1. Value Incommensurability and Vagueness Two items are incommensurable in value iff neither is better than the other nor are they equally good. The Small-Improvement Argument: x +  x ~ y

2. BUT: The Small Improvement Argument does not take into account potential vagueness in value comparisons. We are not willing to say that x is better than y that y is better than x that x and y are equally good (cf. x + ) Perhaps this is because it is indeterminate (= vague) which of these three value relations obtains between x and y? Then it might still be determinate that one of them does obtain, i.e., it might be determinate that x and y are not incommensurable. Thus, the Small Improvement Argument is not quite compelling.

3. John Broome: If vagueness concerning value relations is allowed, incommensurability not only becomes difficult to establish. Its very possibility (within one and the same category) is put into question. For the argument, see below. Broome: Vagueness crowds out incommensurability. (“Is Incommensurability Vagueness”, 1997) My objectives: - critically examine Broome’s argument - show how the intersection modelling can make room for vagueness, along with incommensurability.

4. Setting the scene for Broome’s argument Rather than going through the argument in abstracto, I consider its application in population ethics. If it fails there, it can’t be generally valid. Intuition of neutrality: “We think intuitively that adding a person to the world is very often ethically neutral. We do not think that just a single level of wellbeing is neutral …” (Weighing Lives, 2004, p. 143) Axiological interpretation: Adding a person often does not make the world either better or worse. BUT “There are limits to this intuition of neutrality” (p. 144): - Adding unhappy people to the world makes it worse. - Adding very happy people possibly makes it better. Still, for a sizeable range of wellbeing levels, adding people with levels of wellbeing in that range is neutral: It makes the world neither better nor worse, Call this the neutral range.

5. Setting the scene, cont’d The equal goodness interpretation of the neutrality intuition: The world with an added person with wellbeing within the neutrality range is equally as good as the one without that added person. This interpretation is untenable, if Principle of Personal Good is valid for fixed populations. Principle of Personal Good (PPG): Increasing wellbeing of some without decreasing the wellbeing of others is always an improvement. Example 1: Adam Eve Cain A 3 3  B 3 3 1 levels 1 and 2 within C 3 3 2 the neutrality range By PPG, C is better than B. But then B and C cannot both be equally as good as A.

6. Setting the scene, cont’d The incommensurability interpretation of the intuition: -Adding a person with a wellbeing level within the neutral range does not make the world either better or worse, ceteris paribus. -Nor is the resulting world equally as good as the one without the addition. Broome’s objective: To criticize the neutrality intuition even in this form.

7. Broome’s argument Consider, say, the upper boundary of the neutral range. Reasonable to expect that this boundary is vague. But: No vagueness in the central area of the range. I.e., for some wellbeing level w [in the central area], it is determinate that w belongs to the neutral range, whereas for some wellbeing level w + > w [at the upper boundary], it is indeterminate whether w + still belongs to the neutral range, or is above that range.

8. Example 2:Adam Eve Cain A 3 3  B 3 3 w C 3 3 w + By hypothesis: (1) Indeterminate (C is better than A). (2) Determinate (A and B are incommensurable). (2)  (3) Determinate (A is not better than B). The supervaluationist account of vagueness (following Broome): Determinate = true on all sharpenings of vague predicates Indeterminate = true on some sharpenings, false on other sharpenings Indeterminate   Neither Determinate  nor Determinate  Principle of Personal Good  (4) Determinate (C is better than B). Since betterness is transitive, supervaluationism plus (3) & (4)  (5) Determinate (A is not better than C).

9. Broome’s argument, cont’d Thus, we have: (1) Indeterminate (C is better than A). (5) Determinate (A is not better than C). Broome’s “collapsing principle” for comparative predicates  Symmetry of indeterminacy: For any predicate Fer than, Indeterminate (x is Fer than y)  Indeterminate (y is Fer than x). Applied to better: Indeterminate (x is better than y)  Indeterminate (y is better than x). But then (1) and (5) cannot both be true! Something has to give.

10. Broome’s argument, cont’d If we want to allow vagueness at the boundaries, we must keep (1). So, we should give up (5) Determinate (A is not better than C). But (5) was derived from (2) Determinate (A and B are incommensurable). So, we should reject (2). Analogous argument if the lower boundary of the neutral range is assumed to be vague. Broome’s conclusion: Indeterminacy at any of the boundaries crowds out (determinate) incommensurability in the central area.

11. Criticism: What about Broome’s symmetry of indeterminacy? As applied to betterness: Indeterminate (x is better than y)  Indeterminate (y is better than x). Counterexample (Erik Carlson, Utilitas 2004): Suppose we compare x and y in their excellence as philosophers. x and y are exactly similar in all relevant respects, with the exception of one potentially relevant respect, R. R – rhetorical skills. x has them, but y doesn’t. Suppose it is indeterminate whether R is relevant. If it is, x is a better philosopher than y. If it is not, x and y are equally good philosophers.  Indeterminate (x is better than y), Determinate (y is not better than x). Conclusion:Symmetry of indeterminacy doesn’t always hold.

12. BUT: Is Carlson’s objection applicable to situations in which indeterminacy concerns incommensurability? [rather than equal goodness, as in Carlson’s example] Adam Eve Cain A 3 3  B 3 3w C 3 3 w + The only (potentially relevant) feature in which C differs from A is the presence, in C, of an extra person at wellbeing level w +. Since w + lies at the upper boundary of the neutral range, it is indeterminate whether this feature counts towards making the world better. If it does, C is better than A, if it does not, C is incommensurable with A. (Analogous to rhetorical skills in Carlson’s example.) So: Indeterminate (C is better than A). But: Determinate (A is not better than C). Similar structure as in Carlson’s counterexample!

13. Injecting vagueness into the intersection model for value relations K – a class of permissible preference orderings of the domain of items. [items - possible worlds in the case of population ethics] We can allow the class K be fuzzy at the boundaries. I.e.: K might have a number of admissible sharpenings.  For some pairs of items, it might be indeterminate what value relation (out of 15 possible) that obtains between them.

14. Example:Adam Eve Cain A 3 3  B 3 3 w C 3 3 w + P1P2P3 P4P5 CCCACA ABABBC BAB (gappy orderings ignored) Two sharpenings of K: K1 = {P1, P2, P3}, K2 = {P1, P2, P3, P4, P5}. K1 – w + above the neutrality range, K2 – w + within the neutrality range. Determinate that: - A and B are incommensurable (on a par) - C is better than B - A is not better than C Indeterminate whether: - A and C are incommensurable (true on K2, false on K1). Incommensurability and vagueness are thus fully compatible.

15. Injecting vagueness in another way? We can allow for vague preference orderings, i.e. orderings that themselves have several sharpenings. It does seem possible to have indeterminate preferences. (Note: Indeterminacy in preference is not the same as a gap.) And if indeterminate preferences are possible, then they might be permissible. The former solution: We let the permissibility of preferences to be vague. Now: We let permissible preferences to be vague.

16. Injecting vagueness in another way, cont’d BUT: This second way of injecting vagueness does not make value relations vague. Instead, it expands the set of possible value relations. We now have a larger number of preferential states that are possible with regard to x and y. In addition to states (i) >, (ii) or or < or = obtains. (vii) a state in which indeterminate whether = or / obtains etc 10 such indeterminate states, i.e. 14 preferential states altogether. An atomic type of value relation is obtained by specifying which of these 14 preferential states are permissible. (At least one must be permissible) So, the number of atomic value relations is now 2 14 – 1= 16383.