Belief Function Semantics for Comparative Con rmation Churn-Jung Liau Institute of Information Science, Academia Sinica, Taipei, Taiwan E-mail:
[email protected]
1 Introduction The logic of con rmation is rst proposed by Hempel to study the con rmatory power of a piece of evidence on a hypothesis[3]. However, a paradox is soon found because according to the logic, an observation of white shoes con rms the hypothesis \ All ravens are black". Though Hempel argued that this is not really a paradox by saying \ The impression of a paradoxical situation is not objectively founded; it is a psychological illusion."([3], p. 18), not many philosophers agree with him on this point. Therefore, some alternative formalisms for con rmation logics have been proposed to resolve the paradox. Some of the most well-known resolutions is the selective con rmation theory proposed by Goodman and re ned by Scheer and its relativized version by Grandy[1, 6, 2, 4]. In selective con rmation theory, a piece of evidence con rms a hypothesis and discon rms its contrary in the same time. On the other hand, Popper[5] considers it impossible to con rm a universal scienti c hypothesis. According to his viewpoint, only falsi cation of such statements is possible. Although these viewpoints are radically di erent, they all concern the relationship between evidence and hypotheses. Therefore, it would be interesting to explore the uniform semantic basis of di erent theories of con rmation. In this paper, we will present a general theory of comparative con rmation based on the belief function semantics. The belief function theory is proposed by Dempster and developed systematically by Shafer later, so also known as DempsterShafer theory[7]. A belief function structure(BFS) consists of two nite sets, called evidence frame and hypothesis frame respectively, and a binary compatibility relation between them. When a probability distribution on the evidence frame is given, we can determine the belief and plausibility measures of the hypotheses in the hypothesis frame. We will show that di erent de nitions of hypothesis frame and compatibility relation lead to di erent con rmation theories. So the general setting will provide a ground to understand the di erence among several con rmation theories.
2 Belief Function Structures and Comparative Con rmation
Let L denote a logic language and Lh and Le be sublanguages of L for describing hypotheses and observation reports respectively. Then a comparative con rmation theory (CCT) is a triplet
(; ; ]), where and are nite subsets of Le and Lh respectively, and ] is a compatibility relation between and . We further require that the sentences in (resp. ) are pairwisely inconsistent under the logic L. For each observation report e 2 , de ne C (e) = fh : e]hg as the compatible hypotheses of e. We assume the closed world assumption[8] that requires C (e) 6= ; for all e 2 . Let : ! [0; 1] be a probability distribution on such that Pe (e) = 1 and Qe (e) > 0. We call a positive probability distribution (ppd). Then the corresponding belief and plausibility measures on 2 can be de ned as 2
Bel(H ) =
2
Xf(e) : C (e) H g
and
P l(H ) = 1 ? Bel(H ): Given two ppds 1 and 2 and an observation report e, we say that 2 is an e-increment of 1 if 2 (e) > 1 (e) and 1 (e ) > 2 (e ) for all e 2 n feg. Then we have the following main 0
0
0
de nition.
De nition 1 Let (; ; ]) be a CCT, e be an observation report in , and H be a subset of .
Then 1. e con rms H i for all ppds 1 and its e-increment 2 Bel2(H ) > Bel1(H ) where Beli is the corresponding belief measure of i (i = 1; 2), 2. e discon rms H i for all ppds 1 and its e-increment 2 P l2(H ) < P l1(H ) where P li is the corresponding plausibility measure of i (i = 1; 2), 3. e is neutral to H otherwise. Note in the de nition above, we do not need to specify the ppds really. What we need to know is just the direction of changes of belief measures w.r.t. e-increments of arbitrary ppds, so the de nition is essentially qualitative though it seems resorting to quantitative probability and belief functions. According to the de nition, a valid sentence can not be con rmed. This seems to violate our intuition at a rst glance. However, it is reasonable since a tautology is not empirically signi cant.
3 Three Examples
In this section, let Lh be a rst order language and Le be a set of ground sentences generated from the predicate symbols of Lh and a nite set of new individual constants. We consider the Hempel's black-raven example. Let us use the following notations. h0 = 8x(Rx Bx) e1 = Ra ^ Ba h1 = 9xRx ^ 8x(Rx Bx) e2 = Ra ^ :Ba h2 = 9xRx ^ 8x(Rx :Bx) e3 = :Ra h3 = 8x:Rx; where a is a constant fresh to Lh .
Example 1 (Hempel's logic) De ne the CCT with = fe1; e2; e3g, = fh0; :h0g, and e]h i e j= hfag, where hfag is the development of h for fag([3], p.36). Then it can be easily veri ed that for all ppds , Bel(h) = P l(h) = 1 ? (e2), so it is obviously e2 discon rms h whereas e1 and e3
con rm h. This is essentially the result of Hempel's logic
Example 2 (Popperian view) The and are de ned as above, but the compatibility relation is now given by the logical consistency notion. That is, e]h i e and h is consistent under L. Then for all ppds , Bel(h) = 0 and P l(h) = 1 ? (e2), so e2 discon rms (or falsi es, in the terminology of Popper[5]) h, and e1 and e3 remain neutral to h. This is a naive Popperian viewpoint. It says that universal statement can only be falsi ed but not be con rmed and existential statement can only be con rmed but not be falsi ed.
Example 3 (Selective con rmation) Let and ] be de ned as in Example 2, but = fh1; h2; h3g. Then for all ppds , Bel(h1) = (e1), P l(h1) = 1 ? (e2), Bel(h2) = (e2), P l(h2 ) = 1 ? (e1 ), Bel(h3) = 0 and P l(h3 ) = (e3 ), so e1 con rms h1 but discon rms h2 and h3, etc.
According to the belief function semantics, we can nd that the di erence between Popperian and Hempelian viewpoint is due to the adoption of di erent compatibility relations, while the one between Popperian viewpoint and selective con rmation is due to the available hypothesis set.
4 Conclusion The belief function semantics for con rmation logic shows that con rmation is not an isolated work. It depends on at least three factors, i.e., the possible observation reports, the available competitive hypotheses, and the de nition of compatibility between evidence and hypothesis frames. The di erent viewpoints of con rmation logicians can be traced back to the di erence of these factors. The details and implications of the proposed semantics will be discussed in the full paper.
References [1] N. Goodman. Fact, Fiction, and Forecast. Bobbs-Merrill, Indianapolis, third edition, 1971. [2] R. E. Grandy. \Some comments on Con rmation ans selective con rmation". In A. Brody, editor, Readings in the Philosophy of Science. Prentice Hall, New York, 1989. [3] C. Hempel. \Studies in the logic of con rmation". In C. Hempel, editor, Aspects of Scien c Explanation, pages 3{46. Free Press, New York, 1965. [4] C. T. Lin. Con rmation Theory, Con rmation Logic and Some Applications. The Bu alo Book Co., Taipei, Taiwan, 1990. [5] K. R. Popper. The Logic of Scienti c Discovery. Harper & Row, New York, 1968. [6] I. Scheer. Anatomy of Inquiry. Alfred A. Knopf, New York, 1963.
[7] G. Shafer. A Mathematical Theory of Evidence. Princeton University Press, 1976. [8] P. Smets. \Belief functions". In P. Smets, A. Mamdani, D. Dubois, and H. Prade, editors, Non-Standard Logics for Automated Reasoning, pages 253{286. Academic Press, 1988.
