self-doubts and dutch strategies

This article was downloaded by: [Eindhoven Technical University]On: 20 October 2014, At: 01:08Publisher: RoutledgeInforma Ltd Registered in England and Wales Registered Number:1072954 Registered office: Mortimer House, 37-41 Mortimer Street,London W1T 3JH, UK

Australasian Journal ofPhilosophyPublication details, including instructions forauthors and subscription information:http://www.tandfonline.com/loi/rajp20

Self-doubts and dutchstrategiesJordan Howard Sobel aa University of TorontoPublished online: 02 Jun 2006.

To cite this article: Jordan Howard Sobel (1987) Self-doubts and dutch strategies,Australasian Journal of Philosophy, 65:1, 56-81, DOI: 10.1080/00048408712342771

To link to this article: http://dx.doi.org/10.1080/00048408712342771

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of allthe information (the “Content”) contained in the publications on ourplatform. However, Taylor & Francis, our agents, and our licensorsmake no representations or warranties whatsoever as to the accuracy,completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views ofthe authors, and are not the views of or endorsed by Taylor & Francis.The accuracy of the Content should not be relied upon and should beindependently verified with primary sources of information. Taylor andFrancis shall not be liable for any losses, actions, claims, proceedings,demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, inrelation to or arising out of the use of the Content.

This article may be used for research, teaching, and private studypurposes. Any substantial or systematic reproduction, redistribution,reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access

http://www.tandfonline.com/loi/rajp20

http://www.tandfonline.com/action/showCitFormats?doi=10.1080/00048408712342771

http://dx.doi.org/10.1080/00048408712342771

and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Dow

nloa

ded

by [

Ein

dhov

en T

echn

ical

Uni

vers

ity]

at 0

1:08

20

Oct

ober

201

4

http://www.tandfonline.com/page/terms-and-conditions

http://www.tandfonline.com/page/terms-and-conditions

Australasian Journal of Philosophy Vol. 65, No. 1; March 1987

SELF-DOUBTS A N D D U T C H STRATEGIES

Jordan Howard Sobel

Bas van Fraassen maintains t that anyone who thinks that he may come to be mistakenly certain about something, indeed anyone who entertains doubts concerning the appropriateness of any of his present or possible future credences, can, if he is prepared to make any bet that is fair according to his credences, be drawn into bets on which he will suffer a net loss no matter what. This can seem strange. Most of us have d o u b t s - q u i t e reasonable doubts--concerning credences of other persons, of precisely the sorts that would make us potential victims of clever bookies were we to entertain them about ourselves. In order to be safe f rom clever bookies it seems that any person who is always prepared to 'put his money where his mouth is' must have what he should realise are unreasonably high opinions of his own opinions and think far better of his own credences than he does of those of other persons.

Such high o p i n i o n s - s u c h autobiographical o p i n i o n s - a r e , according to van Fraassen quite indefensible, (p. 255), but then, he maintains, the words that would express such opinions have, as their ordinary pr imary force, the expression of 'avowals ' not opinions, and the 'high avowals ' they would express can be defended in a way.

Avowal, qua avowal, has its own c o n s t r a i n t s . . , if I express my opinion, I invite the world to rely on my i n t e g r i t y . . , my integrity, qua judging agent, requires that, if I am presently asked to express my opinion about whether A will come true, on the supposition that I will think it likely tomorrow morning, I must stand by my own cognitive engagement as much as I must stand by my own commitment of any s o r t . . . I can no more say that I regard A unlikely on the supposition that tomorrow morning I shall express my high expectation of A, than I can today say that on the supposition that tomorrow I shall promise to bring it about that A. (p. 255)

I shall not dispute van Fraassen's pragmatic defence of certain 'high personal avowals' , though I have reservations concerning it. My interest is in related high personal opinions, and what can be said for them even granted that they are generally indefensible. It is important , I think, that these opinions are aspects of a rational ideal. I will, accordingly, at tempt to bring out what is wrong with a person who thinks less than the best o f his credences-- what is wrong with such a person, with the person himself, even if there is nothing wrong with his opinions about himself, or with his expressions of t h e m -

1 van Fraassen (1984), pp. 235-256. Page references, unless otherwise indicated, are to tiffs essay.

56

Dow

nloa

ded

by [

Ein

dhov

en T

echn

ical

Uni

vers

ity]

at 0

1:08

20

Oct

ober

201

4

Jordan Howard Sobel 57

even if, that is, these opinions are all quite reasonable evidentially, and even if his expressions of them are all entirely correct and consistent with his conversational ends. I will attempt to bring out what such a person is missing, what he has to worry about, potential victimisations by clever bookies quite aside. These undertakings occupy Section IV below. Section I contains explanations of Dutch Books and Strategies, and of certain simplifying assumptions. In Section II the condition of suspecting that one may come to be certain but mistaken about some proposit ion is related to another, in ways more general, condition consisting of doubt concerning one's present and possible future credences. Section I I I has in it a p roof that at least semi- Dutch Strategies lie against all persons who have such doubts. And in Section V I indicate how, notwithstanding the validity of an ideal whose attainment would provide security against such doubts and such strategies-- that is, notwithstanding the validity of the ideal detailed in Section I V - the complete realisation of such intellectual perfection is almost certainly not possible for ordinary persons, and is fur thermore not to be pursued at just any cost.

I. Dutch Books and Dutch Strategies Almost any person's degrees of confidence in propositions can be represented by numbers f rom 0 to 1 assigned so that a proposition gets 0 if the person is not at all inclined to believe it, 1 if he is certain of it, and a proposit ion gets a number greater than that assigned to another proposit ion if and only if the person is more inclined to believe it than to believe this other proposition. Let such an assignment of numbers to propositions be a credence function: the credence function for a person b at time t shall be Cr b.

Little is required for the existence of a credence function. A person has one if, for any proposition of which he is certain, he is more inclined to believe it than to believe any proposit ion of which he is not certain, and exactly as inclined to believe it as to believe any other proposition of which he is certain. Believers have credence functions, usually many credence functions. We concentrate on highly opinionated persons who at all times have unique credence functions, and we concentrate further on persons who are at all times certain that they are highly opinionated. These concentrations simplify without compromising our study if, as I think, being highly opinionated and certain that one is are parts of an ideal for intellects. (The ideal I endorse is elaborated in Section IV below.)

A simple bet (a simple money-bet) on a proposition shall be an arrangement in which a person wins an amount if this proposit ion is true, and loses some amount if this proposit ion is false. In a simple bet against a proposition, wins and losses are reversed. To place a simple bet is to arrange to be the person in the bet. Purchasing, at a price x, a ticket that, for some y greater than x, pays y i f p is true, would be a way of placing a simple bet on p with potential win ( y - x ) and potential loss x. To cover a simple bet on [against] a proposition is to place the corresponding simple bet against [on] that proposition.

A simple bet shall be fair for a person if and only if his expected value

Dow

nloa

ded

by [

Ein

dhov

en T

echn

ical

Uni

vers

ity]

at 0

1:08

20

Oct

ober

201

4

58 Self-Doubts and Dutch Strategies

for it is 0. A simple bet on a proposition p, with potential wins and losses W and L, is fair for a person b at time t if and only if

Crb(p)W + Crbt(-p)L = 0.

The simple bet on p

p Wins and losses

T W F L

wherein W, the potential win, is non-negative, and L, the potential loss, is non-positive, is fair if and only if the corresponding bet against p

p Wins and losses

T - W F - L

is fair. A person should be indifferent between placing and covering bets that are fair for him, if he is interested in only their potential wins and losses. Given the satisfaction of a certain familiar condition, we have a useful rule for fair bets: If credences for p and - p of a person b at time t satisfy the complement condition

c ? t ( - p ) = 1 - CrY(p),

then, for any x, the bet

p Wins and losses

T x[1 - CrY(p)] F x[ - CrY(p)]

is fair at t for b. It is clear that given the stated complement condition the expected value for b of this bet is 0.

A conditional bet shall be a win/loss arrangement like a simple bet, except that the arrangement is 'on' only given the bet's condition. For example, a person, instead of simple-betting on a horse in a race, might bet conditionally on it with the understanding that the betting arrangement is in force if and only if the track for the race is dry. Fairness for conditional bets is like fairness for simple ones except that 'conditional credences' rather than simple ones are used in computations of expected values. Conditional credences measure what I call 'possible evidential bearings'. I assume that a contingent proposition that an agent is not certain of can have possible evidential bearings for him on contingent propositions, and that a proposition can have such bearings whether or not he is at all inclined to believe it. We shall confine ourselves to agents for whom possible evidential bearings are measured by unique 'conditional credence functions' with values from 0 to 1.

Evidential bearings, potential evidential bearings, are to be measured by signed differences between conditional and unconditional credences, the evidential bearing o f p on q being measured by the difference, Cr(q /p ) - Cr(q). The possible evidential bearing o f p on q is thus something like what would be the limit of the potential bearing of p on q as Cr(q) approached

Dow

nloa

ded

by [

Ein

dhov

en T

echn

ical

Uni

vers

ity]

at 0

1:08

20

Oct

ober

201

4


0, supposing that Cr(q/p) remained constant: the possible evidential bearing of p on q is, in a sense, 'the maximum possible positive potential evidential bearing of p on q'. I think that only contingent propositions have possible evidential bearings, that they have them only on contingent propositions, and that possible evidential bearings are not limited to things a person is somewhat inclined to believe: persons, when they learn things they were sure were not true, need not be at a complete loss regarding what to make of what they learn. In contrast, possible evidential bearings do seem to be restricted to propositions a person is not certain of. Things a person is certain of have, if anything, actual evidential bearings for him. But I have no theory of such bearings and note only that they could not very well be measured by conditional probabilities. It seems that 'actual evidential bearings', however they are to be understood in exact detail, must be such that distinct certainties could differ in their actual evidential bearings on some single proposition, and this seems a problem for the kind of theory of evidence I favour, given that I suppose that for 'rational' credences, if Cr(p) and Cr(q) are both 1, then Cr(r/p) = Cr(r/q) .

A bet o n p conditional on q with wins and losses W and L is fair for person b at time t if and only if

Cr~(p/q)W + Crbt(-p/q)L = O,

and if the 'conditional complement condition'

Crb(--p/q) = 1 -- Crb(p/q)

is satisfied, then any conditional bet

q p Wins and losses

T T x[1 - Crb(p/q)] T F x[ - Crbt(P/q)] F T 0 F F 0

is fair for b at t. Furthermore, if conditional complement conditions

Crbt( - q) = 1 - Crbt(q)

and

Cr)[ - (pAq)] = 1 - Cr)(pAq)

are satisfied, and conditional credence CrtU(p/q) is related to simple credences thus,

Cr)(p/q) = Crb(pAq)/Crbt(q);

then the conditional bet displayed above is equivalent to a book of fair simple bets, a book of bets each one of which has an expected value of 0. [Proof: It is given that Crbt ( -q)= 1 -CrY(q) and that CRY[-(pAq)l = 1 -Crb(pAq). Thus the following simple bets are fair for b at t.

B(pAq): (pAq) Wins and losses

T x[1 - Crb(pAq)] F x [ - Cr~(pAq)]

Dow

nloa

ded

by [

Ein

dhov

en T

echn

ical

Uni

vers

ity]

at 0

1:08

20

Oct

ober

201

4


B ( - q): - q Wins and losses

T x[Crb(p/q)] [1 - (1 - Crb(q)]

V xICrbt(p/q)l[ - (1 - Crbt(q)]

And since it is given that Crbt(p/q) = Crbt(pAq)/Crbt(q), these bets make a book net wins and losses on which are given in the last column of the table

P q T T F T T F F F

x[ 1 - Cr~(pAq)] x [ - Cr~(p^q)] x[ - Crb(p^q)] X[-- Cr~(pAq)l

B(-q) x[Crbt(p/q)] [Crbt(q) -- 1] x[Cr~(p/q)] [Crkq) - 11 x[Crbt(P/q)] [Crbt(q)l x[Crbt(p/q)][Crbt(q)]

Net wins and losses on the book

x[1 - Crbt(p/q)] x[ - Cr)(p/q)l 0 0

Net wins or losses on this book are in all cases identical with wins and losses on the conditional bet displayed in the text.]

We come now to Dutch Books and Dutch Strategies. A Dutch Book shall lie against a person at a time if there are finitely many bets each of which is then fair for him, on which bets he would suffer a net loss no matter what. A Dutch Strategy shall lie against a person at a time t if there is a rule that at various times, none earlier than t, calls for, or on some condition calls for, bets (simple or conditional) each of which is identifiable entirely in terms of the person's simple and conditional credences at times no later than the time it (the bet) is called for (supposing it is called f o r ) - a rule such that, (i), if it calls on some condition C for a bet at t ime t ' , then the truth-value of C depends only on the person's credence and conditional credence functions for times no later than t ' , and (ii), it follows logically from the person's simple and conditional credence functions at t that (a) only fair bets will be called for, and that (b), no matter what, the person would suffer a net loss on the bets that were actually called for. Books and Strategies that do not guarantee net losses, but do make them possible while making net winnings impossible, shall be semi-Dutch Books and Strategies.

The last ideas we need are those of probability and conditional probability functions. A credence function for a person is a probability function if and only if his credences satisfy two conditions. First, he is certain of every necessary truth: for any proposit ion p ,

(1) [] p-~Cr(p) = 1.

And second, his credences are 'finitely additive' in the sense that his credence for any logically exclusive disjunction is the sum of his credences for its disjuncts: for any propositions p and q,

(2) - O ( p A q ) ~ C r (pAq) = Cr(p) + Cr(q)

I note that, by definition, every credence function is such that

Dow

nloa

ded

by [

Ein

dhov

en T

echn

ical

Uni

vers

ity]

at 0

1:08

20

Oct

ober

201

4


(3) Cr(p) >-_ O.

It is well-known that a Dutch Book lies against a highly opinionated person at a time if and only if his credence function for this time is not a probability function. (To countenance countably infinite books of bets, require countable additivity of probability functions. See Brian Skyrms, Pragmatics and Empiricism, New Haven 1984, pp. 22-23.) Since we are interested mainly in Dutch Strategies that can lie against persons even when Dutch Books do not, we concentrate on persons who are secure from Dutch Books, and thus on persons whosecredence functions are at all times probability functions.

[A brief digression for an open question: Observe that a person b has no inclination to believe a proposition p at time t, if either b is then sure that p is false and so thinks that p is absolutely improbable, or b then has no opinion about p one way or the other. We shall for the moment set the second case aside and concentrate on the parts of credence functions that reflect actual or 'positive' opinions about propositions as distinct from conditions of no opinion. Let the positive-opinion-part of a credence function Cr) be PObt: for any proposition p, PObt(p) =0 if and only if a is at t certain t ha tp is false; and if Crtb(p) ¢ 0, then PO~(p) = Crb(p). Let a principle of probability be false of PO b if and only if this principle has an instance in which only propositions for which PO ) is defined occur, which instance goes into a false statement when throughout 'Pr ' is replaced by 'POt b'. It is clear that a probability theorem, for example, P r (p )_ Pr(pvq), can be false of a positive opinion function even though none of the axioms of probability, ( i ) - ( i i i ) above, is false of this function. This is true of the function PO b = [(A, 1), ([AVBI,0) 1. Let a positive opinion function be closed under negation if it is defined for a proposition if and only if it is defined for its negation: fairs bets are defined on and against all propositions in such a positive opinion function. What is not clear, and what I leave an open question, is whether or not, in every case in which a person's positive opinions are closed under negation, if his positive opinions violate a principle of probability, then a Dutch Book based exclusively on his positive opinions lies against him even if they do not violate any of the three axioms of probability. End of digression for open question.]

A conditional probability function shall be a conditional credence function for a person whose conditional credences and credences satisfy, for any propositions p, q, and r, the following conditions:

(4) Cr(q/p)>_O; (5) [] (p-~q)~Cr(q/p) = 1; (6) [] [p-~ - (qAr)] ~Cr[(qAr)/p] = Cr(q/p) + Cr(r/p); (7) [] (p~q)-~Cr(r /p) = Cr(r/q); (8) Cr[(qAr)/p] = Cr(q/p)Cr[r/(qAp)];

and, to connect credences and conditional credences,

(9) Cr(p) > O-~Cr(q/p) = Cr(pAq)/Cr(p).

Conditional probability functions take as arguments all ordered pairs of propositions. But if Pr~ is a person's conditional probability function, its value

Dow

nloa

ded

by [

Ein

dhov

en T

echn

ical

Uni

vers

ity]

at 0

1:08

20

Oct

ober

201

4


for (q, p) has significance ('evidential bearing' significance), only if both q and p are contingent, and prb(p) ~ 1. Condition (8) is too complicated to be obvious or intuitive as a constraint on consistent possible evidential bearings. It is noteworthy therefore that this condition is only marginally stronger than one like it, but restricted by the antecedent Cr(pAq)>0, which restricted condition is derivable from conditions (1) through (7), and (9). Conditions (4) through (8) are like the axioms of Arthur Burks' Calculus of Inductive Probability. He adds to his axioms a 'definition' of unconditional probability, [] p ~ Pr(q) = Pr(q/p) , and observes that the principle, Pr (p) > 0 ~ Pr(q/p) = Pr(pAq)/Pr(p), which is our condition (9), is a theorem of his calculus, t observe that that 'definition' is a theorem of my system, though not one that has significance for possible evidential bearings. See Arthur Burks, Chance, Cause, and Reason: An Inquiry into the Nature o f Scientific Evidence (Chicago, 1977), pp. 40-41.

We concentrate in what follows on persons whose conditional credence functions are always conditional probabili ty functions, and concentrate further on 'conditionalisers', where a conditionaliser is a person b such that, for any time t and propositions p and q, CrY(p/q)=x if and only if,

if b were at t to ' learn for sure' that q, to be sure that he had learned q for sure, and not to ' learn' about anything of evidential relevance to p independent of the relevance for him at t to it of q, then his credence for p would be 'revised' to x, and revised 'directly'.

( 'Learning that p ' does not in the present context entail that p is true, and 'learning about a thing', though involving a change in one's credence for the thing is not restricted to case in which one becomes certain of it. 'Revising' is here consistent with not changing. And 'directly' means that no more time is to elapse than is required for processing and computation.) We concentrate, as has been said, on conditionalisers. Indeed we concentrate on persons who are at all times certain that they are conditionalisers. Persons who have doubts in this connection, even persons who, notwithstanding such doubts, are in fact perfect conditionalisers, would in some circumstances be vulnerable to Dutch Strategies other than, and more complicated than, those of main concern in this paper. To simplify we thus set aside for now predicaments possible for such persons. Their predicaments, and Dutch Strategies specific to them in certain circumstances, are taken up in the Appendix.

We concentrate, in fact, not only on persons who are secure f rom Dutch Books and f rom complicated Dutch Strategies specific to persons who suspect themselves of being non-conditionalisers, but further on persons who are secure f rom Dutch Strategies of a certain particularly simple sort. For we concentrate not only on persons whose credence functions are at all times probability, functions, but further, on persons who are at all times certain that their credence functions are and always will be probabili ty functions. An agent who had doubts here would be vulnerable to a Dutch Strategy even if his credence functions were always probabili ty functions so that he was never vulnerable to a Dutch Book. A Dutch Strategy against such a person

Dow

nloa

ded

by [

Ein

dhov

en T

echn

ical

Uni

vers

ity]

at 0

1:08

20

Oct

ober

201

4


could call for a bet on the proposit ion that his credences are not, or at some time will not be, probabilities, and, if and when he won that b e t - t h a t bet 'against h imse l f ' - c a l I for a Dutch Book against him (Dutch Books would then lie against him) with a sure loss greater than the amount he had won on that initial bet 'against himself'. We set aside such persons not because their predicaments are not germane to our main concerns, but again in order to simplify our discussion of the predicaments of persons who entertain doubts of still other sorts concerning their present and future credences.

II. Self-Doubts

A Dutch Strategy lies, as van Fraassen maintains, against any person who suspects that he may come to be mistakenly certain about some proposition. In this section we relate such suspicions to another condition of self-doubt. In the next section it is shown that all forms of this second, in ways more general, condition provide occasions for semi-Dutch Strategies, and that some forms of it give rise to (full) Dutch Strategies. The conditions that interest us pertain to a person's credences and probabilities for propositions about his credences and probabilities. Following van Fraassen, we let

cr,U(p) = x

express the proposit ion that person b's credence at time t for proposit ion p is x - that is, the proposition that credence function Crbt has for p the value x. Propositions about probabili ty functions shall be expressed similarly. Highly opinionated persons have definite opinions about such propositions, autobiographical opinions when the proposit ion concerns themselves.

We begin with the condition of a person who suspects that he may come to be mistakenly certain about something: a person b who at t suspects that he may come to be mistakenly certain at t ' about A is in condition

(10) Crbt[- AAcrb-(A) = 1] > 0.

Assuming as we shall that b's credences are probabilities and that he is certain of this we have

(11) Prbt[-AAprb-(A) = 1] >0 .

By principles of probabil i ty and elementary arithmetic it follows that

(12) Prbt[pr~.(A) = 11 x prU,[- A/prb.(A) = 1] > 0,

and

(13) prb[--A/prb.(A) = II >0 .

Inequality (t3) is a case of,

(14) prb[-- A/prbt.(A) =x] > (1 --x),

for x such that 0_< x<_ 1. (For (13), let x be 1 in (14).) It follows from (14) that,

(15) - prbt[ -- A/prbt.(A) = x] < ( x - 1),

and

(16) Prbt[A/prbc(A) = x] < x .

Dow

nloa

ded

by [

Ein

dhov

en T

echn

ical

Uni

vers

ity]

at 0

1:08

20

Oct

ober

201

4


Inequalities (14) and (16) are in fact equivalent: each follows from the other by principles of probability and arithmetic. Inequality (16) is a case of,

(17) Prbt[A/prbt-(A) = x ] # x .

I note that, letting A and ~ be - A and (1 --x) respectively, (4) is equivalent to,

Pr tb[A/prbt(A) = X] > ~.

This equivalence depends on Prbt[prb-(A) =xoprbt-(A)=~] = 1. So there is a sense relevant to the present study in which inequality (16) and the negated identity (17) are equivalent: (16) gives rise to a possible Dutch Strategy against b at t if and only if (17) does.

Above, t is a time later than t. To reach the general condition we seek, this constraint is now relaxed and we require only that t ' not be earlier than t. Any person who is in condition (17) is in violation of what van Fraassen terms the principle of Reflection. Though it is a consequence of (11) that

(18) Prbt[pr t b. (A) = x] > 0,

I emphasise that, given our stipulations for conditional probabilities, (18) is not a presupposition of (17), which entails, in the direction of (18), only o [prbt.(A)=x].

A person of the sort with whom we are concerned is in condition (17), and in violation of Reflection, if his confidence in some proposition, his confidence in it conditional upon his confidence in its coming to be of degree x, is of a degree other than x. Were such a person to 'learn for sure' at t that prbt.(A)=x for some subsequent time t ' , and 'learn' nothing else that was for him independently relevant to A; he would not then, at t, revise his probability for A to what he was sure it was going to be at t ' . Such a person might, for example, find today, in news that tomorrow he was going to doubt that he was passing an examination, a reason for being today quite confident that he was going to pass this examination tomorrow. He might think today that the doubt tomorrow was going to make him k e e n - w a s going to be just the thing to ensure a good performance. Violations of the two-person analogue of Reflection, for distinct b and c,

prb[A/pr~-(A) = x] = x

are, of course, entirely unproblematic and to be expected. That principle could hold for some c only if b considered c to be at t ' an unfailing odds-maker with regard to A. In order to conform to Reflection itself, the one-person principle, a person needs indeed to think better of his own possible credences than he thinks of those of other persons. No one, I suppose, thinks that of anyone else that he is a perfect odds-maker with regard to a// propositions.

A Dutch Strategy lies against any person who is in condition (11) and thinks that he may at some future time be mistakenly certain with regard to some proposition. And any such person violates Reflection,

prb[A/prbt .(A) = X] = X.

In fact, at least a semi-Dutch Strategy lies against any person who violates Reflection, and a (full) Dutch Strategy lies given a certain condition. These wages of Irreflection are demonstrated in the next section.

Dow

nloa

ded

by [

Ein

dhov

en T

echn

ical

Uni

vers

ity]

at 0

1:08

20

Oct

ober

201

4


III. Dutch Strategies against the Irreflective

Suppose that,

(20) Pr~[A/prb.(A)=x] =y ,

wherein t ' is not earlier than t, and x ~ y . There are two cases to consider depending on whether y is less than x, or greater than x. Below, in order to simplify and dramatize, I assume that b is prepared to accept any bet that is fair for him, even though such preparedness is no more essential to the existence of a Dutch Strategy than is the existence of a lurking super-clever bookie, a fiction in which I also indulge.

The basis for the arguments below has two principal parts. First, given (20), bets on A that are conditional on

(21) prb.(A) = x

that are fair for b at t will be based on odds (1 _y):y.2 These bets are of course 'on' if and only if (21) is true. And yet--this is the second part of the basis for the arguments b e l o w - i f (21) is true, simple bets that are fair at t ' will be based on the different odds (1 - x ) : x (different by the hypothesis that x~y) . Supposing (20), a violation of Reflection, then supposing also that the troublesome future credence obta ins -suppos ing that is that (21) is t r u e - b (if prepared to accept any bet that is fair for him) will at t and t ' be prepared to place and cover all bets on A at two different sets of odds. A clever bookie would be able to exploit this difference and arrange for bets on which, i f (21), b would be sure to lose. Furthermore, if

(22) prb[prbt-(A) =X] > 0,

then this clever bookie could have b place a bet on (21), a bet on which b stood to win less than the loss that in that in that case would result from the conditional and simple bets on A described above. A violation of Reflection, doubting in that way some probability that is possible for one, entails the existence of semi-Dutch Strategies, and, conjoined with a positive probability for the doubted possible probability, entails the existence of (full) Dutch Strategies. Here is the proof in detail and by cases.

Case One: y < x.

Since y is less than x, b is at t prepared to give odds-on-A-conditional-on- the-proposition-prbt.(A) = x that are better than, if pr~.(A)=x turns out to be true, b will at t ' be prepared to take on A. Realizing this, a bookie could have b cover at t a bet on A conditional on pr~.(A) =x, or equivalently have b then place a bet on - A conditional on pr b.(A) = x, and then, if prbt .(A) = X turns out to be true, get A to place at t ' a bet on A; a clever bookie could arrange the stakes on these bets in a way that insured a loss for b in case prbt.(A)=x. The conditional bet at t on - A could be,

2 The odds of a bet-what Richard Jeffrey terms its 'betting odds'-is the ratio of possible gain to possible loss. In the case of a fair bet, this ratio is the inverse of the ratio of the probability of gain on it to the probability of loss, which ratio Jeffrey terms the bet's 'probability odds'. See Jeffrey (forthcoming).

Dow

nloa

ded

by [

Ein

dhov

en T

echn

ical

Uni

vers

ity]

at 0

1:08

20

Oct

ober

201

4

66

I. pr~.(A) = x - A

Self-Doubts and Dutch Strategies

Wins and loses

T T [t - (1 - y ) ] =y T F - (1 - y ) F T 0 F F 0

Conditional bet I is fair for b at t: Pr~[A/prbt.(A)=x]=y, and so p r b [ - A /p r ) - (A)=x] = (1 - y ) . The strategy being designed calls for this bet at t and, in case pr~-(A) =x, for the following bet on A at t ' , which bet would then be fair for a.

II. A Wins or loses

T 1 - x F - x

In case prb-(A) =X, conditional bet I would be 'on' , bet II would be placed, and a would suffer the loss,

- ( 1 - y ) + (1 - x ) = ( y - x )

if A is true and the same loss if A is false. (Recall that it is given that y is less than x, and that both numbers are non-negative.) So if pr~-(A) =x, our strategy would lead to a loss no matter what. The strategy constructed to this point is a semi-Dutch Strategy. I f we add to (20) the assumption (22). a (full) Dutch Strategy can be reached by letting our strategy at t call for a bet on (21), to be specific we assume, for some positive z, that

(22 ' ) prb[pr~.(A) =x] = Z,

and let our strategy at t call for the bet,

III . p r b ' ( A ) = x I What b wins or loses

T ] - ( y - x)(1 - z ) F - ( y - x ) ( - z )

This strategy would lead, if prbt.(A)=x is true, to the net loss,

( y - x ) - ( y - x ) ( 1 - z ) : ~v - x ) z .

And it would lead to the same loss if p rb . (A)=x is false, for in that case conditional bet I would be 'oW, bet II would not be placed, and b would lose

- O~ - x ) ( - z ) = Cy - x ) z

on bet III .

Case Two: y > x .

For brevity we assume both (20) and (22 ' ) and construct a (full) Dutch Strategy. A semi-Dutch Strategy based only (20) alone can be reached by deletion.

In this case a clever bookie could get b to bet, at t on A (rather than on - A as in the previous case) conditional on prb-(A)=x, and also to bet at t unconditionally on prb-(A)=x (as in the previous case).

Dow

nloa

ded

by [

Ein

dhov

en T

echn

ical

Uni

vers

ity]

at 0

1:08

20

Oct

ober

201

4

Jordan Howard Sobel

I*,

I I I*.

67

p r b . ( A ) = x A Wins and losses

T T 1 - y T F - y F T 0 F F 0

prb-(A) =X Wins and losses

T F

- (x -y ) (1 - z ) - ( x - y) ( - z)

If prb.(A) =X, the book ie cou ld get b to bet at t ' aga ins t A o r equiva lent ly on - A .

II*. - A Wins and losses

T [1 - ( 1 - x ) ] = x F - ( I - x )

If pr~.(A) = x then bet II* is fair for b at t, for in that case p r b - ( - A ) = (1 -X) . This s t ra tegy wou ld lead to a loss if p r b . ( A ) = x is t rue , for in tha t case

whether or no t A was t rue b would win - ( x - y ) ( l - z ) on be t III* and lose ( x - y ) on bets I* and II* and so suffer the net loss,

( x - y )z .

(I recall tha t y is a s sumed to be greater t han x, and tha t x, y , and z are non- negative.) The s t ra tegy would lead to the same loss i f pr~.(A) = x is false, for in that case cond i t i ona l bet I* would be 'off ' , bet II* would not be p laced , and b would lose - ( x - y ) ( - z ) = ( x - y ) z on bet I l l* .

At least a semi-Dutch S t ra tegy lies aga ins t any pe rson who viola tes Reflection. To keep the r eco rd s t ra ight I recal l tha t Dutch Strategies do not lie against only v io la to rs o f Reflect ion. F o r example a Dutch S t ra tegy will lie against any h ighly o p i n i o n a t e d pe r son who suspects his credences are or at some t ime will be incoherent . A Dutch Strategy will lie against such a person even if he is not in v io la t ion o f Reflection and even i f his credences are always coherent so tha t no Dutch Book 3 ever lies agains t him. A n d , as will be shown in the A p p e n d i x , Dutch Strategies lie in some c i rcumstances agains t persons who suspect tha t they are no t perfect c o n d i t i o n a l i s e r s - even agains t persons who are perfect condi t iona l i se rs in these c i rcumstances .

IV. Reflection and Perfection

One supposes tha t Dutch Books and Strategies exploi t in te l lec tual d e f e c t s - that they lie aga ins t persons on ly given var ious in te l lec tual faul ts . But then there is a p r o b l e m . The p r o b l e m is tha t it seems tha t Du tch Strategies can lie

3 A Dutch Strategy lies, somewhat trivially, also against any coherent and highly opinionated person who is not certain of his probabilities. Let b be coherent and highly opinionated and suppose that (1) Pr~A) = x, and (2) Pr~pr~A) = x] ~e 1. Then, for some y different from x and positive z, (3) Pr~prtb(A)=y] =z. It follows from b's credence function at t, Prt b, that bets on pr~(A) =y, based on the odds (l-z): z are fair, and it follows from this credence function, or more precisely from true propositions about it, that b would lose on any such bet, for it is a truth about Pr~ that pr~(A) #y.

Dow

nloa

ded

by [

Ein

dhov

en T

echn

ical

Uni

vers

ity]

at 0

1:08

20

Oct

ober

201

4


against very reasonable persons precisely because they are very reasonable and realise that in their beliefs they are not dramatically wiser or more temperate than other persons. Reasonable persons will sometimes realise that they are capable of, perhaps destined to, excessive or deficient degrees of confidence in things. I will not impugn the reasonableness of such serf- deprecating suspicions. My plan is instead to make clear that it is not precisely and only on account of them that Dutch Strategies lie against persons, and that, as one expects, these strategies lie against only persons who do not just suspect themselves of intellectual imperfections but who actually are intellectually imperfect. I will contend for aspects of an ideal for intellects, and bring out how violations of Reflection entail imperfections relative to this ideal. Attainment of th{s ideal would provide security f rom Dutch Books and Strategies, but the claims to be made for it are not dependent on this consideration and are based instead on what I take to be intrinsic merits of the ideal condition that recommend it, all considerations of Dutch Books and Strategies quite aside.

As intellects we have a natural and necessary interest in consistency. We have an interest in not believing several propositions not all of which can be true. A fully integrated in te l lec t - a person who was truly of one m i n d - would be perfectly consistent. Inconsistency tends, according to its extent and centrality, to a person's disintegration as an intellect. Connectedly, inconsistencies when made explicit in healthy minds tend to resolve into systems of uncertainties. To the extent, therefore, that we would be in possession of ourselves and know our own minds we cannot be intent on, we cannot make a project of, harboring inconsistencies, and must indeed be committed to the eradication.

Observe that additivity as a constraint on credence functions,

- ~ (pAq)-~Cr(pVq) = Cr(p) + Cr(q),

entails binary consistency as a constraint on credence functions given that these functions are bounded by 1:

o (pAq)~ - [Cr(p) = 1 ACr(q) = 1].

Let a person's opinions be balanced if and only if they are represented by only additive credence functions. What has been said for consistency as part of an ideal for intellects applies without qualification I think to balance. A person's stake in his credences' being balanced is part of his stake in being- in being one integrated mind or intellect. Any person who would be in possession of himself and know his own mind is committed to balance just as he is to consistency.

Logical omniscience, being certain of every necessary truth, and high opinionation, having quite definite degrees of confidence in all propositions, are further aspects of an ideal for intellects, though commitments here are somewhat different f rom those to consistency and balance. It is possible for a person to know fully his own mind while not being logically omniscient or highly opinionated. But these conditions are limits that open minds tend to approach as they have experiences and think about them, and as they think

Dow

nloa

ded

by [

Ein

dhov

en T

echn

ical

Uni

vers

ity]

at 0

1:08

20

Oct

ober

201

4


about the ideas that are involved in their opinions. And so, in so far as a person would be open-minded and prepared to face his world and his ideas, he is committed to logical omniscience and high opinionation as limits to which he is prepared indefinitely to approach. We may sum up aspects of the ideal so far identified by saying that the credences at a time of an ideal intellect would be represented by a probability function, exactly one probability function.

Introspective omniscience, being certain of, and indeed knowing, the state of one's opinions, is a further aspect of the ideal for which I contend. The extent of a person's self-possession is I think itself a partial determinant of his intellectual self. Introspective omniscience is a natural and in a sense necessary end for an intellect, and this is so even if, as I think, it too is an only approachable end. Many aspects of the ideal are only approachable by finite minds. Perhaps even infinite minds could only approach some aspects of the ideal. There are, it seems, ' innumerably many ' p ropos i t ions - the re is, it seems, no proper 'set' of all propositions. (Consider Patrick Grim's, 'There Is No Set of All Truths ' , Analysis, October 1984.) And so there may be more propositions than even an infinite mind could have definite opinions about, though the capacities of such minds are admittedly difficult to ponder.

Concurrent introspective omniscience is a part of an ideal for intellects, and for similar reasons so also I think are omniscience concerning one's past credences and at least selective omniscience concerning one's future ones. A person could hardly aspire to forknowledge of details of his future credences, since these will be due in part to experiences that he has yet to have, but a person can aspire to knowledge of some aspects of his future credence-states, for example, to knowledge of their consistency and balance. An ideal intellect would be in full possession of its past and present credence- states; and in possession of its future ones at least to the extent of knowing of their consistency and balance, and of knowing of its own future introspective omniscience extended as it would be to its then past and present, and selectively to its then future.

We come now to relations of Reflection to intellectual perfection. The synchronic part of Reflection,

prb[A/prb(A) = x] = x,

is, according to van Fraassen (he writes 'I should think'), uncontroversial. (p. 248) He does not say why he is of this view, but, under a certain condition, it is easy to see that synchronic violations of Reflection are not possible for an ideal intellect as so far detailed, and it is possible that this ease is related to van Fraassen's attitude towards synchronic Reflection. Suppose that an ideal intellect is in synchronic violation of Reflection:

(1) b is an ideal intellect (2) Pr~[A/pr~(A) = x] # x

And suppose (this is that 'certain condition') that

(3) Prbt[pr~(A) =x] > 0.

Dow

nloa

ded

by [

Ein

dhov

en T

echn

ical

Uni

vers

ity]

at 0

1:08

20

Oct

ober

201

4


(It is possible that for van Fraassen such conditions are throughout ' to be read'. It is not a part of my framework, but it may be a part of his, that such conditions are presuppositions of conditional probabilities.) Then:

(4) prb[prb(A) = X] = 1 vPrb[pr~(A) = x] = 0 (1) - (5) Pr~[pr~(A) =x] = 1 assumption,ind, arg.

t (6) Pr~(A)=x (1), (5) [ (7) p rb (A)~x (2), (5)

(8) prb[pr~(A) = x] ~ 1 (5)-- (7) (9) Pr)[prbffA) =x] = 0 (4), (8)

Lines (3) and (9) are contradictories, so lines (1) and (2) are, given (3), inconsistent.

A synchronic violation of Reflection addressed to a probability a person is somewhat inclined to think is his, is not possible for an ideal intellect. And, incidentally, it seems (see the argument of the previous section) that only such violations give rise to Dutch Strategies as distinct f rom semi-Dutch Strategies. (It is noteworthy here that even semi-Dutch Books will lie against an ideal intellect, since he will have extreme probabilities for many introspective propositions. Though I think that Dutch Books and Strategies are signs of underlying imperfections, I do not think the same is true of semi- Dutch Books and Strategies.) But what about probabilities such an intellect is sure are not his? Might he not have 'synchronic reflective doubts ' about them? Not, I think, about most of them, for, if, as I now suppose, an ideal intellect would realise he was a conditionaliser, conditions (1) and (2) alone make an inconsistent set for an ideal intellect b, for any A, t, and x such that

b could at t 'learn for sure' that pr~(A)= x, be sure that he had learned for sure that prb(A)=x, and not ' learn' about anything of evidential relevance to p independent of the relevance for him at t to it of prb(A) =x.

(Even at a time when an ideal intellect was sure that the earth was round, it would be an entertainable hypothesis- - that is he c o u l d - 'learn for sure, etc. ' , that his probabili ty for the earth's being flat was 1, but it is at no time even an entertainable hypothesis that an ideal intellect should 'learn for sure, etc. ' , for example, that he is in his credences incoherent, or that, for some A, x, a n d y such that ( x + y ) > 1, [prbt ( -A)=xApr~(A)=y] . ) That (1) and (2) are for such b, A, t, and x inconsistent holds, since, given (1) and (2), b could not at t ' learn for sure that prb(A)=x, etc.', for were he to do so, his probabili ty for A would be x even though he realises that evidential reasons (what for him were evidential reasons) were moving him to revise that probabili ty to some value other than x. Given (1) and (2), were b to 'learn for sure' that prb(A)=x, etc., his probabili ty for A would at t not be 'proport ioned to his evidence'. His probabili ty for A would at t not be what, on consideration, it was as he would know full well destined directly to be, and that sort o f cognitive discordance would not be possible for an ideal intellect, not even momentari ly.

Dow

nloa

ded

by [

Ein

dhov

en T

echn

ical

Uni

vers

ity]

at 0

1:08

20

Oct

ober

201

4

Jordan Howard Sobet 7 l

It has been shown that synchronic violations of reflection are not possible for an ideal intellect b, proposition A, time t, and value x if either (i) Prb[prb(A)=x]>0, or (ii), though prb[prb(A)=x] =0, b could (it is an entertainable hypothesis that b should) at t 'learn for sure' that prbt(A)=x, etc. Other synchronic violations of Reflection would not give rise to full Dutch Strategies, and at least some other synchronic violations of Reflection are, I think, possible for ideal intellects.

Perhaps the synchronic part of Reflection, at least when duly restricted, should be 'uncontroversial ' even though arguing for it is not a trivial task. But if it is plain that synchronic violations of Reflection (at least for probabilities an ideal intellect could 'learn for sure', etc. were his) are impossible for ideal intellects, it is not plain that the same holds for diaehronic violations. Even an ideal intellect can for good reasons change its mind about things, and it may seem that an ideal intellect could suspect that it was going to change its mind about something, and, in this realisation, violate Reflection. In fact, however, not even diachronic violations of Reflection would be possible for an ideal intellect for probabilities an ideal intellect could 'learn for sure', etc., were his. This follows from the synchronic case, and from aspects of the ideal not important for that case. Suppose an ideal intellect b were in diachronic violation of Reflection at t, so that for some subsequent time t ' , and A such that an ideal intellect could at t ' ' learn for sure' that prb(A) =x,

prb[A/pr b.(A) = x] ~ x.

Then, since by the synchronic part of Reflection,

Pr~- [A/prb-(A) =x] =x,

b would by t ' have changed his mind about the reliability of prb-(A) =x. (If it is not then x, then Prbt'[prbt.(A)=x]=0, but , in my framework, the conditional probability in the just displayed line remains defined.) But- -and this is the point on which the present argument p i v o t s - t h a t sort of change of mind would not be possible for an ideal intellect. Since an ideal intellect, b would at t ' , remember the reasons he had t for doubting the possible probability at t ' o f x for A, and, since he is an ideal intellect and thus always 'synchronically reflective', discount them then, at t ' . Indeed, b, since ideal would, at t, anticipate those memories and discountings to come t ' , and being confident of its intellectual conduct at t ' , discount those reasons for doubt already, at t, and thus not then be in diachronic violation of Reflection. A diachronic violation of Reflection of the kind being considered would be either a credence an ideal intellect would realise was unreliable, or an anticipated change in a proposition's evidential potential in a particular connection, and neither would be possible for one who knew himself, and trusted himself, as an ideal intellect would. Such an intellect would be not only always 'synchronically reflective' in relation to credences he could 'learn for sure, etc.', but in relation to these probabilities always reflective both synchronically and diachronically.

An ideal intellect would not only never entertain Reflective Doubts

Dow

nloa

ded

by [

Ein

dhov

en T

echn

ical

Uni

vers

ity]

at 0

1:08

20

Oct

ober

201

4


concerning its current credences, but would have confidence also in all possible future credences. The main reason for this lies in the complete confidence that an ideal intellect would have in its future ideality, and in particular in its future memories of its past opinions and reasons, and in its future respect for these. This confidence would show itself in part in an ideal intellect's perfect conformity to the diachronic part of Reflection, which perfect conformity is I think a condition of an enduring intellect's full integration and self-possession. The key to this integration over time would be an ideal intellect's confidence in its future memories of, and respect for, its past opinions and reasons. Memories, I think, help in a way to make intellects, even if they do not in the way Locke thought help to make persons.

W. D. Fatk has written of an ideal for practical reason and for persons as agents. His endorsements extend with adjustments to the ideal for theoretical reason, and for persons as intellects, for which I contend:

One's own good comprises not only one's states but also the possession of one's self as a mind. One cannot earnestly wish to lose hold of oneself, to be reduced to a shaky mess when in t r o u b l e . . , this preservation of oneself as a capable ego is also something that one may find that one ought to care for when one is [not] . . . inclined to care for i t . . . among the duties of self-preservation is the conscientious man's commitment to live without evading any i s s u e - t o seek out and weigh what cogent reasons would lead him to do, and to submit himself without self-deception or evasion to their d e t e r m i n a t i o n . . . [And] acceptance of the principle of n o n - e s c a p i s m . . , has the most intimately personal reason. It rests on an individual's inmost concern to preserve himself intact as a living and functioning self: mentally in possession of himself and of his world, able to look at h i m s e l f . . , without hiding himself f rom himself. The penalty for slighting this need is his undoing as a person. (Falk (1968), pp. 373-374.)

The involvement of human beings in this practice [of seeking direction f rom cogent reasons] is personal: it turns on their stake in the kind of self-preservation which requires that one should be able to bear before oneself the survey of one's own actions. Responsibly r e a s o n - g u i d e d . . . living exists, in the first place, for the sake of sane and ordered individual being, and not for the regulation of the social order. (Ibid., p. 387)

I hold that a person's commitment to balanced credences, introspective omniscience, and the rest has as its first ground his being as an integrated enduring intellect, and not that relative security f rom Dutch Books and Strategies that it happens to serve, nor even that success in intellectual projects that it may sometimes but, as we shall see, need not always serve.

Dutch Strategies (or things very like them) can of course lie against groups of persons, even against groups of persons no members of which are individually at intellectual fault. This is clear since members of groups need

Dow

nloa

ded

by [

Ein

dhov

en T

echn

ical

Uni

vers

ity]

at 0

1:08

20

Oct

ober

201

4


not be of one mind in their credences, and may entertain reasonable doubts concerning possible credences of other members even if they have no doubts about themselves. Very similarly, Dutch Strategies can lie against individuals who are in one way or another not of one mind, even against persons who are not at any time at intellectual fault, for such persons can at times entertain doubts, reasonable doubts, concerning credences possible for them at other times. However, to the extent that a person does entertain such doubts he is not sure he is of one mind over time, and, notwithstanding his reasonableness at every time, has not realised that measure of intellectual integration and self-possession in which every intellect has a stake and must take an interest.

V. Living with Imperfection Nobody is perfect. I f a wise man is one who proport ions his beliefs to the evidence, then probably no one is perfectly wise. Probably everyone is at least somewhat prone to unproport ioned beliefs and in particular to insufficiently grounded certainties. Thoughtful persons who are honest and objective about themselves realise these things and are consequently in violation of Reflection, and, precisely because of their intellectual pains, vulnerable to Dutch Strategies. What is the proper response to this predicament?

One possibility is that each person should think the very best of his credences- that each person should think far better of himself than he would think of himself were he someone else. The present suggestion could be in part that, in van Fraassen's words, "any accusation of epistemic extravagance i s . . . to be met . . . with the cool judgment 'My credence that A is true, on the supposition that tomorrow I shall accord it credence to degree r, equals r . '" (p. 256) But the suggested course would be difficult if not impossible if it would involve (as it would need to if it is to give protection from Dutch Strategies) not merely saying, but thinking these things, for by hypothesis most of us think less of ourselves than Reflection would have us think, van Fraassen propounds a 'voluntarist interpretation of epistemic judgment ' that 'makes judgment in general, and subjective probability in particular, a matter of cognitive commitment , intention, engagement'. 'Belief', he writes, 'is a matter of the will'. (p. 256) But for the most part belief as distinct from assertion is (in a sense that may be other than the one intended by van Fraassen) not ' a matter of the will'. For the most part beliefs and credences are things experienced-not things we do but things that happen to us and that are subject only indirectly and within severe limits to control and manipulation. A person could try to deceive himself and to reach best opinions about his credences notwithstanding his clear evidence to the c o n t r a r y - a person could try to think cooly to himself, ' I can count on my credences. In every case, my credence for some proposit ion A, on supposition that tomorrow my credence for it shall be some value r, is that value r'. But other things being equal it is always best to proport ion one's beliefs to the evidence and not to self-deceive, and this is especially true when the evidence is as

Dow

nloa

ded

by [

Ein

dhov

en T

echn

ical

Uni

vers

ity]

at 0

1:08

20

Oct

ober

201

4


plain as a person's evidence is apt to be for his propensities to intemperate credences.

We are not intellectually perfect and it would almost certainly be a mistake to try to believe that we are. Perhaps, however, we should at least resolve to be perfect and always to proport ion beliefs to the evidence, and should, consequent to this resolution, come to be in our credences in conformity with Reflection and to think better of ourselves without self-deception. But for most persons such a course is not open, and is in fact closed from what would be its start. I know that I am disposed to possibly msitaken certainties, possibly mistaken certainties that I do not want to suppress, and that I have good reasons for thinking I will not suppress. Sometimes I am sure of my name, and sometimes that the sun is shining, and though I realise that I just might on any given occasion be mistaken about these things I am convinced that short of suicide there is no way in which I couM suppress completely these tendencies to excess. Resolving to be forevermore wise and temperate in all my credences is thus not open to me. Though I could of course say that I so resolved, I could not render myself subjectively resolved, at least not 'just like that ' in the way in which I could decide never to drive after drinking. Let me add another reason for this incapacity. I realise that it is possible, and even allow that it is somewhat probable, that sometimes my opinions are not only not completely under my control but are indeed to a considerable extent under the control of others. And so I know very well that I may on occasion be possessed of unproport ioned beliefs and intemperate credences. Hypnotists and drug therapists could make this so. Propagandists and advertisers could too, especially ones who employed subtle persuasion and subliminal messages. Since I cannot rule out such manipulators as absolutely improbable, I cannot without self-deception sincerely resolve to be always temperate in my credences.

Perhaps the right response to our intellectual predicament is then simply to be as perfect as we can be, and in particular to suppress to the extent that we can our tendencies to credential excess and deficiency. Perhaps intellectual wisdom consists in part at least in proportioning beliefs to the evidence in so far as one can. But I think that not even this is right. For one thing the costs involved in doing everything one could to secure temperature credences might be great, and a wise person weighs costs and allows them to count against projects. It would not, I think be wise for me to do all I can do to make sure that I am never quite certain of my name, even if such efforts wouM promote my intellectual perfection. It is obvious that everything considered it can be reasonable and wise to tolerate tendencies to intemperate credences, and not go to just any lengths to minimise them. In fact it can be wise and reasonable sometimes positively to court intemperate credences. If, to recall a case used by William James, 'I am climbing the Alps, and have had the ill-luck to work myself into a position f rom which the only escape is by a terrible l e a p . . , hope and confidence' may 'nerve my feet', and perhaps everything considered I should, if I can, be quite confident that I can make the leap, evidence, even considerable evidence, to the contrary notwith-

Dow

nloa

ded

by [

Ein

dhov

en T

echn

ical

Uni

vers

ity]

at 0

1:08

20

Oct

ober

201

4


standing. (James (1948), 'The Sentiment of Rationality', p. 27) A person can even have intellectual reasons of a sort for courting intellectually unreasonable credences. This point too was made by James who observed that, though early in an investigation, when the evidence inclines with force neither way, an 'attitude of sceptical balance' is 'the wise one' and unreservably recommended to 'the purely judging mind', yet ' for purposes of discovery such indifference is to be less highly recommended, and science would be far less advanced than she is if the passionate desires of individuals to get their own faiths confirmed had been kept out of the game'. (James, 'The Will To Believe', ppo 101-102, italics added.)

The proper response to our intellectual predicament lies I think not in simply refusing to think less than the best of our credences, nor in resolving to perfect ourselves and to have only proportioned beliefs and temperate credences, nor even in being the best that we can be intellectually and in doing everything in our power to minimise the intemperateness of our credences. The proper response is for a person to learn to live with his imperfections, and to make the best trade-offs. Each person needs to find the compromise that is best for him, and to tolerate and even court intemperate credences in the ways his circumstances, capacities, projects, and psychology recommend. A person has a stake in intellectual perfection, a deeply personal stake, and compromise made here are always degrading in a sense (even if not always in the sense and spirit intended by W. K. Clifford4). They are downgradings of the self relative to the ideal of a fully integrated and self- possessed intellect. But intellectual perfection is not everything, or to be approached at just any cost. Realising this and making appropriate compromises will, for thoughtful and honest minds, entail violations of Reflection. Such minds will not think the best of themselves as intellects, or be prepared cooly and sincerely to say, 'Trust me'.

As we said in the beginning, Dutch Strategies will lie against the reflective and honest; Dutch Strategies will lie against them precisely because of their intellectual virtue and consequent appreciation that no one, themselves included, is intellectually perfect. Perhaps what they should do about that-- about their vulnerabilities to Dutch Strategies- is to look out for clever bookies, and to hedge their bets even to the point of sometimes not betting at all.

Appendix." Dutch Strategies and Conditionalisation

A person has revised his probability for a proposition q by conditionalising on a proposition p if and only if

P r ' (q) = Pr(q/p),

4 'Belief... is desecrated when given to unproved and unquestioned statements, for the solace and private pleasures of the believer.., or even to drown the common sorrows of our kind by a self-deception which allows them not only to cast down, but also to degrade us.' (Clifford (1886), p. 343.) When James quotes from Clifford's essay he includes materials from just before and just after Clifford's references to self-deception and degradation, but not these references themselves. See James, op. cit., pp. 92-93.

Dow

nloa

ded

by [

Ein

dhov

en T

echn

ical

Uni

vers

ity]

at 0

1:08

20

Oct

ober

201

4


where Pr(q/p) is, when the revision in question is begun, his probabili ty for q conditional p , and P r ' ( q ) is his revised probabil i ty for q. Persons who suspect that they would not on the occasion of some 'certain learning' revise their probabilities by conditionalising on the proposit ion ' learned' are vulnerable to Dutch Strategies in some circumstances. Two sorts of circumstances are featured in cases below. In one sort, the proposit ion that could be 'learned for sure' is such that it will be ' learned for sure' if and only if it is true. In the other sort, the proposition that could be 'learned for s u r e ' - let it be p - i s such that, that it had been ' learned for sure' would not be ' further ' relevant to the proposit ion whose probabili ty is considered to be under revision, that is, relevant to it independent of the relevance to it of the proposit ion p itself.

Three times are distinguished in our cases. An 'initial' t ime t*, a closely following time t at which a contemplated possible episode of 'certain learning' would take place, and the time t ' at which all revisions consequent to that episode of 'certain learning' would have been completed. Proposition E shall be the possible, new-found at t, certainty, and proposit ion A the candidate for possible revision. It is assumed for our cases that were b at t to 'learn for sure' that E, he would remain sure of E at t ' -- indeed it is assumed that b is at t* certain of that. It is assumed that

Pr*[pr(E) = ~ p r ' ( E ) = 1] = 1,

and, more usefully, that

( t) Pr*[pr(E)= 1 ~ p r ' ( A ) = p r ' ( A / E ) ] = 1.

It is further stipulated for our cases that in them Pr*(e) is positive, and, for definiteness, that,

(2) Pr*(E) =z.

where z is positive. We consider initially cases in which a person is certain that he would not

conditionalise on some item of possible 'certain learning'. I then indicate how this requirement of certainty can be relaxed to one that calls only for 'sufficient suspicion'.

Case One. Person b is not only certain the he would not in revising his probabil i ty for A conditionalise on E, but certian of exactly what he would do instead. For some x and y such that x # y , it is stipulated that

(3) Pr*(A/E) = x

and that

(4) Pr*[pr(e) = 1 -*pr ' (A) =y] = 1.

Without loss of generality we let x be greater than y. Adjustments to suit the case in which x is less than y will be obvious.

By conditions (1), (3), and (4), person b is at t* sure that, i fpr(E) = 1, then, though bets on A conditional on [pr(E) = 1AE] at odds (1 - x ) : x are fair for him at t*, bets on A conditional on E at the different odds (1 - y ) : y will be fair for him at t ' . Even so, conditions set so far do not entail that b is

Dow

nloa

ded

by [

Ein

dhov

en T

echn

ical

Uni

vers

ity]

at 0

1:08

20

Oct

ober

201

4


vulnerable to a Dutch Strategyfl Vulnerability can be secured by adding the condition that E is a proposit ion that b is at t* sure will be for him certain at t if and only if E is true.

(5) Pr*[E'~,pr(E) = 1] = 1.

Person b could be supposed to be sure that E asserts a possible happening at t, for example, his then having some experience, 6 which will happen if and only if he is then sure it is happening.

For a Dutch Strategy at t* against b we have the betting strategy or rule R which we now detail:

(i) At t* rule R calls for the bet on A conditional on [pr(E)= 1AE],

B*(A/[pr(E) = 1AE]) [pr (E)=IAE] A

T T T F F T F F

Wins and losses

(1 - x ) - - X

0 0

Wins and losses

- (1 - y )

Y 0 0

Part of the condition for this bet's being called for is that p r ' (A/E) = y, which means that if this bet is called for it is fair at t ' , that is, when it is called for.

Clause (iii) does not complete the specification of rule R, and is not at t* a Dutch Strategy against any person b with probabili ty and conditional

A bookie who at t* knew how, i fpr(E) = 1, b would change his odds for bets on A conditional on E, might not at t* know how to arrange for bets on which, i f E, b would be sure to lose. There is I think a gap in a Dutch Book argument that Brian Skyrms has given for conditionalisation: see Skyrms (1975), pp. 192-193, and Skyrms (1984), pp. 24-25. See also Armendt (1980), section 2. Skyrms in recent work on 'dynamic dutch books ' has in effect closed the gap. He studies a 'model ' in which it is clear that a person will learn for sure some member of a partition if and only if this member is the true proposition in the partition: ' . . . the true member of the partition is announced. ' (Skyrms (1985))

6 Condition (5) or something like it is impficit in the Dutch Book argument for conditionalisation that Paul Teller attributes to David Lewis. (Teller, (1980))

B ' ( - A / E ) E A

T T T F F T F F

This bet is fair at t*: since (5), P r * [ E o p r ( E ) = l ] = l , we have Pr*(A/[pr(E) = 1AE]) = Pr*(A/E); and according to (3), Pr*(A/E) = x.

• (ii) At t* rule R calls for the bet on [pr(E)= 1AE],

B*[pr(E)AE] [pr(E)= 1AE] Wins and losses

T (x -y ) (1 - z ) F - ( x - y ) z

This bet is fair at t*: by (5) we have Pr*[pr(E) = 1AE] = Pr*(E), and according to (2), Pr*(E) =z.

(iii) I fp r (E) = 1, p r ' (A) =y , and p r ' ( A / E ) =y; then, at t ' rule R calls for the conditional bet against A,

Dow

nloa

ded

by [

Ein

dhov

en T

echn

ical

Uni

vers

ity]

at 0

1:08

20

Oct

ober

201

4


probabili ty functions Pr* and Pr*( / ). Rather than complete rule R straightaway, however, we will first show that it is nearly a Dutch S t r a t e g y - that under a certain condition it insures a net loss. We demonstrate that if it is true (and not merely as (4) would have it certain) that

(6) pr(E) = 1 -- 'pr ' (A) =yApr ' (A / E) =y ,

then rule R calls for bets on which b would suffer a net loss no matter what. Suppose for argument that pr(E) = 1. Then, given (6), rule R calls for each of the three bets above, and b would be sure to suffer a net loss on these bets. (There is no column in the following table for the proposit ion E, and so it may not be clear that the entries for B ' ( - A / E ) are correct. They are, however, correct given the assumptions currently operating, namely, (6) and that p r (E)= 1; these assumptions entail that ( E ~ [pr(E)= 1AE]).) ~

[pr(E) = 1AE] A

T T T F F T F F

B* [pr(E) = 1AE]

(x-y)(1 -z) ( x - y ) ( 1 - Z)

- ( x - y ) z

- ( x - y ) z

B* (A/[pr(E) = 1AE])

B ' ( - A / E )

- (1 -y) Y 0 0

1--32 --Y

0 0

Net wins and losses

zty-x) zty-x) zty-x) zty-x)

We have assumed for definiteness (and without loss of generality) that x is greater than y. Thus, since z is by (2) positive, z ( y - x ) is negative, that is, a loss. So, given (6), if pr(E) = 1, rule R calls for bets on which b would lose no matter what. Now suppose that p r ( E ) g 1. Then the condition for bet B ' ( - A / E ) is not met, and rule R calls for only B*(A/[pr (E)= 1AE]) and B*[pr(E)= 1AE]. Furthermore, given as we are currently supposing that p r (E )~ 1, bet B*(A/[pr (E)= 1AE]) though called for is 'off', and b would suffer a loss of z ( y - x ) on B*[pr(E) = 1AE], the only bet of the three that under our current assumptions is both called for and 'on' .

We have shown that, if (6), rule R must lead to bets on which b would lose no matter what. But what if not (6)? I f not (6), that is, if

(7) pr(E) = 1A [pr ' (A) ~ yVpr ' (A/E) ~ y],

then bet B ' ( - A / E ) is not called for, the other two bets are called for, and b would not necessarily suffer a net loss on them. Rule R as so far constructed does not insure a loss no matter what. To insure a loss no matter w h a t - t o make rule R a Dutch Strategy against b at t * - w e shall add a bet on proposit ion (6), a bet that if won would return less than z ( y - x ) , the loss that, if this bet is won and thus (6) is true, is, we have seen, guaranteed by the rest o f rule R; and that if lost would cost more than ( l - x ) + ( x - y)(1 - z) = (1 - y) + z(y - x), t he greatest net gain possible on the other two be t s - -B*(A/ [p r (E )= lAE] ) and B*[pr (E)= lAE] ) - - tha t , if this bet

Dow

nloa

ded

by [

Ein

dhov

en T

echn

ical

Uni

vers

ity]

at 0

1:08

20

Oct

ober

201

4

Jordan Howard Sobe! 79

on (7) is lost, rule R calls for. Toward the construction of such a bet on proposition (6), I observe that given conditions (4) and (1) we have

(8) Pr*[pr(E) = 1 ~ p r ' ( A ) = y A p r ' ( A / E ) =y] = 1,

which means that every bet on (6) of the form

pr(E) = 1 ~ p r ' (A) = y A p r ' (A/E) = y

T F

Wins and losses

x(1-1)=0! x(-1)

is fair for b at t*. We now complete the specification of rule R.

(iv) At t* rule R calls for the bet on pr(E) = 1 ~ p r ' ( A ) = y A p r ' ( A / E ) = y,

pr(E) = 1 ~ p r ' (A) =yApr ' (A / E) = y Wins and losses

T 0 F - [(1 - y ) + z C v - x ) + 11

Rule R is a Dutch Strategy against b at t. I f (6) is true, R calls for bets on which b would suffer a net loss of z(y - x ) , and if (6) is false it calls for bets on which b would suffer a net loss of at least 1.

It should be clear that while b's certainty that he would not conditionatise on E, and in particular condition (4), make the final step in the construction of rule R quite unproblematic, something tess than this supposed certainty would have sufficed. How much less would depend on the other numbers in the case (the numbers x, y, and z). The general requirement is that (6) needs at t* to be probable enough to support a fair bet which, if won would return less than the net loss then insured for the other bets that were called for, and which if lost would cost more than could be won by the bets that would then be called for.

Case Two. We begin not with a new sort of case that supports a Dutch Strategy, but with a situation in which I am confident I would not conditionalise. Reflecting on this situation can suggest condit ions-condit ions that are not satisfied in that s i t u a t i o n - that would support Dutch Strategies.

Let G be the proposit ion that there are little green men on the moon, and let g be the proposit ion pr(G) = 1 - t h e proposit ion that very shortly I will be certain that there are little green men on the moon. I do not find that proposition absolutely improbable: P r*[pr (G)= 1] >0 . But I do think that it is much more likely that were I very shortly to be certain that G I would be mistaken, than that I would be right:

Pr*[pr(G) = 1A - G ] > Pr*[pr(G) = lAG].

This means that for me

Pr*[pr(G) = I /G] < 1,

or more succinctly,

Pr*(g/G) < 1.

Even so, if I were now certain that G I might very well very shortly be at least nearly certain that I would very shortly still be certain that I had just been certain that G:

Dow

nloa

ded

by [

Ein

dhov

en T

echn

ical

Uni

vers

ity]

at 0

1:08

20

Oct

ober

201

4

80

or more succinctly,

Self-Doubts and Dutch Strategies

p r ' [pr(G) = 1 ] = 1;

p r ' (g) = 1.

In short, I might very well not revise my confidence in q, on 'learning for sure' that G, by conditionalising on G.

One striking thing about the case just described is that in 'learning for sure' that G I would of course ' learn' something else of relevance to q, something else of independent relevance to g. I would, one supposes, in 'learning for sure' that G, ' learn' (even if not ' for sure') that I had ' learned for sure' that G. I am confident that I would not on 'learning for sure' that G revise my probabili ty for the proposit ion that I had so ' learned' by conditionalisation on G (I would, one supposes, revise that probabili ty 'by introspection'), but one does not suppose that a Dutch Strategy must lie against me, and this seems to be because, that I had so 'learned for sure' that G would be further, and independent o f G, relevant, so that it would be reasonable for me not to revise my confidence in g by conditionalising on G, even if G were the sum total of what I had 'learned for sure', and, though I had also 'learned' g, I had not ' learned it for sure'.

All of which suggests that perhaps Dutch Strategies will lie in some cases in which, for particular propositions, A and E, that I had ' learned for sure' that E would not be further and independently relevant to A. A case that shows that this is so comes f rom Case One by replacing conditions

(2) Pr*(E) = z

and

(5) Pr*[E,~pr(E) = 1] = 1

of that case, by the new conditions

(2/) Pr* [pr(E) = 1AE] = z

and

(5/) Pr*(A/pr(E) = 1AE) = Pr*(A/E) .

Condition (5/) says that, at t*, p r (E)= 1 is for b not further relevant to A, that it is not at all relevant to A independent of the relevance to A of E. (The ideal for (5/) comes f rom section 3 of Brad Armendt 's paper: op. cit.)

Rule R - t h e rule constructed for Case O n e - i s a Dutch Strategy against b at t* in the present case. I note that bet B*(A/[pr(E) = 1AE]) is, in the present context, fair at t* by (5/) and (3), and that bet B*[pr(E)= 1AEI is fair at t* by (2/). The rest of the argument carries over without change.

Work that remains includes relating, to a suitably elaborated ideal for intellects, conditions of persons who, by suspecting that they may not conditionalise, lay themselves open to Dutch Strategies of the kind developed

Dow

nloa

ded

by [

Ein

dhov

en T

echn

ical

Uni

vers

ity]

at 0

1:08

20

Oct

ober

201

4


in this a p p e n d i x . I t is r e l e v a n t to p o s s i b l e d i f f icul t ies o f t h i s w o r k t h a t i t is

not a f e a t u r e o f e i t h e r o f m y cases t h a t w e r e t h e s u b j e c t to l e a r n E fo r su re ,

E w o u l d b e e v e r y t h i n g , o r at l eas t n e a r l y e v e r y t h i n g , t h a t h e w o u l d l e a r n abou t . 7

University o f Toronto R e c e i v e d M a y 1985

REFERENCES Brad Armendt, 'Is There a Dutch Book Argument for Probability Kinematics', Philosophy of

Science 47, I980. W. K. Clifford, 'The Ethics of Belief', Lectures and Essays, Second Edition, London 1886. Richard C. Jeffrey, 'Probability and the Art of Judgement', Experiment and Observation in

Modern Science, eds. P. Achinstein and O. Hannaway, MIT--Bradford Books, forthcoming. W. D. Falk, 'Morality, Self, and Others', Ethics, eds., J. J. Thompson and G. Dworkin, New

York 1968. William James, 'The Sentiment of Rationality', Essays in Pragmatism, ed. A. Castell, New York

1948. Brian Skyrms, Choice and Chance: An Introduction to Inductive Logic (Second Edition), Belmont

1975. . . _ , Pragmatism and Empiricism, New Haven 1984. _ _ _ , 'Dynamic Coherence and Probability Kinematics', typescript, 4 April 1985. Paul Teller, 'Conditionalisation and Observation', Synthese 26, 1980. Bas van Fraassen, 'Belief and The Will', The Journal of Philosophy LXXXI, 1984.

The 'model' Skyrms employs in recent work entails this feature for possible bases for revisions by conditionalisation. He studies rules that 'treat the announced member of the partition as total input'. (Skyrms (1985)) However, Sk;yTms does not depend on this feature of his 'model' when he argues that rules other than conditionalisation provide openings for 'dynamic dutch books'. He relies in his argument on little more than that the thing will be learned for sure if and only if it is true.

Dow

nloa

ded

by [

Ein

dhov

en T

echn

ical

Uni

vers

ity]

at 0

1:08

20

Oct

ober

201

4

self-doubts and dutch strategies

Documents