The last example does not only show that there are theoriesT, evidencesE, and background knowledgesB such that C(T, E, B) = 1. It also illustrates that the degree of confirmation is determined by the proportion of those constant terms ‘t’
which – in case ofP – are in the account ofT inE relative to B to all constant terms ‘t’∈ CB−repr(E); and similarly forLI. The size of the evidenceE in the sense of the cardinality ofCB−repr(E)and its variety or diversity do not matter for power, likeliness, and confirmation.
Is this a point against the approach presented here? I think it is not. The measure of confirmationCdoes not – and is not intended to – measure the overall support there is for a given theory.C(T, E, B)tells us how muchT is confirmed given thatE is all the evidence available andB is the whole background knowl-edge. Whether there is a lot of overall support forT does not only depend on its degree of confirmation byE relative toB; it additionally depends on whetherE is good evidence.
What follows? Do we have to rely on some principle of total evidence telling us that in assessing a given theoryT we always have to consider the total available evidence E, and the total available background knowledge B – at least if we expect the measure of confirmationC to implicitely provide a rule of acceptance for rational theory choice?
I do not think so. IfCis not to provide a rule of acceptance for rational the-ory choice, then allC(T, E, B)is expected to tell us is how muchT is confirmed
by E relative to B – and this it does. If, however, C is to provide such a rule of acceptance for rational theory choice, then there are two possibilities: Either the problem situations this rule is to handle are of the type described in chapter 2, in which caseC does its job, for these problem situations are relative to some evidenceE (and some background knowledgeB). Or else the problem situations this rule is to handle are not relative to some E (and B), but ask which theory to accept independently of the evidence under consideration (and the background knowledge taken for granted).
If, among others, the theory to be chosen should be true in the actual world – and I take this to be one of the features we aim at – then every such problem situation has to be understood as asking which theory to accept with regard to a complete description of the actual world (or at least the total available evidence).
For, after all, (sorry, I am repeating myself), truth is a binary relation between a (set of) statement(s) on the one hand and a world or model on the other, and thus cannot be taken into account without recourse to the world or model whose truth in one is interested in. However, establishing this link is just the purpose of the evidence, and the reason why it is assumed to be true in the actual world. So every problem situation of the latter kind is relative to a complete description of the actual world, whence the second type of problem situation is only a special kind of the first one.
There is a peculiarity of demanding to consider a complete description of the actual world or the total available evidence. The idea behind a rule of accpetance for rational theory choice is to be a guide in deciding which theory to accept with regard to a given evidence and a given background knowledge. If the answer to this question demands of us to collect all the data there are, or even to consider a complete description of the actual world, then we will never be in the position to apply this rule, for we will never have collected all the data there are – nor will we ever possess a complete description of the actual world. So if a rule of acceptance for rational theory choice is to be meaningfully combined with a principle of total evidence, then all this principle can demand of us is to consider all the evidence that is practically available (at a given point of time). The question is whether there is not a better strategy for dealing with all this.
I think there is: In assessing a given theoryT relative to some evidenceE and some background knowledge B, one has to consider not only the degree of confirmation ofT byE relative toB, but must also take into account the “good-ness” of the evidenceE. What the latter consists of, and how it can be measured, is the topic of the last chapter.
Chapter 6
Variety and Goodness of the Evidence
6.1 Introductory Remarks
As already mentioned the measure of confirmation C does not tell us anything about the overall degree of confirmation of a theory, which additionally depends on the “goodness” of the evidence. Similarly, the reliability of the rule (R) of acceptance for rational theory choice of chapter 2 depends not only on the degree of confirmation, but also on the goodness of the evidence, which I take to consist in its size and its variety or diversity.
In this chapter a functionG(·,·,·)is defined on the set of all evidences E, the set of all theories T, and the set of all background knowledges B, and it is argued that, for a given evidence E, G(T, E, B) measures the goodness of E relative to theory T and background knowledge B in the sense of the formers size and variety (diversity). I will reason that the refined measure of confirmation C∗, which is based on C and G, gives an answer to the question why scientists (should) gather evidence, and that it resolves the ravens paradox. The chapter ends with some comments on the reliability of truth indicators.
Intuitively, an evidence is the better, the more data it reports about, the more different classes of facts it consists of, the greater these classes of facts are, the more detailed or accurate they are described, and the more they differ from each other. The concept of evidential diversity or variety of evidence thus clearly de-pends on the notion of a class of facts, in particular, on when two classes of facts count as different ones, on when they (are big and) described in detail, and on
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when two classes of facts differ more from each other than two other ones. A class of facts is construed as a set of individuals mentioned in some evidenceE – respectively a set of constant terms occurring inE– because I take individuals to be ontologically fundamental.
Whether two classes of facts count as different ones depends on the hypoth-esis or theory one is concerned with. Therefore determining the goodness of an evidence E – by determining the number, size, accuracy, and difference of the classes of factsE consists of – involves considering the theory in question. The background knowledgeB has to say something, too, whence the measure of the goodness of evidenceG is construed as a function with three argument places.
For convenience, a fourth argument place is added for the confirmational domains the individuals in the various classes of facts are taken from; but again, strictly speaking the value ofGfor givenT,E, andB is a vector whose length equals the number of confirmational domains ofT andE.
Let us consider why the notion of a class of facts has to be relativised to the hypothesis or theory under consideration. Relative to a theory that claims to account for the colour of people’s hair a black haired man and a black haired woman belong to the same class of facts, whereas a black haired woman and a red haired woman belong to two different classes of facts. On the other hand, relative to a theory about the sexual behaviour of humans the black haired man and the black haired woman belong to two different classes of facts, whereas the black haired woman and the red haired woman belong to the same class of facts – the reason being that the colour of humans’ hair is irrelevant for their sexual behaviour, but relevant for the colour of their hair, whereas the sex of humans is irrelevant for the colour of their hair, but relevant for their sexual behaviour.
Furthermore, enlarging the data may yield that two individuals which belong to the same class of facts relative to a given theory in the old evidence belong to two different classes of facts relative to the same theory in the enlarged evidence, because the new data may be relevant for this theory. For instance, by taking into account the age of humans the black haired and the red haired woman of before, which belong to the same class of facts relative to the theory about the sexual behaviour in the old evidence, will no longer belong to the same class of facts relative to this theory, because their age, which is assumed to be very different, is relevant for and will make a difference in their sexual behaviour.
6.2 (Maximal) Classes of Facts
A class of facts is construed as a set of individuals mentioned in some evidence E. The information we have about these individuals, and on which we can rely in classifying them, is contained in E and the background knowledgeB. Since I want to classify single individuals, and not wholen-tupels, I have to consider one-place predicates instead ofn-ary ones. Finally, as is familiar by now, individuals enter the scence via their names, the constant terms inCB−repr(E).
Let us first consider ann-ary predicate ‘P (x1, . . . , xn)’, where all variables
‘xi’ are of the same sort. Such a predicate gives rise to2n−1·none-place predicates Q1x1. . . Qi−1xi−1Qi+1xi+1. . . QnxnP (x1, . . . , xi−1, x, xi+1, . . . , xn), 1 ≤ i ≤ n, where Qj is an existential quantifier ∃ or a univeral quantifier ∀, 1 ≤ j 6=i ≤ n. By rearranging these quantifiers (changing their order) one gets 2n−1·(n−1)!·n= 2n−1·n!one place predicates from ann-ary predicate (some of them denote the same property, because the order of the quantifiers does not always matter).
Binding argument places with quantifiers is not the only way to get one-place predicates fromn-ary ones. In combination with a set of constant terms ‘c1’, . . ., ‘cm’ of the appropriate sort, ‘P(x1, . . . , xn)’ gives rise tomn−1·npredicates quantifiers occur vacuously. By rearranging the n−1−r quantifiers occurring non-vacuously one thus gets
one-place predicates out of one singlen-ary predicate.
Things get more complicated, when one considers different sorts. I will not show how to get one-place i-predicates ‘P xi’ out of n-ary k1, . . . , kn-predicates
‘Pxk1, . . . , xkn’ and various sets of constanti-terms. I hope the above is suffi-cient to show that this can be done, and that the result is a finite set of one-place i-predicates.
I have argued that whether two individuals belong to the same class of facts depends on the theory under consideration. This appears in the definition of a class of facts by taking as the set of predicatesP Rthe set of predicatesP Ress(T) essentially occurring in theoryT which is to be assessed byE relative toB.1
The restriction to the predicates essentially occurring inT is necessary, be-cause otherwise the set of predicates P R can be chosen arbitrarily (by adding hypotheses which are logically valid and contain occurrences of the predicates one wants to have added).
The ratio behind takingP Ress(T)is that if two individualstandt0(should) belong to two different classes of facts as far as some theoryT is concerned, then this must be due to some property of t that t0 does not have. If every property that can be expressed in terms of the predicates essentially occurring inT is either possessed by botht andt0 or by none of them, then all properties distinguishing betweentandt0are irrelevant forT, whencetandt0cannot belong to two different classes of facts as far asT is concerned. So the predicates essentially occurring in T settle the relevant conceptual space for the classification of the individuals the evidence is talking about.
In the example of before, the predicates ‘male’ and ‘female’ are among the predicates essentially occurring in the theory about the sexual behaviour of humans, whereas the predicates ‘black haired’ and ‘red haired’ do not belong to the essential vocabulary of this theory. Therefore the black haired man and the black haired woman can be distinguished by means of the conceptual framework of this theory, but not by the conceptual framework of the hair colour theory.
All we can rely on in determining the size and the variety – the goodness – of evidenceE, is contained inE or the background knowledgeB. In particular, the
1SoP Ris empty, ifT is logically determined.
information that there is a propertyP theoryT is talking about which is possessed by individual t, but not by individual t0, must be obtained fromE and B. In the definition of a class of facts, this finds its expression in considering whether E and B logically imply P t, where ‘P x’ ∈ P R1, ‘t’ ∈ CB−repr(E), and P R1 is the set of one-place predicates the set of predicates P Ress(T) gives rise to in combination with the constant terms inCB−repr(E).
Based on these considerations we can now define the notion of a (maximal) class of facts.
Definition 6.1 ((Maximal) Class of Facts) Let T be a theory, let E be an evi-dence, let B be a background knowledge, and letDi be a confirmational domain ofT andE (soCB−repr(E)∩Ci is not empty). Let
P R =P Ress(T) = \
T0a`T
P R(T0),
and let ‘P1’ = ‘P1n1xk1, . . . , xkn1’,. . ., ‘Pp’ = ‘Ppnpxk1, . . . , xknp’ be an enu-meration of the predicates inP R, wherep=|P R|. LetP R1i be the set of all one-place i-predicates which result from any of the following one-place i-predicates by rearranging the quantifiers:
Q1xk1. . . Ql−1xkl−1Ql+1xkl+1. . . QnqxknqPqtk1, . . . , tkl−1, xkl, tkl+1, . . . , tknq, 1 ≤ l ≤ nq, where ‘tkj’ = ‘xkj’ or ‘tkj’ ∈ CB−repr(E)∩Ckj (Ckj is the set of constant kj-terms), 1 ≤ j 6= l ≤ n, ‘Pq’ ∈ P R, 1 ≤ q ≤ p, and ‘xkl = xi’ (otherwise one does not geti-predicates).
Let P Ri1 be partitioned into N := 2|P Ri1| sets C1i, . . . , CNi of negated or unnegated one-placei-predicates such that it holds for every suchCji,1≤j ≤N, and every one-placei-predicate ‘P’∈P Ri1:
‘P’ ∈Cji iff ‘¬P’ 6∈Cji. For each of theseN setsCji, letCji
k be thek-th subset ofCji in some enumeration Cji1, . . . , CjiN of itsN subsets. Let ‘t’∈CB−repr(E)∩Ci.
‘t’ respectivelyt belongs toCjik iff it holds for every negated or unnegated one-placei-predicate ‘±P’∈Cji
k,E∪B ` ±P t.
‘t’ respectivelytbelongs maximally toCji
k iff 1. ‘t’ belongs toCji
k, and
2. there is no Cli, 1 ≤ l ≤ N, for which there is at least one Clip ⊆ Cli, The set of all (maximal) classes ofi-factsCFji
k induced byCji
k relative toT, E, andB, for anyjandk,1≤j, k ≤N, is the set of (maximal) classes ofi-factsT, E, andB give rise to.
Let CFji
k be the (maximal) class of i-facts induced by Cji
k relative to T, E, andB, for some set of negated or unnegated one-placei-predicatesCjik, 1 ≤ j, k ≤N.
CFji
k is a non-empty (maximal) class ofi-facts relative toT, E, andB iff CFjik 6= ∅. Otherwise CFjik is an empty (maximal) class of i-facts relative toT, E, andB.
2It is important to demand that there be no such proper superset ofCji
k. Demanding that there be noCli
pwithCli
p `CjikandCjik 6`Clphas the consequence that there may be constanti-terms
‘t’ belonging maximally to more than one set of negated or unnegated one-placei-predicates.
For the definition given, maximal classes ofi-facts are disjoint. For suppose there is a constant i-term ‘t’∈CB−repr(E)∩Cithat belongs maximally to at least two different sets of negated or unnegated one-placei-predicatesC1andC2. Then
E∪B ` ±P t, for every ‘±P’ ∈C1∪C2,
Here the logical consequence relation`between such setsCji
kandCli
pof negated or unnegated one-placei-predicates holds iff this relation holds between the sets
±P t:‘±P’∈Cji
k and
n±P t:‘±P’∈Cli
p
o
, where ‘t’ is a constanti-term.