Abstract
In a recent paper, Okasha imports Arrow’s impossibility theorem into the context of theory choice. He shows that there is no function (satisfying certain desirable conditions) from profiles of preference rankings over competing theories, models or hypotheses provided by scientific virtues to a single all-things-considered ranking. This is a prima facie threat to the rationality of theory choice. In this paper we show this threat relies on an all-or-nothing understanding of scientific rationality and articulate instead a notion of rationality by degrees. The move from all-or-nothing rationality to rationality by degrees will allow us to argue that theory choice can be rational enough.
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Notes
- 1.
For the purposes of this paper we treat preference rankings as strict (asymmetric) relations. This is a standard simplifying assumption made in the social choice literature, and where relevant our results can be modified accordingly to apply to non-strict rankings of theories by virtues. This is of no conceptual importance.
- 2.
Note that usual expositions of Arrow’s result, Okasha’s included, do not include OR as part of the conditions leading to the inconsistency. We decide to do so, as it will be useful to gauge the threat posed by Arrow’s impossibility under OR as opposed to the threat under a natural weakening of it. This weakening is the requirement that the aggregation function deliver a winner, instead of a complete preference ordering. The motivation for this is simple in our theory choice context: after all, according to Kuhn, the winners are those who rewrite the history of science. Where appropriate we refer to this weakened notion with the prefix Condorcet rationality.
- 3.
\(\mathfrak{D}_{m}^{n}\) represents the class of all profiles that can be defined over m virtues and n theories.
- 4.
For the remainder of this paper we will assume there is an odd number of virtues. This means that the output of the aggregation, under pairwise majority, will always be a strict ordering (if an ordering, at all).
- 5.
Assuming all 216 possible ways are equally likely to obtain, the chances of succeeding in rationally choosing the best theory are very low. See a discussion of this assumption further below.
- 6.
Some level of risk aversion is undoubtedly rational. Consider the following scenario: you are offered two bets. Bet 1 gives you the chance to win 100& with probability 1. Bet 2 gives you a chance to win 200& with probability.5 and 0& otherwise. Choosing Bet 1 in this instance does not seem irrational, and in fact, many people will do so. However, as the probability of winning 200& in Bet 2 increases, Bet 2 becomes more appealing, and fewer people will avoid it. There seems to be a point at which choosing Bet 1 over Bet 2 becomes irrationally cautious (if this still doesn’t appeal to your intuition, consider Bet 3 with probability.9 of winning 1,000,000& and 99& otherwise). In this sense, a scientist refraining from using pairwise majority, as this rule fails in 1 out of 216 cases appears irrationally cautious.
- 7.
The only restriction we place on Pr is that it assigns a non-zero probability to every profile in \(\mathfrak{D}_{m}^{n}\). The motivation for this restriction is the same as the motivation for UD. If a theory choice function is rational only if it is defined over every profile in a domain, then all profiles are considered as ‘live options’. Assigning to any profile a zero probability of occurring would undermine this.
- 8.
In the social choice literature this is known as the ‘impartial culture’ assumption Gehrlein (1983). This assumption is discussed in more detail below.
- 9.
These correspondences rely on our restriction on Pr stated in Footnote 7 above.
- 10.
The values in Table 26.1 have been calculated in Mathematica 10. Please contact the authors if you wish to consult the notebooks used.
Table 26.1 The μ-rationality of pairwise majority for n theories and m virtues - 11.
For a more sophisticated discussion (in the context of social choice) of the results in this table, as well as for a general formula for approximating the probability of a cycle given any number of voters (odd) and any number of alternatives, see DeMeyer and Plott (1970).
- 12.
The reason for this is that performing these calculations is a computational demanding task and some of the older papers did not have the technical means of obtaining all results.
- 13.
Interestingly, it has been proven that as the number of voters tends to infinity, any deviation from impartial culture will reduce the probability of majority cycles, as long as the deviation isn’t in favour of a distribution that assumes the Condorcet paradox from the start (Tsetlin et al. 2003). However, since we are working in a context with a finite, relatively small number of virtues (voters) we cannot rely on this result to motivate impartial culture here.
- 14.
We are grateful to an anonymous referee for encouraging us to motivate this assumption.
- 15.
We are grateful to an anonymous reviewer for pointing out that if this involves Bayesian statistical model selection that the criterion might not be accuracy, but rather posterior probability. However, as Okasha (2011, pp.105–110) notes such an approach to model selection is immune from the Arrovian challenge.
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Marcoci, A., Nguyen, J. (2017). Scientific Rationality by Degrees. In: Massimi, M., Romeijn, JW., Schurz, G. (eds) EPSA15 Selected Papers. European Studies in Philosophy of Science, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-53730-6_26
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