# Mere addition paradox

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The mere addition paradox, also known as the repugnant conclusion, is a problem in ethics, identified by Derek Parfit, and appearing in his book Reasons and Persons (1984). The paradox identifies an inconsistency between four seemingly true beliefs about the relative value of populations.

## The paradox

Consider the four populations depicted in the following diagram: A, A+, B− and B. Each bar, within a population, represents a distinct group of people, whose size is represented by the bar's width and whose happiness is represented by the bar's height. Unlike A and B, A+ and B− are thus complex populations, comprising two distinct groups of people. (For simplicity, we might imagine that everyone in a group has exactly the same level of happiness, although this is not essential to the argument. We might instead imagine that the height of a bar represents the average happiness within that group of people.)

How do these four populations compare in value? Let's start by making comparisons between pairs of populations.

First, it seems that A+ is no worse than A. This is because the people in A are no worse off in A+, while the additional people who exist in A+ are better off in A+ compared to A. (Arguably, existence is good for these additional people, assuming that they have lives which are worth living and preferable over non-existence.)

Second, it seems that B− is better than A+. This is because B− has greater total and average happiness than A+.

Finally, B seems equally as good as B− as the only difference between B− and B is that the two groups in B− are merged to form one group in B.

Put together, these three comparisons entail that B is better than A. (If y is no worse than z and x is better than y it follows that x is better than z.) However, when we directly compare A and B, it may seem that B is in fact worse than A.

Thus, we have a paradox—the mere addition paradox—because the following intuitively plausible claims are jointly inconsistent: (a) that A+ is no worse than A, (b) that B− is better than A+, (c) that B− is equally as good as B, and (d) that B is worse than A.

## Criticisms and responses

Some scholars, such as Larry Temkin and Stuart Rachels, argue that the apparent inconsistency between the four claims just outlined relies on the assumption that the "better than" relation is transitive. We may resolve the inconsistency, thus, by rejecting the assumption. On this view, from the fact that A+ is no worse than A and that B− is better than A+ it simply does not follow that B− is better than A.

Torbjörn Tännsjö argues that we must resist the intuition that B is worse than A. While the lives of those in B are worse than those in A, there are more of them and thus the collective value of B is greater than A. Michael Huemer also argues that the Repugnant Conclusion is not repugnant, i.e. that we must revise the intuition that the conclusion is repugnant.[1]

## Alternative usage

An alternative use of the term mere addition paradox was presented in a paper by Hassoun in 2010.[2] It identifies paradoxical reasoning that occurs when certain statistical measures are used to calculate results over a population. For example, if a group of 100 people together control \$100 worth of resources, the average wealth per capita is \$1. If a single rich person then arrives with 1 million dollars, then the total group of 101 people controls \$1,000,100, making average wealth per capita \$9,901, implying a drastic shift away from poverty even though nothing has changed for the original 100 people. Hassoun defines a no mere addition axiom to be used for judging such statistical measures: "merely adding a rich person to a population should not decrease poverty" (although acknowledging that in actual practice adding rich people to a population may provide some benefit to the whole population).

This same argument can be generalized to many cases where proportional statistics are used: for example, a game sold on a download service may be considered a failure if less than 20% of those who download the demo then purchase the game. Thus, if 10,000 people download the demo of a game and 2,000 buy it, the game is a borderline success; however, it would be rendered a failure by an extra 500 people downloading the demo and not buying, even though this "mere addition" changes nothing with regard to income or consumer satisfaction from the previous situation.

In addition, there is consideration of a mere subtraction paradox in addressing Economic Inequality. In an unequal society or world, simply removing the rich people and their resources would technically result in equality, yet nothing would improve for the poorer people. This creates questions over whether or not "inequality" is the correct issue to consider.