Abstract
Most of the literature on belief change presupposes a theory of rationality according to which rational agents are optimizers: they chose the best among the feasible epistemic options available to them. But when epistemic values are indeterminate this view of rationality is not applicable. In this situation it is still rational to maximize, i.e., to deem an option as choosable when it is not known to be worse that any other Sen (1997). In this article we present some basic results about rationalizability in terms of maximization rather than optimization.
In recent work Arl)o-Costa (2006) I propose to maximize the categorical preference that arises when one takes the set of shared pairs from all the permissible value orderings. This categorical preference need not be complete but it can be maximized. One possible shortcoming of this procedure is that in many cases the set of shared agreements might be empty. In this case one might use a different approach in cases of indeterminacy. This approach (first proposed in a different form by Isaac Levi in Levi (1986)) has two parts, or tiers. The idea is to first take as admissible any option that is deemed as maximal by some permissible ordering. This is the first tier of the decision rule. Of course, this can generate an admissible set that is not a singleton. To untie ties, one then uses an auxiliary notion of security (qualitatively represented by an additional security ordering). This is the second tier of the decision rule.
The problems of fully characterizing the two-tier decision rule procedure via conditions on choice functions is open. We survey here some attempts to solve this problem and consider applications in the theory of belief change under indeterminacy. If one limits ones attention to the first-tier decision rule there are results by M. Aizerman and A. Malishevksi characterizing a choice function that corresponds to the decision rule Aizerman and Malishevski (1981). This result is applied to axiomatize a belief revision operator where indeterminacy is resolved via the first-tier decision rule with an empty security order. The techniques used to correlate conditions on choice functions with belief revision axioms extend the usual methods first investigated by Hans Rott in Rott (2001).
We conclude with a discussion of applications in philosophy of science. In partic-ular we consider an approach proposed by Sidney Morgenbesser and Arnold Koslow in an unpublished paper Morgenbesser and Koslow (2008). This paper contains various important insights related to the problem of indeterminacy in the process of theory change and theory choice. Most of these insights are related to applications in the philosophy of science. We compare our proposal to the one offered in Morgenbesser and Koslow (2008) and consider some of the questions that are left open in this manuscript.
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Notes
- 1.
The usual procedure in the belief revision literature is to define a selection function γ as a function that returns a non-empty output when applied to a non-empty remainder set. This makes implicit requirements on the marking-off relation ≤. For example, the relation cannot be irreflexive.
- 2.
- 3.
- 4.
Much of the presentation here follow the one introduced in Arló-Costa and Pedersen (2009b). In both cases the section introduces the reader to material that is well known in the belief revision literature.
- 5.
The “g” in \((\ast 7g)\) is for “Gärdenfors” (Rott 2001, p. 110).
- 6.
For a belief set K and a sentence ϕ, a remainder set \(K\bot\varphi\) is the set of maximal consistent subsets of K that do not imply ϕ. Members of \(K\bot\varphi\) are called remainders. Thus, in the AGM framework, a belief set K is fixed, and for every sentence ϕ such that \(\varphi\notin{\textup{Cn}({\emptyset})}\), \(\gamma(K\bot\varphi)\) selects a set of remainders of \(K\bot\varphi\). The situation in which \(\varphi\in{\textup{Cn}({\emptyset})}\) can be handled as a limiting case at the level of the selection function Alchourrón et al. (1985) or at the level of the revision operator.
- 7.
In Alchourrón et al. (1985, pp. 517–518), a relation ≥ is defined over remainder sets for a fixed belief set K, and (Eq≥) is called the marking off identity:
$$\gamma(K\bot\varphi)=\{B\in K\bot\varphi : B\geq B^{\prime}\textup{for all }B^{\prime}\in K\bot\varphi\}.$$ - 8.
Now let \((X,\mathcal{S})\) be a choice space. We call \(\,\mathbb{K}\,\) finitely additive if it is closed under finite unions; we call \(\,\mathbb{K}\,\) subtractive if it is closed under relative complements; and we say that \(\,\mathbb{K}\,\) is compact if for every \(S\in\mathcal{S}\) and \(I\subseteq\mathcal{S}\), if \(S\subseteq\bigcup_{T\in I}T\), then there is some finite \(I_{0}\subseteq I\) such that \(S\subseteq\bigcup_{T\in I_{0}}T\). If γ is a selection function on \((X,\mathcal{S})\), we call γ finitely additive (subtractive, compact) if \(\,\mathbb{K}\,\) is finitely additive (subtractive, compact).
- 9.
The parenthetical notation used here indicates that a is preferred to c and c to b.
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Acknowledgments
I would like to thank specially Arnold Koslow for sharing with me an unpublished version of a paper he wrote in collaboration with Sidney Morgenbesser (discussed in last section of this chapter). I understand that this paper is currently under submission. So, perhaps the published version will be slightly different from the manuscript I read. In any case, I think that most of the discussed aspects of the original version of the paper are quite important and deserve a philosophical discussion.
I would like to thank also Arnie Koslow for an enlightening philosophical discussion of the contents of the unpublished manuscript. Finally I would like to thank Paul Pedersen who read the entire manuscript and made useful comments as well as an anonymous referee.
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Arló-Costa, H. (2011). Indeterminacy and Belief Change. In: Girard, P., Roy, O., Marion, M. (eds) Dynamic Formal Epistemology. Synthese Library, vol 351. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0074-1_10
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