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B iff A+B is abnormaL3
The abnormal propositions are the ones a priori equivalent to tradiction and the a rions are the ones a priori equivalent to the A
B f P ( A I A + B ~ .and = . ~q(BIA+B) = 1, since.additivity were normal. We can divide this equivalence relatron into its two conjuncts:
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DEFINITION: A P> B iff P(AIA+B) = 1 What we have here is a new non-tcvial relationships between propositions. De Finetti suggested relations of local comparison of this type.4 Here are a few facts about these notions that come in handy as we try to use them:
P> is transitive, and if A logically implies B then B b A. If A+B is normal, then A b B means that A is com aratively superior to B, in the sense that A is certainly true le supposition that one but not both are the case. But if A+B and B certanly false, on t! is abnormal then the relationship A P> B amounts to A
B. The ri ht readin for " b " is therefore "is su~eriorto or a p,riori equivalent to". To be brief, owever, f l l just say "A is superior to B for "A P> B , and ask you to keep the qualifications in mind.
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6. The Propositions in the 'Body of Science' The propositions whlch are a priori, in the above (very non- absolute sense) at a given stage of science are those whose contraries are all abnormal. There is a weaker condition a proposition K can satisfy: namely that any normal proposition wluch im lies K is superior to any that are contrary to K. Consider the following conditions and efinitions:
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(Al) Normality: (A2) Superiority: (A3) Contingency:
K is normal If A is a non-em ty subset A of K while B and K are disjoint, then A > B the complement U-K of K is normal
B
We can restate the "Superiority" condition informally as follows: Superiority: the alternatives K leaves open are all superior to any alternative that K excludes. We can deduce from these conditions something reminiscent of Carnap's "regularity" (or Shimony's "strict coherence"): (A4) F'inesse:
all non-empty subsets of K are normal.
What we now have available as candidate for the 'body of science' is the famil of propositions which are logically implied by a certain privileged family, which Twill characterize as follows: DEFINITION: K is a core proposition (for P) iff K satisfies (A1)-(A3). Note that the a priori pro ositions satisfy (Al) and (A2), thou h definitely not (A3), but rather its op osite. iowever, all the a prioris are among t e propositions implied by these cores. ?n fact:
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If K is a core pro osition then P(KIU) = 1;if K is a core proposition then A P> K iff K implies A; i ? ~is a core proposition and A is a priori then K is a subset of A. To characterize the full body of science we need take into account the extreme ossibility of there being no core propositions. In that case we still want the a prioris to e part of what science says. (Tlus corres onds to what I have called the "Zen rmnds": states of opinion in which notlung is fully {elieved if it is subjectively possible to withhold belief.)
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DEFINITION: A belongs to the body of science (at a stage characterized by P) iff A P> K for some core proposition K; or is no core proposition, and A is a priori for P. i i)ii) there The following conditions are equivalent: (a) A belongs to the body of science (characterized by P) (b) Some roposition J which is either a priori or a core proposition is such
that A$> J
(c) A is implied either by an a priori or by a core proposition (for P) Intuitively we could also render these notions, a bit less rigorous1 , as follows: A core pro osition is a roposition K such that: (a) K and its comp ement are both normal; (by K does not eave open any abnormal alternatives; (c) any alternatives left
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open by K are superior to any alternatives K excludes. We can add: the body of science consists of exactly those propositions which are im lied by core propositions (or are a p~?ov).As before, a roposition is here called an 'a! ternative left open by K exactly if it is non-empty an implies K.
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7. Below the Tip of the Iceberg What is the advantage of thinkin of the bod of science in this way? Its members are all propositions which have abso Ute robabi$ty 1 at that stage (i. e. conditional on the tautolog ), but the converse will not told in eneral We should now hope to add: this body o?science, though possibly infinite an not finitely axipmatizable, is guaranteed to be consistent, and moreover, its members belong to a hierarchical structure which could guide retrenchment if any of its members have to be given up.
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Not every 2-place probability function may in fact give such a nice result if we take it to encapsulate scientific opinion. As usual, formalization will allow the entry of monsters and miscreants among the described flora and fauna. What matters, however, is that we have available the resources for characterizin scientific opionion in such a favorable way, that we are not forced by paradoxes eit er to give up our way of descnbing science, or to declare science to be such as to fit our Procrustean mold.
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in the body of science are clustered: core proppsition K (if there are any at intersection of {A: A P> K), and that these clusters form a chain, If K, K' are belief cores, then either K is a subset of K' or K' is a subset of K. This result is crucial for both the consistency and the hierarchical structure of that body of propositions. Writing K* for the intersection of all the belief cores, we conclude that if A is in the body of science then K* implies A. But is K* itself a core proposition? Does it have 100% robability? Is it even non-em t ? This is the problem of transfinite consistency of ull belief in our new settin . &$at we can prove is that the intersection of a non-empty countable family of belie cores is a core proposition.
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The significance of this result may be challenged by noting that the intersection of countably man sets of measure 1 also has measure 1. So how have we made progress with txe transfinite lottery paradox? In four ways. The first is that in the representation of scientific opinion we may have a "small" family of.belief cores even if robability is continuous and there are uncountably.many propositions with probabi ity 1. The second is that no matter how large a chain is, its intersection is one of its members if it has a first.(= "smallest") element. The third is that the following is a condition t pically met in spaces on which probabilities are defined even in the most scientifical y sophisticated applications:
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(*) Any chain of propositions, linearly ordered by set inclusion, has a countable
subchain with the same intersection.
Of course, if (*) holds and there is at leet,one core proposition, then. (by the above noted result) the intersection of all core propositions is also a core proposition. Fourthly, we will (see below) be able to describe an especially nice class of models ("lex~cograph~c probability"). For these we can prove that the intersection of the,core propositions, if any, is always also a core proposition . There are no countability restrictions there.
8. To What Extent Does Scientific Belief Guide Scientific Opinion? There are models in which there are no core propositions at all. For example, if we take Lebesgue measure m on the unit interval, and tnvially extend it to a
two-place function by P(AIB)=m(AB)/m(B) if defined and P(AIB) = 1 if not (though A,B in domain of m). Then eve unit set (x} is in the domain and is abnormal. Therefore there is no set all of w ose subsets are normal, and hence no core propositions. The absence of core propositions in our resent example derives from its triviality, and not from the continuity. All maxim a? probability propositions are here on a par,with the tautology; this does not seem to me a good candidate for representing scientific opinion. At the other extreme from this example?there is the Very Fine case of a probability function P for which every non-empty set is normal.
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DEFINITION
P is core covered if the union of the core propositions equals U.
In that case, P is Very Fine. For let A be any non-empty proposition; there will be some core proposition K such that KA is not empty, hence normal, thus malung A normal.
in section 3 furnishes us with a relatively simple example of this sort.T he Recall t at P is there constructed from the series p pl, ...,p,,... where the whole robabilit mass of p is concentrated (and evenly distrituted) on the natural numbers P10n ,.... l&+9}. In tgis example, the core propositions are exactly the sets
These core propositions clearly cover U; P is belief covered and Very Fine. Indeed, the core propositions are well-ordered. Define the belief remnants
Clearly pi = P( R.); for example, pl = P( (10,...,191) probabilities conditiona1,on belief remnants (beliefs remaining upon retrenchment to a weaker core) determne all probabilities in this case: P(-IA) = P(-IA n Ri) for the first i such that P(AIRi)>O This says quite clearly that (in this case) belief guides opinion, for robabilities conditional on belief remnants are. so to speak. all the conditional probalilities there are. The body of science, at any iven stage, should in my opinion be viewed as having this sort of structure (even i such a probability function is too complete, recise, and idealized a model to be ver realistic). That is, we should be able to see [ow, within science (and possibl re&tive to additional background beliefs characterizin a given scientific comrnunityy the representable possibilities left open are graded wit{ respect to probabilities, and how essentially the same sort of structure emerges under suppositions which may or may not be contrary to the pro ositions currently unconditionally asserted (let alone those currently having maxima~probability).
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9. A Large Class of Models I will define a class of models such that P satisfies principles 1-11 iff P can be represented by one of these models, in the way to be explained. A model begins with a sample space S=
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if A is in F then FA = {E n A: E in F}; thus FA is a field on A. For each such field designate as PA the set of probability measures defined on FA. (When A is empt FA = (A} and PA is empty.) The restriction o f a member of PA to a subfield with B a subset of A. w ~ l be l designatbe the union of all the sets PA, A in F. ed p F B Finally let
PE
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A model M will consist of a sample space S ,as above, and a function n defined on a subset of F, with range in PS. That is, n associates some probability measure on some subfield with certain progositions (These will be the normal propositions,) I will abbreviate %(A)" to "nA . The function n is subject to the fol owing conditions: (MI) nA(A) is defined and positive. M2 If nB A) is defined and positive, then n A is defined M3 If nB A) is defined and positive, then nAIF(A n B) is proportional to nBIF(A n B).
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This does not entail that if nB(A n B) > 0 then nA(A n B) > 0, because the pro ortionality constant can be 0 (in which case nA ives 0 to all members of F(A n &). It is easy to see what the constant of proportion a? ity has to be: If nB(A) is defined and positive, then nAIF(A n B) : nBIF(A n B) = nA(A n B) : nB(A n B) Finally we define what it means for one of these functions to represent a two-place function: DEFINITION: Model M = <S,n> with S=
If P is a two-place probability measure satisfying the principles 1-11, then P is
represented by a model.
Having established this re resentation result, we now look for easil constructed models, for illustration, re utation of conjectures. and exploration o examples.
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DEFINITION: Model M = <S,n> with S=cU,F> is lexicographic iff there is a sequence (well-ordered class) SEQ of 1-place robabilit measures defined on the whole of F, such that nB(A) = q(A n B)& B) for tKe first member q of the sequence SEQ such that q(B) > 0; nB is unde ined when there is no such q.
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(The term "lexicographc" is used similarly in decision theory literature; see Blume et al., 1989, 1991.) The members of SEQ correspond to the probabdities conditional on belief remnants (see discussion above). We will say that nA comes before nB in SEQ exactly when the first q in SEQ such that (A)>O comes before the first q in SEQ such that (B)>O It is easily checked that = <S,n> is a model. Spefifically, if,A is a subset of B %en nB,will not come after nA, since whatever measure ass1 ns a posiuve value to A wdl then assign one to B., Neither can n* come.after nB if n ~ ( * f>0;in that case nA = nB. Consequently condition (M3) is easdy venfied: the proportionahty constant = 1. It is now v e y eas to make up e~amples,of.2~place probability measures. Just take two or three or indee any number, finite or infinite, of ordinary probability measures and well-order them. We can.readi1 construct lexlcographic models to show that in general not all propositions with pro ability 1 are members of the body of science (i. e. implied by any core pro osition). A special exam le, whose eystence depends on the axlom of choice is ths: et SEQ contsun all one-p ace probablli measures defined on given domain F. In that case, the only abnormal proposition is t e empty set (the self-contradiction). Also the only a prior1 is the tautolog . Short of this, we could of course have a sequence which does .not contain literally a 1 the definable probability measures, but contains all those which give 1 to a given set A. In that c q e , all propositions other than Athat imply A are normal. Let us call P Very Fine on A in such a case. (The case of P Very Fine on U was already called "Very Fine" above.) Note that one of the defining conditions of a core proposition K was that P had to be Very Fine on K.
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In a lexicographic model, the intersection of all core propositions, if any, is always a core proposition too. Since this does not depend on cardinality or the character of the sample space, the result adds significantly to the previous theorems. APPENDIX. Previous Literature
Notes l 4 l technical details for this paper can be found ,in my "Fine- grained opinion, conditional robability, and the logic of full belief", in the Journal of Philosophical Logic (1993. The present adaptation of those results is offered on behalf of the contention that general epistemolo y and scientific methodolo y are each other writ small and writ large respective y (not necessarily in that or er).
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2 ~ propositions y I mean the semantic content of statements; the same proposition can be expressed by many statements. I am not addressing how opinion is stored or communicated. 3 ~ r o mthis point on I shall drop the ubiquitous "for P" unless confusion threatens, and ust write ' apriori", "abnormal", etc, leaving the context to specify the relevant 2-p ace probability measure.
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4 ~ Finetti e (1936); this idea is developed considerably further, with s ecial reference to zero relative fre uency, in Vickers (1988), sections 3.6 and 5.4. he relation here defined is slightly gifferent from the so-named one in my (1979) for convenience in some of the proofs.
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References Birkhoff, G . (1967), Lattice Theory, 3rd ed. Providence: AMS. Blume, L., Brandenburger, A., and Deckel. E. (1991a), "Lexicographic probabilities and choice under uncertainty", Econometrica 59: 61-79. Blume, L. (1991b), "Lexicogra hic probabilities and equilibrium refinements", Econometrica 59: 8 1-9f. De Finetti, B. (1936), "Les probabilites nulles", Bulletin des sciences math&matiques: 275-288. - - - - - - -. (1972), Theory of
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Field, H. (1972), "Logic, meaning and conceptual role", Journal of Philosophy 74: 374-409.
Hajek, A. (1992), "The conditional construal of conditional probability". Ph.D. Diss. Princeton University. Harper, W.L. (1976), "Rational belief chan e, Popper functions, and counter-factuals" in C. Hooker and W. Harper (efs.) Foundations ofProbability Theory, V O ~ .1: 73-112. Kappos, D.A. (1969), Probability Algebras and Stochastic Spaces. New York.
Levi, I. (1980), The Enterprise of Knowledge. Cambridge: MIT Press.
Maher, P. (1990), "Acceptance without Belief' PSA 1990, volume 1, 381-392.
McGee, V. (1994), "Learning the impossible", in (Skyrms, 1994).
Popper, K. (1959), The Logic of Scientific Discovery. New York: Basic Books
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Skyrms, B. and Eels, E. (1995), Probabilities and Conditionals: Believ Revision and Rational Decision. New York: Cambridge University Press, 1994. Vickers, John M. (1988), Chance and Structure. Oxford: Oxford University Press. van Fraassen, B.C. (1979), "Foundations of probability: a modal frequency interpretation", in G. Toraldo di Francia (ed.) Problems in the Foundations of Physics (1979), pp. 344-387.
- - - - - - - - - -. (1981aJ "Probabilistic semantics ob'ectified: I Postulates and logics" Journal of hilosophical Logic 10: 371-494. - - - - - - - - - -. (1981b), "Probabilistic semantics objectified: 11. Im lications in probabilistic model sets" Journal of Philosophical Logic 10: 49 -5 10.
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- - - - - - - - - -. (1995), "Fine-grained opinion, probability, and the logic of belief', Journal of Philosophical Logic, 24 (1995), pp. 349-377.
