Abstract
The article transfers an argument of Pattanaik and Xu on ranking opportunity sets to tragic choices and the so called “numbers problem”. We characterize conditions that make the numbers count. This in itself will not resolve any problem relevant to the ongoing ethical debate but should shed some fresh light on it by forcing participants to state specifically which of the assumptions (axioms) should give way for what reasons.
This chapter has been published in Homo Oeconomicus 26 (2), 2009.
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Notes
- 1.
In fact this may not be much of an ‘assumption’ as long as we have not required anything about the comparison. It is practically implied by any ranking activity.
- 2.
That R forms an ordering is, of course, a substantial assumption. A binary relation R on Π(X) × Π(X) is an ordering, if and only if for all A \( \in \) Π(X): A R A holds (reflexivity of the ranking R) and for all A, B \( \in \) Π(X): A R B or B R A holds (completeness of the ranking R) and for all A, B, C \( \in \) Π(X): A R B and B R C ⟹ A R C holds (transitivity of the ranking R). The proof of the central theorem, however, requires only transitivity. The other properties of an ordering are then implied by the axioms.
- 3.
Even if we are ethical non-cognitivist we may still use this axiom as a constitutive characteristic of moral discourse. Generalization in ethics is, of course, classically discussed in Singer (1971), with respect to utilitarianism in Hoerster (1971/1977), while its relation to the very concept of morals is analyzed in Singer (1973).
- 4.
It can be easily seen that the first axiom along with the premise that there is one individual whom it is better to rescue than not and transitivity implies axiom 2.
- 5.
Pattanaik and Xu (1990) in ranking sets of objects (rather than sets of human individuals) assumedfor all x, y \( \in \) X, x ≠ y, {x, y} P {y}.
- 6.
The start of the induction in the proof by Pattanaik and Xu is n = 1.If #A = #B A I B is implied by Indifference between Singletons. In our case we have to deal with #A = 0, too.
- 7.
One could also assign a certain value to each person and rank sets of individuals with respect to the sum of the values assigned to the persons in each set. For the case of ranking sets of opportunities this proposal was modeled by Ahlert (1993).
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Acknowledgments
Hospitality of the economics department of the University of Hamburg is gratefully acknowledged. We are particularly indebted to Mathew Braham, Manfred Holler and Frank Steffen. Comments by Yongsheng Xu on an earlier version helped to clarify some statements. Our general interest in the problem was triggered and inspired by Weyma Lübbe who is Germany’s leading expert on the numbers issue. In this chapter we rely strongly on her writings. Particularly useful were Lübbe (2004, 2005).
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Ahlert, M., Kliemt, H. (2013). Necessary and Sufficient Conditions to Make the Numbers Count. In: Holler, M., Nurmi, H. (eds) Power, Voting, and Voting Power: 30 Years After. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35929-3_34
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