DISCUSSION

R E P L Y TO P R O F E S S O R F R E U D E N T H A L

I will first make a few remarks on specific points, and then attempt to clarify

what I take to be the most serious difference of opinion that divides us.

I. DEDUCTIVE CLOSURE

Partly in the interest of generality, and partly because it reflects an important

truth, I did not impose deductive closure on rational corpora. But of course I

did not stipulate that deductive closure should fail for rational corpora, so

that it is perfectly open to anyone to suppose, as an additional constraint on

rational corpora, that they are deductively closed. If we do this, biconditional

chains, identity chains, etc., collapse immediately into biconditionals,

identities, etc., and the system applies, as it stands, to the result. In fact, this

approach constitutes merely a simplification, with a consequent loss of

generality, of the original system. In reality, I think that the full deductive closure of a rational corpus would

be flatly inconsistent. But this is not to say that large parts of it might not be

considered as the deductive consequences of a single finite conjunction of

statements; it is quite open to us to suppose that we have adequate grounds

for accepting such conjunctions, without being committed to accepting the full deductive closure of everything we accept. This seems to correspond to

what Freudenthal recommends, when he suggests that we should undo the restriction on deduction but only "with the greatest care". I take it that he

does not want to impose full and automatic deductive closure, but only as much as seems reasonable in a given context. What this amounts to should of course depend on our state of knowledge concerning that particular context.

I think we have no strong conflict of intuitions concerning deductive closure. We do have a strong conflict of intuitions concerning the value of the doubtful contraptions and artifices I introduced in this connection; I shall

return to this conflict of intuitions shortly.

Synthese 36 (1977)493-49"8. All Rights Reserved. Copyright © 1977 by D. ReidelPublishing Company, Dordrecht-Holland.

494 DISCUSSION

II. CLOSURE CONDITIONS ON REFERENCE CLASSES

Consider a newly minted coin, checked for balance, of which we justifiably

believe that it will land heads half the time. Consider more specifically, the

fifth toss. The fifth toss is a member of the set of tosses; it is a member of the

tosses whose ordinal number is prime; it is a member of its unit class; and so

on. It is also, let us say, a member of the set of tosses performed on Tuesdays,

a member of the set of tosses performed in Rochester, a member of the set of

tosses performed by me on March 5, 1976 in Rochester, New York.

To arrive at the probability that this toss yields heads, we must assign it to

a good reference class, and, in the example at hand, we want to assign it to

the set of tosses of this coin, of which we know that half yield heads.

Consider, first, intersections. The intersections of the class of tosses of this

coin and the various other classes mentioned above are such that the

frequency of heads is not known. One of the conditions on Randomness

stipulates that when possible we should choose a reference class about which

we have relatively precise information. None of these intersections will

therefore interfere with our choice of the set of tosses of this coin as a good

reference class (at least, under ordinary circumstances). We may further

consider such intersections as: the set of tosses of this coin and the set of

tosses yielding heads. This too is a perfectly good reference class, and it is

furthermore one about which we have very precise knowledge of the

frequency of heads; but it is a class of which we do n o t know that the toss in

question belongs (ordinarily). So again, such intersections do not interfere

with the desired randomness of the fifth toss in the set of tosses of that coin.

The story is quite different with respect to unions. Let H be the set of

tosses of this coin yielding heads, and a be the fifth toss. What is to prevent a

from being a random member of the union of H and the unit set of a? We

know that a belongs to this set; we even know that the frequency of heads in

this set is either unity or very close to it. To be sure, we know that the

frequency of heads among tosses in general is different from 1 - but

ordinarily when there is a conflict of this sort between frequencies in two

reference classes, and we know that one is included in the other, we should

use the smaller. Why shouldn't we apply the same principle to this case?

Or suppose we stipulate that all atomic reference classes should be

infinite - that will rule out a reference class defined in terms of theuni t set

DISCUSSION 495

of a. But then consider the union of H and the set of tosses whose ordinal

number is prime. Both components of the union are infinite (or potentially

infinite); a is known to belong to the union. And again the frequency of heads is very close to 1.

In general, it is possible to construct a reference class with any

characteristics whatever, to which any object in question belongs, so long as we are allowed to use any set-theoretical technique whatever in the construction. There may be other ways of eliminating these classes from

consideration than to rule out unions, but that is the procedure that struck me as most natural.

The difference between unions and intersections is roughly the following:

by constructing 'artificial' unions, we can obtain a spurious precision in our probabilities. By considering unnatural intersections, we are led to spurious

vagueness. But the vagueness is ruled irrelevant by the rules of randomness,

and thus can be handled naturally within the system.

III. ACCEPTING STATISTICAL STATEMENTS

Freudenthal is quite right when he says that the question of the way in which

statistical statements enter the rational corpus cannot be evaded by formal arguments. In my book I was concerned with two of the ways: statistical

inference (by way of fiducial inference, confidence intervals, Bayesian

inference) and by way of set theory (certain statistical statements, e.g., that

most subsets of a given set are representative with respect to a given property, are set theoretical truths). Of course this does not constitute a complete

account of scientific inference, but that is another matter, and one that I

considered beyond the scope of my book. One has to start somewhere.

IV. THE DISEASE EXAMPLE

There are a number of things I would like to say concerning the disease example. The first of them is that it is indeed a somewhat awkward example

for my theory, though it is fairly clear that in real life it might also be a somewhat awkward example. I confess I do not follow all the details of Professor Freudenthal's comments on this example - in particular I do not understand the difficulty he sees when the two intervals I1 and 12 are

496 DISCUSSION

identical. But I do understand some of his remarks, and feel that in fact they

can be represented in my system. One suggestion he makes is that if no other evidence is available, we could weaken the statistical knowledge to the point where there was no longer any conflict between the intervals. This can be

represented in my system easily enough: to operate with a stronger level of practical certainty will result in weaker statistical statements being acceptable

into the rational corpus of that degree of practical certainty. And it may be that in his way the conflict can be eliminated.

But the most important suggestion is that it is an unnatural idealization to

suppose that 'no other evidence' is available. Freudenthal suggests a list of

factors that might be known to be relevant in our body of knowledge

concerning the disease and the tests, but of course his list is not to be taken

very seriously. What should be taken seriously are the considerations that physicians familiar with the disease and the tests would offer, and I am sure

that Freudenthal would be happy to agree to this. The question is how that

information should be taken account of, and how, in fact, 'one will act to the

best of his knowledge'. I have tried to make some suggestions in my book. I

know of no one else who has made specific suggestions as to how a body of

knowledge should be used in this sort of situation. As Freudenthal implies,

we will be more likely to get help from looking at the practice of statistics

than from looking at what statisticians say we ought to do; but I fear that

the practice of statisticians or of those who use statistics is neither so uniform nor so simple that we can suppose there are no real puzzles and problems

there.

V. A POINT OF CONFLICT

This brings me to my main point, and a point of genuine conflict between us:

Professor Freudenthal thinks it is a dreadful mistake to frame things in formal

languages, and that we should rather pursue our methodological studies by examining what scientists do. I agree wholeheartedly that we should look at

what scientists do, but I do not think that whatever passes at the moment for

current practice is sacred. First, fashions change in statistics as in other

things: what is current practice at one time will be frowned upon at another. Second, even at a given time the practice of different groups and different

individuals is not uniform and not consistent. Third, the practice of most

DISCUSSION 497

scientists represents, in effect if not explicitly, the adoption of a particular methodological point of view which may, considered generally, be quite

indefensible- in fact it may be possible to trace it back to one of those abstract formal philosophical positions which Professor Freudenthal deplores.

Fourth, difficult as methodology is, it would be surprising if no improve-

ments were possible in the methodologies employed by practicing scientists. It is here that I think formal languages can play an important role. Let us

consider mathematics. Mathematical systems can be presented in formal

languages, with explicit formation rules, explicit rules of proof, and so on. In

general, of course, mathematicians don't d~ this, and it would be a dreadful waste of time and paper if they were to do so. But the possibility of doing it nonetheless serves a useful function. It means that certain disputes - say a dispute about whether or not a certain proof is valid - can be definitively

settled. It is unlikely that the disputants will ever have to go back to the basic

formal language and formal system representing that part of mathematics

which is at issue, but they can go back toward it just as far as they need to,

and it exists there as the ultimate standard of validity.

Similarly, in the statistical case, the formal language lurks in the

background and performs a function even if it need never be called upon

explicitly. The function is not quite the same, because it is not clear that it

can often settle a dispute definitively, but it does provide a framework for the

dialectic which should eventuate in agreement. The rules of randomness in

my system, for example, were designed with that quite explicitly in mind.

Scientist A thinks that X is the appropriate reference class and Scientist B thinks that Y is the appropriate reference class. What kinds of considerations

can they bring to bear which indicate how to choose? Put the question in the formal framework, and progress toward an answer is assured by the system.

Leave the matter at the level of intuition, and we may be left with conflicting

intuitions, and no hope of resolution. This is a particularly important

consideration in the realm of statistical inference, where, vast as the areas of

agreement among statisticians may be, there are also vast areas of disagree-

ment, not merely on the theoretical and abstract level, but on the down to earth level of the interpretation of actual experimental results in science. We should agree to disagree regarding the effectiveness of radical mastectomy in mammary cancer only as a last resort. I believe that formal languages can help to provide a framework in which the dialectic of argument can approach a

498 DISCUSSION

resolution. Freudenthal, I suspect, thinks that the time wasted in developing

formal languages and artificial contraptions could more profitably be spent in

other ways. I don't know which of us is right, but (obviously) I have chosen to bet on formal languages.

University of Rochester HENRY E. KYBURG, JR.