interpersonal comparisons of the good

8/8/2019 Interpersonal Comparisons of the Good

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Draft as of Sept 2010. Please do not cite without authors permission.

Interpersonal Comparisons of the Good:Epistemic not Impossible.

Mathew Coakley

AbstractTo evaluate the overall effects of any policy or institutional choice we needsome way of comparing the benefits and losses to those affected: we need tomake interpersonal comparisons of the good/welfare. Yet skeptics have worriedeither: (1) that such comparisons are impossible as they involve an impossible

introspection across individuals; (2) that they are indeterminate as individuallevel information is compatible with a range of welfare numbers; or (3) thatthey are metaphysically mysterious as they assume either the existence of asocial mind or of absolute levels of welfare when no such thing empiricallyexists. I argue, however, that we should treat this as an epistemic problem that is as a problem of forming justified beliefs about the overall good based onevidence about the good of individuals and that if we do so, these critiquespotentially fail.

I show what such beliefs would be, set out the axioms that justify them,and discuss what this means for the evaluation of social and economic policiesand political institutions.

In the Annex I prove that, for any non-dogmatic well-defined credence

function, accepting Arrows Independence of Irrelevant Alternatives logicallyentails the possibility of incoherent beliefs. If we seek justified beliefs aboutoverall welfare or the overall good, Arrows Impossibility Theorem should notbe accepted as validly characterizing our task.

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Interpersonal Comparisons of the Good:

Epistemic not Impossible.

IntroductionA longstanding and prominent way of evaluating social, political, and eco-nomic policies and institutions is by how they net affect people. If one policybenefits those affected more than another then, on an overall-good based moraltheory, this potentially makes that policy morally preferable1. And as everyactual conceivable collective policy or institutional change will have some whobenefit and some who lose we need a way of comparing these loses and benefits:we need some way of making interpersonal comparisons of the good, what iswithin welfare economics termed interpersonal comparisons of welfare2. Yet,aside from familiar generic moral skepticism - notably that of amoralists, ego-ists and nihilists - there are three prominent skeptical concerns that have been

raised about such interpersonal comparisons in particular.Firstly, there is the possibility-critique: that such comparisons cannot

be made, specifically as doing so would require introspectively experiencingthe internal mental phenomena of multiple individuals. It would, in a sense,require us to get into the heads of those affected by policies or institutions,something we cannot do. Robbins, for instance, justifies his skepticism byobserving that Introspection does not enable A to measure what is goingon in Bs mind, nor B to measure what is going on in As. There is no wayof comparing the satisfactions of different people.3 The possibility critique

1The most well-known moral theory to do so is, of course, consequentialism, roughly wherethe right (policy / action /institution) is that which best promotes the overall good, but the

importance of the concept of the overall good extends well beyond this: some deontologicaltheories, for instance, hold that, outside those acts that are morally forbidden, the promotionof the good may be morally desirable, even when not morally required.

2Within moral philosophy the term the good is frequently used; within economicswelfare is near ubiquitous. The good is a b etter term for any general discussion as itdoes not assume that welfare is the right account, but - given the widespread use of welfare -in particular examples I often revert to that where it better coheres with the extant literature.This is not meant to imply any necessary support for welfarism. The one term, however,I try to mostly avoid throughout is utility. This has a clear and now deeply entrenchedmeaning in p ositive economics: it represents an agents behavior. Alas, if we use this toalso refer to their welfare we risk consistently having the two conflated. Arrow, for instance,writes that if any course of behavior can be explained by a given utility function, it hasbeen amply demonstrated that such a course of behavior can be equally well explained byany other utility function which is a strictly increasing function of the first. If we cannot

have measurable utility, in this sense, we cannot have interpersonal comparability of utilitiesa fortiori. He here as also elsewhere simply runs descriptive utility (sentence one) andnormative utility (sentence two) together. Given how embedded utility is as a descriptiveconcept this paper avoids it in normative contexts so this conflation is not encouraged.Kenneth Arrow (1963), Social Choice and Individual Values. New York: John Wiley andSons, at p9.

3Lionel Robbins (1962), An Essay on the Nature and Significance of Economic Science,London: MacMillan. p140. And see chapter VI for the general discussion.

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holds that while introspective comparisons of alternatives by an individualare possible, the mere existence of multiple minds renders such comparisonsimpossible across multiple individuals.

Secondly, there is the determinacy-critique. This typically assumes thatonly ordinal individual information is possible, that for instance we can onlyknow that one arrangement is better than another for an individual, but notby how much. As such there are infinite possible numerical values that couldbe attached to these rankings that would be consistent with them when ag-gregating the good of multiple individuals, and therefore our overall rankingwill be indeterminate except for when everyone has two options ranked non-conflictingly (in that there are not two people one of whom has A rankedhigher than B and one of whom B higher than A). According to Jevons, thekey problem is thus that the susceptibility of one mind may, for what weknow, be a thousand times greater than that of another. But, provided thatthe susceptibility was different in a like ratio in all directions, we should never

be able to discover the profoundest difference. Every mind is thus inscrutableto every other mind, and no common denominator of feeling is possible4. Onenotable consequence of this is that we cannot know that some individual isnot what Nozick has dubbed a utility monster5 - satisfying their preferencesactually contributes much more to overall welfare than that of others eventhough this fact is not observable. As a result, so the argument holds, wecannot have determinate knowledge of overall welfare.

Thirdly, and perhaps most fundamentally, there is the metaphysical-critique: that interpersonal comparisons are meaningless as they assume theexistence of an entity - the overall good or social welfare - whose onto-logical or empirical status is mysterious: what does it even mean to think

that the overall good or social welfare exists? An individuals welfare is po-tentially based on facts about the mind / brain of the individual in question(her likes, dislikes, values, emotions etc). But overall welfare would require,so the critique suggests, either some multi-mind object that simply does notexist or on the existence of absolute amounts of welfare for a individuals ineach situation even though this cannot be measured or empirically verified.This concern seems ultimately to lie behind much skepticism. Arrow statesthat The viewpoint will be taken here that interpersonal comparisons of util-ities has no meaning and, in fact, that there is no meaning relevant to welfarecomparisons in the measurability of individual utility.6 This is justified forhim both by the determinacy concern and fundamentally that it seems to

make no sense to add the utility of one individual, a psychic magnitude in hismind, with the utility of another individual7. The worry is that interpersonalcomparisons require either a social mind or absolute welfare values, neither of

4Stanley Jevons (1871), Theory of Political Economy. London: MacMillan. p21.5Robert Nozick (1974), Anarchy, State and Utopia. New York: Basic Books, p41.6Arrow, Social Choice and Individual Values, p97ibid p11

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which seem to empirically exist, and thus that the overall good is, in a sense,metaphysically mysterious.

All three types of skepticism are important and interesting in their ownright - not as mere instances of general moral skepticism - as they typicallyhold that while claims about the good of individuals are both possible andmeaningful - via perhaps their preferences, what they value, their happiness,their well-being etc - there is simply no way to correctly, determinately andmeaningfully aggregate such claims, and thus to compare such claims. Skep-tics need not deny that someone being robbed, made unemployed, tortured,losing income, suffering disability or living in penury may be bad for them.What they deny is that when what is bad for some individuals clashes withwhat is good for others when some people benefit and some lose from somechange either that we can be in a position of knowledge to compare thesechanges directly across individuals (the possibility critique), that we can usethis information to gain determinate knowledge of overall welfare (the deter-

minacy critique) or that the overall good or welfare meaningfully exists (themetaphysical critique).

In the first part of this paper I am going to try to show why both the pos-sibility and determinacy critiques should be rejected, and then, subsequently,why as a result the metaphysical critique may fail too. The means of doingso will be by treating the task of making interpersonal comparisons as anepistemic problem. To illustrate what I mean by this, consider the following.

In a room there are three large balls of string: one red, one blue, one green.Some students go in, one by one, and cut off a length of each string such thateach student has one piece of red string, one blue, one green. Initially, bystipulation, we know nothing else about the students at all. Here are two

scenarios:

1. We ask the students to indicate which is longer, their red orgreen strings.

2. We ask the students to indicate which is longest, middle andshortest, their red, green or blue strings.

In each of these cases, once we know their individual string length information,can we use this to form justified - not infallible, but justified - beliefs aboutwhich of the color of cut strings is longer overall? If the answer in each ofthese cases is yes, and we can set out the epistemic principles that permit

this, then we can use those very same principles to form justified beliefs aboutthe overall good of different alternatives given either partial or full ordinalindividual information. We can, in other words, make justified interpersonalcomparisons of the good given any conceivable set of individual information8.

8Ordinal information can be inferred from cardinal information, thus if we can show thepossibility and meaningfulness of interpersonal comparisons given ordinal information wehave shown the possibility and meaningfulness of them given cardinal information too.

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If true this undercuts the possibility and determinacy critiques. The first partof this paper discusses how and why.

In doing so, it reframes the normal debate by making four key claims.Firstly, that the problem of interpersonal comparisons is fundamentally epis-temic: it is a problem of how to use evidence about the good of individuals toform justified beliefs about the overall good. As such it seeks justified beliefs,not infallible ones. This is all we need.

Secondly, that in order to settle the problem of interpersonal comparisonsof the good we do not need to specify what the correct account of the individ-ual good is: whether it is well-being, happiness, preferences, valued-ends orwhatever. All we need to determine is, given such an account, how to use evi-dence about the good of individuals to form justified beliefs about the overallgood. By doing so our problem is greatly simplified and the key issues moreeasily identified.

Thirdly, that merely by virtue of seeking justified beliefs we must rejectArrows Impossibility Theorem as correctly characterizing our task, for oneof its key premises the Independence of Irrelevant Alternatives logicallyentails the possibility of incoherent beliefs (a result I prove in the Annex).No credible credence function or account of epistemic rationality would acceptsuch a premise, and we should not.

Fourthly, if we do think of the overall good as epistemic, then we mayremove a range of metaphysical concerns over the ontological status of thisgood. The overall good, I will suggest, is simply a product of combining allknowable information about the good of individuals in a way that is justified,that is in a way that is unbiased and coherent. It does not require the existenceof a social mind nor that absolute individual welfare values be empiricallyreal. An epistemic solution to the possibility and determinacy critiques wouldpotentially also therefore have the resources to challenge the metaphysicalcritique.

This paper forms the initial part of a larger project whose underlying goal isto shift the focus of academic enquiry from identifying policies that are pareto-efficient, to directly evaluating policies by their overall impact. Aside fromavoiding the well-known biases toward higher-income individuals of efficiencymeasures (and the distorting effects of implicitly adopting productive efficiencyas norm to identify research questions) doing so opens up an extremely broadresearch agenda spanning politics and economics. If we can make interpersonalcomparisons, and we have a model of the empirical consequences of policies,we can evaluate those policies, whether relating to political reform, healthcare,taxation, social policies, international relations, the law, the environment andmuch, much more. The aim of the larger project is to establish how in principleand in practice this might be done.

The paper proceeds as follows. In part one I try to illustrate the logic ofthe overall approach using simple string and word-length analogies. In part

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with absolutely no evidence for this we would violate unbiasedness in givingRoberts information more weight than that of others (it might incidentally bethat he took much less). In the string case our intuitions may be nebulous andnot always support this: perhaps you think male students tend to cut more or

less string and thus would want to give Robert more or less weight. But in thewelfare/good case unbiasedness is a much more compelling general conditionas it may be sufficiently justified in either of two ways: methodologically ormorally.

The first broad justification of unbiasedness is to see it as a basic method-ological principle, roughly as a generalisation of something like Harsanyisprinciple of unwarranted differentiation whereby If two objects of humanbeings show similar behavior in all their relevant aspects open to observation,the assumption of some unobservable hidden difference between them mustbe regarded as completely gratuitous hypothesis and one contrary to soundscientific method10. This is a broadly evidentialist view of empiricial method

- that the epistemic justification of a belief is determined by the quality of thebelievers evidence for the belief11 - though would also be straightforwardlyentailed by Bayesianism with uniform priors12.

Secondly, however, unbiasedness may be sufficiently justified not only bya methodological principle but also by a ethical one, by the principle of equalmoral concern: if we show bias towards the good of some individuals withoutany good-based reason for so doing then this discrimination against otherscannot be justified. All humans are worthy of equal moral concern, underthis principle, and this means that the comparative weight we give them inevaluating what happens to them should be equal. The skeptic, in arguing forbiasedness, therefore minimally has to show the falsity of both the method-

ological and moral justifications of unbiasedness.Positive supervenience in the string case entails that beliefs about the

overall lengths of particular string colors should be based on beliefs about thelengths of the particular bits of strings, such that if we find out that Jackhas a red string longer than his blue we should be more confident the red isoverall longer than if we had learned that Jack had identical length strings, orhis blue longer than his red. In the good or welfare case it means that beliefsabout the overall amount of good or welfare are based upon beliefs about theamount of good or welfare of all the individuals. Positive supervenience may,

10John Harsanyi (1955), Cardinal Welfare, Individualistic Ethics, and Interpersonal Com-parisons of Utility, The Journal of Political Economy, 63 (4), pp 309-321 at p317.

11Earl Connee and Richard Feldman (2004), Evidentialism, Oxford University Press: Ox-ford, UK, at p83. Evidentialism would, prima facie, treat informationally identical evidenceas supporting the relevant hypotheses identically.

12It is important to note that the type of multi-parameter cases where uniform Bayesianpriors are problematic, such as that of forming priors over a factorys cube sizes, do notapply in the case of beliefs about the good/welfare since this represents only one parameter.Uniform priors for the Bayesian are thus easily constructed. For the famous cube examplesee Bas C van Fraassen (1989), Laws and symmetry. Oxford: Clarendon, p303.

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on some accounts, simply be a consequence of defining the overall amount ofgood / length of the red (or whatever) string as the sum of the particularamounts of good / bits of red (or whatever) string, and asserting that variousconstraints on justified beliefs necessary follow from this. I include it as a

separate principle however for clarity.

Unbiasedness and positive supervenience permit us to form justified beliefsabout the overall lengths of the strings based on any information about thestudents individual string lengths and, using the same logic, do so for theoverall good. I formalize, generalize and prove this result below, but first letus consider an important refinement.

Case two: multiple ordinal information, the relevance of independent alterna-tives, and the rejection of Arrows Impossibility Theorem.

Consider three different students, about whom we know nothing, who cut threepieces of string each, one red, one blue, one green. We learn that two have red

longer than blue and one has blue longer than red. As a result we justifiablybelieve that the red is overall longer than blue (or we are justified in having ahigher credence that R>B than that B>R). If we now learn about how theirgreen strings compare, is it possible we should change our beliefs about thered and blue overall? Assume we think no, as we believe the following:

The Independence of Irrelevant Alternatives (IIA): beliefs aboutwhether A overall represents more or less of some quantity than Bshould solely be derived from evidence about the relative amountof the quantity in, or comparing, specific instances of A and Balone.

In other words: your beliefs about which string - R or B - is overall longershould be determined solely by evidence about how individual bits of R andB string compare. This sounds plausible. But it is epistemically irrational asit may force one to hold incoherent beliefs.

Consider again that we learn that two students have red longer than blue,and one blue longer than red.

Student 1: r > b Student 2: r > b Student 3: b > r

If this is all the evidence we have then we would be justified in believing thatred is longer than blue overall (or we are justified in having a higher credencethat R>B than that B>R). If not we violate either unbiasedness or positivesupervenience. We now learn that their red and blue strings compare to theirgreen one as follows (and this comprises all the evidence we have):

Student 1: g > r > b Student 2: r > b > g Student 3: b > g > r

If the Independence of Irrelevant Alternatives is correct, we would still bejustified in believing red is longer than blue as we have learned nothing newabout the comparative lengths of individuals red and blue strings. But we are

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also now apparently justified in believing green is longer than red, and thatblue is longer than green (to see why, imagine we simply learned the overallinformation in a different order). Our overall beliefs are now incoherent. Theseare not beliefs about a supposedly mysterious property social welfare, if we

adopt IIA we are going to potentially have incoherent beliefs about somethingas un-mysterious as the lengths of bits of string. If we want to have justifiedbeliefs about string lengths, or welfare, we should thus reject IIA. Hence why, ifwe seek justified beliefs about welfare or the overall good, Arrows ImpossibilityTheorem should not be accepted as validly characterizing our task.

In fact, any credible credence framework or account of epistemic justifica-tion would violate IIA, and I prove this result in the Annex.

The overall logic is extremely familiar, and underlies various voting para-doxes, most famously that attributed to Condorcet. The right way, I think,to therefore interpret the importance of Arrows Impossibility Theorem is asa generalized demonstration of the impossibility of forming justified beliefs

about overall welfare if we adopt the Independence of Irrelevant Alternatives.It is an implicit argument, as such, against one of its major axioms. It does notpose any difficulties for the notion of social welfare, rather it usefully highlightsthe wrong way to think about the problem.

It is noteworthy, I think, that Arrow primarily discusses and motivatesthis condition in terms of voting (and how the winner of a vote should notdepend on eliminating another candidate, such as one who dies) even as theoverall discussion is framed in terms of welfare13. It is thus worth stating theobvious point that social welfare bears no necessary connection to a justifiedaccount of how best to structure a vote in committee or polity. For sometheories, the latter may be instrumentally justified by the former, but even

then the question of what is social welfare is distinct to the question of howto choose and institute a decision rule that best promotes it. Arrow unfor-tunately implicitly runs both projects together, and it is this that underpinsthe impossibility result. Neither project however - of evaluating the welfare ofdifferent arrangements based on individual level information, or of structuringand instituting a normatively justified decision procedure - is impossible, butthey are distinct in nature: the first is epistemic and methodological, the sec-ond is a question of institutional or procedural design. As such their requisitenecessary conditions will be different. If we form some amalgam of the twowe may indeed produce various elegant impossibility results, but the wronginference from this is the impossibility of either task considered separately,

as they should be. Evaluating the effects of social and economic policies is adistinct project to deciding on rules of association; if we conflate the two we

13See Arrow, Social Choice and Individual Values, pp26-28. His initial paper was, reveal-ingly, titled explicitly about welfare, see Kenneth Arrow (1950), A Difficulty in the Conceptof Social Welfare, Journal of Political Economy, 58 (4), pp328-346, but then the IIA is in-tuitively justified by reference to election candidates with no mention of welfare at all, seep337.

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risk being severely misled14.

One additional thing perhaps worthy of note is that the IIA would be epis-temically irrational as a condition on beliefs about overall welfare even if itwas validly a condition on individual welfare information. In applying it tocollective situations there is a variant of the fallacy of composition being com-mitted: the conditions we might wish to apply to individuals in how evidenceof their welfare relates to various options need not be the same as that appliedto evidence about overall welfare. For instance, consider that we ask a groupof students to write down on a piece of paper their birth-city, the species oftheir first pet and their favorite authors surname. They then sum up the let-ters in each, and rank them by number e.g. Cambridge, Dog, Melville wouldgive a ranking of city > author > pet. How someone ranks their pet comparedto how they rank their city should not be affected by whether their author isthe longest, middle or shortest word: the IIA seems a valid condition, in thiscase, on the individual word-rankings.

When we learn the rankings of individuals however, we should reject IIA.Consider we pick four students at random, and want to know which of themcombined has more letters their authors overall or cities overall.

Student1: Author > Pet > CityStudent2: Author > Pet > CityStudent3: Pet > City > AuthorStudent4: Pet > City > Author

Given IIA, as two students have authors longer than their cities, and two theopposite, by unbiasedness we would seem to be justified in not favoring eitherthe theory that the authors overall contain more letters than the cities orthe theory that the cities contain more than the authors. But if we also seekjustified beliefs about how the pet words relate to the two other categories thenwe are going to need to reject this condition, else we will believe the pets andauthors contain the same overall amount of letters, that the authors and citiescontain the same overall amount of letters, and simultaneously that the petscontain more letters than the cities. Other alternatives must be relevant. Itcannot be that forming justified beliefs here is correctly impossible: we are notdiscussing anything mysterious, we are discussing word lengths, a decidedlyprosaic topic.

14The project of forming justified beliefs about social welfare is also, obviously, quite dis-tinct from the question of whether a conception of democratic legitimacy as expressing acoherent majority will or group judgement can overcome these paradoxes, or not be under-

mined even if in principle they might accrue. For the skeptical case see William H. Riker(1982), Liberalism against Populism: A Confrontation between the Theory of Democracyand the Theory of Social Choice. San Francisco: W. H. Freeman. For an attempted norma-tive conception of the general will that can meet this see Joshua Cohen (1986), An EpistemicConception of Democracy, Ethics, 91 (1): 26-38, and for an argument that we may be ableto ameliorate the impact of the problem see John Dryzek and Christian List (2003), SocialChoice Theory and Deliberative Democracy: A Reconciliation, British Journal of PoliticalScience, 33(1): 1-28 .

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Even if we cannot know the actual numbers of letters in all the words, wecan still use evidence of the rankings of the students to from justified beliefsabout which contains more words overall: the cities, authors or pets (in theabove example, given absolutely no other evidence: the pets). Presumably

there will be some underlying probabilities depending on where one does thisand if we could obtain evidence of these then that could bear on our credencestoo. But if all the information we have is the individual rankings in thiscase it might not be, but in the welfare/good case it is then we can stillform justified beliefs overall. The string/word to welfare/good analogy isthus inexact. It is stipulated in these examples that direct individual levelinformation is all the evidence we have, as that is true in the good/welfarecases. However, we probably have multiple rather diffuse intuitions about pets,authors and cities. These examples, of strings and words, are thus meant to beillustrative of the underlying logic, even if some additional implicit backgroundevidence about dog vs. kangaroo prevalence perhaps requires ignoring. (A

tighter analogy might be if the example also involved students in an unknownforeign language and place.)

The overall epistemic approach simply says that we should use evidenceabout individual amounts (individual words/strings/good/welfare) to formcoherent beliefs about overall amounts (overall words/strings/good/welfare),and do so in a manner that treats all unbiasedly (with equal moral concern).

2. Formalizing and proving the approach

We can formalize the above discussion in either a credence framework or usingthe concept of expected value (the two yield functionally identical results, butone may be easier to use than the other in particular contexts). To do so, weneed to set just three values. Firstly, we need to decide whether the identity ofindividuals should affect their relative weight in the overall credence functionor calculation of expected values. Secondly, we need to decide in principlewhat credence to give to the theory B>R relative to the theory R>B,or relative expected value to give to B and R, before we have any evidencewhatsoever. Thirdly, if we learn of an individual that for them R has more ofthe quantity than B that r(i)>b(i) then we need to decide how this shouldaffect our overall credences or expected values compared to had we learneddifferent information. Unbiasedness and Positive Supervenience allow us todo all three.

Here are three basic axioms in a credence framework:Let Cr (r(i)>b(i) / R>B) represent our credence that r>b for individual i werewe to learn that R>B overall. Let h* = R>B, R=B or Rb(i), r(i)=b(i), or r(i)


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Cr (e*i / h*) = Cr (e*j / h*) for all individuals i, j

2. Pre-evidential indifference PI:Before any evidence, Cr (B>R) = Cr (BG) for all R, B, G;

3.Equal positive supervenience EPS:For all R, B and all individual level information where g=r and 0B ) = x Cr (r(i)>g(i) / R=B);x Cr ( r(i)>g(i) / Rg(i) / R=B),15

Here are the same axioms in an expected value framework:

Let E [R] represent the expected value of any arrangement R before the evi-dence, and E[R / e] represent the expected value of R after learning e.

1. Equal treatment ET:For all individuals i,j and arrangements RE[R / e*(i)] = E[R / e*(j)] for all individuals i,j and arrangements R

2. Pre-evidential indifference PI:Before any evidence E [R] = E [B] for all R, B.

3. Equal positive supervenience EPS :For all R and all individuals i, j, where y>0;E[R / r(i)>b(i)] = E[R] + yE[R / r(i) = b(i)] = E[R]E[R / r(i) < b(i)] = E[R] y

Equal treatment and pre-evidential indifference stem from unbiasedness: tobe biased towards one arrangement represents an implicit bias towards thosefavored by that arrangement, to give the information of some individuals more

weight than that of others is to be biased toward them. Equal positive super-venience is based the fact that the overall amount of something is comprisedof the amount in its constituent parts. Rejecting equal positive superveniencewould mean, for example, that the more students you learned had red stringslonger than their blue ones, the more confident you would become that theblue strings were overall longer (violating positive supervenience) or that yourbeliefs about string lengths would be affected by the identities of individualsrather than information about their strings (violating unbiasedness).

Finally, for the credence framework we need to adopt a credence function.Here, I will simply use Bayesianism, as the theory is currently the most widelyused and accepted such framework. One notable feature of any such potential

framework would be that our beliefs be coherent, in that if X entails Y ourcredence (or belief) in X should not be greater than our credence (or belief)in Y. For instance, if we learn e1, then e2, and e1 and e2 entails e3 then

15For the case where g=b we can simply treat this as two bits of independent information(about r and about b), and as a result (1/x2) Cr ( r(i)>b(i) / R>B ) = Cr ( r(i)>b(i) /R=B) = x2 Cr ( r(i)>b(i) / R


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epistemic approach: we use evidence to form justified beliefs. In the welfarecase I discuss subsequently how true beliefs might in fact simply be justifiedbeliefs given all possible evidence.

What is particularly striking about this overall result is how minimal arethe conditions we require to produce it. If we seek beliefs that are unbiasedand coherent (positive supervenience is a type of coherence requirement) thenwith solely ordinal information the above result is uniquely justified.

A brief note on comparisons with cardinal information

Though we do not need to do so to form justified beliefs about the overall good,extending these results to cardinal evidence is relatively straightforward, andin fact can be done one of two ways. Firstly, we can directly use unbiased-ness to determine what we are going to assume is the comparative averagewelfare/good of different individuals (if - without any evidence to justify it -we assume one individuals average welfare is higher than anothers then we

violate unbiasedness). That is: .

The cardinal epistemic welfare/good principle : If we have no evi-dence that an unbiased super-set of options would differ, then weare justified in believing that the average welfare/good of the setof options for all individuals is the same.

Secondly, however, we can generate the above result if we simply convert (ourtotal) cardinal evidence to ordinal evidence and then use the previous ordinalepistemic welfare/good principle18. To see the logic, think perhaps aboutlearning that some students red string is a third as long as her green. Well,

we can capture that information ordinally by thinking of each of her stringsas compared to a hypothetical hundred strings the first of which is 1% of hercombined string length, the second is 2% of the combined string length andso on (thus her red string is longer than twenty four other strings, and shorterthan seventy five, her green is longer than seventy four, and shorter thantwenty five)19. The end result is that we can make comparisons using cardinalevidence either by directly applying unbiasedness or simply converting suchevidence to hypothetical ordinal evidence and using that.

There is a great deal one would want to say about, and do with, suchcomparisons - showing how alternative suggestions (such as the zero-one rule)

18In fact some theories hold that cardinal information can only meaningfully be gained

from ordinal information, such as the lotteries over outcomes individuals would rationallychoose. See notably John Von Neumann & Oskar Morgenstern (1944), The Theory of Gamesand Economic Behavior. Princeton: Princeton University Press.

19There are rounding issues in using a hundred strings, as some marginal ordinal infor-mation might be lost, but if so we could simply use an arbitrary fixed large number - aslonger as we assert the same arbitrary number for all individuals then what the number iswill not matter to the resulting welfare/good beliefs. One thing important to note is that itis combined string lengths that are converted.

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violate basic epistemic principles20; deriving overall welfare estimates fromevidence of the good of income and leisure and applying them to our basicmacro-economic policy models; evaluating political and institutional choicesin ideal and actual political systems and so on. Once we have justified beliefs

about the overall good, we can begin to apply them to a breathtaking array ofactual political and economic choices and problems. The primary aim of thispaper, however, is to show how skepticism about interpersonal comparisons ofthe good/welfare may be overcome, as such skepticism has somehow ended upa remarkably widespread and entrenched default assumption. The focus here isthus on explicating and defending comparisons given only ordinal information,as this is the case most favorable to the skeptic (from any cardinal set ofinformation one can infer ordinal information, thus if comparisons based onordinal information are possible, determinate and meaningful then so are onesgiven cardinal information). Though the cardinal case is perhaps the mostinteresting, the ordinal is sufficient to illustrate the basic epistemic logic, and

the ways in which the epistemic approach may be mis-interpreted. It is tothese I now turn.

3. Three fallacies to avoid

You are a peasant. Consider the following options:

1. The king takes all your property.

2. The king takes three-quarters of your property, but has to workfor a week digging turnips.

3. The king takes half your property.

We could collect information on how these compare and then might find ofthem that the king has more welfare in 1 than 3, and more in 3 than 2 (he isvery wealthy already, and really hates digging turnips). You have more welfarein 3 than 2, and more in 2 than 1. If this is all we know then it might seem wewould be justified in believing that 1 represents more overall welfare than 2.But this seems absurd. Which it is: the belief that 1 represents more welfarethan 2 is unjustified because it violates unbiasedness, in that unbiasednesscovers both how we treat individual-level information, and how we select andstructure it. More specifically, even if the above options were the three optionswe had to choose between, if we use them alone to structure the evidence inso doing we would be committing the first of three fallacies, namely:

The unjustified domain fallacy: is that beliefs about the overallgood (or welfare) are justified even if the choice-domain specifica-tion that underpins them is itself unjustified.

20This is often attributed back to Isbell and assigns to an individuals most preferredoption a 1, and to their least a 0, and normalizes accordingly. John Isbell (1959). On theenumeration of majority games. Mathematical Tables and Other Aids to Computation, Vol13, pp21-28

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The issue here relates to how we select the different options about which weseek and incorporate evidence. If we (wrongly) think of welfare by analogywith voting, then the domain appears naturally to be the candidates or choices,and overall beliefs justified by the individual information about these. In

contrast, however, for the epistemic approach the options are selected anddistinguished based on our theory of the good and the evidence. The keypoint is this: if we have to choose between a set of options it is the alternativesabout which we can gain evidence that represent the potential domain whichwe use to form justified overall beliefs, not merely the options at hand. Evenif we are only choosing between red and green strings, the blue strings canbe informative as to that choice, and all such string information could bepotentially included. Hence why for justified beliefs we need to seek out andinclude a set of evidence that can be justified by unbiasedness and our theoryof the good, not necessarily simply by the options under consideration.

The unjustified domain fallacy is very closely related to a second way we

can go wrong in using the epistemic approach, namely:

The ignored evidence fallacy: is that a method of forming be-liefs about the overall good (or welfare) is unjustified due to theimplausibility of the beliefs it produces where the justification ofthese beliefs is based on the exclusion of the very evidence thatunderpins the judgment of their implausibility.

For example, consider someone arguing that slave holders prefer to keep adomestic slave, such a slave prefers to be free, and thus if this is all theinformation we have we are justified in believing that either arrangement is of

equal welfare, thus the epistemic approach must be wrong. This is fallaciousas the correct judgment about the conclusions implausibility stems from thevast, rich and detailed evidence and understanding we have about what itis like to be a slave, what slaveholding entailed, what the effects of it wereas an institution, and how much of what is most valuable in human life wasdenied the slave. As social beings in a rich cultural environment we have anincredibly detailed and complex understanding of a wide range of patterns ofhuman valuation, of suffering, hope, pain, dignity, freedom from domination,the enjoyment of security, the determinants of social respect, the ability todevelop ones potential and shape ones life and so on. If we ignore vast swathesof evidence we will reach conclusions that, with this evidence in mind, seemunjustified. But it is the ignoring of the evidence that is at fault, not theattempt to use evidence to form justified beliefs.

There is thus a certain rhetorical misdirection in the common habit offraming the key illustrative question concerning social welfare as how to com-pare two individuals tastes (often over food or drink) when such cases haveperhaps the richest information, the greatest cross-individual variance, andrepresent the least pressing need to evaluate political and economic institu-

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tions and their policies. The ignored evidence fallacy is likely to easily surfacein such cases.

It is also for a similar reason that this paper explicitly separates out twoquestions. (1) What comprises evidence of the good of individuals? (2) How,in principle, given an account of the good of individuals, should we combineevidence thereof into beliefs about the overall good? If we fail to treat these asdistinct then disagreement about 1 will risk fueling disagreement about 2 aseach theorist will, from the other theorists perspective, have ignored evidence(the relevant evidence that arises from the different accounts).

Finally there is:

The absolute comparison fallacy: that skepticism about ordinalor cardinal epistemic interpersonal comparisons of the good/welfarecan be justified by reference to claims about absolute levels of theindividuals good/welfare.

That is, it might be that in a range of situations we think that peoples welfareis of a similar absolute level, that for instance we think the welfare level ofeveryone sleeping (without dreaming) or being dead is identical. However, ifwe use the epistemic approach based on only ordinal or cardinal informationthen these values might come to differ: someone who would knowingly andconsistently take a higher risk of death for a large range of goods assumingthey live could be assigned a lower level of welfare to death than someonewho was much more relatively death averse21. But if we think being deadmust represent the same welfare for all people then the epistemic approachseems to have given us the wrong answer. Such an argument does not support

skepticism about the good/welfare however, as it assumes the very falsity ofsuch skepticism.

The epistemic principles outlined earlier are based on only having ordinalor cardinal information: we know how options compare, but not their absolutevalues. If we assert such absolute values for some options however then ourprevious comparisons will not reflect that information. But this is not a prob-lem, it is great: we can make direct interpersonal comparisons. For example,consider the previous example where everyone writes down their birth city,favorite author and species of first pet. We learn how the number of letters ineach of these compare for each individual, and form beliefs about the overallnumber of letters in the city, author and pet words combined. Now some-

one points out that everyone was born in the same city and yet our previouscomparisons had implicitly assigned a lower value to the cities of individu-als who had more letters in their pet and author words than those who hadthe most letters in their city words. This is mis-premised as a criticism aswhat we had previously were justified beliefs given that we could not directly

21I assume in this case that peoples welfare is partially based on what they value, andthat what they would choose is evidence of what they value.

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compare peoples city words. If we assert such absolute comparisons then wewill need to update our beliefs, but also we can now make direct interper-sonal comparisons: if your author word is longer than your city, my pet isshorter than my city, and we find out we have the same city word, then we

can know that your author contains more letters than my pet. We can makedirect comparisons. It is true that our previous comparisons justified basedon only ordinal or cardinal information will now be unjustified, but this is nota skeptical-supporting observation, quite the opposite: it is predicated on thefalsity of skepticism, it is predicated on us making direct comparisons (suchas assuming a common number of city letters, or common absolute level ofwelfare for sleeping or death).

The point of emphasizing all three fallacies is to try to preempt the confla-tion of localized disagreement over a particular conclusion based on the epis-temic approach with generalized counter-arguments applying to the approach.If we choose biased domains, ignore evidence, or assert absolute comparisons

then, having done so, we will risk conflict with epistemically justified com-parisons implicitly based on unbiased domains, included evidence or absentabsolute comparisons. This, however, is an argument for evaluative consis-tency and the updating of ones beliefs, not skepticism.

4. Metaphysical implications

The point of making the analogies with both string and word lengths was tohighlight how we can use well established epistemic principles in order to com-bine individual level information about various quantities into beliefs aboutthe overall amount of various quantities. There is one way this may be mis-leading however: both the length of string and the overall number of lettersin words are directly empirically verifiable properties in that they have valueswhich we justifiably believe exist. There are eight letters in Melville, not nineor two. If we think that welfare is like that, in that there can be some empiri-cal truth to Bob somehow having 57 (rather than any other number) units ofwelfare in situation A then we can still form justified beliefs and make, by im-plication, interpersonal comparisons. As such the determinacy and possibilitycritiques fail. But welfare need not be like this, and in this section I wantto suggest why, and why as a result there is potentially nothing necessarilymetaphysically mysterious about overall welfare or the overall good.

Consider the hypothetical case where we know all there is possible to knowabout the relative good or welfare of every possibility for each of the individ-uals affected: we have complete cardinal information about how much betteror worse each possible scenario is for each individual. Now, if we assumethat there is some overall good/welfare that actually exists as the sum ofindividual-specific values that themselves empirically exist then it is possiblethat in using the individual level information to gain justified beliefs aboutoverall good/welfare we would still have false beliefs, notably that some indi-

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viduals may just be more important for the overall good in ways that cannotbe known based on any information about them. In this case in combiningtheir individual level information in a way that was unbiased we could formjustified but false beliefs.

This is, what we might label, physical realism about the good22. Thephysical realist position is that the overall good exists in the same way thatthe number of letters in the previous sentence exists as the sum of its elevenconstituent words. If we make certain assumptions about how this good re-lates to the good of individuals we can still use evidence about the good ofindividuals to form justified beliefs about the overall good (and this is whatwe did, for instance, with string). Such beliefs are however justified but notnecessarily true. For instance, if all the evidence we have is that sentence Ahas eighteen words and sentence B has three words, then we will be justifiedin believing A has more letters overall. But we could be wrong. It is possiblesentence B is Counterrevolutionaries protest antidisestablishmentarianism

while sentence A is I go up and hop in a big bag and he says I am a big baghopper. It is possible, as the evidence does not prove the theory, that wehave justified beliefs that are false. For the physical realist the same is true ofthe overall good: there are true empirical facts about the absolute level of anindividuals welfare (for instance) such as that in situation X Meghna has awelfare of 21 even though we cannot ever conceivably gain direct knowledgeof this number, nor indeed can Meghna. But it exists. Overall welfare is thesum of these numbers.

The alternative, I would like to tentatively call, for want of a better term,epistemic realism about the good23. The epistemic realist position would bethat the overall good is a construct that is created by combining information

about the good of individuals in a justified manner. For the epistemic realist,once we have an account of justification then the true overall good is thatwhich we would be justified in believing were we to know all possible informa-tion about the relative good of individuals. There is nothing more, needs tobe nothing more, and as such there is nothing ontologically mysterious: theempirical status of Jack having a welfare of 57 in A is simply an epistemicconstruct, it is not that an omnicognisant being could look into Jacks headand count 57 welfare units or measure his welfare to be 57.

22I focus on this view as it appears to be what the metaphysical critique implicitly hasin mind. It seems to be a strong version of a cognitivist, truth-apt and non-error theory, onethat conceives of the good very much in the same way as it conceives of empirical entities.Hopefully the label physical realism manages to capture this.

23The broader meta-ethical theory being invoked here - epistemic realism - has someparallels with what is often called Kantian constructivism about reasons (roughly, consid-erations for an agent in favor of one action or another), but does not necessarily accept thefocus on reasons instead more naturally being about moral beliefs. For a very clear discus-sion of reasons-based variants see Sharon Street (2008) Constructivism about Reasons, inOxford Studies in Metaethics, vol. 3, ed. Russ Shafer-Landau (Oxford: Clarendon Press,2008).

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The basic epistemic realist position would perhaps be:

1. There are true empirical facts about how different arrangementsrelate to individuals.

2. A justified theory of the good/welfare would supervene on thesefacts to yield facts about how, for each individual, different ar-rangements relatively compare in terms of their good/welfare.

3. Combining all possible such facts in a justified manner wouldyield facts about the overall good/welfare.

4. We can use individual level evidence to choose between theoriesabout the overall good/welfare and, in principle, were we to believeall such potential evidence and reason correctly, we would have truebeliefs about the overall good/welfare.

For the epistemic realist, the overall good is constructed from various epistemicaxioms and is justified by virtue of the axioms being justified, rather than inreference to an empirical social mind or meta-mind empirical entity of theoverall good. It is possible to think of alternative axioms we could rejectunbiasedness in favor of a racist axiom or similarly adopt the Kaldor-Hickscompensation criteria for instance. But so long as we are justified in rejectinggiving extra moral and epistemic weight to some particular race or those onhigher incomes, then beliefs about the overall good based on unbiasedness willbe justified.

The point of introducing epistemic realism here is not to demonstrate thatit is the right meta-ethical stance, and indeed all I have done is to sketch it,

not to defend it. The point instead is to show that there is nothing necessarilymetaphysically mysterious about the overall good, epistemic realism being one(potentially among many) theories to be able to make sense of the overall goodwithout requiring the existence of a social mind or of empirically real absoluteindividual welfare levels.

5. Revisiting skepticism about the good

For the epistemic approach, the possibility critique is addressed simply byvirtue of the possibility of having evidence about the good of individuals: ifwe dont need to get into the minds of individuals to be able to make mean-ingful statements about their good or welfare, then when using epistemic prin-

ciples to combine this evidence we dont need to do so either. The possibilityof individual-level evidence about the good entails the possibility of justifiedbeliefs about the overall good.

The determinacy critique, that we cannot have determinate knowledgeof overall welfare, is addressed in two ways. Firstly, the epistemic approachsimply denies that determinate knowledge is the necessary goal: what weseek are justified beliefs, for these allow us to evaluate actions, policies and

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institutions, the reason for seeking interpersonal comparisons in the first place.For the determinacy critique rests on the following mistaken reasoning:

Premise : The evidence does not deductively entail a unique hy-

pothesis about overall welfare.

Conclusion: We cannot form determinate justified beliefs aboutoverall welfare.

Now, if we assume a physical realist account of the good, then the premisewill be almost always true. But even if so the conclusion is unwarranted.Indeed, if the general reasoning were correct then it would potentially entailthe impossibility of empirical method in general, for it is rare for evidence toprove a general theory and on some accounts it is normally impossible24.

Consider for instance if the determinacy critique was generally valid. Weundertake a survey of what we believe to be a representative third of thehouseholds in a town, and find out that eighty percent of them own a car.Should we believe that a majority of the towns households own a car? Well,if we seek determinate knowledge, understood as having a theory proved by theevidence, then we can conclude no such thing. But of course this is not all weseek, we want justified beliefs. Just as when we do any social science research,we are almost never going to be gaining infallible determinate knowledge, butrather if we are methodologically careful, use reliable procedures, and gathergood evidence we may form beliefs that are evidentially justified.

The second way the determinacy critique may be addressed is that, ifepistemic realism about the good is correct, then the overall good simplyis that represented by justified beliefs given all possible evidence and thedeterminacy critique cannot even arise as the gap between justified beliefs andtrue beliefs is, in principle at least, bridge-able. If epistemic realism is correct,as we gain more evidence justified beliefs will be expected to actually convergeon true beliefs and, were we to gain complete evidence, represent true beliefswith no possible indeterminacy.

The ultimate problem for the possibility and determinacy critiques is thatthe conditions that supposedly justify them - our lack of access to the actualvalues of the various individual quantities, or inability to experience valuesacross individuals - are exactly the conditions that hold in the string andword examples (and many, many, many more). A skeptic is potentially forcedto either accept that determinate justified beliefs about string, word-lengths,

welfare and the good are all possible; or that they are all not. And if thelatter, then they have seemingly embraced a very wide-encompassing epistemicskepticism covering the physical world, not one addressed at interpersonalcomparisons of the good in particular.

24This is an enabling premise of falsificationism, see Karl Popper (2002), Conjectures andRefutations: The Growth of Scientific Knowledge. Routledge, London.

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Finally, as noted previously, if the possibility and determinacy critiques canbe addressed, the metaphysical critique loses much of its force and, if epistemicrealism about the good is correct, it is removed. In a sense, the metaphysicalcritique rests on a mis-analogy. The individual good is potentially based on the

brain / mind of the individual involved. But there is no reason to think thatthe social good therefore needs to assert some social mind or indeed that thesocial good needs to have the same properties as we might expect of that of asingle mind. Groups are not unitary sites of decisions based on a unified mentalprocess, they do not have beliefs, they do not have preferences. If we think theydo then not only are we going to encounter a range of problems in describingthese (such as over intransitive majority preferences, and in justifying exactlywhy a group mind is conflatable with a majority preference), but skeptics mayrightly wonder what this mind is made of, what empirical facts about thesocial brain it supervenes upon. We need, however, no such thing to formjustified beliefs about the overall good, and if epistemic realism is correct we

can fully account for both the meaningfulness and ontological status of thisgood.

Viewed through an epistemic paradigm, skepticism therefore looks prob-lematic. This result should not really be surprising for consequentialists atleast, as the theory requires the meaningfulness of interpersonal comparisonsin order to be of practical prescriptive relevance. There has thus always beensomething odd about skepticism being what List rightly characterizes as thecurrent orthodox view in welfare economics25. For it is virtually impossibleto say anything about overall welfare without interpersonal comparisons, afact that is brutally unavoidable, despite multiple prominent results often be-ing mis-interpreted as indicating otherwise26. There has quite plausibly never

been a pareto-improving economic or political policy or institutional changeundertaken by any state, at any time, at any point, in all of recorded history,and likely never will be. There are always winners and losers in terms of theirpersonal welfare. A welfarism based on Pareto rankings must either idealizehuman agents or causal chains to the point where they are barely recognizable,or opt for productive efficiency as a duplicitous welfare proxy and by doingso abandon seeking an unbiased normative justification27. There is something

25See Christian List (2003), Are Interpersonal Comparisons of Utility Indeterminate?,Erkenntnis 58, pp229-260

26Most notably the First and Second Fundamental Welfare Theorems, Kaldor-Hicks com-pensation criteria.

27Hammond (p201) rightly, but perhaps over-kindly, characterizes using monetary value

as an implicit welfare proxy as almost certainly unethical. Alas, cost-benefit analysesun-weighted by income are still common, usually deploying the misleading euphemism ofefficiency, which actually means Pareto efficiency, not welfare or normative efficiency. Manyplausible slave states are Pareto efficient. Pareto efficiency is not an intrinsically desirablefeature. It just happens to be a feature of the ideal state of affairs (it is perfectly possiblethe average Pareto efficient state has lower welfare than the average Pareto inefficient state).Peter Hammond (1991) Interpersonal Comparisons of Utility: Why and How They Are andShould Be Made, in Jon Elster and John Roemer (eds.) (1991), Interpersonal Comparisons

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odd about being officially committed to evaluating policies by how they mightimprove human welfare and at the same time denying the very possibility ofthe comparisons that render it an achievable or meaningful goal. It is a bitlike professing to be a medical researcher while maintaining it is impossible

to compare and evaluate treatments or in principle gain evidence about whichworks better.

Fortunately such a negative conclusion is not warranted, for the problemof interpersonal comparisons is ultimately a problem of how we justifiablycombine evidence of individual amounts into beliefs about overall amounts,something that is both a tractable epistemic problem (assuming a basic ac-count of justification), and one with intriguing implications for how we mightunderstand the ontological status of the overall good.

6. Conclusions

The main argument of this paper has been that, if we treat it as an epistemicproblem, then beliefs about the overall good, and by implication interper-sonal comparisons of welfare, are possible, determinate and meaningful. Withthis positive conclusion in mind, I would like to conclude, however, with twoqualifications.

Firstly, that we should be wary of interpreting the arguments put forwardhere as showing that evaluating the good of different choices in practice isboth easy and straightforward. This is not necessarily the case. For theargument has been that, given an account of the individual good, forming justified beliefs about the overall good is both possible and meaningful. Yetthis assumes an account of the individual good. This may considering only

goods, services, leisure and income be relatively tractable. But if we theninclude life expectancy, other determinants of quality of life, the effects onpotential beings who may and may not exist, and endogenous changes to thefuture good of individuals, then a range of difficult problems arise28. Theepistemic approach can allow us to make progress even without resolving allthese difficulties, in that it at least allows us to form justified beliefs giventhe evidence and theory of the good we have, and as such evaluate a widerange of social and economic policies by their impact on those affected. Butour confidence in such conclusions will be increased the more confident we can justifiably become in the theory of the individual good underpinning them.It will also, obviously, additionally be affected by the quality and range ofour evidence about the causal effects of different policies and institutions: thebetter evidence we can gain, the better will be our overall evaluations. Thisis potentially a formidable task.

of Well-Being. Cambridge University Press: Cambridge.28For a range of practical problems that are very clearly set out, see John Broome (2006)

Weighing Lives. Oxford University Press: Oxford. General temporal difficulties are famouslyoutlined in Derek Parfit (1984), Reasons and Persons. Oxford University Press: Oxford.Especially ch8, 16, 17, 18, 19;

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Secondly, nothing in the epistemic approach per se entails that morallyright actions, policies or institutions are those that best contribute to theoverall good. Other potential good-based alternatives, for instance, would givegreater weight to those worse off (prioritarianism29), to maximizing the good

of those with the least (maximin30) or give some intrinsic weight to equality ofoutcome irrespective of whether it maximized the overall good31. What thispaper seeks to establish is the feasibility and meaningfulness of the possibilityof evaluating actions, policies and institutions by how they contribute to thegood of all. For some theories this will establish their moral desirability (suchas with a welfarist consequentialism). But not for others. Deciding betweenthem requires substantive argument. The epistemic approach merely showswhy one stance, that based on the overall good, is eligible for consideration,despite the skepticism that it has often attracted.

Given that the policies and institutions we adopt impact great numbers ofhuman beings, being able to evaluate the overall consequences of alternatives

seems a hugely important task. The argument of this paper is that it is inprinciple possible.

29

See Derek Parfit (1997) Equality and priority, Ratio, vol. 10, no. 3, pp. 202-221.30This, when applied to income in general, is sometimes described as a Rawlsian welfarefunction. The attribution is inexact: Rawls was quite clear that maximin would be appliedto primary goods, and that that should be done only after securing equal basic libertiesfor all, and then ensuring that offices are open to fair equality of opportunity. Thus thetype of social function that Rawls theory would suggest would plausibly be much richer andsignificantly different from one solely focused on maximizing the income of the worst off.John Rawls (1999), A Theory of Justice. Belknap Press: Cambridge, MA.

31Larry Temkin, Inequality, (Oxford: Oxford University Press: 1996)

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Annex: Proof that accepting Arrows Independendence of IrrelevantAlternatives (IIA) is epistemically irrational.

Bayesianism violates IIA, as would any credible credence framework, so long

as it is well-defined (credences in mutually exclusive theories are additive),coherent (we are not more confident in a proposition than in any propositionthat is logically entailed by it) and non-dogmatic (possible theories are notbelieved impossible). Formally:

Axiom 1: Additivity. If h1 and h2 are mutually exclusive (such that h1h2) then Cr (h1 h2)=Cr(h1) + Cr(h2)

Axiom 2: Coherence. If X Y then Cr(Y) Cr(X)

Axiom 3: Non-dogmatism. Before evidence, we assign some positive cre-dence to all possible theories: Cr(h)>0 so long as h is not a logical contradic-tion (h = A A)

From Coherence we can derive a further condition of upper and lower bound-edness (our credence in any theory is at least as great as our credence in logicalfalsehoods, and not greater than our credence in logical truths):

Boundedness: Cr(h h) Cr(h) Cr(h h) for all h.32

For convenience we can map all credences to the unit scale, such that Cr(h h)=1, and I do so in what follows. [We already know that Cr(h h)=0from A1 as h h h therefore Cr(h h h) = Cr(h h) + Cr(h).]Mapping to a specific interval is convenient, but not strictly required, as wecould simply express all credences as fractions of our credence in all logicaltruths.

Finally, under the Independence of Irrelevant Alternatives, if one learnsabout an individual that r>b, r=b, or rG shouldnot change. That is:

Independence of Irrelevant Alternatives (IIA): Cr(B>G / b(i)>r(i))= Cr(B>G / b(i)=r(i)) = Cr(B>G / b(i)G) for all overallamounts B, G, and individuals i.

If A1,A2, A3 then IIA must be false. That is, for any non-dogmatic well-defined credence function, adopting Arrows Independence of Irrelevant Alter-natives entails the possibility of incoherent beliefs.

32

Proof: As (h h) h therefore Cr(h h) Cr(h). As h (h h) thereforeCr(h) Cr(h h).In fact, given Additivity, not only does Coherence entail Boundedness, but the converse

is true too. Proof: As Y (Y) therefore Cr(Y) + Cr(Y) = Cr(Y Y) [From A1].If X Y then Cr(X Y) = Cr(X) + Cr(Y) [From A1].Thus Cr(X Y) + Cr(X Y) = Cr( (X Y) (X Y) ) = Cr(Y Y)Therefore Cr(X) + Cr(Y) + Cr(X Y) = Cr(Y) + Cr(Y)As Cr(X Y) 0 [From Boundedness] therefore if X Y then Cr (Y) Cr (X) QED.

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Proof

Define the overall quantity of something as the sum of its individual amounts.

Def 1: Let R = r(i) + r(j)..... + r (n) for all individual instances i, j....n

Before any evidence, consider three possible overall quantities, A, B and C

Case one: There exists any two credences such that Cr(A>B) = Cr(A>C) orCr(A Cr(A>C); (ii) Cr(A>B) < Cr(A>C); (iii)Cr(A Cr(AC B=C) = Cr(A>C)Cr(AC)Cr(AB B=C) < Cr(A>C B=C)iii.Cr(A Cr(Ac(j); .. b(n)>c(n)] for all individuals i, j .... n

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As a result Cr(B>C) = 1 Cr(BC) = 0 [Def 1, A1, A2]

Under IIA, Cr(AB); thereforeCr(A=B / E) = Cr(A=B). And Cr(A C); therefore Cr(A=C / E) = Cr(A=C)

Now, as:Cr(A>B) = Cr(A>C) = Cr(A Cr(A Cr(A>C) Lemma 1.

But as Cr(P) + Cr(P) = Cr(P P) and Cr(BC) = 0:

Cr(AB B>C) > Cr(P P)/2;Cr(AC B>C) > Cr(P P)/2;

That is Cr(A>C) > Cr(AC) and Cr(A Cr(AB) Lemma 2 [A1, A2]

We have incoherent beliefs. (From the combination of Lemmas 1 & 2, Coher-

ence is violated: AC entails AC). QED.That is, no matter what ones initial credences, so long as they are non-dogmatic and well-defined, then adopting IIA entails the possibility of inco-herent beliefs. (And while this result requires three or more arrangements tohold, if there are less than three arrangements then the IIA itself is irrelevant:there are no independent alternatives).

Arrows Independence of Irrelevant Alternatives should not be a conditionupon beliefs about how the overall amount of something relates to its con-stituent components if such beliefs are to be justified, and as such should notbe a condition on how we form beliefs about overall welfare or the overallgood.

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interpersonal comparisons of the good

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