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A Solution to the Mere Addition Paradox

 

by Socrethics   First version 2014   Last version 2018

 

 

 

 

 

Table of Contents

 

Abstract

 

1.  Introduction

2.  The Notion of a Life Worth Living

3.  The Conversion of Quantity into Quality

4.  The Metric of the Hedonistic Scale

5.  Conclusion

 

References 

 

 

 

 

 

Abstract

 

 

Starting point

The Mere Addition Paradox was identified by Derek Parfit [Parfit 1984, Chapter 19]. It is characterized by the so-called Repugnant Conclusion, which says that (measured by the value of total welfare)

- a large population with a minimal average welfare can be ethically better than

- a small population with a high average welfare.

For a description and analysis see On the Buddhist Truths and the Paradoxes in Population Ethics.

 

The Mere Addition Paradox was originally presented as a two parameter model with the parameters average welfare and population size [Parfit 1984, Chapter 19] [Arrhenius, 2000]. In this paper we treat three characteristics of the paradox – which are usually summarized in the term axiology – as additional parameters. These three parameters are

- The notion of a life worth living (chapter 2)

- The conversion of quantity into quality (chapter 3)

- The metric of the hedonistic scale (chapter 4)

 

 

Type of problem

How can the Repugnant Conclusion be avoided?

 

 

Result

1. Direct Solution: Modify the utilitarian conversion of quantity into quality, so that it becomes impossible, to convert very low qualities of life into very high qualities.

2. Indirect solution: Completely discard the population size. The justification for this approach is based on the view of an impartial observer and the corresponding negative utilitarian metric. From the perspective of an impartial observer the sign of global welfare is uncertain.

 

The endeavor to find a “better than” relation in terms of welfare and population size” is reminiscent of classical utilitarianism, which started with the slogan “the greatest happiness for the greatest number”.

- The latter project ended with the calculation averages (indices) on the basis of surveys (note that the GDP measures economic welfare and not happiness); see Short History of Welfare Economics.

- The former seems to arrive at the same result. The best population is the one with the highest average life satisfaction, calculated on the basis of surveys.

In both cases the view of an impartial observer asks for a negative utilitarian metric.

 

 

 

 

 

1.  Introduction

 

 

Starting point

The Mere Addition Paradox was identified by Derek Parfit [Parfit 1984, Chapter 19]. It is characterized by the so-called Repugnant Conclusion, which says that (measured by the value of total welfare)

-        a large population with a minimal average welfare can be ethically better than

-        a small population with a high average welfare.

For a description and analysis see On the Buddhist Truths and the Paradoxes in Population Ethics.

 

The Mere Addition Paradox was originally presented as a two parameter model with the parameters average welfare and population size [Parfit 1984, Chapter 19] [Arrhenius, 2000]. In this paper we treat three characteristics of the paradox – which are usually summarized in the term axiology – as additional parameters. These three parameters are

1.      The notion of a life worth living (chapter 2)

2.      The conversion of quantity into quality (chapter 3)

3.      The metric of the hedonistic scale (chapter 4)

 

 

Type of problem

How can the Repugnant Conclusion be avoided?

 

 

 

2.  The Notion of a Life Worth Living

 

 

Semantics

For the purpose of this paper the terms happiness, (positive) welfare, quality of life and life satisfaction can be treated as synonyms.

The term suffering accordingly stands for uncompensated suffering [Fricke, 18] and is a synonym for negative welfare.

 

 

Competing notions

A possible approach to remove the Repugnant Conclusion consists in revising the classical utilitarian notion of a life worth living [Stanford, chapter 2.4]. In this paper we treat such revisions as a separate parameter:

         Extreme asymmetry means

o   every life is worth living or 

o   no life is worth living

         Moderate asymmetry means

o   a life is worth living, if it reaches a certain level of happiness or

o   a life is worth living, if it does not fall below a certain level of suffering.

         Symmetry corresponds to the classical utilitarian definition of a life worth living. According to this definition there is a level of welfare, at which the value of a life is neutral [Broome 2004, 142]. Above this level a life is worth living, below it is not worth living. A neutral life has the value zero on the hedonistic scale [Broome 2004, 257].

 

Indirectly the notion of a life worth living also determines the value assigned to non-existence:

-        If every life is worth living, then non-existence is the worst case

-        If no life is worth living, then non-existence is the best case

 

The value given to non-existence in turn determines the kind of happiness that is pursued [Contestabile 2010, 107]:

-        If non-existence (of the ego) is associated with the worst case, then it makes sense to pursue the biological kind of happiness.

-        If non-existence (of the ego) is associated with the best case, then it makes sense to pursue the meditative kind of happiness.

 

 

Extreme asymmetry

Intuitions about non-existence are driven by the interest to avoid suffering/frustration and the (conflicting) interest to survive [Contestabile 2010, 109-111].

1.      The intuition that suffering has a higher moral value than non-existence is e.g. defended in hospitals, where the prime interest is to avoid death. Since these hospitals do not know lives with negative welfare, death is given the value zero (see Quality-adjusted life year, Wikipedia). Even the most horrible life is considered to be worthy of preservation (Fig.2, left hand side) [Contestabile 2014, 299-300]. According to the hospital ideology the will to live creates a positive feeling, which compensates suffering. More realistically, however, the patient only has a choice between two evils:

a.      The suffering caused by illness, injuries, age-related morbidity etc.

b.     The suffering caused by the imagination of a painful death and non-existence.

2.      The complete opposite can be found in negative preference utilitarianism [Contestabile 2014, 307-308] where even an almost perfect life is not worth living. This is called the Reverse Repugnant Conclusion (Fig 2, right hand side) [Stanford, chapter 2.4].

 

Fig.2                                           

Hospital                                               Negative Preference

            axiology                                                   Utilitarianism

1.      The hospital axiology is counter-intuitive, because even nearly maximum suffering is given a positive value. This tolerance of suffering makes sense, however, in a “biological” axiology which assigns the maximum negative value to non-existence. The “biological axiology” is driven by the interest to survive under all circumstances. It is usually represented by religious physicians who consider life to be holy.

2.      Negative Preference Utilitarianism is counter-intuitive, because even almost perfect happiness is given a negative value. This degradation of happiness makes sense, however, in perfectionist Buddhism which assigns the maximum positive value to non-existence [Contestabile 2010, 107]. Perfectionist Buddhism is driven by the interest to experienceeternal bliss” (Brahman) after death. The Brahman concept originates in Hinduism and was later adopted by some forms of Buddhism [Fowler, 34]. Interestingly Buddhists – in contrast to some Hindu sects – never followed a suicide cult [Beckwith, 85]. But they pursued a meditative lifestyle and adhered to the ethical ideal of childlessness.

The hospital axiology is closer to the majority’s intuition, because it has a biological root.

 

Above axiologies are characterized by two Repugnant Conclusions that have to do with the relation between quantity and quality.

1.      If the hospital axiology is applied to population ethics, then there are only populations with positive totals, so that the (positive) Repugnant Conclusion applies (Fig.3, left hand side) [Contestabile 2010, 105].

2.      The mirror image of this axiology can be found in negative preference utilitarianism, where the prime interest is to avoid frustrations. If there are only populations with negative totals, then the Negative Repugnant Conclusion applies (Fig.3, right hand side) [Broome 2004, 213-214].

 

Fig.3

Moderate asymmetry

Attempts have been made to mitigate the repugnant conclusions:

         The so-called critical level utilitarianism [Blackorby, 2004] [Contestabile 2010, 110] maintains that it is unethical to create new lives that fall below a critical level of welfare. Lives below this critical level are given a negative sign. In this case the Repugnant Conclusion is mitigated, but the Negative Repugnant Conclusion aggravated (Fig.4, left hand side).

         If, conversely, an axiology tolerates a certain level of suffering by giving it a positive sign, then the Negative Repugnant Conclusion is mitigated, but the Repugnant Conclusion aggravated (Fig.4, right hand side).

 

Fig.4

Symmetry

We repeat the characteristics of extreme asymmetries, in order to compare them with symmetry:

         In the hospital axiology every life has a positive sign (Fig.2, left hand side).

It is characterized by an extreme version of the Repugnant Conclusion (Fig.3, left hand side).

         In Negative Preference Utilitarianism every life has a negative sign (Fig.2, right hand side).

It is characterized by an extreme version of the Negative Repugnant Conclusion (Fig.3, right hand side).

In classical utilitarianism happy lives have a positive sign and suffering lives a negative sign. As a consequence both kinds of repugnant conclusions apply. The classical utilitarian setting in Fig.5 avoids the aggravated forms of the repugnant conclusions and represents a kind of compromise or equilibrium [Broome 2004, 213-214, 264].

 

Fig.5

 

 

 

3.  The Conversion of Quantity into Quality

 

In this chapter we adopt the so-called Cantril Ladder for the investigation of the Repugnant Conclusion. The Cantril Ladder is used in the Gallup world poll and delivers, among others, input to the World Happiness Report. It has the following characteristics:

-        The notion of a life worth living is asymmetric (there are no lives with negative numbers).

-        The metric of the hedonistic scale is a linear point scale ascending from 0 to 10. Step 0 means that the scale knows neutral lives, in contrast to the hospital axiology, which only knows lives worth living (see chapter 2).

 

The remaining parameters then are

1.      The conversion of quantity into quality

2.      Average welfare

3.      Population size

 

 

Conventional utilitarian conversion

Reports on the basis of the Cantril Ladder necessarily generate positive totals and avoid the problem of whether (and how) suffering can be compensated by happiness. The conversion of quantity into quality is accordingly simple:

 

Example:

Given a utilitarian logic

-        1 person with the highest quality of life (10 points) can be replaced by

-        10 persons with a very low quality of life (1 point)

because total welfare is the same:

 

 

1

person

 

10

1

9

1

8

1

7

1

6

1

5

1

4

1

3

1

2

1

1

1

 

 

 

 

 

 

 

 

 

 

 

can be replaced by

 

 

 

 

 

 

 

 

 

 

 

 

 10 persons

1

1

1

1

1

1

1

1

1

1

 

In other words: the aggregated welfare of 10 unfortunate persons is ethically equivalent to the welfare of a single, fortunate person. This is a pocket size example of the Repugnant Conclusion.

 

 

Conversion with downgraded quantity

Obviously it is (too) easy to replace quality by quantity. Classical utilitarianism has an inflationary effect, because it is easier to increase the population size than the average welfare. Above conversion of quantity into quality, however, is not God-given. It stems from welfare economics, where welfare is aggregated by adding amounts of money. In population ethics, however, we have to deal with general welfare and the conversion has to be adapted to the corresponding intuitions. In the following example we use a so-called exponentiation with base = 2 for the conversion of quantity into quality:

-        2 persons on level 1 can replace 1 person on level 2 (21 = 2)

-        4 persons on level 1 can replace 1 person on level 3 (22 = 4)

etc.

-        512 persons on level 1 can replace 1 person on level 10 (29 = 512)

In other words: 512 persons are required instead of 10 persons (as above), in order to replace 1 person on the highest quality level: the ethical value of quantity relative to quality is downgraded.

 

What is the best value for the base?

-        The lower limit is 1. With 1n = 1 we get the conventional utilitarian conversion.

-        The upper limit is about 10. With 109 = 1 billion we are in the range of the world population.

There is a conflict of interest between

-        perfectionists, who tend towards base = 10, because they prefer a higher quality of life, even at the price of a shrinking population.

-        expansionists, who tend towards base = 1, because they prefer larger populations, even at the price of a lower quality of life.

[Contestabile 2010, 109-110].

In tiny populations it makes more sense to support the expansionists and in huge populations it makes more sense to support the perfectionists. We can account for that by making the base dependent on the population size. If the expansionists cannot completely overrule the perfectionists, then the Repugnant Conclusion disappears.

 

 

Negative total welfare

An analogous approach can be used to compare populations with negative total welfare. In this case the conflict between quantity and quality takes the following form:

-        the interest to reduce negative total welfare (negative quantity) versus

-        the interest to reduce the negative average welfare (negative quality).

 

Populations with negative total welfare occur

         in catastrophic scenarios within the classical utilitarian axiology (Fig.5 right hand side) and

         in negative utilitarianism, see Antinatalism and the Minimization of Suffering and [Contestabile 2010, 110].

 

 

 

4.  The Metric of the Hedonistic Scale

 

In this chapter we adopt the classical utilitarian notion of a life worth living.

 

The remaining parameters then are

1.      The conversion of quantity into quality

2.      The metric of the hedonistic scale

3.      Average welfare

4.      Population size

 

 

Asymmetric scales

In classical utilitarianism the hedonistic scale is linear and symmetric. In this chapter we investigate the influence of asymmetric scales. Terms are used as follows:

-        We use the term prioritarianism, if a symmetric scale is combined with a prioritarian weighting function.

-        We use the term negative utilitarianism, if the hedonistic scale is asymmetric in the first place.

Prioritarianism and negative utilitarianism, as understood in this paper, are functionally equivalent. In both cases suffering has more weight than happiness. For details see Antinatalism and the Minimization of Suffering.

 

 

Example 1

Imagine a two-person society: its only members are Jim and Pam. We compare the following two histories:

-   In society 1, Jim's well-being level is 44 (blissful); Pam's is -32 (hellish); overall well-being is +12.

-   In society 2, Jim's well-being level is 4; Pam's is 2; overall well-being is +6.

Prioritarians would say that society 2 is better or more desirable than society 1 despite being lower than society 1 in terms of overall well-being. Bringing Pam up by 34 is weightier than bringing Jim down by 40. Prioritarianism is arguably more consistent with commonsense moral thinking than utilitarianism when it comes to these kinds of cases, especially because of the prioritarian's emphasis on compassion. It is also arguably more consistent with common sense than radical forms of egalitarianism that only value equality (adapted from Prioritarianism, Wikipedia).

Different Prioritarian Welfare Functions correspond to different degrees of compassion.

 

How can we valuate welfare in such a way, that society 2 in above example is preferable to society 1?

Roughly, the idea is that we should maximize welfare, but gains in welfare matter more, the worse off people are, and losses in welfare matter less, the better off people are (…). Another way to express this intuition is to say that the marginal value of welfare is diminishing [Arrhenius 2000, 106]. One can achieve this result by applying a strictly increasing concave transformation to the numerical representation of people’s welfare before summing them up [Arrhenius 2008, 8]

Strictly increasing concave transformation means the following: The lower the welfare of an individual is relative to the others, the more weight it gets in the accumulation and vice-versa [Holtug, 13]. The total of all weights (percentages) equals 1 (respectively 100%). The following weighting function works with 3 levels of welfare:

 

 

life satisfaction

= welfare

welfare

%

moral weight

%

blissful

+25 till +50

14.25

neutral

-25 till +25

28.50

hellish

-25 till -50

57.00

Total

100

approx. 100

 

 

Under these premise positive values greater than 25 are cut into half and negative values smaller than -25 doubled:

-   In society 1, Jim’s well-being becomes 44/2 = 22, Pam’s -32*2 = -64 and overall well-being -42.

-   In society 2, overall well-being is unchanged +6.

Society 2 is now clearly preferable to society 1.

 

 

In this example we see how the “better than” relation in population ethics is influenced by the metric of the hedonistic scale.

-        With the classical utilitarian (symmetric) scale population 1 is better than population 2

-        With the negative utilitarian (asymmetric) scale population 1 is worse than population 2

For another example how the weight given to suffering affects the comparison of two populations with the same number of people see [Contestabile 2010, 111].

 

 

Example 2

The following table shows

1.      how total welfare can turn negative, if an exponential weighting function is applied

2.      how the measurement of life satisfaction with a 5-point scale can be converted into a cardinal scale.

 

 

life satisfaction

= welfare

welfare

%

x = welfare in units of 20%

moral weight

f(x) =

exp(-x)

w = moral weight

%

z =

(x times w)

times

(persons)

z

highly

positive

+30 till +50

+2

0.135

1.2

(+2.4) (300)

+720

moderately

positive

+10 till +30

+1

0.370

3.3

(+3.3) (500)

+1650

neutral

 

-10 till +10

0

1.000

8.7

(0) (100)

0

moderately

negative

-30 till -10

-1

2.718

23.5

(-23.5) (90)

-2115

highly

negative

-50 till -30

-2

7.388

63.3

(-126.6) (10)

-1266

 

 

Total 100

 

11.611

100.0

Total 1000 persons

-1011

 

 

Exponential weighting functions have e.g. been proposed by [Lumer]. In this example there are many more happy people than suffering ones, but (because of the exponential function) total welfare turns negative (-1011).

 

 

The impartial observer

The solution of the Mere Addition Paradox, which is sketched in chapter 3, is a solution of the conflict between quantity and quality. From the perspective of an impartial observer, however, the major theoretical problem is not quantity versus quality, but positive versus negative total welfare. An impartial observer would replace the classical utilitarian (symmetric) scale by a negative utilitarian (asymmetric) scale [Contestabile 2016]. Depending on the metric, the total welfare of a population can be positive, negative, or zero. Note that the notion of a life worth living is the same as in classical utilitarianism, only the metric for the hedonistic scale is different.

 

Following a pictorial representation of the example 2 above:

-        Surveys on subjective life satisfaction use symmetric scales so that total welfare is positive (Fig.6, left hand side)

-        With an asymmetric scale total welfare can be negative (Fig.6, right hand side).

 

Fig.6

The moral weight of suffering

       

 

           metric 1                                                             metric 2

   classical utilitarian                                             negative utilitarian

 

There are, of course, trivial cases where the sign of total is known and where the method of chapter 3 has to be applied. But as soon as cases of extreme suffering are involved, it is unclear if – seen from an impartial view – the aggregated happiness can compensate the aggregated suffering. This is true in particular, if we associate the term population with today’s nations. Given this uncertainty, it makes sense to disregard the population size and focus on the average welfare. This means that the conversion of quantity into quality falls off the agenda as well. The Mere Addition Paradox disappears, although indirectly (by a side effect) and not by a solution of the original problem (as in chapter 3).

 

 

Average utilitarianism

Average utilitarianism is the most popular axiology among welfare economists [Arrhenius 2000, 53]. A change in the population size

-        is ethically good, if the average increases

-        is ethically bad, if the average decreases

Average utilitarianism has several theoretical deficiencies [Arrhenius 2000, 54-57], but they are all based on the assumption that the sign of total welfare is known. The crucial point remains the metric, which is used to compare happiness with suffering. As long as the sign of total welfare is uncertain, we should actually construct the hedonistic scale in such a way that total welfare becomes zero. From then on we can observe in which direction the average moves. But no matter, if the tendency is positive or negative, the uncertainty about the sign remains, so that it is still justified to neglect the population size.

 

 

The revised index

One could argue that – as long as the task is only to observe the changes of the average – one could as well operate with current indices like the OECD Better Life Index or the Satisfaction with Life Index. But there is a major difference with regard to ethical priorities. The current indices suggest that the traumatic suffering of a minority can easily be compensated by the happiness of the majority. The revised index, in contrast, operates with an asymmetric hedonistic scale, which makes it hard to compensate suffering by happiness (see Negative Utilitarianism and Justice). Under these premises the most efficient way to improve total welfare is to reduce the worst cases of suffering with the highest priority.

 

 

 

5.  Conclusion

 

There are two ways to remove the Repugnant Conclusion:

1.      Direct solution: Modify the utilitarian conversion of quantity into quality, so that it becomes impossible, to convert very low qualities of life into very high qualities.

2.      Indirect solution: Completely discard the population size. The justification for this approach is based on the view of an impartial observer and the corresponding negative utilitarian metric. From the perspective of an impartial observer the sign of global welfare is uncertain.

 

The endeavor to find a “better than” relation in terms of welfare and population size is reminiscent of classical utilitarianism, which started with the slogan “the greatest happiness for the greatest number”.

-        The latter project ended with the calculation of averages (indices) on the basis of surveys (note that the GDP measures economic welfare and not happiness); see Short History of Welfare Economics.

-        The former seems to arrive at the same result. The best population is the one with the highest average life satisfaction, calculated on the basis of surveys.

In both cases the view of an impartial observer asks for a negative utilitarian metric.

 

 

 

References

 

1.      Anderson Ron (2012), Human Suffering and Measures of Human Progress, Presentation for a RC55 Session of the International Sociological Association Forum in Buenos Aires, Argentina

2.      Arrhenius Gustav (2000), Future Generations, A Challenge for Moral Theory, FD-Diss., Uppsala University, Dept. of Philosopy, Uppsala: University Printers

3.      Arrhenius Gustav (2008), Egalitarianism and Population Change, in A.Gosseries & L.Meyer (eds.) Intergenerational Justice, Oxford UP

4.      Arrow Kenneth J. (1966), Social Choice and Individual Values, Wiley, New York

5.      Beckwith Christopher I. (2015), Greek Buddha: Pyrrho's Encounter with Early Buddhism in Central Asia, Princeton University Press, Princeton

6.      Blackorby Charles, Bossert Walter, Donaldson David (2004), Critical Level Population Principles and the Repugnant Conclusion, in The Repugnant Conclusion, Essays on Population Ethics, Kluwer Academic Publishers

7.      Broome John (2004), Weighing Lives, Oxford University Press, New York

8.      Contestabile Bruno (2010), On the Buddhist Truths and the Paradoxes in Population Ethics, Contemporary Buddhism Vol.11, No.1, 103-113, Routledge, London

9.      Contestabile Bruno (2014), Negative Utilitarianism and Buddhist Intuition, Contemporary Buddhism, Vol.15, Issue 2, 298–311, Routledge, London

10.  Contestabile, Bruno (2016), The Denial of the World from an Impartial View, Contemporary Buddhism Vol.17, No.1, 49-61, Routledge, London

11.  Fowler Merv (1999), Buddhism: Beliefs and Practices, Sussex Academic, Brighton

12.  Fricke Fabian (2002), Verschiedene Versionen des negativen Utilitarismus, Kriterion Nr.15, p.13-27

13.  Holtug Nils (2004), Person-affecting Moralities, in The Repugnant Conclusion, Essays on Population Ethics, Kluwer Academic Publishers

14.  Lumer Christoph (2005), Prioritarian Welfare Functions, in Daniel Schoch (ed.): Democracy and Welfare, Paderborn: Mentis

15.  Parfit Derek (1984), Reasons and Persons, Clarendon Press, Oxford

16.  Stanford Encyclopedia of Philosophy (2016), The Repugnant Conclusion