No Evidence Without Relevance

At the beginning of this chapter it has been argued that in order for coherence to be indicative of truth in the actual (or some other) world, coherence has to be relativised to this world. This has been done by relativising the coherence of a set of statementsT to an evidenceE, which is assumed to be true in the actual world.

In a certain sense, this is foundationalist coherentism.

As an evidence is in general no complete description of the actual world, coherence w.r.t. the evidence is, properly speaking, not indicative of truth in the actual world – if it is truth indicative at all – but indicative of truth in mod(E).

The idea behind the concept of coherence w.r.t. the evidence can be sketched as follows:

Idea 1 (Informal Characterisation of Coherence w.r.t. E) Two statementsh₁and h₂ cohere with the world or the data, if their conjuntionh₁ ∧h₂ says something about the world or the data which is not already said by one ofh₁, h₂alone.

Two statementsh₁ andh₂ cohere the more with the world or the data, the more their conjuntionh₁∧h₂says about the world or the data which is not already said by one ofh₁, h₂ alone.

This relation of coherence is symmetric in the sense that h₂ and h₁ cohere with the world or the data, ifh₁andh₂do. It is stipulated that two statements logically contradicting each other do not cohere with the world; their degree of coherence w.r.t. the data is minimal. Furthermore, the evidential statements describing data about the world have a special status: They are epistemically distinguished in the sense of assumption 1.4 respectively 4.1. A more difficult question is whether it makes sense to call a single statement coherent with the data.

evidenceEis the smaller, the greater the variety or diversity ofE. As the conditional probability p(T |E)ofT givenEis the greater, the smallerp(E), it follows that, other things being equal, the degree of confirmation ofTbyEis the greater, the greater the variety ofE– the other things beingp(T)andp(E|T).

This is clearly seen in case T logically impliesE, for herep(T |E) = ^p(T)_p(E), which is the greater, the smallerp(E), providedp(T)is held constant.

As already noted, by choosing the “right” prior distribution one can explain nearly everything;

for instance, thatT is more confirmed byE, if the weather is nice than if it is not, for on sunny days one is inclined to assign high priors toT and low priors toE, whereas on rainy days it is the other way round.

The idea of coherence w.r.t. the evidence as informally characterised above is similar to the concept of relevance of Sperber/Wilson (1995) according to which

[a]n assumption is relevant in a context if and only if it has some contextual effect in that context.⁵⁵

Here,

[...] the various types of possible contextual effects [include]: contex-tual implications, strengthenings, and contradictions resulting in the erasure of premises from the context.⁵⁶

The important concept of a contextual implication is defined as follows:

A set of assumptions P contextually implies an assumption Q in the context C if and only if (i) the Union of P and C non-trivially implies Q, (ii) P does not trivially imply Q, and (iii) C does not non-trivially imply Q.⁵⁷

Without restrictions, the idea of above results in triviality in the sense that any two statementsh₁ andh₂ (none of which logically implies the other) cohere, because there is always something the conjunctionh₁ ∧h₂ says which is not already said by one ofh₁, h₂ alone – namely the conjunctionh₁ ∧h₂. In order to avoid this, Sperber/Wilson (1995) restrict the consequences of the union P∪C – in our case:

the consequences of the conjunctionh₁∧h₂ – to non-trivial logical implications involving only elimination rules:

A set of assumptions P logically and non-trivially implies an assump-tion Q if and only if, when P is the set of initial theses in a derivaassump-tion involving only elimination rules, Q belongs to the set of final theses.

Another possibility⁵⁸ is to restrict the consequences of the conjunction h₁ ∧h₂ to relevant (consequence-) elements in the sense of Schurz (1998) respectively Schurz/Weingartner (1987), and to consider

RE(h₁∧h₂)\(RE(h₁)∪RE(h₂)).

55Sperber/Wilson (1995), p. 122.

56Sperber/Wilson (1995), p. 115.

57Sperber/Wilson (1995), p. 107-108.

58Note that restricting the consequences ofh1∧h2to content parts in the sense of Gemes (1994c) and (1997a) is not sufficient, forh1∧h2is a content part ofh1∧h2, for any two statementsh1

andh2. For more on the notion of a content part see below.

A third way of solving the problem that any two statements h₁ and h₂ none of which logically implies the other cohere is not to let it arise at all: This is the case if, for a given statementh, to say something about the world or the data means to account for some entitytmentioned in some evidenceE.⁵⁹

Definition 4.1 (Account) LetT,B, andSbe (not necessarily finite) sets of wffs, let E be an evidence, and let ‘t’ be a constant term occurring in E. T accounts for t respectively ‘t’ in E relative toB iff there is a finite and non-redundant⁶⁰ D⊆D_E(t)and a wffA∈Dsuch that

T ∪B ∪(D\ {A})`A.

The set of all constant terms ‘t’ accounted for byT inErelative toBis called the account of T inE relative toB; it is denoted by ‘A(T, E, B)’.

The set of all constanti-terms ‘tⁱ_l’ inA(T, E, B)∩C_ess(E)for which there is noj < lsuch that

1. T accounts fortⁱ_j inE relative toB, and 2. S∪E `tⁱ_j =tⁱ_l,

is called theS-representative of A(T, E, B). It is denoted by ‘AS−repr(T, E, B)’.⁶¹ If T consists of a single wff h, ‘A(h, E, B)’ and ‘AS−repr(h, E, B)’ are written instead of ‘A({h}, E, B)’ and ‘A_S−repr({h}, E, B)’, respectively.

In order for the problem of above to arise it would have to hold that for any state-ments h₁, h₂ (not logically implying each other), every evidence E, and every background knowledgeBthere is at least one constant term ‘t’∈C(E)such that h₁∧h₂acounts for ‘t’ inErelative toB, buth₁does not, andh₂ does not either.

Clearly, this is not the case – it suffices to give an example of two statementsh₁, h₂ (not logically implying each other), an evidenceE, and a background knowledge B such that it holds for every constant term ‘t’∈ C(E): If h₁∧h₂ accounts for

‘t’ inE relative toB, then so does one ofh₁, h₂alone.⁶² In this sense there is no evidence without conclusion-relevance.

59Insofar as the notion of accounting for is defined in terms of relevant elements, this way is subsidiary to the second one of restricting the consequences to relevant elements.

60Non-redundancy should avoid triviality. Ast =tis a relevant element ofE, for any ‘t’ and anyE, there is always a finite (but redundant) set of relevant elements ofE– namelyD={t=t}

– and a wffA∈D(namelyt=t) such thatT∪B∪(D\ {A})`A.

61The representative should avoid that an entityt with more than one name is counted more than once.

62h1=∀xF x,h2=∀xGx,E={F a, Gb}, andB=∅do the job.

In the following the distinction between the constant terms and the entities denoted by them is handled loosely, if no confusion can arise. Before turning to the measure of coherence w.r.t. the evidence, let me introduce a notion which will provide useful below: Power.

Definition 4.2 (Power) LetT andB be (not necessarily finite) sets of wffs, and letE be an evidence. The power of T forE relative toB,P(T, E, B), is given by the following equation:

P(T, E, B) = |AB−repr(T, E, B)|

|CB−repr(E)| .⁶³

IfT consists of a single wffh, ‘P(h, E, B)’ is written instead of ‘P({h}, E, B)’.

The power functionP is discussed to a greater extent in the chapter on loveliness and likeliness. For the moment it suffices to note thatP is a power searcher which is formally handy for finite sets of statementsT andB.