts' The Journal of Symbolic Logic 28 (1963), 135-142. (The original source for the central argument of the knowability paradox.)

*Kaplan, D. and Montague, R. (1960) `A Paradox Regained', Notre Dame Journal of Formal Logic 1: 79-90. (Searches for a truly paradox reading of the announcement in the Surprise Examination Paradox, and finds one that makes the announcement analogous to a Liar statement.)

*Kaplan, M. (1981) `A Bayesian Theory of Acceptance', The Journal of Philosophy 78: 305-30. (Develops an account of acceptance in terms of degrees of belief that avoids the lottery paradox.)

*Kyburg, H. (1961) `Conjunctivitis', Probability and the Logic of Rational Belief, Middletown, Conn.: Wesleyan University Press. (The original source of the lottery paradox.)

*Makinson, D.C. (1965) `The Paradox of the Preface', Analysis 25: 205-7. (The original source of the preface paradox.)

*Meltzer, B. (1964) `The Third Possibility', Mind 73: 430-433. (Argues that the students' reasoning is at fault for assuming the Law of Excluded Middle.)

*Nozick, R. (1981) Philosophical Explanations, Cambridge: Harvard University Press. (A view on which knowledge is not closed under deduction, and knowledge does not distribute over conjunction.)

*O'Connor, D.J. (1948) `Pragmatic Paradoxes', Mind 57: 358-9. (The original published version of the Surprise Examination Paradox.)

Pollock, J. (1986) `The Paradox of the Preface', Philosophy of Science 53: 246-258. (Presents a solution to the preface paradox that arises out of artificial intelligence work on defeasible reasoning.)

*Quine, W.V.O. (1953) `On a So-Called Paradox', Mind 62: 65-7. (Argues that the announcement could be coherent even if given only one day in advance, thereby proposing a skeptical solution to the Surprise Examination Paradox.)

*Shaw, R. (1958) `The Paradox of the Unexpected Examination', Mind 67: 382-4. (Claims that the solution to the paradox is found in distinguishing two readings of the announcement, only one of which can be used to deduce that no such examination can be given and which is self-referentially incoherent.)

Williamson, T. (1993) `Verificationism and Non-Distributive Knowledge', Australasian Journal of Philosophy 71: 78-86. (Argues that knowledge must distribute over conjunction.)