A Power Searcher and a Truth Indicator

The basic ideas behind the definitions of the functionsP andLI are due to Carl Gustav Hempel, and can be found in his Studies in the Logic of Confirmation (1945) under the headings of the prediction criterion and the satisfaction criterion, respectively.

It is crucial that these functions are only defined, if the evidential domains and the domains of proper investigation overlap. Any domain which is among both is called a confirmational domain (ofT andE). Though the definitions are stated in semantic terms, they are purely syntactic, because the domains are only distinguished by means of the different sorts of variables and constants occurring inT,E, andB.

The evidential domains and the domains of proper investigation overlap whenever there is an essential occurrence of ani-variable, but no occurrence of a constanti-term inT, and no occurrence of ani-variable, but a constanti-term es-sentially occurring inE, for some sortiof variables and constants. The domains

1For a definition of the concept of sensitivity to diversity considerations in the sense of some functionsC_LO,LIandGsee chapter 6.

of T, E, and B are also only distinguished by the sorts of variables and con-stants occurring in T, E, and B. Strictly speaking, these “domains” are domain variables taking domains as their values.

So referring to the domains of T, E, and B, D₁, . . . , D_n, is just another way of referring ton different sorts of variables and constants occurring inT, E, andB. The functionsP andLI therefore have three argument places; the confir-mational domains are uniquely determined byT andE. Technically, the valueP takes on for givenT,E, andB, is a vector whose length equals the number of

con-firmational domains ofT andE, and is of the form:hP(T, E, B;D₁), . . . ,P(T, E, B;D_c)i, whereD₁, . . . , D_c are the confirmational domains ofT andE. The claim thatP

is a power searcher means that conditions (1)-(3) in the definition of searching power formod(E)are satisfied by P(T, E, B;D_i)for every confirmational do-main D_i of T and E, for allT, E, and B. Similar remarks apply to LI and its being a truth indicator.

The functionP is already familiar from the chapter on coherence w.r.t. the evidence.

Definition 5.1 (Confirmational Domain) LetT be a theory with domains of proper investigationD₁^T, . . . , D^T_m, letE be an evidence fromD₁^E, . . . , D_n^E, and letD_ibe a domain (with correspondingi-variables and constanti-terms).

D_i is a confirmational domain of T and E iff D_i is among both the evi-dential domains of E, D^E₁, . . . , D^E_n, and the domains of proper investigation of T, D₁^T, . . . , D^T_m; i.e. iffT contains an essential occurrence of ani-variable, but no occurrence of a constant i-term, and E contains an essential occurrence of a constanti-term, but no occurrence of ani-variable.

Definition 5.2 (Power) LetT be a theory, letE be an evidence, letB be a back-ground knowledge, and letD_i be a confirmational domain ofT andE (with cor-respondingi-variables and constanti-terms).

The power of T for E relative toB inDi,P(T, E, B;Di), is given by the following equation:

P(T, E, B;D_i) = |AB−repr(T, E, B)∩C_i|

|CB−repr(E)∩Ci| , whereC_i is the set of constanti-terms.²

2C_B−repr(E)∩Ci6=∅, becauseDiis a confirmational domain ofTandE, and hence among the evidential domains ofE.

The functionLI has not been dealt with so far.

Definition 5.3 (Likeliness) Let T be a theory, let E be an evidence, let B be a background knowledge, and letD_i be a confirmational domain ofT andE (with correspondingi-variables and constanti-terms).

The likeliness of T w.r.t. E andB inD_i,LI(T, E, B;D_i), is given by the following equation:

LI(T, E, B;D_i) = maxLI(T, E, B;D_i)

|CB−repr(E)∩Ci| ,³ providedE∪B 6` ⊥, where

maxLI (T, E, B;D_i) := max{|C∩CB−repr(E)|:C ⊆C_E,B,i, E `Dev_CE,B,i(B)→Dev_C(T)^o,

C_E,B,i := C(E∪B)∩C_i = C_i(E∪B), C_i(X)is the set of constant i-terms occurring inX, andC_i is the set of constanti-terms.

Concerning likeliness in domainD_i, it is important to note that only thei-variables in T are replaced by the constant i-terms of C in the development of T for C, Dev_C(T); thek-variables,k 6=i, occurring inT and the quantifiers binding them remain unchanged (cf. definition 1.11).

The following theorems yield thatP andLI satisfy the formal and material conditions of adequacy.

Theorem 5.1 (P Is a Formally Handy Power Searcher) P(·,·,·),P(·,·,·) :T × E ×B → <, is a power searcher which is non-arbitrary, comprehensible, and com-putable in the limit, provided for everyE ∈ E and every ‘t’∈ C_ess(E)there is a contingent⁴ A∈RE(E)with ‘t’∈C(A).

More precisely, P is formally handy, and for any theories T andT⁰, every evidence E, every background knowledge B, and every confirmational domain D_i ofT andE, and ofT⁰ andE:

1. P(T, E, B;D_i)≥0,

2. ifT ∪B `E, thenP(T, E, B;D_i) = 1, and

3Cf. the preceding footnote.

4Contingency should rule outt=t, which is a relevant consequence of anyE.

3. ifT⁰ `T, thenP(T⁰, E, B;D_i)≥ P(T, E, B;D_i),

provided for every E ∈ E and every ‘t’ ∈ C_ess(E) there is a contingent A ∈ RE(E)with ‘t’∈C(A).

Theorem 5.2 (LI Is a Formally Handy Truth Indicator) LI(·,·,·),LI(·,·,·) : T × E × B → <, is a truth indicator which is non-arbitrary, comprehensible, and computable in the limit.

More precisely,LI is formally handy, and for any theoriesT andT⁰, every evidence E, every background knowledge B, and every confirmational domain Di ofT andE, and ofT⁰ andE: IfE∪B 6` ⊥, then

1. LI(T, E, B;Di)≥0,

2. ifE∪B `T, thenLI(T, E, B;Di) = 1, and

3. ifT⁰ `T, thenLI(T⁰, E, B;D_i)≤ LI(T, E, B;D_i).

If the proviso in theorem 5.1 does not hold for some constant term ‘t’∈CB−repr(E), for someE andB, then noT can account for ‘t’ inE relative toB. The proviso is satisfied, if (1)RE(E)`E; or if (2)Eis minimally observational in the sense that for every ‘t’ essentially occurring inE(and thus for every ‘t’∈CB−repr(E)) there is at least one contingent statement Acontaining only one predicate occur-rence such that ‘t’ ∈ C(A) and E ` A. (Any such statement A is a relevant element of any evidence E logically implying A. There is just one predicate occurrence that can be replaced, whence substituting a logically determined pred-icate for it would yieldE inconsistent.)

The term ‘minimally observational’ arises from the following consideration:

One may define an evidence to be observational just in case it consists only of (possibly negated) atomic statements, because – so it may be argued – we do not observe (negative or) disjunctive properties, but only whether some entity has a property (whether some entities stand in some relation); disjunctive (and negative) properties are not observed, but inferred.

Any (possibly negated) atomic statement has only one predicate occurrence, and thus is of the required form. But other statements – e.g. Popperian Basissätze of the form ‘At space-time point kthere arex1, . . . , xn such thatA[x1, . . . , xn]’⁵ – do not have the form of (possibly negated) atomic statements; nor do they imply

5Cf. Popper (1994), p. 66ff.

such statements. However, they usually entail a statement with just one predicate occurrence.

The proviso is superfluous, if CB−repr(E) is restricted to those constant terms ‘t’ for which there is at least one contingent relevant elementAofE with

‘t’ ∈ C(A). The reason for not doing so is that I conjecture that the proviso is satisfied anyway – for lack of mathematical skill I just cannot prove it.