4.3 Foundationalist Coherentism
4.3.5 Properties of Coh
As is obvious from the examples of the last subsection, the values ofCohdepend heavily on the formulation ofT. In particular,Cohis not closed under equivalence transformations ofT.
Theorem 4.6 (No InvEquTrans ofT forCoh) For every evidenceE, and every set of wffsB there are theoriesT andT0 such that
T a`T0 and Coh(T, E, B)6=Coh(T0, E, B), provided there is at least one theoryT withCoh(T, E, B)6= 0.
Theorem 4.7 (Coherence Versus Power) LetT be a finite set of wffs, letE be an evidence, and letBbe a set of wffs. IfT∪B∪E 6` ⊥andAB−repr(∅, E, B) = whereP is closed under equivalence transformations ofT andB.
Remember,P is not only closed under equivalence transformations ofT andB; it is also searching power and formally handy for finite sets of statementsT andB.
What do the above theorems tell us? There are at least the following four interpretations.
1. The definition of coherence (w.r.t. the evidence) respectively its measure Cohis not adequate – e.g. because the relation of accounting for is mono-tone w.r.t.T andB.65
2. The concept of coherence (w.r.t. the evidence) can be dispensed with. The concept of power (for the evidence) is sufficient and has the advantage that its measure P is closed under equivalence transformations of T, whence theories T may be defined as sets of models mod(T) without restricting oneself to some “canonical” formulation ofT.
3. It does not make sense to call a single statement – as the conjunctionVh∈T h – coherent w.r.t. an evidenceE, because the concept of coherence (w.r.t. an evidence) makes only sense, if several statements (propositions) are consid-ered.
4. The set of statements T whose coherence w.r.t. some evidenceE is to be assessed, has to be formulated in some special, perhaps uniquely determined way.
65That is, ifTaccounts for ‘t’ inErelativeB, then so does everyT0logically implyingT; and T does so relative to everyB0 logically implyingB. This is not the case forE, because`crelis not monotone.
Obviously, (3) and (4) are intimately related. If one considers these four alterna-tives as the only serious interpretations of the above theorems; if one adoptsCoh as measure of coherence w.r.t. the evidence; and if one does not already give up the concept of coherence at this point of the discussion, then (3) and (4) are the only possible alternatives.
How, then, has the set of statementsT be formulated? Intuitively,T should be formulated naturally in the sense of being split up into its smallest (content) parts. There are at least two approaches to this end: The first is based on Schurz’
notion of a relevant (consequence-) element.66
1. The formulation of T has to be an irreducible representation of T, i.e. a non-redundant setIof relevant elements ofT such thatI a`T.
The second is based on Gemes’ notion of a content part.67
2. The formulation of T has to be a natural axiomatization of T, i.e. a finite set of wffsAsuch that
2.1 Aa`T,
2.2 every wffh∈Ais a content part of (the conjunction of all wffs in)A, 2.3 there is no content partch of some wffh∈Asuch thatA\ {h} `ch,
and
2.4 there is no finite set of wffsA0satisfying (2.1)-(2.3) with|A0|>|A|.68 However, if Coh has to be closed under equivalence transformations ofT, then neither (1) nor (2) is viable, for there are theories T1 and T2, evidences E, and background knowledgesBsuch that bothT1andT2are irreducible representations and natural axiomatizations ofT1, and such thatCoh(T1, E, B)6=Coh(T2, E, B) – this is shown byT1 andT2of example (3) of the preceding subsection.69
66Cf. Schurz (1991a), (1998), and Schurz/Weingartner (1987).
67Cf. Gemes (1993), (1994c), and (1997a).
68The fourth clause is added by Gemes in a footnote – cf. Gemes (1993), p. 483. Without it the concept of a natural axiomatization is of no help here, for the conjunctionV
h∈Ahof all wffs of a finite set of wffsAsatisfying clauses (2.1)-(2.3) for a given set of wffsT also satisfies (2.1)-(2.3) forT.
69If ‘↔’ is not eliminated, and the notion of content part is formulated as follows:
For any wffsAandB: Bis a content part ofAiff (i)AandBare contingent, (ii) A ` B, and (iii) there is no wffC such thatA ` C,C ` B,B 6` C, andCis formulated in the vocabulary ofB,
Nevertheless, suppose these cases are only very rare, so that for the most partCoh(T0, E, B) is the same for all irreducible representations or natural ax-iomatizationsT0ofT. Then the measure of coherence w.r.t. the evidenceCohcan be made invariant under equivalence transformations ofT by defining it in one of the following two ways:
Cohirr(T, E, B) = max{Coh(T0, E, B) :T0 ∈I(T)}, Cohna(T, E, B) = max{Coh(T0, E, B) :T0 ∈NA(T)}.
One could, of course, take some other function instead of the maximum func-tion. Note that it would not be of help to consider all sets of statements T0 withT0 a` T, for then one would nearly always – wheneverT is a theory and AB−repr(∅, E, B) = ∅ – consider the singleton {Vh0∈T0h0} containing the con-junction of all statements of some finite axiomatizationT0 ofT. Though this does not hold of the minimum function, a similar problem arises in this case, for one would have to deal with the set of all statements logically following fromT.
A question not yet answered is whether for every set of statements T of LP L1= there is at least one irreducible representation or natural axiomatization of T. For the propositional calculus, the answer is affirmative for irreducible representations: In Schurz/Weingartner (1987) it is shown that for every statement AofLP C there is a statementA0such thatAa`A0 andA0 `crel A0.70
Suppose, however, all these problems can be dealt with in a satisfying way.
Is coherence w.r.t. the evidence under these assumptions indicating truth in the actual world? The answer is no, for it is not even indicating truth inmod(E): For a given evidenceE, there are always theoriesTEand background knowledgesBE such that
E∪BE `TE and Coh(TE, E, BE) = 0
– which violates the second clause of the definition of indicating truth inmod(E).
Still, one may argue that although the feature of interest usually ascribed to coherence is that of being truth indicative, this is not what coherence should do in case of the assessment of theory by evidence relative to background knowledge.
then onlyT5 and T6, but neither T1 norT2 of example (3) are natural axiomatizations of T1. However, for this notion – which is the one Gemes (in personal correspondence) favours –T2and T3of example (4) are sufficient to show thatCohis not closed under equivalence transformations ofT, even ifThas to be a natural axiomatization of itself (based on the notion of content part just stated).T1of this example is an irreducible representation, but no natural axiomatization of itself (in second sense of this footnote).
70Cf. Schurz/Weingartner (1987), p. 58.
Here the job of coherence w.r.t. the evidence is not to indicate truth inmod(E), for some evidenceE, but to be sensitive to loveliness and likeliness in the sense of some power searcher LO and some truth indicator LI. However, this does not either hold of Coh– even if it is assumed that there is exactly one canonical formulationFT for every set of statementsT.
Theorem 4.8 (No SensLoveLike ofCoh) For every power searcher LO, every truth indicator LI, and every evidenceE there is a theory TE and a background knowledgeBE such that it holds for any sets of wffsT andB, and every evidence E0: IfT a`TE,E0 a`E, andB a`BE, then
1. T ∪B `E0, and thusLO(T, E0, B) = 1, 2. E0 ∪B `T, and thusLI(T, E0, B) = 1, and 3. Coh(T, E0, B) = 0.
I conclude that if Coh captures to some extent the concept of coherence (w.r.t.
the evidence), then the latter has to be given up as indicator of truth in the actual world, provided the second clause of the definition of indicating truth inmod(E) is adopted as minimal requirement for any truth indictorf: If evidenceEtogether with background knowledge B guarantees (in the sense of logical implication) the truth of some theoryT, then the degree to whichf indicates the truth ofT in mod(E)relative toB is maximal.
I conclude further thatCohis no adequate measure of confirmation, because it is not sensitive to loveliness and likeliness in the sense of any power searcher LO and any truth indicator LI – even if it is assumed that there is exactly one canonical formulationFT for every set of statementsT.
In the next chapter I will therefore pursue the second approach to a solution of the problem of a quantitative theory of confirmation: First, to define for ev-ery (primary) confirmational virtueV a functionfV (·,·,·)such thatfV (T, E, B) measures the degree to which (primary) confirmational virtue V is exhibited by T, E, andB, for every theoryT, every evidenceE, and every background knowl-edgeB; and then to define the measure of confirmationC as a function of (some of) these functionsfV.
Chapter 5
Loveliness and Likeliness
5.1 Recapitulation
This chapter contains the definition of a power searcherP(·,·,·),P(·,·,·) :T × E × B → <, and a truth indicatorLI(·,·,·),LI(·,·,·) :T × E × B → <, which together determine the measure of confirmationC(·,·,·).
We know from theorem 3.1 that there are lots of power searchers and truth indicators. As has been noted there, what is needed are a power searcher and a truth indicator which are formally handy, i.e. non-arbitrary, comprehensible, and computable in the limit. Arbitrariness will be avoided by defining two single functions (without parameters that can be chosen arbitrarily); comprehensibility will be achieved by purely syntactical definitions in the terms ofP L1 =andZF; computability in the limit will be a consequence of these definitions.
Before defining these functions remember that non-arbitrariness, compre-hensibility, and computability in the limit are formal conditions of adequacy for any formal theory, in particular, any quantitative theory of confirmation intended to implicitely provide a rule of acceptance for rational theory choice. Sensitivity to (and only to) the confirmational virtues is a material condition of adequacy for any quantitative theory of confirmation. The second approach to a solution of the problem of a quantitative theory of confirmation defines the measure of confirmationC as a function of (some of) the functionsfV (·,·,·)measuring the confirmational virtuesV. The formal conditions of adequacy for these functions are inherited from those for C. The material conditions of adequacy are those of chapter 3: The functionfLOwhich measures loveliness has to be a power searcher;
the functionfLI which measures likeliness has to be a truth indicator.
107
There will not be any functions for the derived confirmational virtues. These enter only when the resulting measure of confirmation C is considered. C has to be sensitive to the primary and derived confirmational virtues, i.e. sensitive to loveliness and likeliness in the sense of some power searcher LO and some truth indicatorLI – sensitivity to simplicity considerations and unimpressability by redundancies (and invariance under equivalence transformations of T) being consequences of this.
In the next chapterCis combined with a functionG(·,·,·), whereG(T, E, B) measures the “goodness” of evidenceE in relation to theory T and background knowledgeB. The reason for this is thatC does not and is not intended to take into account that evidence which is varied or diverse is better than evidence which is uniform or homogenous; and thatE is the better, the more information it con-tains. C(T, E, B)only tells you how much T is confirmed byE relative to B, ifE is all the evidence available. The refined measure of confirmationC∗(·,·,·) which is the result of combiningCandGcan be shown to be sensitive to diversity considerations in the sense ofC andG.1 Before continuing, let me note thatGis independent ofC, and may be combined with any measure of confirmation – or coherence (w.r.t. the evidence).