Abstract
This chapter elaborates on foundational issues in the social sciences and their impact on the contemporary theory of belief revision. Recent work in the foundations of economics has focused on the role external social norms play in choice. Amartya Sen (Econometrica 61(3): 495–521, 1993) has argued that the traditional rationalizability approach used in the theory of rational choice has serious problems accommodating the role of social norms. Sen’s more recent work (Language, world and reality 1996, pp. 19–31; Econometrica 65 (4): 745–779, 1997) proposes how one might represent social norms in the theory of choice, and in a very recent article Walter Bossert and Kotaro Suzumura (Social norms and rationality of choice, preprint, 2007) develop Sen’s proposal, offering an extension of the classical theory of choice that is capable of dealing with social norms.
The first part of this article offers an alternative functional characterization of the extended notion of rationality employed by Bossert and Suzumura (Social norms and rationality of choice, Preprint, 2007). This characterization, unlike the one offered in (Social norms and rationality of choice, Preprint, 2007), represents a norm-sensitive notion of rationality in terms of a pure functional constraint unmediated by a notion of revealed preference (something that is crucial for the application developed in the second part of this article). This functional characterization is formulated for general domains (as is Bossert and Suzumura’s characterization) and is therefore empirically more applicable than usual characterizations of rationality. Interestingly, the functional constraint we propose is a variant of a condition first entertained in Carlos Alchourrón et al. (The Journal of Symbolic Logic 50(2): 510–530, 1985) in the area of belief change.
The second part of this article applies the theory developed in the first part to the realm of belief change. We first point out that social norms can be invoked to concoct counterexamples against some postulates of belief change (like postulate (*7)) which are necessary for belief change to be relational. These examples constitute the epistemological counterpart of Sen’s counterexamples against condition α in rational choice (as a matter of fact, Rott (Change, choice and inference: A study of belief revision and nonmonotonic reasoning 2001) has showed that condition α and postulate (*7) are mutually mappable). These examples are variants of examples Rott (Synthese 139(2): 225–240, 2004) has recently presented. One of our main goals in this article consists in applying the theory developed in the first part to develop a theory of norm-inclusive belief change that circumvents the counterexamples. We offer a new axiomatization for belief change and we furnish correspondence results relating constraints of rational choice to postulates of belief change.
We wish to thank Isaac Levi, David Makinson, and Teddy Seidenfeld for valuable comments on earlier versions of this chapter.
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Notes
- 1.
The relationship between formal theories of belief change and non-monotonic reasoning has been well examined in Gärdenfors and Makinson (1994) and Makinson and Gärdenfors (1991). For arguments calling into question the claim that the AGM theory of belief change and the Rational Logic of Kraus et al. 1990 are two sides of the same coin, see Arló-Costa (1995).
- 2.
Occasionally Rott claims in addition that his formal results should be interpreted as a formal reduction of theoretical rationality to practical rationality or, less ambitiously, as a way of utilizing the theory of choice functions as a more primitive (and secure) theory to which the theory of belief revision has been reduced (e.g., see Rott (2001, pp. 5–6, 142, 214)). It is unclear whether these claims hold independently of the formal results presented by Rott. Erik Olsson (2003) offers criticisms along these lines. Isaac Levi (2004b) argues that the reduction is not a reduction to a theory of choice per se. In addition, Levi (2004b) offers an analysis of belief change where the act of changing view is constructed as an epistemic decision. The main technical tools Levi uses are not taken from the theory of choice functions but from other areas of decision theory.
We will appeal to some of the techniques Rott (2001) uses, but we do not claim that we are offering a reduction of belief change to the theory of choice or a reduction of theoretical rationality to practical rationality. As the reader will see, nevertheless, the mathematical techniques Rott (2001) exploits have a heuristic value to discover interesting postulates regulating belief change when social norms are relevant.
- 3.
There are many possible responses to Sen’s examples. One option could be to redefine the space of options in such a way as to tag y as the option of ‘taking the last apple from the plate’ (see Levi 2004a, for an analysis along these lines). Sen himself seems ambivalent regarding the analysis of his own examples. On some occasions he seems to think that redefinitions of this sort are feasible, yet on other occasions he has argued that these type of redefinitions make the principles of rational choice empty. In general, maneuvers of this kind tend to be blind with respect to the role of social norms in reasoning. We prefer here to take norms at face value as Sen does in many of his writings.
- 4.
A non-exhaustive list of influential articles in belief change and non-monotonic reasoning which presume relationality: Alchourrón et al. (1985), Alchourrón and Makinson (1985), Rott (1993), Rott and Pagnucco (2000), Hansson (1999), Arló-Costa (2006), Makinson (1989), Kraus et al. (1990), Lehmann and Magidor (1992), Gärdenfors and Makinson (1994).
- 5.
In fact, the influence of social norms and more generally menu dependence even threatens the presumption that selection functions are pseudo-rationalizable (see Moulin, 1985, for a discussion of this notion).
- 6.
Following Rott (2001), we do not require that \({\cal S}\) or \(\gamma({\cal S})\) consists solely of nonempty sets. This approach allows for more generality.
- 7.
Herzberger also writes, ‘Under the natural interpretation of \(\gamma(S)\) as a set of solutions to the problem S, the value \(\gamma(S)=\emptyset\) earmarks a decision problem that is unsolvable by the function γ; and so the domain \({\cal S}\) bears interpretation as the class of all decision problems that are solvable under the given choice function’ (Herzberger 1973, p. 189, notation adapted).
- 8.
There are alternative notions of rationalizablity one might find appealing. For example, we might instead say that a binary relation R on X rationalizes γ if for every \(S\in {\cal S}\) such that \(\gamma(S)\neq\emptyset\), \(\gamma(S)=\{x\in S: xRy\textrm{ for all }y\in S\}\).
- 9.
For a binary relation R on X, \(R^{-1}: = \{(x,y)\in X\times X:(y,x)\in R\}\).
- 10.
However, every selection function rationalized by a complete binary relation is M-rational. (A binary relation R on X is complete if for every \(x,y\in X\), either xRy or yRx.) To see why not every G-rational selection function is M-rational, we present an example from Suzumura (1976, pp. 151–152).
Consider a selection function γ on a choice space \((X,{\cal S})\), where \(X: = \{x,y,z\}\), \({\cal S}: = \{\{x,y\},\{x,z\},X\}\), \(\gamma(\{x,y\}): = \{x,y\}, \gamma(\{x,z\}): = \{x,z\}\), and \(\gamma(\{x,y,z\}): = \{x\}\). Then γ is rationalized by a reflexive binary relation ≥ defined by \(\geq: = \{(x,x),(y,y),(z,z),(x,y),(x,z),(y,x),(z,x)\}\). However, γ is not M-rationalizable, for otherwise, if > is an asymmetric binary relation for which \((X \times X)^{-1}\) rationalizes γ, then since \(\gamma(\{x,y\})=\{x,y\}\) and \(\gamma(\{x,z\})=\{x,z\}\), it follows that \(x\ngtr y\) and \(x\ngtr z\), but since \(\gamma(\{x,y,z\})=\{x\}\), it follows that \(y>z\) and \(z>y\), contradicting that > is asymmetric.
- 11.
Actually, Sen’s Property β is more pervasive than Sen’s Property β + (Sen, 1977, p. 66). Condition β demands that if \(S\subseteq S^{\prime}\) and \(\gamma(S^{\prime})\cap \gamma(S)\neq \emptyset\), then \(\gamma(S)\subseteq \gamma(S^{\prime})\) (Sen 1971, p. 313). Condition β + entails condition β, and in the presence of condition α, condition β and condition β + are logically equivalent.
- 12.
Condition α, also known as Chernoff’s Axiom, should not be confused with another important condition, the so-called Independence of Irrelevant Alternatives (Arrow, 1951, p. 27). See Sen (1977 pp. 78–80) for a vivid discussion of the difference between these two conditions. See also Ray (1973) for another clear discussion of this sort.
- 13.
Recall the notion of rationalizability briefly discussed in footnote 8. It can be shown that a selection function is rationalizable in the sense of footnote 8 just in case it satisfies the following condition: For every nonempty \(I\subseteq{\cal S}\) and \(S\in{\cal S}\), if \(S\subseteq\bigcup_{T\in I}T\) and \(\gamma(S)\neq\emptyset\), then \(S\cap(\bigcap_{T\in I}\gamma(T))\subseteq \gamma(S)\). This illustrates how one can modify coherence constraints for other notions of rationalizablity.
- 14.
Theorem 3.6 is stated within a more general framework that we introduce in Section 8.3. A direct proof of Theorem 2.5 proceeds in a way unlike the proof in Richter (1971). This is primarily because the results in Richter (1971) concern what is called the V-Axiom. Condition \(\gamma \textrm{R}_{\infty}\) is not discussed in Richter (1971).
- 15.
The minimal conditions needed for a proof can be gathered from the proof in Richter (1971, Theorem 3, p. 34). Although the proof in Richter (1971) does not itself establish Theorem 2.6, a careful inspection of the proof in Richter (1971) should make it clear that some assumptions of the theorem associated with this proof can be weakened. Indeed, condition \(\gamma_{1>\emptyset}\) is not discussed in Richter (1971) or to our knowledge anywhere else in the literature on choice functions. As we have indicated, selection functions are assumed to be regular in Richter (1971). See footnote 14.
- 16.
Interestingly, it seems that neither Rott nor AGM noticed that condition \(\gamma \textrm{R}_{\infty}\) is necessary and sufficient for rationalizability over general domains (i.e., domains for which no restrictions are imposed, such as closure under finite unions or compactness). Even more interesting is that to our knowledge, condition \(\gamma \textrm{R}_{\infty}\) has not appeared anywhere in the literature on choice functions. In particular, it appears that no one has explicitly pointed to a connection between condition \(\gamma \textrm{R}_{\infty}\) and rationalizability.
- 17.
Postulates (\(\substack{\cdot \\[-3pt] -} 7\)) and (\(\substack{\cdot \\[-3pt] -} 8r\)) are supplementary postulates of belief contraction (see [AGM85] and [Rott93]). For a fixed belief set K and contraction function \( \substack{\cdot \\[-3pt] -} \), postulate (\( \substack{\cdot \\[-3pt] -} 7\)) demands that \(K\substack{\cdot \\[-3pt] -}\varphi\cap K\substack{\cdot \\[-3pt] -}\psi\subseteq K\substack{\cdot \\[-3pt] -}(\varphi\wedge\psi)\), while postulate (\(\substack{\cdot \\[-3pt] -} 8r\)) requires that \(K\substack{\cdot \\[-3pt] -}(\varphi\wedge\psi)\subseteq\textup{Cn}(K\substack{\cdot \\[-3pt] -}\varphi\cup K\substack{\cdot \\[-3pt] -}\psi)\).
- 18.
- 19.
The ‘g’ in (*7g) is for ‘Gärdenfors’ (Rott, 2001, p. 110).
- 20.
Rott labels this postulate \((\ast8vwd)\) in Rott (2001, p. 110).
- 21.
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 (Rott, 1993).
- 22.
In Alchourrón et al. (1985, pp. 517–518), a relation ≥ is defined over remainder sets for a fixed belief set K, and \(\textrm{Eq}_\geq\) is called the marking off identity:
$$\gamma(K\bot\varphi)=\{B\in K\bot\varphi : B\geq B^{\prime}\textrm{ for all }B^{\prime}\in K\bot\varphi\}.$$ - 23.
Rott’s results (2001) show much more. For example, Rott shows that condition α corresponds not only to posutlate (*7), but also to postulate \((\substack{\cdot \\ -} 7)\) of belief contraction (which requires that \(K\substack{\cdot \\ -}\varphi\cap K\substack{\cdot \\ -} \psi\subseteq K\substack{\cdot \\ -} (\varphi\wedge\psi)\)) [Rot01, pp. 193-196] and to rule (Or) of non-monotonic reasoning (which demands observance of the following: From \(\varphi |\sim\chi\) and \(\psi |\sim\chi\), infer \(\varphi\vee\psi |\sim\chi\)) (Rott 2001, pp. 201–204).
- 24.
Sen’s Property β + was mentioned in footnote 11 because it has been given special attention in connection to rationalizablity in belief change and non-monotonic reasoning (e.g., in Rott 1993, 2001, and Arló-Costa 2006). It is known that condition β + corresponds to belief revision’s rationality postulate (*8), which requires that \(K\ast\varphi\subseteq K\ast(\varphi\wedge\psi)\) whenever \(\neg\psi\notin K\ast\varphi\) (see Rott 2001, p. 198). But since condition β is more pervasive in the study of rational choice, one might ask the following question: What rationality postulate corresponds to condition β? The second author has shown elsewhere (Pedersen 2008) that condition β corresponds to postulate \((\ast 8\beta)\), which demands that if \(\textup{Cn}(K\ast\varphi\cup K\ast (\varphi\wedge\psi))\neq \textrm{For}({\cal L})\), then \(K\ast \varphi\subseteq K\ast(\varphi\wedge\psi)\).
- 25.
In fact, as Rott (2004) points out, Aizerman’s Axiom is also violated (see Section 8.2, for a statement of this condition). This coherence constraint corresponds to postulate (*8c) via Rott’s correspondence results (2001, pp. 197–198). (Postulate (*8c) demands that if \(\psi\in K\ast\varphi\), then \(K\ast\varphi\subseteq K\ast(\varphi\wedge\psi)\).) This means that pseudo-rationalization is precluded (see footnote 5).
- 26.
David Makinson suggested to us in a private communication that there might be cases of pure menu dependence, where the choice depends on the content of the menu, irrespective of the context in which it is offered. It is unclear whether there are pure cases of this sort. It seems difficult to find examples that do not depend on the reliability of the information source used to extend the menu. In any case, the distinction between pure and impure cases of menu dependence seems worthy of further analysis.
- 27.
The notion of expectation here should not be confused with the notion of expected utility in rational choice. Whereas expected utility concerns expectations of the values of various outcomes, here expectation concerns beliefs about the world (see Gärdenfors and Makinson 1994, p. 5). Perhaps the only dissenting voice regarding this point is Levi who in [Lev96] treats expectations as cognitive expected value. We use expectations here in the first epistemic sense of the word.
- 28.
Other supplementary postulates are violated as well, such as postulate (*8c) (see footnote 25). No basic postulates are violated in this example.
- 29.
Again, Levi traces a sharp distinction separating full beliefs and expectations. In Levi (1996) he distinguishes between the ordinary versions of the postulates of belief change, like postulate (*7), and inductively extended versions of these postulates. And he has pointed out that the inductively extended version of postulate (*7) fails to hold. So, we assume that this would be his preferred explanation of examples like the one offered by Rott. Perhaps the distinction holds even if one does not buy Levi’s theory of induction, and if one uses a different theory of induction instead. So the issue of what counts as a counterexample to well-known principles of belief formation sanctioned by AGM depends on a previous understanding of the notion of expectation as opposed to full belief. While Levi’s strategy might work (assuming that one buys his notion of expectation) in the cases where menu dependence is the main mechanism, it is unclear whether this strategy applies to cases where the main underlying mechanism is determined by the use of social norms. More about this will be discussed below.
- 30.
Here we adopt a notational convention: the expression \((a, \neg b, \neg c, \neg d,\neg e)\) denotes the possible world for which a is true and the rest of the items – b, c, e, d – are false.
- 31.
A possible solution of compromise between Levi’s inductive approach and the use of norms in examples of this sort could be to say that we are dealing with norms that induce or generate expectations. But in this case, these expectations do not seem to be those from a theory like Levi’s but rather the purely epistemic expectations usually used in theories of belief change. Still, we can see a norm-sensitive operator of belief change as the composition of two operators: one classical AGM operator plus an inductive operator sanctioning an inductive jump made possible by the underlying norms. Most of what follows can be seen as a positive theory about this operator–-where we provide new axioms that the operator should obey.
- 32.
A norm warrants a belief ψ for acceptance if the norm makes the acceptance of ψ permissible. The warranted beliefs are the beliefs permitted by the norm.
- 33.
Of course, it would have been sufficient to specify that * satisfies only postulates (*2) and (*6), without specifying that * satisfies (*1).
- 34.
When social norms do not have any real on influence belief formation, postulate \({\Box\raisebox{-1.5pt}{*}}\) ι is satisfied.
- 35.
As before, we adopt the notational convention that (\(a, \neg b, \neg c, \neg d,\neg e)\) denotes the possible world for which a is true and the rest of the items – b, c, e, d – are false.
- 36.
Condition \(\gamma_{1>\emptyset}\) should be imposed as well, but the central condition in the result is \(\gamma \textrm{R}_{\infty}\).
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Arló-Costa, H., Pedersen, A.P. (2010). Social Norms, Rational Choice and Belief Change. In: Olsson, E., Enqvist, S. (eds) Belief Revision meets Philosophy of Science. Logic, Epistemology, and the Unity of Science, vol 21. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9609-8_8
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