1. Value Relations Ruth Chang’s four ways in which items can be comparable in value: Better WorseEqually GoodOn a Par Mozart vs Michelangelo. Multidimensionality of value comparisons Joshua Gert, Ethics 2004: Value comparisons are normative assessments of preference. (A form of FA-analysis) Gert’s contribution: Two levels of normativity  Parity can be distinguished from equal goodness. Standard worry: parity vs incomparability. An extension of Gert’s approach does the trick.

2. Plan of the talk Gert’s idea is attractive. But not the way he formally develops it, using an ‘interval’ modelling: Comparisons of the intervals of permissible preference strength. This modelling is inadequate. Instead, I propose an ’intersection’ modelling  a taxonomy of all binary evaluative relations (15 types!) Problems with this modelling. A new interpretation of the intersection modelling

3. Ruth Chang, Ethics 2002, Making Comparisons Count 2002 To prove: parity ≠ equal goodness The Small-Improvement Argument: x +  x ~ y Previous proposers of this argument: de Sousa (1974), Broome (1978), Sinnott-Armstrong (1985), Raz (1985/86)

4. To prove: Parity is not incomparability Chang: The Unidimensional Chaining Argument x ~ y.. z 2 z 1 z 1, z 2, …., x – a series of small unidimensional (one-respect) improvements. Possibly: different respects in different steps. Assumption: A small unidimensional improvement won’t make comparability disappear.[Sorites?] Thus, by induction: If z 1 is comparable with y, so must be x. Remaining tasks: To distinguish parity from vagueness and from gaps in evaluative knowledge.

5. Gert’s analysis of betterness In the manner of “Fitting Attitudes”- analysis of value. FA-analysis: An object x is valuable iff it is fitting to have a pro- attitude towards x. ‘fitting’, ‘appropriate’, ‘correct’, ‘ought’, ‘required’, ‘reason’, etc. - the normative component in FA-analysis. For better, the relevant pro-attitude is preference: Brentano: An item is better iff preferring it is correct. Gert: An item is better iff preferring it is rationally required. Unlike for Brentano, for Gert preferences are choice dispositions rather than emotions. Implicit assumption: Epistemic familiarity with the alternatives.

6. Two items x and y are equally good iff what is rationally required is indifference with regard to x and y, i.e., equi- preference. Gert’s main innovation: Two levels of normativity. The normative component can have a stronger or a weaker form. Required vs PermissibleOught vs May x and y are on a par iff it is (i) rationally permissible to prefer x to y, and (ii) rationally permissible to prefer y to x. (Gert’s own definition demands more, but incorrectly so.)

7. Incomparability This framework needs to be extended to allow for incomparabilities. Preference - a disposition to choose. Indifference - yet another choice disposition: Being equally prepared to make either choice But: For some pairs of items, a person might lack a choice disposition. Preferential gap. Preferential gap  indifference. In case of a gap, the situation is typically viewed as being internally conflicted: Reasons on both sides, but the agent cannot (or does not) balance them off.

8. Incomparability – cont’d Two items are incomparable iff it is rationally impermissible to prefer one to the other or to be indifferent. Are incomparabilities possible? Certainly, between items from different ontological categories. (Say, x is a state of affairs and y is a person.) But what if x and y belong to the same category of objects? It may well be rationally permissible to have no preferential attitude regarding x and y. [weak incomparability] But can it ever be required? Sophie’s choice?

9. Comparability? Weak sense: Two items are comparable iff they are not incomparable. Strong sense: Two items are fully comparable iff a preferential gap with respect to them is rationally impermissible. Parity is a form of comparability according to Chang. But in what sense? Strong or weak? Either solution is possible: x and y are on a par iff (i) it is rationally permissible to prefer x to y and (ii) it is rationally permissible to prefer y to x. If x and y in addition are fully comparable, they are fully on a par.

10. Gert’s interval modeling The strength of one’s preference for an item can be measured. [interval scale?] The range of rationally permissible preference strengths with respect to an item x: [x min, x max ]. Example: 40 x 30 10 y 5 Permissible to prefer x to y and vice versa, or to equi- prefer. [Parity]

11. Gert’s interval modeling, cont’d The Range Rule: x is better than y iff x min > y max. The weakest permissible preference for x is stronger than the strongest permissible preference for y.

12. Objections to the interval model (i) Equality in value will be rare. Equal goodness iff indifference is required. But: if x  y, indifference requires intervals of length zero (i.e. points). In particular, we don’t have equal goodness even when (ii) Incomparability can’t be modeled. If the intervals for x and y don’t overlap, preference for one of the items is required. And if they overlap, even at one point, indifference is permissible.

13. Objections to the interval model, cont’d (iii) Certain betterness structures can’t be represented Australia + $100x+y+ South Africa + $100 Australiaxy South Africa So, what’s gone wrong? Is the weakest permissible preference for x+ stronger than the strongest permissible preference for x? Surely not! Instead, the situation seems to be like this: x+ x

14. Source of the trouble: Interval modeling lacks resources to specify permissible combinations of preference strengths. (Similar objection to the interval representation of indeterminate subjective probabilities. Let x and y be some independent propositions, whose probabilities intuitively are on a par, and let x + and y + be the disjunctions ’x or z’ and ’y or z’, respectively, where z is some third independent proposition that is much less probable than both x and y. The probability of such a disjunction is slightly higher than the probability of its first disjunct.) Can we do any better?

15. Intersection modeling Holism: Instead of looking at each item separately, we consider the class, K, of permissible preference orderings of a domain I. The orderings in K need not be representable by cardinal measures of preference strength. They might not even be complete rankings: gaps are allowed. However, orderings in K are at least ‘partial’ in this sense: preference-or-indifference is a quasi-order (transitive and reflexive)

16. Intersection modeling – cont’d Betterness is the intersection of preferences in K: x is better than y iff x is preferred to y in every ordering in K. The Australia - South Africa example: P1P2P3 x+y+x+ y+ xyx y y+x+ yx

17. Intersection modeling – other value relations x and y are equally good iff they are equi-preferred in every K-ordering, incomparable iff every K-ordering contains a gap as regards x and y, on a par iff x is preferred to y in some K-orderings and dispreferred to y in some K-orderings. And so on.

18. Does this model add anything to the original analysis? Very little. Which is OK! It adds, though, a possibility to derive formal features of evaluative relations from formal requirements on permissible preference orderings. Example: Permisible preference is transitive  Betterness is transitive.

19. Taxonomy of binary value relations 123456789101112131415 > ++++++++ < ++++++++ = ++++++++ / ++++++++ BWEPPPPI

20. Comments to table Each column - one atomic type of a binary evaluative relation. Plus signs mark preferential relations between two items that are permissible in a given type. Four kinds of preferential relations: preferring (>), dispreferring (<), indifference (≈), and a gap (/). Example: Type 7 allows for all preferential relations except the gap. The number of types = the number of ways to pick out a non-empty subset out of set of four preferential relations. Columns - atomic types. Groupings of atomic types - types in a broader sense. Examples: Weak incomparability: types 8 – 15 Parity (types 6 -9). So, parity is not quite a fourth type of cmparability. It is not an atomic relation, unlike B, E and E. Also, while parity is incompatible with incomparability (I), items that are on a par can be weakly incomparable (types 8 and 9).

21. Do all the logically possible types represent ‘real’ possibilities? Maybe not. Possible requirement: Whenever two items are on a par, indifference is permissible. This would exclude types 6 and 9. A further possible requirement: Preference gaps are permissible in cases of parity. This would exclude type 7. Then parity would reduce to type 8. Such extra requirements, based not on logic but on considerations of rationality, might substantially narrow the space of possibilities.

22. Consequentialist principles An action is right iff its outcome … [1] is at least as good as the outcome of every alternative action. BUT: If parity is possible, such action might not be available. [2] is not worse than the outcome of any alternative action. Every finite alternative set will contain at least one such action. BUT: Can an action be right if every permissible preference ordering ranks it below some alternative action? [3] is top-ranked as compared with other alternatives in at least one preference ordering in K. (top-ranked = not dispreferred to any alternative) Proposal 3 is intermediate in strength between Proposals 1 and 2. Every finite alternative set contains at least one action that satisfies [3]. Unlike [1] and [2], which are framed in terms of value relations, [3] requires explicit mention of K.

23. Money Pump Even proposal 3 leads to a problem: It legitimizes money-pumping. Suppose that x + is better than x and that both these items are on a par with y. Then it is ok (= right) on proposal 3 to exchange x + for y, ok to exchange y for x and ok to pay  to exchange x for x +. In this pump some steps are not prescribed (unlike as in the standard money pump), but all steps are ok (= right). Solution: Right choices in a sequence must all follow the same permissible preference ordering. In every permissible ordering x + is preferred to x. But then if such ordering y is at least as preferred as x +, y must be preferred to x.  Money pump is avoided. But proposal 3 turns out to be not quite correct (ignores consistency between choices in a sequence).

24. Money pump, cont’d But even this solution encounters problems in the presence of preferential gaps. If it’s ok to exchange y for x in case there is a preferential gap between them, then we are again saddled with a possible money pump, if there is a permissible preference ordering in which there is a gap between x and y and between x + and y. It seems then, that - in the presence of preferential gaps - an action might have to be classified as being neither right or wrong. In particular, if our choice is between two incomparable actions, then neither of them should be considered as being right (or wrong).

25. Some problems with my modelling of value relations CONCEPTUAL TRUTH It’s problematic that formal properties of value relations (such as transitivity of betterness or of equal goodness) depend on requirements on permissible preference orderings. Conceptual truths should not be dependent on normative requirements (even if these requirements happen to express fundamental rationality constraints).

26. Problems with my modelling, cont’d INCOMPARABILITY AND PREFERENTIAL GAPS It’s problematic that incomparability comes out as such a demanding notion. It’s questionable whether it can ever be that case that a preferential gap with respect to two items is not just permitted but positively required.

27. Problems with my modelling, cont’d REISNER’S PROBLEM To motivate the possibility of preferential gaps, i.e. the possibility of the absence of choice dispositions, I assumed that the relevant choice dispositions should be reason-based. This makes it possible to claim that sometimes such dispositions are absent (when the conflict of reasons is not adjudicated by the agent). BUT: A reason-based disposition to choose x rather than y will involve a thought that x is better than y. HOWEVER: How can it be permissible to consider x as better than y if x is not better than y, as when they are on a par? Is this problem serious? Can’t balancing of reasons yield an outcome that is seen as optional by the agent? She might consider the weights she assigns to different reasons as being optional to some extent.

28. A new interpretation of the intersection model Give up the idea that preferences are dyadic attitudes. It isn’t obvious that the FA-theorist must analyze dyadic value relations in terms of dyadic atitudes. Go back to Gert’s approach to some extent, but retain holism. For each item x, we can specify the degree to which x is favoured by the agent. The agent prefers x to y = she favours x to a higher degree than she favours y. THUS: Preferences aren’t dyadic attitudes on this approach but instead comparative relations of strength between monadic attitudes.

29. Formal interpretation D - the set of possible degrees of favouring, ≻ - the ordering relation on D ( d ≻ d’ – d is a higher degree of favouring than d’.) Assumption: ≻ is asymmetric and transitive. Let F be the class of permissible favouring assignments with respect to a given domain of items. A favouring assignment, f, is an assignment of degrees of favouring in D to the items in the domain I. f: I  D x is preferred to y in f = df f(x) ≻ f(y). x and y are equi-preferred in f = df f(x) = f(y).

30. Formal interpretation - cont’d Some pairs of degrees of favouring might be such that neither degree is higher than the other. Therefore, a permissible assignment f might be gappy: f(x) ≠ f(y) but neither f(x) ≻ f(y) nor f(x) ≻ f(y). This might be the case when the ways in which x and y are favoured are sufficiently different. A preference ordering on I – as previously- specifies for each pair x, y of items in I whether x is preferred to y, equi-preferred with y or neither. Each f induces a preference ordering on I. K - the class of permissible preference orderings - is now defined as the class of orderings induced by the favouring assignments in F.

31. Formal interpretation – cont’d Note: Two different favouring assignments in F might induce the same preference ordering. Example: f and f’ might differ in that each item is more highly favoured in f’ than in f, but if f’ is a monotonical transformation of f, then they both induce the same preference ordering.

32. Note: Similar kind of approach can also be used for indeterminate subjective probabilities. f, f’, … in F are then different permissible assignments of degrees of belief to propositions.

33. How does the new interpretation solve our problems with the intersection model interpreted in the old way? CONCEPTUAL TRUTH Conceptual truths about formal properties of value relations are now derivable from the properties of the ordering relation ≻ (= ’higher than’) on degrees of favouring. That this relation is transitive is itself a conceptual truth and not a rationality constraint.

34. INCOMPARABILITY AND PREFERENTIAL GAPS When x and y are very different items (say, x = Mozart and y = Mother Theresa), permissible ways of favouring these items might be so different that degrees to which they are favoured are incommensurable on every permissible favouring assignment. This gives us incomparability: preferential gap on every permissible preference ordering.

35. REISNER’S PROBLEM This problem doesn’t arise if preference for x over y is not a dyadic attitude with regard to x and y, but instead a comparison between monadic attitudes directed to x and to y, respectively. If preference is a comparative relation of strength between monadic attitudes of favouring, preference for x over y need to involve any thought that x is better than y. Indeed, it need not involve any conscious comparison of the two items.