Abstract
The expansion or revision of false theories by true evidence does not always increase their verisimilitude. After a comparison of different notions of verisimilitude the relation between verisimilitude and belief expansion or revision is investigated within the framework of the relevant element account. We are able to find certain interesting conditions under which both the expansion and the revision of theories by true evidence is guaranteed to increase their verisimilitude.
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Notes
A finite Lewis sphere model is a sequence SL = (Si: 0 ≤ i ≤ n) of subsets of possible worlds, called spheres Si, which satisfy the nesting condition S0 \( \subseteq \) S1 \( \subseteq \) S2 \( \subseteq \)…etc. SL is centered iff S0 = {w*}, where w* is the actual world (cf. Lewis 1973; note that for infinite Lewis' sphere models one has to require closure under infinite union and intersection of spheres).
This fact is intuitively clear; a proof of it is found in lemma 5 of Schurz and Weingartner (2010).
Note that the concept of a “completely false theory” cannot be applied to Popper’s account which identifies conjunctive parts with arbitrary consequences, because every theory A has some true consequences (e.g., a ∨ truth, for all a ∈ A).
In Schurz and Weingartner (2010) this result is taken to show that Zwart and Franssen's impossibility result in (2007) does not apply to the relevant elements account.
If contractions are applied to sets of relevant elements, forming intersections of equally best maxichoice contractions is not sufficient, because it is not guaranteed that the disjunction of two sentences in which two maxichoice contractions of Ar by \( \neg \)e disagree is in the intersection (while this is guaranteed if maxichoice contractions are applied to deductively closed sets). One would rather have to add disjunctions to those intersections. For example, assume Ar = {\( \neg \)p1, \( \neg \)p2, p3}, e = p1∨p2, then two possible maxichoice contractions are (Ar\\( \neg \)e)1 = {\( \neg \)p1, p3} and (Ar\\( \neg \)e)2 = {\( \neg \)p2, p3}, and their disjunction would be {\( \neg \)p1∨\( \neg \)p2, p3}. More generally, we define the maxichoice-disjunction-operation applied to a maxichoice contraction (Ar\\( \neg \)e)1 as follows: identify some element a1 which is in (Ar\\( \neg \)e)1 but not in some other maxichoice contraction (Ar\\( \neg \)e)2 and some element a2 which is in (Ar\\( \neg \)e)2 but not in (Ar\\( \neg \)e)1, and replace a1 in (Ar\\( \neg \)e)1 by the disjunction a1∨a2 (under the proviso that a1∨a2 is still relevant in the so replaced set; otherwise simply omit a1 from the set). Every set of relevant elements which can be obtained from an iterative application of maxichoice-disjunction-operations to a maxichoice contraction of Ar by \( \neg \)e is called a disjunctive maxichoice contraction. The extended version of def. (13) would then (in a first attempt) run as follows:
(13*) (A − \( \neg \)e) : = Cn(Ar\\( \neg \)e), where Ar\\( \neg \)e is a simple or disjunctive maxichoice contraction of Ar by \( \neg \)e.
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Acknowledgments
For valuable help I am indepted to Theo Kuipers, Erik Olsson, Graham Oddie, Roberto Festa and Gustavo Cevolani.
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Schurz, G. Verisimilitude and Belief Revision. With a Focus on the Relevant Element Account. Erkenn 75, 203–221 (2011). https://doi.org/10.1007/s10670-011-9291-1
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DOI: https://doi.org/10.1007/s10670-011-9291-1