Basic and Refined Nomic Truth Approximation by Evidence-Guided Belief Set Revision

Abstract

Straightforward theory revision, taking into account as effectively as possible the established nomic possibilities and, on their basis induced empirical laws, is conducive for nomic truth approximation. The question this paper asks is: is it possible to reconstruct the relevant theory revision steps, on the basis of incoming evidence, in the terms of so-called belief revision and, more specifically, in the terms of belief set revision, also called the AGM-approach? Assumin g exclusion theories, a positive answer will be given in two rounds, first for the case in which the initial theory is compatible with the established empirical laws, then for the case in which it is incompatible with at least one such a law.

Strongly revised version of: “Basic and refined nomic truth approximation by evidence-guided belief revision in AGM-terms ”, Erkenntnis, 75, 2, 2011, pp. 223–236. Acknowledgements: The author wishes to thank Roberto Festa for the occasion to present this paper in 2009 at a symposium in Trieste, Gustavo Cevolani , Gerhard Schurz and Wlodek Rabinowicz for very useful comments, and the Netherlands Institute for Advanced Study (NIAS), Wassenaar, for providing again paradisiacal conditions for research and writing, including linguistic correction by Anne Simpson .

Appendix

It must be noted that the set-theoretic part of the refined approach sounds more intuitive when we take the original and refined theories as inclusion or representation theories. However, it does not seem plausible to give a belief revision interpretation in AGM-terms of them. In the next chapter we show, in a very general way, that the belief base approach to belief revision (due to Sven Owe Hansson ) leaves perfectly room for at least a basic account for two-sided theories , and hence for the extreme cases of purely exclusion and purely inclusion theories.

For completeness, we like to tell here the story that results when we transform (the formulations of) the last part of Sect. 14.3 in terms of inclusion theories, indicated by changing symbols from P, P* and P^r _R(S) to M, M* and M^r _R(S). Recall that an inclusion theory M makes the claim “M⊆T”, but we will not make a notational difference between the set and the theory. The definition of refined truth approximation for (pure) inclusion theories reads then (see Chap. 6).

• Definition: M* is refined at least as truthlike as M iff

• (ir) ∀x ∈ M z ∈ T r(x,z) → ∃y ∈ M* s(x,y,z)

• (iir) ∀y ∈ M* − (M ∪ T) ∃x ∈ M − T ∃z ∈ T − M s(x,y,z)

It is again easy to check that (i^r) is a strengthening of (i^b) of the corresponding basic definition and that (ii^r) is a weakening of (ii^b). (i^r) roughly says that every comparable pair of structures, one of M and one of T, has an ‘intermediate’ in M*. In other words, M* represents what is to be included, i.e., T, at least as precisely as M. It is this clause that seems intuitive more plausible than the corresponding clause for exclusion theories. (ii^r) states that if M* − (M ∪ T) is at all non-empty, which is excluded in the basic case, these structures are ‘useful’. The definition reduces to the basic one when s is trivial.

Whereas the corresponding basic revision M^b _R(S) is again easily seen to be basically at least as truthlike as M, the refined revision M^r _R(S) is again not necessarily at least as truthlike as M in the refined sense. Hence, there is again even more reason to turn to successfulness.

• Definition: M* is refined at least as successful as M , relative to R/S, iff

• (i^r-sf) ∀x ∈ M z ∈ R r(x,z) → ∃y ∈ M* s(x,y,z)

• (ii^r-sf) ∀y ∈ M* − (M ∪ S) ∃x ∈ M − R ∃z ∈ S − M s(x,y,z)

The Refined Success Theorem tells now again that, assuming correct data, ‘refined at least as truthlike’ entails ‘refined at least as successful’. Again the proof is not difficult. Similar to the basic case, the consequence of the theorem is that being persistently more successful in the refined sense is conducive for refined truth approximation.

The final crucial question now is whether the refined revision M^r _R(S) of M by R/S is at least as successful as M in the refined sense. In that case it would be potentially conducive for refined truth approximation for it may become persistently more successful in the refined sense and hence conducive for refined truth approximation. This happens to be the case according to the following:

• Main Theorem (inclusion): M^r _R(S) is refined at least as successful as M, relative to R/S.

Let us look at the specific claims:

• (i^r-sf-wrt M^r _R(S)) ∀x ∈ M z ∈ R r(x,z) → ∃y ∈ M^r _R(S) s(x,y,z)

This is trivial, for R is a subset of M^r _R(S) and r(x,z) → s(x,z,z) is a (plausible) minimal s-condition.

• (ii^r-sf-wrt M^r _R(S)) ∀y ∈ M^r _R(S) − (M ∪ S) ∃x ∈ M − R ∃z ∈ S − M s(x,y,z)

This is also trivial, for M^r _R(S) is a subset of S, hence M^r _R(S) − (M ∪ S) is empty.

In view of the Refined Success Theorem we may now abductively conclude that if M^r _R(S), if kept fixed, remains persistently more successful than M in the refined sense in the light of new evidence, this is due to being closer to the truth than M in the refined sense. Of course, we will then even be more interested in the revision taking also all later evidence into account.

In sum, as far as set-theoretic aspects are concerned, the refined revision story for inclusion theories is easy to tell, and it is at least as intuitively plausible as that for exclusion theories, but how to give an AGM-interpretation of the a basic and refined account for such theories?