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Abstract
The Preface Paradox, first introduced by David Makinson (1961), presents a plausible scenario where an agent is evidentially certain of each of a set of propositions without being evidentially certain of the conjunction of the set of propositions. Given reasonable assumptions about the nature of evidential certainty, this appears to be a straightforward contradiction. We solve the paradox by appeal to stake size sensitivity, which is the claim that evidential probability is sensitive to stake size. The argument is that because the informational content in the conjunction is greater than the sum of the informational content of the conjuncts, the stake size in the conjunction is higher than the sum of the stake sizes in the conjuncts. We present a theory of evidential probability that identifies knowledge with value and allows for coherent stake sensitive beliefs. An agents beliefs are represented two dimensionally as a bid ask spread, which gives a bid price and an ask price for bets at each stake size. The bid ask spread gets wider when there is less valuable evidence relative to the stake size, and narrower when there is more valuable evidence according to a simple formula. The bid-ask spread can represent the uncertainty in the first order probabilistic judgement. According to the theory it can be coherent to be evidentially certain at low stakes, but less than certain at high stakes, and therefore there is no contradiction in the Preface. The theory not only solves the paradox, but also gives a good model of decisions under risk that overcomes many of the problems associated with classic expected utility theory.
Introduction
A historian writes a history book containing many facts, each of which, as a diligent researcher, she is fully justified in believing are true. The historian, being epistemically modest, acknowledges in the preface to the book that, in spite of her most careful and diligent research, she believes that there is at least one false statement in her book. Given the fallibility of human knowledge and the large number of facts in the book, the statement in the preface might also be considered justified. So here lies the paradox: it seems that she is fully justified in each of a set of propositions taken individually yet at the same time she is justified in believing the conjunction of the set of propositions to be false.
The Preface paradox dramatizes a problem at the very heart of evidential justification. Generally speaking we have many beliefs that we are fully justified in believing given our evidence. These beliefs are settled for us, and our evidence gives us the authority of testimony. Full justification is the greatest degree of justification and can be equated with evidential certainty. When a belief is fully justified, it requires no further justification and any further evidence for the belief is redundant. In this paper we will assume that fully justified belief is a necessary condition for knowledge and also for evidence for further conjectures and generalisations. In terms of evidential probability, when a belief is fully justified by the evidence, then the belief has an evidential probability of 1.
But many of our fully justified beliefs are justified on non deductive grounds. We believe things on the basis of past experience, inductive generalisation, testimony and other more or less reliable grounds. There is always a possibility that our fully justified beliefs could be mistaken. In other words, even our fully justified beliefs are fallible. If we take all of our fully justified beliefs as a whole, we can recognise that there is some probability that some of them are mistaken; yet if we take any of our fully justified beliefs individually, we again find that it is fully justified and evidentially certain. The uncertainty seems to be an emergent feature of the whole, which vanishes in the detail. The Preface paradox dramatizes this more general problem for evidential justification.
For example, for most practical and scientific purposes a visual observation is sufficient for evidential certainty. If a lab technician takes a thermometer reading at noon and reads that the temperature is 30 degrees, then he is thereby fully justified in believing that the thermometer read 30 degrees at noon (within an interval that takes into account the relevant level of precision). For the purposes of statistical inference, this reading can be used as a data point and will be accorded an evidential probability of 1. But it is not the case that such observations are infallible. If we were to ask the question whether it is possible that a person could take the wrong reading from a thermometer, then we must surely answer yes, and furthermore admit that there is likely to be a non-zero frequency of such misreadings. So when inquiring into the temperature, the observation counts as evidence and is accorded a probability 1; but when inquiring into the reliability of the entire class of observations of this kind, the accuracy of the observation is accorded a probability of less than 1. A change in inquiry, it would appear, can lead to a change in the evidential probability.
The Preface paradox could also shed some light on the problem of whether and to what degree a generalisation is confirmed by its instances. We could see the conjunction of all the propositions in the book as the generalisation that all statements in the book are true. Each statement in the book can then be seen as an instance of this generalisation that confirms the generalisation to some degree. We could then use Bayes theorem to attach a posterior probability to the generalisation on the evidence that all the statements in the book are true. Using this methodology the paradoxical nature of the Preface is clearly revealed. Call the generalisation that all the facts in the book are true book, and call the evidence formed from the fully justified individual propositions in the book facts. The posterior probability of the generalisation is then:
P(Book | Facts) = {P( Facts | Book) P(Book)} / P(Facts)
The problem is that the prior probability that book is true surely ought to be the same as the prior probability that all the facts are true. This is because the content of book is the same as the content of all the facts since book is simply defined as the generalisation that all the facts are true. And the conditional probability that all the facts are true given that book is true must surely be 1 for the same reason. Given all this then the posterior probability of the book given the facts comes out as 1 in all cases where the prior probability of the book is greater than zero. The paradox then is why the Preface statement: There is at least one false statement in the book can seem reasonable to us, when its negation: Every statement in the book is true, has a posterior probability of 1.
We can argue that the Preface statement is reasonable in betting terms. Suppose that the book was published and endorsed as a reference book by many great institutions. Supposed it contained the statement that the battle of Hastings was fought in 1066. It would be then possible to settle bets on the date of the battle of Hastings by looking it up in the book. But it would not be possible to settle bets on the accuracy of the book itself by checking each fact in the book by looking it up in the book. So while the book does not contain any extra facts that are not in the book; it does appear to fully confirm each claim in the book, without fully confirming a statement to the effect that every statement in the book is true. However, if the reliability of the book is in question, then it is no longer obvious that looking up the battle of Hastings in the book is sufficient evidence to settle bets on the date of battle of Hastings.
Contextualism
We might frame this in terms of a shift in the context of enquiry. When enquiring into the date of the battle of Hastings, the evidence that went into the book provides a full justification, but when enquiring into the epistemic standards of the book itself, then the same evidence no longer provides a full justification for ostensibly the same fact.
This may remind us of contextualist ways of meeting the sceptical challenge (Cohen 1988, De Rose 1995, Lewis 1996). The sceptical challenge can be illustrated through G.E. Moores famous proof of the external world. In an ordinary context of enquiry, it seems that Moore is fully justified in believing that he has two hands on the evidence produced by waving his hands in front of his face. But he cannot produce such fully justifying evidence that he is not a brain in a vat. In fact, he can produce no evidence whatsoever that he is not a brain in a vat. But a fully justified belief that he has two hands seems to provide a deductive proof that he is not a brain in a vat. The sceptical challenge, then, is to explain why it is that he knows he has two hands but that he doesnt know that he is not a brain in a vat, given that if he is a brain in a vat, then he does not have two hands. It seems to follow logically that he does not know that he has two hands after all.
Lewiss response to the sceptical challenge can be put like this: In order to know that p your evidence has to eliminate every possibility that ~p. But every has a contextually variable scope. In a philosophical sceptical context every can include the brain in the vat hypothesis; whereas in another, more ordinary context, every does not include the brain in a vat hypothesis. So G.E. Moore is right in that, ordinarily, he is fully justified in believing he has two hands; but he cannot use this as proof that he is not a brain in a vat, because as soon as the sceptical hypothesis is raised, he no longer knows that he has two hands.
We can see how this might work in the Preface paradox. The enquiry as to whether there are any false statements in the book creates a shift in context that makes it necessary to attend to the possibility that at least one statement is false. But in the case of the Preface paradox, this does not seem so much a solution, as a restatement of the problem. The problem is why are we fully justified at the particular level, but not fully justified at the general level?
Threshold Theory
One obvious solution to the Preface paradox already in the literature (Hawthorne & Bovens 1999; Sturgeon 2008) is to have a threshold theory of justified belief. The idea is that in order to be fully justified it is not necessary to be certain, where certainty is a justified degree of belief 1. Instead, the threshold for fully justified belief is some degree of belief below 1. In this case the probability of the conjunction can be very low while the probability of the conjuncts is very high. This solution is very appealing because our intuition tells us that the greater the number of facts in the book, the greater the probability that the Preface is true. For example, let us suppose all the facts in the book were probabilistically independent and have a probability of 0.9 which is, suppose further, the threshold for justified belief. If there were any more than 21 propositions in the book, then the probability that the Preface was true would exceed the threshold of 0.9. (Remember, the Preface is true if one or more of the propositions is false, so with 21 propositions the probability that the Preface is true is 1 0.921, which is greater than 0.9.)
This solution is neat and valid. However there are two problems with the threshold theory that I will mention; firstly any threshold will run into the lottery paradox. A consequence of the threshold theory is that there is a lottery such that one is fully justified in believing that each ticket in the lottery will lose before the draw. But it is certain that one ticket will win. Another way of putting this problem is that any threshold will lead to a failure of modus ponens. Any threshold will allow that one is fully justified in all the premises of a valid argument without being fully justified in the conclusion or worse, fully justified in believing the conclusion is false. Without going into too much detail, this seems like a bad result.
Secondly, there seems little motivation to set the threshold at any particular level. Any threshold seems arbitrary. Furthermore, once you take away full justification as a measure of certainty, then it becomes unclear how any probability judgement is to be interpreted. If fully justified statements arent certain, then it is hard to see how anything can be certain beyond logical tautologies. Once certainty is abandoned, then lesser degrees of belief lose their meaning.
However, rather than relying on these general arguments against threshold theories, we will simply allow that a threshold theory could account for many versions of the Preface paradox; but no threshold theory can solve a version of the preface where each fact in the book is evidentially certain, or in other words, where each fact has an evidential probability 1. In betting terms, the preface paradox can be stated thus: how can the Historians evidence justify settling bets on each statement in the book, but not justify settling bets on the statement that every proposition in the book is true? The threshold theory fails to solve this version of the paradox because the threshold theory works by denying that the evidence is sufficient to settle bets on the individual propositions.
Stake Size Sensitivity: A sketch of a solution.
The solution we present relies upon the claim that the degree to which a proposition is justified by a body of evidence is sensitive to what is at stake in believing the proposition. Importantly, whether or not a proposition is fully justified by a body of evidence can depend on what is at stake, so it is possible that the same evidence can fully justify a belief at low stakes but not fully justify the same belief at high stakes. We assume that fully justified belief is a necessary condition for knowledge, so it follows that it is possible that whether one knows that p on the basis of a fixed set of evidence can also depend on what is at stake.
Given stake size sensitivity, then the paradox is solved by showing that the book is much higher stakes than the sum of the stake sizes on the individual propositions, while the evidence remains constant. So it can be reasonable to be certain of the low stakes propositions, but less than certain of the high stakes conjunction.
The stake size sensitivity claim has many proponents, both contemporary and historical. However the expression of stake size sensitivity depends on the general theory of mind and evidence in which it is imbedded.
Economic Theory
In economic theory, the stake size sensitivity claim can be expressed in terms of discounting the expected utility of a prospect in proportion to the risk. A high risk prospect will have an expected value that is lower than its mean return derived from past performance. The magnitude of the discount is based on the variance in the past performance, and risk is defined in terms of variance, which is a measure of how far the outcome deviates from the mean. Variance, and its relatives, standard deviation and absolute deviation, are arguably the best measures of second order uncertainty (Hansson 2009). Higher stakes gambles have a higher variance given equivalent exposure. It is the common practice in the markets to discount due to risk in this way, and risk itself has become a tradable commodity. All the stake size sensitivity claim amounts to in this context is that discounting due to risk as measured by variance is rational.
Understood in this way, stake size sensitivity has a long history going back to Daniel Bernoulli (1738). Sadly, the phenomenon of the lower expected utility of risky prospects has been associated with the diminishing marginal utility of money, which is not obviously related to stake sensitivity in the epistemological sense and therefore could be mistaken for a competing explanation. The idea is that differences in utility are not linearly related to differences in money, specifically, differences between large sums of money have less impact on utility than the equivalent differences between small amounts of money, which has the effect that people tend to undervalue large gambles because the utility odds are worse than the money odds.
However there are several decisions under risk that cannot be explained by either a straight forward diminishing marginal utility curve, nor a more complex non linear expected utility curve like those introduced by Friedman and Savage (1948). These are principally the Ellsberg problem (1961) the Allais problem (1953) and the reflection effect and framing effects discovered by Kahneman and Tversky (1979). The latter posited decision weights, which function as degrees of belief in terms of calculating expected utility, but, according to Kahneman and Tversky, do not conform to the axioms of probability (1979 p.280). However, if the theory allows for subjects to have a bid-ask spread, so that their degree of belief varies depending on whether they are buying a prospect or selling it, then all these phenomena can be explained by the theory that bid-ask spreads vary with epistemic factors like variance and weight of evidence. In other words, the less evidence you have for a prospect, the greater the difference between your bid price and your ask price. Bid-ask spreads can easily be measured in the markets and are the subject of a number of studies (Mian 1995; Frank & Garcia 2009). They have been shown to co-vary with variance, transaction volume and market volume in a way that is consistent with stake size sensitivity.
Analytic Epistemology
In the setting of analytic epistemology, stake size sensitivity is the claim that whether a subject knows that p, or alternatively, is rational to fully believe that p, depends in part on the practical interests of the subject. As a consequence, it is possible that two subjects could have the same evidence for p yet differ in that the one in a low stakes situation knows that p while the other in a high stakes situation does not know that p; or alternatively the one in low stakes could be justified in believing that p whereas the other in high stakes is not justified in believing that p (Fantl and McGrath 2002; Hawthorne 2004; Stanley 2006; Weatherson 2005).
Naturalised epistemology
Another context in which stake size sensitivity finds expression is in naturalized epistemology. If epistemology is viewed as a natural science akin to psychology, then it becomes a significant question as to whether peoples and other animals beliefs actually are sensitive to stake size. If the answer to this question turns out to be yes, then this is enough to posit the hypothesis that beliefs ought to be stake size sensitive if they are to fulfil their proper biological function. The latter normative claim is a little harder to establish from the data alone, but Ruth Millikan at least explicitly states that knowledge is sensitive to stake size from a naturalist perspective.
"Unless the stakes are high, we do not require of the person who "knows that p" because he has been told by so-and-so that he be ready to place extremely high stakes on the truth of p.
(Millikan 1993 p. 253)
Stake Size Sensitivity by whatever name solves the paradox
Given stake size sensitivity, we can solve the preface paradox by showing that the proposition that every statement in the book is true is at higher stakes than the sum of the propositions taken individually given the same interests. Therefore, it is possible that the statements in the book are fully justified taken individually, but not fully justified taken as a whole. The reason for the change in stake size is clear enough. The doubtful conjunction has a great deal more informational content than the fully justified conjuncts. There are more ways the conjunction could be wrong, and so there is a greater investment in believing the conjunction.
A betting model of beliefs.
In order to be able to argue for this solution to the preface, we must sketch a sufficiently precise model of beliefs. We will use a straightforward betting model of action in which it is assumed that we act on the basis of our beliefs, and these actions resemble bets insofar as their success depends on our beliefs being true. The betting model therefore defines degree of belief as an expected value calculator, such that you calculate the expected value of an action by multiplying the stakes on the action by the degree of belief in success (Ramsey 1926 p.16).
In order to have a betting model of belief, we need to have a measure of value. The sense of value we intend is that which allows a subject to fulfil an essential function and thereby flourish. This is quite different from Ramseys scale of goods and bads, which is derived from the subjects preferences. Our measure of value assigns to a subject a state of wealth, and can measure the differences in states of wealth on a scale that preserves intervals. States of wealth are objective in that being in a certain state of wealth does not entail that you believe that you are in that state of wealth. States of wealth must always be positive, since we assume that whenever a subject reaches a zero state of wealth, then the essential function ceases. So if the subject is a business, then it folds, if it is a person then he dies, and if it is a species then it becomes extinct. We equate value with knowledge, so that it is deliberate increases of value through labour which typifies the appropriation of property and the accumulation of wealth. The most important states of wealth in terms of a bet on p are B, the state of wealth if p, and the bet is won; C, the state of wealth if ~p and the bet is lost; and A, which is the state of wealth of the subject given by the addition of the bet to his holdings. A is therefore the expected value of his holdings given by the degree of belief in p and will be no lower that C and no higher than B.
A subjects degree of belief can be expressed as a function of the price (A C) of a chance to win the stakes (B C) if p is true. The subjects degree of belief that p in these conditions is equal to the ratio price/stakes, or (A C)/(B C). As has long been established, price/stakes ratios conform to the laws of probability; which shouldnt be surprising considering that probability theory was developed in the context of pricing bets. We can see straight away that the price should never exceed the stakes, so that the price/stakes should be less than or equal to 1. Furthermore, assuming the stakes are positive, there is little sense in having a negative price. This means that price/stakes is always between 1 and 0 inclusive.
Stakes can be negative as well as positive. A bet with negative stakes is one where the value B is lower than the value C, so that winning the bet is in fact a loss. In this case the price will also be negative, since the value A will now be lower or equal to value C and higher or equal to B. Negative stakes bet prices can be thought of as selling prices, or ask prices. For example a bet at negative stakes 10 on ~p at price/stakes 1/10 would be one where you sold a promise to pay 10 if ~p for the ask price of 1. This would have identical consequences to a positive stakes bet where you buy a promise of 10 if p for a bid price of 9. Either way, you gain 1 if ~p and lose 9 if p. Every positive stakes bet can likewise be viewed as a negative stakes bet on the negation.
A feature of stake sensitivity is that it can be coherent to have a different ask price at negative stakes from your bid prices at positive stakes. Coherence demands that the ask prices are higher than the bid prices, and the difference in ask bid is called the bid-ask spread. The width of the bid-ask spread can then be used to measure uncertainty in the first order probability judgement. It follows from this characterisation of negative stakes, that a positive bet on p at price/stakes x/U is identical to a negative bet on ~p at price/stakes (x-U)/-U. From this follows a basic law of probability, in its stake sensitive form:
P(p)U = 1 P(~p)-U
The subscripts indicate the stake size. The unique stake size sensitive feature here is that this law only holds when the stakes have the same absolute value, one negative and the other positive
Ramsey assumed that degrees of belief are stake size invariant when proving the laws of probability, and this assumption allowed him to apply these laws across stake sizes; but we are allowing that degrees of belief are stake sensitive. This is a foundational difference so we have to go back to basics. Using bets as a measure of belief, we can do various mathematical operations on bets on different propositions and thereby establish logical relations between degrees of belief.
We can add bets together. A bet on p at a/U added to a bet on q at b/U where a and b are the prices and U is the stake size, results in a bet with a price a + b which pays U if p, U if q and 2U if both p and q. This composite bet has three possible end states rather than two and so is not reducible to a price/stakes bet on any proposition. Consequently the sum of these two bets does not represent or measure a degree of belief in any proposition.
We can also subtract bets by subtracting the values of the price and the stakes. So, using the example above, we could subtract a bet of c/U on (p & q) and derive the first law of probability, the law of disjunction. The resulting bet would have a price of a + b c and result in U if p OR q. This bet would be identical in consequence to a bet of (a + b c) / U on the disjunction p OR q. Thus the law of disjunction:
P(p OR q)U = P(p)U + P(q)U P(p & q)U.
It should be noted with great attention that the law of disjunction does not hold unless the stakes are the same. If the stakes on p and the stakes on q were different sizes, then the identity would not hold at all, and the bet on the left would have different consequences from the bet on the right, whatever the prices. The bet on the right does not necessarily represent a bet on a single proposition at all unless the stakes are kept constant.
Of course, there are constraints on degrees of belief across stake sizes. The one important constraint I shall call the minimum constraint.
MINIMUM. If S has P(p)U= x, then S has P(p)V e" Ux/V whenever V > U.
This is the rule that you can t prefer a bet at smaller stakes to a bet at larger stakes for the same price.
Proof: Suppose S had P(p)U = x and P(p)V = y such that y <Ux/V. The Dutch bookie would simply give S a bet on p at stakes U in exchange for a bet on p at stakes V for a small fee of xU - yV, and, if p was true, pay the subject U out of his winnings V, leaving the bookie with a profit of xU yV if ~p and V U if p.
The beauty of allowing stake size variation is that is it much more realistic and does not assume that paying a small price for a small bet somehow commits you to paying a much larger price for a much larger bet at the same odds. If this counterintuitive commitment to stake size invariance is dropped, then it is impossible to be Dutch Booked on bets at different stake sizes provided that one conforms to the minimum constraint. This allows for considerable variation of degrees of belief across stake sizes without incoherence. The important freedom it allows is to be able to have greater degrees of belief at lower stake sizes and to therefore be able to have a lower bid price from ones ask price, which allows for probabilistically coherent bid-ask spreads.
The second law of probability is the law of conjunction. For the law of conjunction we need to introduce a new operation on bets, which is a multiplication of bets. To multiply two bets together, you bet the entire stake of the first bet as the price on the second bet. For example, if you were to multiply a bet on 6 on the first role of the die by a bet on 6 on the second role of the die, both at price/stakes 1/6, then you would place 6 price on the second bet for stakes 36 on condition that the first bet won, otherwise nothing. Multiplying these two bets together would result in winning 36 iff two consecutive sixes were thrown, for a price of 1. We can see that this is identical to a bet at price/stakes 1/36 on a double six. Notice that the second bet is a conditional bet, which is a measure of conditional beliefs. A conditional bet p given q is a bet on p that is only valid if q. If not q, then the stakes are returned and there is no bet. So as we can see, a bet on p at price/stakes a/U multiplied by a bet on q | p at price/stakes U/X is identical in consequence to a bet on p & q at price/stakes a/X where X = U/P(q | p)X.
So the stake sensitive law of conjunction is that:
P(p&q)X = P(p)U P(q | p)X , where X = U/ P(q | p)X
The important feature, unique to stake size sensitive probability, which allows us to solve the Preface Paradox, is stake size escalation. The stakes on the conjunction are greater that the stakes on the unconditional conjunct by a factor of 1 over the conditional probability. On our theory of evidential probability, the stakes are equivalent to the value of the informational content of the proposition. Stake size escalation reflects the increased informational content in the conjunction over the conjuncts. The degree of stake size escalation is therefore dependent not on the probabilities of the propositions in the conjunction, but on the conditional probabilities between them. If knowledge that p makes it highly probable that q, then knowledge that p & q is not going to be a great deal more useful than knowledge that p on its own. Whereas, if p makes it highly unlikely that q, then knowledge of the conjunction p & q is going to be a lot more informative and valuable than the knowledge that p by itself. For example, knowing that someone is both pregnant and a woman is hardly more valuable than simply knowing that they are pregnant; whereas knowing that someone is pregnant and a man is much more valuable than simply knowing that they are a man.
The stake sensitive law of conjunction can be iterated many times with the stakes on the conjunction escalating exponentially with each additional conjunct. Since each conditional bet in the chain is at a higher stake size, the conditional probabilities will not remain constant, so that the ordering is important when deriving the conditional probabilities. Conditional probabilities are taken to be primitive and stake size sensitive in this theory, so the conditional probability on a proposition is derived directly from the stakes and the evidence according to the formula we will give below.
The stake sensitive law of conjunction allows that you can have a lot lower degrees of belief in the conjunction than in the product of the degrees of belief in the conjuncts without being probabilistically incoherent. Because the stake size is necessarily higher on the conditionals when the conjuncts are multiplied together, it is within the minimum constraint to accept odds on the individual conjuncts, but refuse the same odds when they are chained together conditionally in a multiplier bet. For example, it is perfectly invulnerable to a Dutch book to buy two disjoint bets on consecutive die rolls at price stakes 1/6 but to decline a bet on double six at 1/36. This follows because in order to multiply the bets together, you have to accept the second bet at stakes 36 rather than stakes 6, and there is nothing in nature that requires you to accept the same odds at this higher stake size.
The epistemological consequence of this is that the same evidence can rationally lead to a higher degree of belief in the conjuncts than their product in the conjunction, which is the result required to make the Preface coherent, because it allows the special case where one may coherently have a degree of belief 1 in the conjuncts, but less than 1 in the conjunction.
In order to show how this works in the epistemological context we now need a measure of the value of evidence. The value of evidence relative to a particular bet is a measure of how much the evidence changes the expectation on a bet. Because the value of evidence is measured in this way, the value of evidence can be both negative and positive, depending on whether it is evidence for or evidence against. This concept, the value of evidence, is closely related to Turings score (see Gillies 1990); I.J. Goods (1950) weight of evidence and Poppers (1974) severity of test. My chief inspiration for this measure, however, is Ramsey in his obscure note The value, or weight, of knowledge (Galavotti 1991) in which he presents a means for calculating how much it is worthwhile to find out a piece of knowledge. The problem with both Popper and Goods measures is that they both ignore stake size completely and frame their measures purely in terms of change in probability. This means that their measures cannot allow for stake size sensitivity.
Our measure of the value of evidence can be understood as a measure of the weighting of evidence for and against a proposition. We can use the metaphor of the scales of justice. Once we have a measure of the value of evidence we can form odds by taking the ratio of the value of evidence for to the value of evidence against. So for example if the value of evidence for p was 9 units and the value of evidence against p was 1 unit, then the odds would be 9 : 1. In this picture, stake size sensitivity would amount to increasing the relative weighting of evidence against as the stakes get higher. So in this example, the degree of belief on this evidence at zero stakes would be 0.9, but at higher stakes the value of evidence for would carry less weight relative to the evidence against and so the stake sensitive probability would be less that 0.9.
We propose to formalize this stake sensitivity by equating the value of knowledge to the value of the stake size. This is metaphysically profound and amounts to identifying knowledge with value. We can get a sense of the meaning of this identity if we consider that knowledge is a necessary condition for successful deliberate action, and deliberate action is a necessary condition for the creation of value. These are not empirical statements, but a priori statements that follow from the concepts of knowledge, success, action and value.
Another way to put this identity is to say that the value of a bet you know you will win is equal to the entire stake. If C is the state of your wealth if not p, and B the state of wealth if p, then evidence in favour of p is counted up from C, since the value of the bet can be no less than C. Knowledge that p is therefore when the value of the evidence increases the value of the bet from C all the way up to B and therefore is equal to B C, which is the stake size. This assumes that the value of the bet if one had no evidence at all that p would be equal to C, and therefore the degree of belief that one has in a proposition at positive stakes for which one has absolutely no evidence is 0. In case there is no evidence either way, then the degree of belief at positive stakes is 0 in both p and ~p, which in betting terms means your bid price is zero and your ask price is 1 times the stake, making your bid - ask spread the maximum of 1.
Lets call the value of evidence in favour of p kp. On our theory, the expectation of a bet is equal to C + kp, and the degree of belief justified by the evidence is equal to kp/U, unless kp>U, in which case the degree of belief is 1. The value of knowledge and evidence works both ways, so that when a bet is settled by, for example, a direct observation, the value of the evidence given by that observation is equal to U. This observation will then add U to the value of evidence in favour of propositions of type p in similar circumstances. In the case where the evidence in favour of p exceeds U, then kp/U > 1 and we have a surplus of evidence. In this case, the belief in p is justifiably certain and has a degree of belief 1, since degrees of belief can by necessity be no higher than 1. But the surplus remains significant, both as an indication of the strength of certainty at stakes U, and as measure of the level of certainty (Blamey 2008) which is the value kp, and can be defined as the maximum stake size at which p is justifiably certain given the evidence.
Of course, this is only half the story. There is also negative evidence, evidence in favour of ~p. Negative evidence works its way down from B towards C. So, given no evidence whatsoever in favour of ~p, the value of a bet on p is B, and the degree of belief in p is 1; whereas if we know that ~p, then the value of the bet on p is C and the degree of belief in p is 0. The expectation on a bet on p is therefore equal to B + k~p where k~p is negative, and the degree of belief in p will equal 1 k~p/U.
So far we have only talked about cases where the evidence is one sided. In the case where there is evidence both for and against p, we will take the total value of evidence K to equal the sum of the absolute values kp + k~p, which is the total value of evidence relevant to p. Bigger values for K will mean that the probability judgement has a lot of evidence behind it and wont be so sensitive to stake size.
The mixed evidence probability is given by the odds ratio C + kp : B + k ~p. In this odds ratio all values represent states of wealth or information and are therefore positive. The resulting mixed probability is then given as:
PK(p)B - C = (C + kp)/(C + B + K) given that k~p > 0, kp > 0
PK(~p)C - B = (B + k~p)/(C + B + K) given that k~p >0 , kp > 0
And to re cap, one sided probabilities are given as:
If k~p = 0, kp > 0, then PK(p)U = K/U, or 1 if K > U
If kp = 0, k~p > 0, then PK(p)U = (U - K) /(U), or 0 if K > U
A careful consideration of this formula will show that it preserves the stake size sensitive laws of probability and therefore always ensures that those with the same state of evidence cannot be Dutch booked, whilst allowing stake size sensitivity (Blamey 2011). It also allows that one can be certain with relatively little evidence at low stakes provided that there is no evidence against the proposition. Further more, the formula gives a bid ask spread that gets tighter as the value of evidence gets greater relative to the stake size, and gets wider the greater the stake size relative to the value of evidence. This last property shows that the formula gives a good representation of how uncertainty devalues prospects, and therefore gives a good account of the Ellsberg paradox and the Allais problem, as well as fairly accurately modelling bid-ask price structuring in markets.
On this model of belief and evidence, the updating rule of evidence is very simple. A new piece of evidence in favour of p is simply added to the existing knowledge in favour of p. If there is no negative evidence in relation to a specific bet this is simply to add the value of evidence directly to the price. So if we already had kp and discovered ke, then, relative to a bet on p at stakes U, the new degree of belief would simply be (kp + ke) / U, or 1 if kp + ke > U.
The solution
The paradox is solved then by a stake sensitive theory of evidential probability which measures evidence in terms of increase in expectation. The same increase in expectation at high stakes raises the probability to a lesser degree than the same increase in expectation at lower stakes. We identify knowledge with value by equating the stakes on a proposition with its value in terms of informational content. Given this, we have shown that conjoining propositions escalates the informational content exponentially by 1 over the conditional probability between the conjuncts. Therefore, evidence sufficient for certainty in each of the propositions may only be sufficient to raise the probability of the conjunction by a small amount.
To conclude we will go through a formal example which contains an element of fantasy. The historian has written a book of two parts, each part containing ten exciting previously undiscovered historical claims. She is satisfied that she has enough evidence to prove beyond reasonable doubt to an educated audience that each claim is true. But before she publishes she decides to take the ten claims from Part I to the (fictitious) Old Historians Club and bet on them prior to revealing her evidence. She is fully confident that the OHC will settle in her favour once she shows them her evidence. This kind of bet happens all the time at the OHC, and a committee has been set up to decide on fair odds given the state of knowledge at the club and to assessing the new evidence and deciding whether such bets are settled. The OHC committee assesses that each proposition in Part I has a probability of at stakes 100 and offers her 50/100 per bet. She takes all the bets, lays out 500 and expects to win back a profit of 500. Sure enough, when the committee sees her evidence, they decide that each bet is settled in her favour. This makes her evidence valued at 50 per proposition, totalling 500.
Flushed with success, she returns to the club with a new proposal. Her evidence for the ten propositions in Part II is qualitatively the same, and since she is now confident that the OHC considers that level of evidence sufficient to settle bets, she decides to increase her profits by betting sequentially on a multiplier bet. She bets 50 on the first proposition and bets 100 on the second proposition conditional on winning the first. She then bets on the third proposition conditional on the conjunction of the first and second and so on. Formally, the bet looks like this, (p1)100 (p2 | p1)200 (p3 | p1&p2)400 (p10 | p9 & p8 & p7 & p6 & p5 & p4 & p3 & p2 & p1)51200.
In effect she is committing the hubris of betting at huge stakes that everything she has written in Part II is true. We can perhaps see why it is that the informational content of the conjunction is far greater than the sum of its parts. It allows for stake escalating strategies, which are not possible if the facts are taken on an individual basis.
To her chagrin, the committee decide that her evidence does not support every claim in the book and she loses her 50 price. In the letter they tell her that they have a stake size variable theory of probability, and according to this theory, her evidence raised the probability of Part II from 1/1024 to 11/1024, making the expectation on her bet 500. The judgement was broken down as follows:
Value of evidence for Part II already available to the OHC . KPIIold = 50.
Informational content/stake size KPII = 51 200.
Evidential probability at stakes 51 200 PKPIIold(PII)51 200 = 1/1024
Value of new evidence KPIInew = 500
Updated value of evidence 550
Evidential probability at stakes 51 200 . PKPIInew(PII)51 200 = K/U = 11/1024
Summary: Although the evidence provided supports the claims in Part II taken on an individual basis at stakes of 100, we do not feel that the evidence supports Part II in its entirety at the much higher stakes of 51 240.
The views expressed in the letter from the OHC are coherent, in that they cannot be Dutch booked even though they assign different probabilities to the same proposition at different stake sizes.
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