In the foregoing sections I have appealed to an intuitive understanding of the love-liness or power a theoryT in relation to an evidenceE and a background knowl-edgeB. I think that any adequate measure LO(T, E, B)of the loveliness of T andB forE should be searching power in the following sense:
Definition 3.1 (Searching Power) LetE be an evidence. A functionf(·, E,·), f(·, E,·) : T ×E × B → <, is searching power for mod(E) iff it holds for any theoriesT and T0, and every background knowledgeB: IfT ∪B 6` ⊥ and T0 ∪B 6` ⊥, then
1. f(T, E, B)≥0,
2. ifT ∪B `E, thenf(T, E, B) = 1, and 3. ifT0 `T, thenf(T0, E, B)≥f(T, E, B).
A functionf(·,·,·),f(·,·,·) :T × E × B → <,is a power searcher ifff(·, E,·) is searching power formod(E), for every evidenceE.
The notion of searching power can, of course, be generalised to any sets of state-mentsT, E, andB respectively functionsf with domains℘(LP L1=). However, it will turn out that the restriction to theories T, evidences E, and background knowledgesB is necessary in order for several theorems to hold.
The first and second condition set lower and upper bounds, respectively, for the values a power searcher can take on, where the second condition in addition tells one that the power ofT forErelative toB is maximal, ifT andB guarantee (in the sense of logical implication) that E is true. The third condition is a con-dition of monotonicity saying that the power ofT0 for E relative to B is greater than or equal to the power ofT forE relative to B, ifT is logically implied by T0. That is, power or loveliness increases with logical strength.
A consequence of the third condition is that every power searcher LO is closed under equivalence transformations ofT. More precisely:
IfT ∪B 6` ⊥andT0 a`T, thenf(T, E, B) = f(T0, E, B),
for any theories T, T0, every evidence E, every background knowledge B, and every power searcherLO(·, E,·)formod(E).
The intuitive understanding of likeliness I have appealed to in the last sec-tion is made precise by demanding of any measureLI(T, E, B)of the likeliness
of theoryT relative to evidenceE and background knowledgeB to be indicating truth in the following sense:
Definition 3.2 (Indicating Truth) Let E be an evidence. A function f(·, E,·), f(·, E,·) : T ×E× B → <, is indicating truth inmod(E)iff it holds for any theoriesT andT0, and every background knowledgeB: IfE∪B 6` ⊥, then
1. f(T, E, B)≥0,
2. ifE∪B `T, thenf(T, E, B) = 1, and 3. ifT0 `T, thenf(T0, E, B)≤f(T, E, B).
A functionf(·,·,·),f(·,·,·) :T × E × B → <, is a truth indicator ifff(·, E,·) is indicating truth inmod(E), for every evidenceE.
This definition can, of course, also be generalised to any sets of statements. As mentioned before, the restriction to theories T, evidences E, and background knowledges B is necessary in order for several theorems to hold. In particular, this is the case for the truth indicativeness of the likeliness functionLI presented in chapter 5.
The first and second conditions set again lower and upper bounds, respec-tively, for the values a truth indicator can take on, where the second condition in addition tells one that the likeliness of T relative to E and B is maximal, if E and B guarantee the truth ofT. As in the previous case, the third condition is a condition of monotonicity saying that the likeliness of T relative to E andB is greater than or equal to the likeliness of T0 relative to E and B, if T0 logically impliesT. In other words, likeliness decreases with logical strength.10
A consequence of the third condition is that every truth indicator LI is closed under equivalence transformations ofT.
There are many power searchers and truth indicators.
Theorem 3.1 (Power Searcher and Truth Indicator) LetT,E, andBrange over wffs ofLprop(instead of theories, evidences, and background knowlegdes, respec-tively, which are sets of wffs ofLP L1=) in the definitions of searching power and
10One might want to add the condition that likeliness increases with the logical strength of the background knowledgeB, i.e.
if B0`B, then LI(T, E, B0)≥ LI(T, E, B).
In my opinion this is inadequate, because new background information may even lead to the refu-tation of a theory. A similar remark applies to the definition of searching power.
indicating truth. Then it holds for every contingent wffEand every strict (uncon-ditional) probabilityp(·):
1. p(· |E∧ ·)is indicating truth inmod(E).
2. i(·, E,·) := 1−p(· ∧ · | ¬E)is searching power formod(E).
3. i0(·, E,·) := 1 −p(· | ¬E∧ ·) is searching power for mod(E), if it is defined, i.e. if¬E∧B 6` ⊥.
What is needed are not only two functionsLOandLI which are searching power and indicating truth, respectively. In addition these functions have to be formally handy, i.e. non-arbitrary, comprehensible, and computable in the limit. Arbitrari-ness will be avoided by defining two single functions; comprehensibility will be achieved by purely syntactical definitions in the terms of P L1 = and ZF; and computability in the limit will be a consequence of these definitions.
Let me stress that the measure of confirmationCshould not be both search-ing power and indicatsearch-ing truth, for such functions are constant.
Theorem 3.2 (Truth Indicating Power Searchers Are Constant) Let E be an evidence, and letf(·, E,·), f(·, E,·) : T ×E × B → <, be searching power formod(E).
Iff(·, E, B)is indicating truth inmod(E), then it holds for every theoryT and every background knowledgeB withE∪B 6` ⊥:f(T, E, B) = 1.
The measure of confirmationCshould be sensitive to loveliness and likeliness; it should balance between these two conflicting concepts of confirmation.
If the likeliness ofT relative toE andB equals the likeliness ofT0relative toE0 andB0, then the degree of confirmationC(T, E, B) ofT by E relative to B should be greater than the degree of confirmation C(T0, E0, B0) of T0 by E0 relative toB0just in case the loveliness or power ofT andBforEis greater than the loveliness or power ofT0andB0forE0. Similarly, if the loveliness or power in the first case is equal to the loveliness or power in the second case, then the degree of confirmation should be greater in the first case if and only if the likeliness is.
Furthermore, confirmation should be minimal just in case loveliness or likeliness is minimal; and it should be maximal if and only if both are maximal. This is expressed in the following definition.
Definition 3.3 (Sensitivity to Loveliness and Likeliness) LetLO(·,·,·),LO(·,·,·) : T × E × B → <, be a power searcher, and letLI(·,·,·),LI(·,·,·) :T × E × B →
<, be a truth indicator.
A functionf(·,·,·), f(·,·,·) : T × E × B → <, is sensitive to loveliness and likeliness in the sense of LO and LI iff it holds for any theories T and T0, any evidences E andE0, and any background knowledgesB and B0, where X =hT, E, BiandX0 =hT0, E0, B0i:
1. IfLI(X) =LI(X0)6= 0, thenf(X)≥f(X0)iffLO(X)≥ LO(X0), 2. ifLO(X) =LO(X0)6= 0, thenf(X)≥f(X0)iffLI(X)≥ LI(X0),11 3. f(X) = 0iffLO(X) = 0orLI(X) = 0, and
4. f(X) = 1iffLO(X) = 1andLI(X) = 1.
It is straightforward that sensitivity to loveliness and likeliness in the sense of some power searcherLOand some truth indicatorLI is sufficient for invariance under equivalence transformations ofT.