weight

BARBARA DAVIDSON AND ROBERT PARGETTER

W E I G H T

Received 4 June, 1985)

1. INTRODUCTION

I am given a symmetrical-looking coin; I toss it ten times, obtaining five heads

and five tails. What (epistemic) probabil i ty should I give to it landing heads

on the next toss? Other things being equal, the answer is clearly one-half.

Similarly, had I tossed it one hundred times with the same half-and-half

result, the answer would have been one-half. But although the answer con-

cerning the probabi l i ty would have been the same, surely something would

have gone up sharply; but if not the probabil i ty , what exactly? Again, as

we get more information from the opinion polls, our estimate of the Prime

Minister's probabi l i ty of being re-elected may rise, fall, or remain constant ,

but s o m e t h i n g - and, therefore something e l s e - typical ly rises throughout

the process of gathering more information. J. M. Keynes put it this way:

[a]s the relevant evidence at our disposal increases, the magnitude of the probability of the argument may either decrease or increase, according as the new knowledge strength- ens the unfavourable or the favourable evidence but something seems to have increased in either case - we have a more substantial basis upon which to rest our conclusions.1

We will follow Keynes in using the term 'weight ' for this something: the

probabil i ty of heads next time based on one hundred past tosses is weightier

than that based on ten tosses, the probabi l i ty of re-election based on all the

polls is weightier than that based on the first couple, and so on.

One's first reaction to the problem of weight is to think that it is easy,

that a little reflection will reveal its analysis. We will argue, however, in w 2

that every extant account, or gesture at an account, faces major objections.

In w we propose and comment on our own account, and in w we defend

it. But first some more preliminaries to help us to focus on our target, and

firm up our pre-analytic intuitions.

The weight of a probabi l i ty is clearly a function of the evidence on which

it is based. What, for instance, makes the probabil i ty of re-election based on

Philosophical Studies 49 (1986) 219-230. (~) 1986 by D, Reidel Publishing Company

220 B A R B A R A D A V I D S O N A N D R O B E R T P A R G E T T E R

all the polls weightier is precisely that it is based on all the polls, rather than

on the first couple. It is not, though, a function of the reliability of the

evidence or of whether the evidence is a person's total evidence. The evidence

about the result of the ten tosses or of the first couple ofpoUs might be.cer-

tain, be clearly my total evidence at the time, and it might not be in dispute

that my probability judgments based on them were the right ones given the

evidence, yet these judgments would still not be as weighty as those later ones

based on the one hundred tosses or on all the polls.

What matters, rather, is the amount of evidence. As a rule, the more evi-

dence the more weight. But only as a rule. Increase in weight is associated with a decrease in the need for more evidence. The low weight of the probabil-

ity judgment about the Prime Minister's chances of reelection based on the

first couple of polls is linked to the evident need for more evidence; the high

weight of the probability judgment about heads next time based on one

hundred tosses is linked to the fact that such evidence is sufficient or close to

sufficient. Now, as a rule, more evidence reduces the need for yet more evi-

dence, but there are exceptions. Here is a case described by Isaac Levi.

A physician might want to find ou t whether McX has disease D or E, which call for different therapies. Relative to the evidence available to him, he feels justified in diag- nosing D. Subsequently, new evidence is obtained that casts doubt on that diagnosis, but without being decisive in favour of E. The amount of relevant evidence has increased; but would it not be plausible to say that the need for new evidence increased after the increase of relevant evidence? ~

Or consider the case where a coin is tossed one hundred times with half the

tosses landing heads and half landing tails. We then learn that the distribution

of heads and tails was very far from random, and indeed that a clear trend

had started to emerge halfway through the tosses, although we do not learn

the direction of the trend. Here we have more evidence, the same probability

(0.5) of heads, but clearly less weight.

What we are talking about here, of course, is the relationship between

weight and increasing amounts of relevant evidence. If we learn that the coin- tosser's name begins with 'R' , neither probability nor weight will be affected.

Evidence that changes probability is relevant evidence. But so too is evidence

that changes weight, even if it leaves probability unaffected. Evidence is

relevant if it changes either probability or weight. As the relevance of a body of evidence will also depend on what proposition the evidence is taken to be evidence for, weight cannot be determined by any intrinsic property of that body of evidence.

W E I G H T 221

One desideratum, accordingly, in framing an account of weight is that it explains why, as a general but not universal rule, increase in relevant evidence

leads to increase in weight. A second desideratum relates to action. Many 3 have noted a connection

between the weight of a probability and the rationality of acting on that probability. Other things being equal, we would be happier betting on a horse race according to odds resulting from the accumulation of a significant body of evidence, rather than on the odds based, say, simply on the number of horses running. Again, we would prefer to bet on the coin landing heads next time after observing one hundred tosses than after observing ten tosses.

The connection here is most immediately between weight and rationality of acting, not between amount of evidence and rationality of acting. For the earlier examples where, against the general rule, increase in amount of evi-

dence does not lead to increase in weight are equally examples of exceptions

to the universality of the principle that action becomes more rational as

amount of evidence increases. Thus, if you learnt that a clear trend had started to emerge in the results of the coin tossing, you would try hard to

delay betting. Nevertheless, the rationality of acting on a probability depends on more than its weight. For instance, the rationality of betting depends on how much money I have, the odds available, and the cost of obtaining more evidence, as well as on the weight of my probability judgements concerning the various horses. The relevant desideratum, then, is that an account of weight explain the way in which the weight of a probability is a factor in

determining the rationality of acting on that probability. We will see later how our account of weight explains our two desiderata;

how it is that, as a rule, increase in relevant evidence increases weight, and

how it is that weight is a factor in the rationality of acting on a probability. But, first, why are we unhappy with extant accounts of weight?

2. E X T A N T A C C O U N T S O F W E I G H T

As the number of tosses goes up, it is very natural to think that one's probability of heads on the next toss gets, or is likely to get closer to something, perhaps the "true" probability or the truth of the matter; at any rate our intuitive response is that we are progressing in the sense of homing in, or probably homing in, on something worth homing in on. Think of a punter

collecting more and more information relevant to which horse will win the

222 B A R B A R A D A V I D S O N AND R O B E R T P A R G E T T E R

Melbourne Cup, she certainly thinks she is getting closer to something worth getting closer to. Why put in all that work otherwise?

Suppose H is the~ttypothesis we are interested in, and PI(H), P2(H) are, respectively, probabilities of lesser and greater weight. Perhaps, PI(H) is the probability of Phar Lap winning relative to a punter's total evidence afterone day o f investigation or the probability of heads relative to the results of five tosses, while P2(H) is the probability relative to the evidence gathered after a week of investigation or relative to the results of one hundred tosses. There are four obvious ways of trying to capture the intuition we have just been discussing: (i) say that P2(H) is closer to the truth than PI(H); (ii) say that P2(H) is probably closer to the truth than PI(H); (iii) say that P2(H) is closer to the true probability than PI(H), or, finally, (iv) say that P2(H) is probably closer to the true probability than PI(H). All four ways face major objections.

What might plausibly be meant by saying that P2(H) is closer to the truth than PI(H) is that if H is true then, P2(H) is closer to 1 than is PI(H) (that is, P2(H) >PI(H)) , while if H is false then P2(H) is closer to 0 than is PI(H) (that is, P2(H)< PI(H)). Thus closeness to the truth may be measured by closeness to the value, 1 or 0, as the case may be. Now we have seen that weight may go up, or down, without the probability value changing. Suppose we know that exactly this has happened in this example. That is, PI(H) = P2(H). Then we know that PI(H) and P2(H) must be exactly the same distance

from the truth. Hence P2(H)'s greater weight cannot be a matter of its being closer to the truth. Nor can it be a matter of its being probably closer to the truth. Suppose P~(H) = P2(H) = 0.6. How probable is it that PI(H) is close to the truth? Well, there is 0.6 probability that P~(H) is 0.4 from the truth,

and a 0.4 probability that it is 0.6 from the truth. And exactly the same is true for P2(H). What probability can we give to P2(H) being 0.4 from the truth other than 0.6, and to its being 0.6 from the truth other than 0.4?

The same example shows that it is wrong to explicate weight in terms of closeness to the true probabi l i ty- however that is analysed. If PI(H) and P2(H) are the same, they are the same distance from everything, even from potential mysteries like true probabilities.

It remains, however, to show what is wrong with the explication of weight in terms of probable closeness to the true probability. PI(H) and P2(H) are based on different bodies of evidence, say, el and e2, respectively, and so it might be suggested that, even if PI(H) and P2(H) are the same, a person in possession of the larger body of evidence, e2, is entitled to a greater confidence

WEIGHT 223

that the true probability is near to the common value of PI(H) and P2(H),

say, as before, 0.6, than someone in possession of el. Both give 0.6 as their best estimate of H's true probability, but differ in how much more probable this estimate is than its rivals, such as that H's true probability is 0.4.

But what is the true probability of H, in this context? It cannot be the

subjective probability of H, obviously. It would be absurd to think that a

person in possession of e 2 rather than el makes a better estimate of that! But equally it cannot be something like the rational degree of belief to have

in H given one's total evidence at the time. A person whose total evidence is el is just as able as one whose total evidence is e2, to calculate (however exactly one is supposed to do it) the rational degree of belief to have in H given their total evidence. 4 The estimate of the person with e2 is no more likely to be correct or near to correct than the estimate of the person with e 1 as their total evidence. If anything, the fact that e 2 is larger than el makes the chance of a slip a little greater!

Nor can the true probability be what might be called the final probability,

the probability when all the evidence is in. That is, it is wrong to analyse weightier probabilities as ones that are probably closer to the final probability. For the final probability may be specified as the probability relative to

all the evidence available in practice or all that is available in principle. If the former, we get an account of weight too dependent on the contingencies

governing the availability of evidence. The repealing of Freedom of Informa-

tion would not make probabilities weightier! Or, again, our punter calculating Phar Lap's chances on a small body of evidence does not have her probability

made one with maximum weight by being locked in a sealed room, although it will then be true that her probability is certainly the same as her final probability in the sense of the probability based on all the evidence available

to her in practice. Alternatively, if the final probability is the probability

relative to all the evidence available in principle, then, if Determinism is true, it will be 0 or 1. But now consider a modification of our earlier case where PI(H) = P2(H), namely, where they both equal 0.5. They may still differ in weight, yet neither is probably closer to the final probability than the other, both being certainly 0.5 away from it. The same objection applies if true

probability is interpreted as objective chance. It becomes impossible for a determinist to allow that probabilities of 0.5 based on different bodies of evidence may differ in weight, s

Finally, the true probability of H might, at least for the purpose at hand,

224 B A R B A R A D A V I D S O N AND R O B E R T P A R G E T T E R

be interpreted in terms of long run relative frequency. What goes up when weight goes up is the probability that the probability value is near to the long run relative frequency. 6 But which long run relative frequency? It turns out that answering this question raises difficulties precisely paralleling those for frequency interpretations of probability. But, without covering this only too familiar ground, we can see that there is a special difficulty for this proposal as one for analysing weight. Sometimes the weightier probability is clearly the

worse guide to long run frequency. I want to bet on County Boy in the next race. I start with the information that he is one horse in a field of ten, and

accordingly start with a probability of 0.1 of his winning. I then gather information to the effect that he is on a par in terms of basic ability with his rivals but that special factors favour him in this race. I then give him a

probability of 0.3 of winning. The second probability is clearly the weightier, but the first is clearly the better guide to his long run frequency of winning.

3. O U R A C C O U N T O F W E I G H T

Our suggestion is that the weight of a probability is a matter of how hard it

is to change it. The easier it is to change a probability, the less its weight; the harder it isto change it,the greater its weight. How do we change probabil-

ities? By doing things - which might range from, at one extreme, staying put with our eyes and ears open, to, at the other, carrying out a detailed series of expensive experiments over half a century - which result in the determination that some proposition, previously uncertain, is either true or false. We ask Fred and either discover that his results accord with ours or discover that they do not; we consult Chambers Encyclopedia and discover that Charles I was beheaded in 1642, or we discover that he was not beheaded then; and so on and so forth. Having determined that E, or that E, we then conditionalize on whichever it is of E or E that obtains. The size of the change effected in P(H) is accordingly either [P(H) - P(H/E)[ or [ P ( H ) - P(H/E)[. How changeable or shiftable, then, is P(H)? Clearly t w o factors are involved. One is the size of I P ( H ) - P(H/E)[ and of [ P ( H ) - P ( H / E ) [ . The other is the value of P(E) and of P(E). A probability i s highly shiftable, that is, very easy to shift, if there is some E such that both E has a high probability and conditionalising on E significantly changes the (former) probability. A probability is very hard to shift if any evidence which would significantly change the probability is itself high improbable, or, alternatively, any evidence with a

WEIGHT 225

high probability is such that conditionalising on that evidence does not significantly change the probability. Thus the shiftability of P(H) is high (low), relative to (E, E) , when both P(E) and [ P ( H ) - P ( H / E ) [ are high

(low). Accordingly, we suggest P(E). I P(H) - P(H/E) [ as a measure of the shiftability of P(H). We do not need to keep separate track o fP(E) . I P(H) - P(H/E)I , as by the probability calculus P(E). [ P ( H ) - P ( H / E ) I =P(E) .

I P(H) - P(H/E) I. This gives us a measure of the shiftability of P(H) with respect to { E, E };

our interest though is in shifts in probability which are within our power at a time. The shifts we could effect. Say that a determination concerning { E, E} is within our power if there is something we could do at the time which would lead to either our knowing that E or to our knowing that E. Our suggestion is that how hard it is to shift P(H) depends on the value of P(E). I P (H) - P (H/E) I for the various determinable { E, E) . The larger its value the easier it is to shift or change P(H). Accordingly, we explain weight by the

values [1 - P ( E ) . [ P ( H ) - P(H/E)[] takes for the various determinable (E, E) . They are what matters for weight.

Typically, there will be a number of such values corresponding to a number of determinable pairs of evidential propositions and their negations. Hence, often the answer to whether one probability is weightier or harder to shift than another will be that it is weightier relative to this investigation which would determine as between (A, A), but not relative to that investigation which would determine as between { B, B}; that is, it might be that P(A). I P I ( H ) - P I ( H / A ) I < P ( A ) . IP2(H)-P2(H/A)I , while P(B). [ P I ( H ) - PI(H/B) I > P(B). [ P2(H) - P2(H/B) I �9 There will be cases where a probability is weightier or harder to shift than another relative to all determinable { E, E ). Perhaps a probability of heads on the next toss based on a thousand tosses as opposed to one based on ten tosses is an example. (Remember it is only determinable (E, E} that need to be considered, and the determination must be possible prior to determination of H itself.) In any case, as we will see later, the notion of weight relative to determinable (E, E) is all we will need in order to explain what we ought to be able to explain.

Here is how our account looks applied to some examples. Consider a variant on our earlier coin tossing example. I toss C six times,

getting three heads, and so assign to the proposition, H, 'C will land heads when tossed in exactly five minutes time' a probability of 0.5. Intuitively, this probability has a low weight. Why? Because there are a number of


propositions which are such that (i) I can determine their truth value before the toss takes place in five minutes, that is, I can add them or their negations to my stock of knowledge, (ii) so doing would make a significant difference to the probability I give H that is, P(H/E) and P(H/E) are significantly different from P(H), and (iii) P(E) and P(E) are not unduly small.

For instance, I could toss C three more times before the crucial toss in five minutes time; thus 'Were C to be tossed three more times, all tosses would land heads' is a proposition whose truth value I can determine. The probability of getting three heads out of three tosses is not unduly low (one could hardly give it a value less than 0.125, the value the assumption of independence yields), and the probability of heads in five minutes time would be noticeably raised by getting the three heads. For one would then have six heads out of nine tosses, that is, 66.6% heads. By contrast, had C originally been tossed fifty times giving twenty-five heads, the new figure would be twenty-eight tosses out of fifty-three tosses, that is, 52.8% heads. The probability of heads in five minutes based on fifty tosses, half being heads, is much harder to shift than the probability of heads in five minutes based on six tosses. Thus the lower weight of the latter.

Similar points apply in the horse racing example. If I have a lot of information about Phar Lap, the possible impact of some new possible item of information is minimised. The sheer bulk of what I already know will tend to blanket the change learning something new might make. Thus we can explain the general connection between amount of information and weight noted at the beginning. The more I know, other things being equal, the closer P(H)

and P(H/E), for some possible new E, will be. Of course, there will always be some E which is such that P(H) and P(H/E) are very different. But, if weight is high, either E is very improbable given what I known, or else I cannot determine which of E or E is true - {E, E} is not determinable by me prior to the time at which H is determined.

The latter possibility is why the truth of determinism does not affect what we are saying. If determinism is true, there will be some proposition, X, giving the state of the world in great detail as of now, such that, for any proposition, H, about the future with a probability other than 0 or 1, P(H) is very different from both P(H/X) (= 1) and P(H/X) (= 0). Further, neither P(X) nor P(X) need be small. But all this does not make the probability of H highly shiftable, and so of low weight; because we cannot determine which of X or X is the case by the relevant time. It is beyond our power to find out

W E I G H T 227

beforehand which is true. It is easy to find out Phar Lap's jockey's position

on the championship ladder, or what happens when a coin is tossed, but impossible to find out the state of the world as of now, and the relevant laws of nature, in enough detail to determine just what will happen in the

future.

Our account of weight in terms of shiftability explains two facts we noted early on. The fact that, in special cases, additional evidence can reduce weight,

and the intuitive appeal of accounts of weight in terms of probabilities

homing in on some 'true' probability.

Although, as a rule the more evidence we have the smaller the affect of

possible further evidence - t h a t is, large amounts of evidence have, as we

have seen, a damping effect - t h e r e are exceptions. A new piece o f informa-

tion can give new relevance to a possible still newer item. This is how our

account of weight explains the exceptions to the rule that more evidence

equals more weight. For instance, to draw on an earlier example, two addi-

tional tosses after one hundred is neither here nor there, at least by comparison

with two extra tosses after five. But if I now learn that a clear trend was

�9 starting to emerge near the end of the one hundred tosses, the situation is

totally changed. The result of the two additional tosses may give me crucial

information on the direction of that trend. The extra information about

the existence but not the direction of an emerging trend increases the change

conditionalizing on, for example, the extra two tosses being heads makes.

If increase in weight is decrease in shiftability, as weight goes up - typical-

ly, through the accumulation o f evidence - we will get probabilities that are

harder and harder to move around when new investigations are carried out.

Hence, our probabilities will tend to converge. I will start with a certain

probability for Phar Lap winning based on very little information. This probability will move around quite a bit as more evidence comes in. But, as time goes on, the movements will get smaller and smaller. It will be just as if I were converging on some independent, true probability. Hence, the recal-

citrant intuition.

4. W E I G H T AND A C T I O N

We observed early on that there is a close connection between weight and

action. As a rule, weightier probabilities are better ones to act on. We all

feel happier betting on a race after getting all the available information


rather than just some of it. The connection, though, is an 'as a rule' one, as we saw. If the stakes are minor or if the cost of raising a probability of low weight to one of high weight is too high, it may be perfectly rational to act on a probability of low weight. If I am making a small bet, or if the charge for a relevant piece of information is high by comparison with what I stand

to win, it may be perfectly rational for me to bet using a probability of Phar Lap winning which is of low weight. As we put it earlier, weight is a factor in making it rational to act on a probability.

How can we justify an 'as a rule' connection between the weight of a probability and the rationality of acting on it? How can we show that probabilities that are hard to shift are good to act on? Our approach will be to exploit a well-known result concerning how getting more evidence may increase expected utility.

Suppose I face a choice of which of two actions, A and B, to perform. Let $1 . . . . . Si, ..., Sn be a partition of states of affairs which individually determine what would happen whichever action is performed. 7 Suppose, also, that my probability distribution over the Sis is independent of whether I choose to do A or B. Then the expected utility of A, U(A), = NP(Si). V(SiA); similarly, U(B) = 2P(Si). V(Si B). Suppose, further, that U(A) > U(B). Then the well known result is that if { E, E} is such that either UE(A ) < UE(B) or V~(A) < Ui~(B), where VE( - ) = U ( - ) with P ( - ) replaced by P( - /E) , then U(A) must be less than the expected utility of first determining the truth value of E, and then performing whichever of A and B has higher expected utility taking the new information into account, provided there is not too great a cost attached to getting the new information. That is, if the truth value of E can be determined cheaply and such determination may reverse which of A and B has the greatest expected utility, then overall expected utility is maximised by determining the truth value of E prior to deciding which of A and B to perform. (This consideration will apply for each determinable pair { E, E} that can be determined prior to the time at which the decision of which of A and B to perform must be made.) Suppose it is UE(A ) < UE(B ) which obtains. Then the gain in expected utility = P(E). [UE(B)- UE(A)]. That is to say, the gain depends on P(E) and on how much conditionalizing the probabilities in U(A) and U(B) on E changes U(A) from being greater than U(B) to being less than U(B). Consequently, the harder it is to shift the P(Si)s, that is, the weightier the P(Si)s are, the less the gain in utility of determining which of { E, E} obtains before acting, and so the more

WEIGHT 229

rational it is to act on the P(Si)s one has at the time. Weight is an ingredient in the rationality of acting on a probability by being an ingredient in whether one should get more information before acting on a probability.

An important question remains, however. We have seen the significance of weight when deciding between a number of courses of action. But surely weight matters even when I am not considering using my probabilities in deciding between courses of action having various possible pay-offs. Suppose I am simply interested in whether or not Phar Lap will win; I have no inten- tion of putting money on the race. Don't I typically seek more evidence in an effort to obtain a weightier probability concerning him, even so? Weight matters in the project of pure enquiry, as well as in practical decision-making.

This is true. But this is because we put a positive value on believing the true, and a negative value on believing the false. When I am interested simply in whether or not Phar Lap will win and not in what bets to make, it is still true that I want to believe what is true, and want not to believe what is false. If that much was not true, I would not seek weightier probabilities, or, indeed, any evidence at all. Why should I bother?

Suppose V(T) and V(F) are the values I give believing what is true and believing what is false, respectively, and that no other values are relevant concerning H. Then, the expected utility of believing H, U(bel. H), = P(H). V(T) + P(H). V(F); while U(bel. H) = P(H). V(F) + P(H). V(T). And, as one would expect, U(bel. H) > U(bel. H) iff P(H) > 0.5. Now, arguing in essential- ly the same way as before, if, say, U(bel. H)>U(bel . H), but UE(bel. H ) < UE(bel. H), we can show that the expected utility of fh'st determining the truth value of E, and then believing H or believing H according to which then has highest expected utility, must be greater than that of believing without further ado, provided (as before) that there is no cost associated with determining which of { E, g } obtains. More particularly, if U(bel. H) > U(bel. H), while UE(bel. H) < UE(bel. ]~) -- and so, P(H) > 0.5, while P(]~/E) > 0.5 - then the gain in expected utility = I V ( T ) - V(F)]-. P(E). [1 - 2 P ( H / E ) ] . Hence, inter alia the greater P(E), and the more P(H/E) falls below 0.5, and so, below P(H), the greater the gain in expected utility of first determining which of {E, E} obtains, if that is possible without (too great a) cost. Hence, we have the same kind of result as before. The less the weight, or, equivalently, the greater the shiftability, of the probability of H, the more the need to get additional evidence before acting in the sense of believing or disbelieving H. Hence, even in the project of pure enquiry, getting more

230 B A R B A R A DAVIDSON AND ROBERT P A R G E T T E R

evidence may be act ing in accord wi th the max im o f increasing expec ted

ut i l i ty . And the value o f weighty probabil i t ies lies in the correspondingly

lower value a t tached to acquir ing more in fo rma t ion before acting.

S. CONCLUSIONS

Our overall contention, then, is that our account of weight in terms of shiftability explains the phenomena that an account of weight ought to

explain.S

NOTES

t j .M. Keynes, A Treatise on Probability, MacMillan, London, 1952, p. 71. 2 Isaac Levi, Gambling with Truth, M.I.T., Mass., 1973, p. 142. 3 E.g., Keynes and Levi. 4 As Keynes in effect notes, op. cit., this point is particularly clear if you accept his view that we are dealing here with logical relations between propositions. The same applies, of course, if you accept Rudolf Carnap's view in Logical Foundations of Probabil- ity, University of Chicago Press, 2nd ed. 1962, that the relation is analytic. 5 We assume the common though not entirely non-controversial view that all objective chances are 0 or 1 if Determinism is true. See, e.g., David Lewis, 'A subjectivist's guide to objective chance', reprinted in his Philosophical Papers, vol. I, Oxford, 1984. 6 Cf., Carnap, op. cit., p. 555. 7 If you follow David Lewis, 'Causal decision theory', Australasian Journal of Philosophy, 59, 1 (March 1981), pp. 5-30, see p. 12, the S i would be dependency hypotheses; but an argument of broadly the following kind can be, and usually is, based on non- causal decision theory. See, e.g., Paul Horwich, Probability and Evidence, Cambridge University Press, 1982, and the references therein. 8 We acknowledge a great debt to Frank Jackson for his assistance in the preparation of this paper, and for many suggestions that have been incorporated in it. Many of the issues discussed were raised in his unpublished 'Inductive arguments and probability' (1973).

University o f Wollongong, New South Wales, Australia 2500

and

La Trobe University, Bundoora Victoria, Australia 3083.

weight

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