Amartya Sen - Incompleteness and Reasoned Choice 2004

8/6/2019 Amartya Sen - Incompleteness and Reasoned Choice 2004

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AMARTYA SEN

INCOMPLETENESS AND REASONED CHOICE

The subtitle of Isaac Levis book, Hard Choices, explains the nature of the

problem that he addresses in that classic work: Decision Making under

Unresolved Conflict.1 We have learned greatly from Levis analyses of

why, despite our best efforts, the valuational conflicts that we face may

not always be fully resolved when the point of decision making comes,

and how we may nevertheless use systematic reasoning to decide what

one should sensibly do despite the presence of unsettled conflicts. Indeed,

through a variety of contributions stretching over several decades, Isaac

Levi has powerfully illuminated the challenges of decision making in the

presence of imperfect information, conflicting evidence, divergent values,

discordant commitments, and other sources of internal dissension.2

I seize the wonderful occasion of celebrating Isaac Levis work and ac-

complishments by presenting a series of observations on rational decision

making with incompletely resolved internal dissensions. In that context, I

comment also on the nature and use of incomplete valuational orderings

and the ways and means of extending their reach. I pursue these issues in

the form of addressing a series of questions. Sometimes I draw on Levis

work, and at other times, I comment on differences that we may still have.

As will be obvious, even when we disagree, my understanding of theseissues is strongly influenced by Levis thinking.

1. IS NONCOMMENSURABILITY THE PRIMARY REASON FOR

UNRESOLVED CONFLICT?

Noncommensurability of different types of values is often seen as a reason

indeed the main reason for incompleteness of an overall (all things

considered) ranking. Indeed, the belief that commensurability is neces-

sary for arriving at a completely ordered ranking of different optionsseems to have considerable following in the literature. Since I have always

had some problem in grasping the reasoning behind that belief, I should

perhaps try to articulate where my difficulty lies.

Synthese 140: 4359, 2004.

2004 Kluwer Academic Publishers. Printed in the Netherlands.


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44 AMARTYA SEN

Commensurability of two distinct objects stands for their being measur-

able in terms of each other. Noncommensurability is present when several

dimensions of value are irreducible to one another.3 In the context of

evaluating a choice, commensurability requires that in assessing its results,we can see the values of all the relevant results in exactly one dimension

measuring the significance of distinct outcomes in a common scale so

that in deciding what would be best, we need not go beyond counting the

overall value in one homogeneous metric. Since the results are all reduced

to one dimension, we need do no more than checking which option will

give us how much of the one good thing to which every value is reduced.

We are, certainly, not likely to have much difficulty in choosing

between two alternative options both of which offers just the same good

thing, but one offers more than the other. This is an agreeably trivial case,

but the belief that whenever the choice problem is not so trivial, we must

have great difficulty in deciding what we should sensibly do seems pe-

culiarly feeble (it is tempting to ask, how spoilt can one get!). Indeed, ifcounting one set of real numbers is all we could do, then there would not

be many choices that we could sensibly and intelligently make. Whether

we are deciding between buying different commodity baskets, or making

choices about what to do on a holiday, or deciding whom to vote for in an

election, we are inescapably involved in evaluating alternatives with non-

commensurable results. Noncommensurability can hardly be a remarkable

discovery in the world in which we live. And it need not, by itself, make it

very hard to choose sensibly.

For example, a fine mango may give us nutrition as well as some palatal

or olfactory pleasure, whereas buying the record of a good song may offer

a very different reward (not immediately reducible into the dimensions ofthe other), and given a budget constraint we could quite possibly face the

choice of having one or the other. This involves choosing between non-

commensurable results. And yet we may have no great difficulty in opting

for the mango when immensely hungry or starved, and going for the song,

when well endowed with tasty food but short of melodious entertainment.

The choice need not be hard to make in many situations, despite the non-

commensurability involved. The distinct dimensions of values may not be

reducible into one another, and yet there may be no problem whatsoever in

deciding what one should sensibly do when our priorities or weights over

these values are clear enough.

Making choices with noncommensurable rewards is like speaking in

prose. It is, in general, not particularly hard to speak in prose (even if M.Jourdain in Molieres Le Bourgeois Gentilhomme may marvel at our ability

to perform so exacting a feat). But this does not negate the recognition


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INCOMPLETENESS AND REASONED CHOICE 45

that speaking can sometimes be very difficult (for example, when one is

overwhelmed by emotions), but that is not because expressing oneself in

prose is in itself arduous. The presence of noncommensurable rewards only

indicates that choice decisions will not be trivial (reducible just to countingwhat is more and what is less), but it does not at all indicate that it is

impossible or even that it must be particularly difficult.

What we have to ask is: why are some choices hard? This must be

the case when the diverse values involved are both in conflict and difficult

to weigh relative to each other, that is, when (as Isaac Levi discusses with

such care) it is hard to resolve their conflicting pulls. The exercise of

overall evaluation may not only involve more than counting, the process

of non-trivial aggregation may be complex and challenging in some cases

in a way it may not be in others. To say that noncommensurability is the

primary cause of incompleteness amounts to missing the specific causes

of incompleteness in favour of an ever-present precondition that has little

discriminating relevance. To use an analogy, it would be like saying that wefeel hungry primarily because we have a stomach. Certainly, it is hard to

explain hunger without presuming something about the stomach, but it is

not particularly useful to concentrate on the existence of the stomach in the

human body in trying to explain hunger in the world. While the stomach is

generically involved, we are likely to get more help in explaining hunger if

we try to examine instead how difficult it may be for a person particularly

a poor person to fill his or her belly. Similarly, even though the presence

of noncommensurable results is generically involved in incompleteness,

we have to investigate whether it is difficult or easy to weigh the

different types of values involved and to resolve their conflicts.

2. IS THERE ULTIMATELY ONLY ONE SOURCE OF IMPORTANCE?

Noncommensurability does not in itself take us very far in understanding

the nature of the decisional problem to be addressed. It is, therefore, in

many ways a pity that so much attention has been heaped in the literature

on the existence of noncommensurability as such, which is an omnipresent

and ordinary feature of the world, compared with the distinct reasons

that make value conflicts so serious in some cases and not at all in oth-

ers. But perhaps there is a different, more dialectical, explanation of why

several perspicacious thinkers have chosen to emphasize the importance of

noncommensurability. The attention paid to noncommensurability is partlya critique of the belief often implicit that there must ultimately be

only one source of significance. By focusing on noncommensurability, the

critics of such reductionism assert something of importance regarding the


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46 AMARTYA SEN

absence of a single standard of value.4 Indeed, the hold of the tradition

of reducing everything to one homogeneous virtue may explain why the

mundane recognition that there are different values that are not reducible to

each other has been seen as worth asserting. Even a humdrum cognizancemay have a positive and indeed a constructive role in the dialectics of

conceptual diagnosis.

Adam Smith complained more than two hundred years ago about the

tendency of some philosophers (he had separated out Epicurus, but clearly

had others in mind as well, including circumstantial evidence indicates

his friend, David Hume) to look for a single homogenous virtue in terms

of which all values could be explained:

By running up all the different virtues to this one species of propriety, Epicurus indulged

a propensity, which is natural to all men, but which philosophers in particular are apt to

cultivate with a peculiar fondness, as the great means of displaying their ingenuity, the

propensity to account for all appearances from as few principles as possible. And he, no

doubt, indulged this propensity still further, when he referred all the primary objects of

natural desire and aversion to the pleasures and pains of the body.5

There are indeed schools of thought which insist, explicitly or by implic-

ation, that all the appearances of value must be reduced ultimately to a

single source of importance. This claim about the nature of value is of-

ten supplemented by the further thought, in which commensurability does

come in, that in order to be able to choose rationally, all values must be

reduced to one. Evidently, the protagonists of this view believe that human

beings can count but cannot evaluate.

The counting freaks (if I may call them that, without intending any

disrespect) include some but not all utilitarians. There is indeed a

version of utilitarian reasoning that takes the form of arguing that there

is no noncommensurability at all in what ultimately matters, to wit utility.

In the example considered earlier, if we contingently have no difficulty

in choosing between the mango and the song, it is because (the argument

runs) both can be judged by their respective ability to generate utility. In

the last analysis (the argument insists), we count, not judge.

I shall not scrutinize here this particular argumental loop, but only note

that utility may be far from a homogenous magnitude (a recognition that

had not escaped that great utilitarian, John Stuart Mill), and also that there

can be other reasons for choice (other than the pursuit of utility), no mat-

ter how utility is substantively defined. If, however, utility is not defined

independently, but only as the real-valued representation of the binary re-lation underlying choice behaviour (as is common in modern economics),

then utility is not only not one good thing, it is not a thing at all, but a

mere phantom of representation. Furthermore, even a consistent phantom


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may not exist when choice behaviour is non-binary. I have discussed these

issues elsewhere, and will desist from pursuing them further here.6

3. HOW USEFUL IS LEVIS GENERAL CONCEPT OF

V-ADMISSIBILITY IN UNDERSTANDING THE DEMANDS OF

RATIONAL DECISION MAKING WITH UNRESOLVED CONFLICTS?

V-admissibility, which is a central concept in Levis investigation, is in-

deed very useful. Even though I shall presently argue (in Section 5) that

particular questions can be raised about some applications of the idea of

V-admissibility that Levi endorses, let me first discuss why the concept is,

I think, important and helpful.

In a set of options (let me call it the menu), the V-admissible options

are those that have not been prohibited by the agents value commitments

from being chosen by the agent.7 They are, thus, admissible relative tothe agents valuations of the feasible options as better or worse, all things

considered. The concept of V-admissibility introduces, in two distinct

ways, a useful gradualism in the process of systematic choice. First, if

an option is not V-admissible, then it would not be sensible to choose it,

but if it is V-admissible, there may still be further questions to be asked

as to whether it is optimal or not. In being definitive in exclusion but only

permissive in terms of inclusion, it makes the choice process capture an

important asymmetry. Weeding out the clearly unchoosable options can

be a good way to begin, but even when the rotten ones have been weeded

out, the remainder may call for further and closer attention.

Second, V-admissibility itself is a parametric concept: the criteria of

admissibility are not already ingrained in the very idea of V-admissibility

and they have to be additionally defined, and can be varied. As the criteria

are more and more specified, the constraints imposed by V-admissibility

can be gradually tightened. Sequential tightening can be helpful in coming

to grips with hard choices.

Let me illustrate by recharactering Levis investigation in the form of

four distinct steps associated with the idea of V-admissibility (drawing on

different parts of Levis work).

(1) Single-dimensional valuation. It is, as discussed already, trivial to

decide what to do when the different results are all fully commensurable.

We can, in this very special case, simply count our way to checking

which option or options are the best (that is, have most of the onegood thing to which everything else is reduced). With single-dimensional

valuation, V-admissibility is both trivial and decisionally definitive: in one

step we can reject all the rejectable alternatives and go straight to what we


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48 AMARTYA SEN

should choose.8 If this does not work and mostly it will not then we

must get into non-trivial exercises.

(2) Congruent multi-dimensionality. When distinct and disparate values

all move together, in the same direction with each other, the ranking ofoptions can simply follow the shared ordering of all the distinct value com-

mitments. As Levi notes, when the agents value commitments generate

no conflict but constrain the agent to evaluate his feasible options in an un-

equivocal manner, the agent is obligated by those commitments to restrict

his choice to one of the feasible options which is optimal according to the

mandated ranking.9 The V-admissible alternatives are those that could not

be eliminated on the ground of their being inferior to some feasible option

in terms ofeach of the value commitments involved. This is also a simple

enough case, but it does not presuppose any commensurability at all.

(3) The Weighted Average Principle: So far nothing more than ordinal

ranking of each value commitment has been invoked, but we must now go

beyond that. Let v1, . . . , vn be the n numerical valuational functions thatgive the value of each option according to each of the n value commitments

respectively. It is assumed that each of these vi is cardinally measurable

unique up to a positive affine transformation.10 Levi confines attention now

to the weighted sum of all the vi as reflecting the aggregate value v of the

respective options, for a set of non-negative weights (w1, . . . , wn).11 So we

have: v = wi vi, and depending on the weights we choose for the respect-

ive value commitments, we get a corresponding set of aggregate values of

each option. If we had a uniquely specified set of such weights (w1, . . . ,

wn), then of course the valuational exercise would be over. But so long as

the conflict is unresolved, this could not be presumed. So, at this stage,

Levi concentrates attention on the entire class of non-negative weights.However, some options may end up having a lower aggregate value than

another under every permissible set of non-negative weights, and in that

case, the uniformly lower valued alternative can be eliminated as being

non-admissible. This way the afforced criterion of V-admissibility elim-

inates some non-admissible options in the menu.12 This is a critically

important and innovative step in Levis reasoning, and I shall examine the

pros and cons of going this way in Section 5, when I specifically scrutinize

the weighted average principle.

(4) Categorical preference and optimality. It is possible to go beyond

valuational conflicts when aggregate valuations are in conformity with

each other, at least in part. For example, ifx is strictly preferred in value to

y according to all permissible valuation rankings, then x is categoricallypreferred to y.13 The notion of categorical preference is useful in checking

admissibility, but it can also in some cases with luck help to identity an


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optimal choice. When it turns out that there exists an option x in the menu

such that for every permissible valuational ranking, x is at least as good as

every other alternative in the menu, then x is declared to be V-optimal.

If, furthermore, the rankings are such that any option y in the menu thatis regarded as indifferent to an optimal option x is also optimal (but y is

definitely not optimal when x is ranked above y), then we get to the more

regular idea of full optimality rather than only V-optimality.14

The possible existence of optimal or V-optimal options can make de-

cisions easier to take despite unresolved conflicts, but the broader idea

of V-admissibility gets us, in general, part of the way, even when we do

not arrive at a categorically justified optimal decision. In the gradualist

approach investigated by Isaac Levi, the role of each of the finely defined

concepts is explored and explained.15

4. IS THERE AN ALTERNATIVE SYSTEMATIC APPROACH TO RATIONALDECISION MAKING UNDER UNRESOLVED CONFLICTS?

There is, and indeed, to a great extent, Levis strategy can be seen as a

reasoned departure from an older approach that focuses on maximization

based on a possibly incomplete ordering derived from the intersection of

different value commitments. That approach can be seen in terms of its two

constitutive components, viz. maximization and intersection.16 I shall call

it (not terribly imaginatively) intersection maximization.17

The idea of maximization as choosing an option that is no worse than

any other feasible option can work with incomplete orderings. To qualify

as maximal an option need not be shown to be at least as good as all the

other feasible options only that it is not strict worse than any. The idea

of maximality in this broad sense has been well formalized by Bourbaki,

Debreu and others.18 If decision making is based on maximization (rather

than optimization choosing an option that is at least as good as every

other option), it is not a requirement that all value conflicts be resolved

before a reasoned choice can occur. The incomplete ordering to be used for

maximization can be derived on the basis of the intersection of the different

valuational rankings. Even when unresolved conflicts exist, an incomplete

ordering can be identified that ranks two options in the overall ranking in a

certain way if and only if those options are ranked in the same way by all

the different values involved. So, if x is ranked above y according to each

of these values, then x is ranked overall above y. Similarly, if x and y tiein terms of each of these values, then x and y are taken to be indifferent

overall. The intersection partial ordering generated in this way can then

be used, along with the approach of maximization, for systematic choice


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50 AMARTYA SEN

despite the presence of unresolved conflicts (which can make the overall

ranking incomplete).

The combined use of maximization and intersection is basically iso-

morphic with Levis identification of a mandated ranking related toV-admissibility before we get to his weighted average principle.19 Since

the older approach does not invoke the weighted average principle, we

can perhaps ask what alternative principle does it use? The answer is noth-

ing: it stops right there with maximization based on the intersection partial

ordering. In this sense, intersection maximization is an ineloquent theory (I

shall presently comment on the strategic use of this lack of eloquence). The

maximal set is the set of all options that are undominated by the inter-

section partial ordering. It can be shown that maximization, in this broader

sense (broader than optimization, and also as it happens, broader than

V-admissibility with Levis weighted average principle) yields many im-

portant and rather far-reaching properties in the discipline of choice,

and it can be defended both on analytical and substantive grounds. 20 Thegeneral approach of maximization can also incorporate various additional

features that were not part of the old structure of maximality identified by

Bourbaki and Debreu: such as incorporating actions of agents as a part of

the comprehensive outcome of choice, which allows us to re- examine

the dividing line between consequentialist and deontological reasoning.

Various other properties can also be optionally imposed by utilizing the

capacious format of maximization, and making use of the room for further

articulation left open by the ineloquent form of that approach.21

We have to examine whether Levis addition of weighted average prin-

ciple and its implications for categorical preference can also be sensibly

added on to the basic structure of intersection maximization. I take up thatissue next.

5. IS THE CRITERION OF WEIGHTED AVERAGE PRINCIPLE FO R

V-ADMISSIBILITY PERSUASIVE AND ACCEPTABLE?

There are certainly good arguments in favour of using some additional

structure, like Levis weighted average principle, to extend the reach of

intersection maximization. There is, first, a manifest need here, and second,

Levis particular constructive proposal has some conspicuous merit.

I begin with the issue of need and the motivation for Levis departure.

The maximal set based on intersection can be quite large. Consider Levisexample of a hapless office manager, called Jones: I shall call her Ms.

Jones. She wants to hire a secretary who is a good typist and a good

stenographer as well (and thus has two value commitments), and con-


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siders three candidates: Jane, Dolly and Lilly. Jones finds that in terms of

typing skill, Jane is better than Dolly and Dolly superior to Lilly, but in

stenographic competence, Lilly is better than Dolly, who is better than

Jane. The intersection of these two rankings is empty, and intersectionmaximization based on these two orderings would suggest that all three

are maximal.

Levi, not surprisingly, sees this as hopelessly inarticulate. He wants to

ask such questions as whether Dolly, the middle skilled in each, is the

second best typist and second best stenographer, or merely the second

worst typist and second worst stenographer. Dolly is, of course, all

these, in ordinal terms, but Levi wants to be able to get and use more

information about where Dolly figures vis-a- vis Jane and Lilly. This takes

Levi to cardinal valuational functions vi and then to the weighted aver-

age principle. With this motivation for wanting a more informed and

through that a more articulate choice, I am in total sympathy.22 The issue

to be examined is whether the particular procedure developed by Levi isadequate.

The weighted average principle is a way of cutting down the

intersection-based maximal set by eliminating some options that would

have a lower value than an alternative option in terms of all non-negative

weights that can be placed on the cardinal valuational functions associated

with the respective value commitments. The Levi approach goes beyond

the older maximization approach both (i) by introducing cardinal meas-

urement associated with each value commitment, and (ii) by using the

strategy of weeding out any option as V-inadmissible if it gets dominated

by all possible weighted aggregate values by some other feasible option (in

terms of the cardinal representation chosen). This is where the departurecomes, and we have to ask whether it is convincing.

The second argument, which takes us beyond motivation, is that

Levis reasoning certainly has considerable plausibility (which as I shall

presently discuss is not the same thing as its being, all things considered,

convincing). The plausibility can be illustrated by giving valuational num-

bers to the two skills of the three candidates each. Consider the following

valuations:

Candidates Typing skill Stenographic skill

Jane 10 1

Dolly 2 2

Lilly 1 10

Given the cardinal characteristics of this measurement, any positive affine

transformation of these numbers will serve as well.


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52 AMARTYA SEN

We can add up the two numbers of each with any set of non- negative

weights. But it is clear that no matter what these chosen weights are, Dolly

will get trumped by either Jane or Lilly, since Dolly is nearly as bad a typist

as Lilly and nearly as bad a stenographer as Jane. To see this consider thepossibility that Dolly gets a higher total score than Jane. In that case the

weights must be such that the one- point advantage in stenography that

Dolly has over Jane (with her mark of 2 against Janes 1) outweighs the

8-point advantage that Jane has over Dolly in typing (with their respective

figures of 10 and 2). So the weight on stenographic skill must be at least

8 times as great as on typing skill. But, then, clearly Lilly will beat Dolly

hollow with her 8-point advantage over Dolly in stenography, despite her

one-point deficit vis-a-vis Dolly in typing. If Dolly beats Jane, then Lilly

beats Dolly, and it can also be readily checked that if Dolly beats Lilly,

then Jane will beat Dolly. Dolly is, thus, not V-admissible, given Levis

weighed average principle. The reasoning not only takes us beyond the

purely ordinal comparisons of intersection maximization, it also has someevident appeal.

Though I have not been retained by luckless Dolly, let me now argue,

first, against this particular conclusion, and then, against the principle of

weighted average in general. The central issue is this: can we do an overall

evaluation of the candidates without asking how the importance of the

two skills may vary as we consider different levels of achievement of the

candidates? Suppose that a skill level of 1 in typing, which Lilly has, does

not make it possible for letters to be typed well enough to be despatched,

but level 2, which Dolly has, makes that possible, though it is nowhere near

the superb level of typing skill that Janes mark of 10 indicates. Suppose

also that a skill level of 1 in stenography, which Jane has, leads to totalchaos in the office (say that a third time please, Ms. Jones), whereas

Dollys level 2, modest as it is, prevents that, even though the stenographic

work would be massively better with a skill level of 10, which Lilly enjoys.

Faced with these considerations, Ms. Jones may drop Jane to prevent chaos

in office, and she may also let Lilly go so that some letters can actually

get typed. Dolly satisfies the qualifying level in each skill, and may well

be chosen on these grounds (despite her V-inadmissibility in terms of the

weighted average principle). I rest my case for Dolly or more accurately

for not giving Dolly the boot without checking what the importance of the

different levels of skill are for the choice at hand (and not just how much

skill there is).

I turn now to the more general issue of the acceptability of the weightedaverage principle. Underlying my scepticism is a question about the inter-


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pretation of the valuational numbers related to each value commitment.

What do the numbers given by vi stand for? Do they:

(1) measure the levels of accomplishment in field i, or

(2) measure the importance of these accomplishments for the purpose for

which the choice decision is being taken?

Levis language suggests that he takes the former interpretation, so that the

numbers represent the levels of skill of Dolly and others. If so, we still have

to ask how valuable is that skill for the decision at hand, and this cannot be

answered independently of the levels of that skill and the presence of other

factors that allow or facilitate or hinder the use of that skill. For example,

Janes excellent typing skill (mark 10 no less) can be quite valueless

in terms of Ms. Joness decisional problem if the woeful nature of her

stenographic skill makes her an impossible holder of the office position in

question. We need not only a valuation of the respective skills (and moregenerally, of the extent of accomplishment in each value commitment), but

also an evaluation of the contingent importance of the exact level of skill

(or accomplishment) given everything else that is involved in the decisional

picture.

If, on the other hand, the second interpretation is taken, then we have

to ask how can the importance of a skill (or more generally, of a value

accomplishment) for the decision at hand be represented independently of

other factors in the choice? How can we, for example, determine that the

choice- context importance of Janes typing skill, in an additive framework,

is invariably 10 for this office job, no matter whether she can do anything

else at all that may also be a part of the job in question? Something or

other is clearly missing in this framework, and the operation of weighted

addition, which leads the way to the weighted average principle, can be

deeply problematic.

Perhaps an analogy from social choice theory can help to clarify the

issue. Consider a welfarist framework, in which the social value of a state

of affairs is seen as a function of the welfare levels of all the people in

that state.23 We can think of each persons well-being as a kind of value

commitment, in terms of an analogy with Levis framework. To do this

exercise, we need:

(1) to evaluate each persons well-being (measured on its own);

(2) to make interpersonal comparisons of the values of different personswell-being (putting them in a comparable scale);

(3) to decide on the weights to be put on increments or diminutions in each

persons well-being respectively vis-a-vis those of others (reflecting


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54 AMARTYA SEN

what importance we want to attach respectively to distributive and

aggregative considerations).24

These distinct exercises cannot be combined in one assignment of vi

functions (unique up to a positive affine transformation) which are thensimply added together with non-negative weights. No matter how we

choose the values ofvi functions, something or other would not have been

addressed. I would argue that a similar lacuna exists in the framework

of V-admissibility with vi functions, limited to the class of fixed affine

transformations and fixed non-negative weights. Judging the decisional im-

portance of the accomplishments of different value commitments requires

more than a weighted averaging of the different accomplishments.

6. CAN WE USE V-ADMISSIBILITY WITHOUT THE WEIGHTED

AVERAGE PRINCIPLE?

Even though the particular proposal of using the weighted average prin-

ciple with V-admissibility seems problematic that does not negate the

motivation that led Isaac Levi to seek an extension of the reasoning under-

lying intersection maximization. The maximal set given by the intersection

of all the value commitments can be unhelpfully large, and it is sensible

enough to try to see how that set can be reduced in size to give more bite

to the decisional process. Dolly may be hard to eliminate from the list of

three through the weighted average principle in particular, but that need not

indicate that all three must be seen as fine appointees merely because the

intersection partial ordering is empty. Levis motivation in seeking more

stringent criteria in the form of tighter requirements of V-admissibility is

in general just right.

Consider a more general specification of the problem of going beyond

intersection maximization through V-admissibility, by combining Levis

motivation with formal investigations pursued in social choice theory. 25

We follow Levi in considering valuation functions vi related to each value

commitment, but do not necessarily constrain the uniqueness properties

of each vi to the class of positive affine transformations. Depending on

the type of values involves, the class may be wider (and the extent of

measurability correspondingly less, e.g., ordinal), or narrower (and cor-

respondingly have more measurability, e.g., ratio scale), and it is possible

also to consider various intermediate possibilities of a hybrid kind.26 Wecan proceed from there to consider aggregation functions v = fk(v1, . . . ,

vn), and impose invariance conditions that reflect the particular extents

of measurability and any comparability restrictions that we may want to


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impose.27 We can consider a number of such aggregation functions fk, with

k = 1, . . . , m, in a class F. If for some feasible y, for all fk in F, v(x) is less

than v(y), then x is not V-admissible.

This general structure can accommodate Levis weighted average prin-ciple if the valuation functions vi are cardinal and if the aggregation

functions fk in F are restricted exactly to the class of weighted aver-

ages. Similarly, by choosing ordinal vi functions and an ordinal class

F of aggregation functions fk, we can get back to simple intersection

maximization. But there are a great many other possibilities as well, and

V-admissibility can be used to consolidate the minimal articulation of in-

tersection maximization, and then to go beyond that to the extent that is

permitted by our actual ability to marshall information and to scrutinize

our assessments. Unresolved value conflicts leaves open many possibilities

of methodical use of decisional reasoning, and V-admissibility can serve

as a general format that accommodates as much articulation as we can

reasonably justify.

7. IS INCOMPLETENESS ALWAYS TENTATIVE, AWAITING

RESOLUTION?

In the preceding discussion the focus has been on reducing incomplete-

ness as much as possible. But what about any remaining incompleteness?

Should we see it as an embarrassment, or as a defect that calls for ways

and means of total elimination. I would argue that the answer must depend

on the nature of the remaining incompleteness.

In many cases the incompleteness is best described as tentative. It

awaits resolution (with more information, or deeper analysis, or closer

scrutiny, or whatever), whether or not the resolution actually occurs. This

kind of incompleteness has to be contrasted with the idea of assertive

incompleteness.28 This category separates out cases of incompleteness in

which the lack of completeness is positively asserted, yielding statements

such as x and y cannot be ranked. It is radically different from incom-

pleteness that is tentatively accepted, while awaiting or working for

completion. The partial ranking may simply not be completable, and

may not even be ideally completed. Rather, incompleteness may be the

right answer in these cases.

It is useful to distinguish between three different types of assertive

incompleteness related to the source of the incompletability involved.First, when the inquiry concerns a specific field of ethical or decisional

judgment (such as justice), the assertive incompleteness may relate to

the domain of that type of judgment (for example, the reach of a theory


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56 AMARTYA SEN

of justice). The recognition that incompleteness of judgments can be a

constituent part of the definite conclusions advanced by a complete theory

of justice can be quite momentous for practical reason.

Indeed, even a complete theory of justice can yield and assert incomplete rankings of justice. Affirmed incompleteness (e.g., x and y

cannot be ranked in terms of justice) may be as much a definite conclu-

sion as the decision that x and y can be ranked in terms of justice (and

in particular that, say, x is more just than y, or that x is just, whereas y

is not). However, it must also be noted that even when it is assertively

concluded that x and y cannot be ranked in terms of justice, this does not

entail that they cannot be ranked in terms of some other type of ethical

or political concern. There is a distinction here between this kind of field-

specific conclusion (important as it may be) and more ambitious claims of

assertive incompleteness for all things considered decisions.

Second, even for an all things considered evaluation, a partial ranking

(or a partial partition) may be a crucial assertion as the end product at a par-ticular stage of a multi-stage exercise. A theory may assert incompleteness

for a well-defined purpose, leaving room for a possible extension through

an appropriate subsequent stage. If some decisional issues are not decid-

able by, say, general (or foundational) ethical reasoning, the incomplete

ranking emerging from general ethical reasoning may be an appropriate

assertion for that general stage. This may perhaps be supplemented by

a subsequent stage using other procedures (for example, some kind of a

democratic decision mechanism) that extend the general ethical reasoning

by choosing between alternative courses of action that are all consistent

with general ethical reasoning.29 The affirmation of incompleteness at one

stage of the process may be critically important for the necessity andviability of the next stage.

Third, aside from field-specific and stage-specific assertive incomplete-

ness, there is the possibility of an unqualified assertion of incompleteness.

It is possible that incompleteness may be a durable and definitive part

of the end product of all things considered and all stage included

evaluation. It may be as far as we can proceed with reasoned discrim-

ination, given the information that we can conceivably have. If so, the

incompleteness will not await completion at a later stage or over a wider

field of reference, and will yield such statements as: x and y definitely

cannot be ranked for decisional purposes.30 There is a need to see as-

sertive incompleteness as a conceptual category of its own. For example, a

fine-grained discrimination between two value commitments with exactlyspecified weights may simply be beyond the reach of reasoned ethical

scrutiny, or may demand information that may not only be contingently


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lacking but also impossible to obtain even in principle. For example, it

can be argued that interpersonal comparisons of well-being, though by no

means impossible, may not take us all the way to invariance conditions

that yield one-to-one correspondence of everyones well-being numbersvis-a-vis each other.31

I end by noting that the recognition of the possibility of assertive incom-

pleteness does not reduce in any way the value of scrutiny and investigation

aimed at reducing the extent of tentative incompleteness through continued

scrutiny of unresolved conflicts. Nor, for that matter, does it rule out the

possible usefulness of challenging whether a putative claim of assertive

incompleteness is indeed justified. Assertive incompleteness no less than

tentative incompleteness is well within the domain of inquiry and jus-

tification which Isaac Levi has done so much to clarify for us. We know

from Isaacs work how magnificently capacious that domain is.

NOTES

1 Isaac Levi, Hard Choice: Decision Making under Unresolved Conflict, Cambridge

University Press, Cambridge (1986).2 Isaac Levi, Gambling with Truth, Knopf, New York (1967); Information and Inference,

Synthese 17, 369379 (1967); On Indeterminate Probabilities, Journal of Philosophy 71,

391 418 (1974); The Enterprise of Knowledge, MIT Press, Cambridge, MA (1980); Hard

Choices (1986); Conflicts and Inquiry, Ethics 102, 814834 (1992); The Covenant of

Reason, Cambridge University Press, Cambridge (1997). Levi has been concerned with

decisions in epistemology as well as practical reason. The focus of this essay is, how-

ever, exclusively on the latter, even though some of the basic issues have relevance to

epistemology as well.3 The Covenant of Reason (1997, p. 236).4 As Isaac Levi explains, when the perspective on inquiry and justification is combined

with a rejection of a single fixed standard of value, the key elements of the pragmatism of

Peirce and Dewey which I admire are identified ( The Covenant of Reason, p. 218).5 Smith, The Theory of Moral Sentiments, revised edn., VII.ii.2.14 (1790). Republished,

Clarendon Press, Oxford (1976, p. 299).6 See Choice, Welfare and Measurement, Blackwell, Oxford (1982); Harvard University

Press, Cambridge, MA (1997); On Ethics and Economics Blackwell, Oxford (1987); and

Rationality and Freedom, Harvard University Press, Cambridge, MA (2002).7 Levi, Hard Choices (1986, p. 10).8 I am abstracting here from the problem of infinite sets in which even a complete ordering

need not necessarily yield a best or an optimal alternative. That raises a different range

of issues, with which I shall not be concerned in this essay. In the present investigation,we may as well assume that the set of options is finite, so that a complete ordering will

always identify a best alternative. Indeed, even an acyclic but complete ranking will do

that; on this and related results, see my Collective Choice and Social Welfare, Holden-

Day, San Francisco (1970). Republished, North-Holland, Amsterdam (1979, Chap. 1),


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58 AMARTYA SEN

and Choice Functions and Revealed Preference, Review of Economic Studies 38 (1971);

Hans Herzberger, Ordinal Preference and Rational Choice, Econometrica 41 (1973); and

Levi, Hard Choices (1986).9 Levi, Hard Choices (1986, p. 9).

10 An affine transformation is of the form: a + b.vi, and the positivity of such a trans-formation refers to b being positive. On the characteristics and uses of aggregations based

on cardinal values (with uniqueness up to positive affine transformations), see my Col-

lective Choice and Social Welfare (1970); Kotaro Suzumura, Rational Choice, Collective

Decisions and Social Welfare, Cambridge University Press, Cambridge (1983); Levi, Hard

Choices (1986).11 The weights w1, . . . , wn are not only non-negative each, but also they must sum to 1.12 Levi, Hard Choices (1986, Section 5.4, pp. 7779). See also the motivating discussion

on pp. 6977.13 This may require us to go beyond the weighted average principle. An extreme case

would be one in which there is a unique set of appropriate weights ( w1, . . . , wn). But

categorical preferences over a subset of alternatives can arise in many different ways, and

may even be precipitated, in some cases, by the weighted average principle.

14 Levi, Hard Choices (1986, pp. 8355). Levi discusses these structures in the context ofexamining values revealed by choices, but they throw light also on the movement from

values to choices, and not merely from choices to revealed values.15 There are many other structural features which Levi has investigated, which I shall not

examine in this essay.16 Combined use of maximization and intersection can be found in many exercises in

public economics. See for example A. B. Atkinson, The Measurement of Inequality,

Journal of Economic Theory 2 (1970); Amartya Sen, On Economic Inequality, Clarendon

Press, Oxford (1973); extended edition, with a joint Annexe with James Foster (1997).17 In On Economic Inequality (1973) it was called the intersection approach, taking

maximization for granted.18 N. Bourbaki, Elements de Mathmatique, Herman, Paris, and Theory of Sets, Addison-

Wesley, Reading, MA (1968); Gerard Debreu, The Theory of Value, John Wiley, New York

(1959).19 Levi, Hard Choices (1986, p. 9).20 On this see my Maximization and the Act of Choice, Econometrica 65 (1997).21 Some of these possibilities have been explored in my Maximization and the Act

of Choice (1997) and Consequential Evaluation and Practical Reason, Journal of

Philosophy 97 (2000).22 I have a natural sympathy here for various reasons, including the fact that I had a similar

motivation in trying to move social choice theory towards accommodating more informa-

tion on cardinality and interpersonal comparability of well-being and also including more

cognizance of freedoms and liberties, in Collective Choice and Social Welfare (1970).23 This is, of course, a very limited model of social choice, but adding further consid-

erations, such as freedoms, liberties or rights, will not make the problem at hand any

easier.24

For the distinctions involved see my Collective Choice and Social Welfare (1970) andChoice, Welfare and Measurement (1982).25 For assessments of alternative contributions to the ways and means of using more in-

formation in social evaluation, see my Social Choice Theory, in Kenneth Arrow and


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Michael Intriligator (eds.), The Handbook of Mathematical Economics, North-Holland,

Amsterdam (1986); and Kenneth Arrow, Amartya Sen and Kotaro Suzumura (eds.), Social

Choice Re-examined, Elsevier, Amsterdam (1997).26 See my Collective Choice and Social Welfare (1970, Chaps. 7 and 8).

27 The form offk may or may not be additive.28 I have discussed the notion of assertive incompleteness in Maximization and the

Act of Choice, Econometrica 65 (1997); and in Justice and Assertive Incompleteness,

Rosenthal Lectures (Lecture 2), Northwestern University Law School (1998).29 The fact that despite my arguing for the use of the capability perspective in comparing

individual advantages and my attempt to highlight the relevance of some basic capabilities

in particular (for example in Well-being, Agency and Freedom: The Dewey Lectures

1984, Journal of Philosophy 82 (1985)), I have failed to specify a fixed list of distinct

capabilities (with specified weights or other ways of prioritization) has been the source

of some chastisement I have received. However, if such a fixed list with fixed priorities and

fixed weights were indeed arrived at by general ethical reasoning, it is not clear to me how

this would be consistent with the democratic process of setting priorities and precedence.30 Assertive incompleteness must not be confused with an assertion of indifference. If it is

claimed that x and y cannot be ranked, then that is what it says, not that they can be rankedas equals. Indeed, in some ways incompleteness is the opposite of indifference. Consider

two possible claims: (1) x is at least as good as y, and (2) y is at least as good as x. If x

and y are indifferent, then both (1) and (2) are true, whereas if their ranking is assertively

incomplete, then (1) and (2) are both denied. On this distinction, see my Collective Choice

and Social Welfare (1970, Chap. 1), and Maximization and the Act of Choice (1997).31 I have tried to discuss the class of limited though possibly quite extensive

informational discrimination in Interpersonal Aggregation and Partial Comparability,

Econometrica 38 (1970), and in Collective Choice and Social Welfare (1970), and also

in On Weights and Measures: Informational Constraints in Social Welfare Analysis,

Econometrica 45 (1977).

Trinity College

Cambridge, CB2 1 TQ

U.K.

and

Harvard University

Cambridge, MA 02138

U.S.A.


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Amartya Sen - Incompleteness and Reasoned Choice 2004