**5. A Counterexample** **to the Sixth Impossibility Theorem**
What if the Crucial Assumption is weakened, to merely state that (**L**, **≿**) has a fine-grained S-discrete suborder? This assumption is trivial, given that (**L**, **≿**) is itself fine-grained, since the differences between adjacent levels in an S-discrete suborder of (**L**, **≿**) can, indeed, be arbitrarily small. However, it is not difficult to show that the sixth impossibility theorem is false if the Crucial Assumption is thus weakened.
To this end, let us use our toy theory of welfare, in either its S-discrete or its dense variant. Given fine-grainedness, the weakened version of the Crucial Assumption is satisfied. Further, we assume that superior as well as inferior welfare can be added across lives. Thus, letting (*h*_{A}, *l*_{A}) and (*h*_{B}, *l*_{B}) represent the total amounts of superior and inferior welfare in populations *A* and *B*, respectively, *A* contains more welfare than *B* iff *h*_{A} > *h*_{B}, or *h*_{A} = *h*_{B} and *l*_{A} > *l*_{B}. Finally, we assume that a population is, other things equal, better than another iff it contains more welfare.^{33}
Let us verify that this population axiology, which we may label “non-Archimedean totalism”, satisfies all of Arrhenius’s adequacy conditions. That the axiology satisfies Egalitarian Dominance and Weak Non-Sadism is obvious. To see that Weak Quality Addition is satisfied we need only note that adding any number of lives with very high welfare to a population *X* always makes the resulting population better than *X*, according to our axiology, whereas adding lives with very negative welfare plus any number of lives with very low positive welfare always makes the resulting population worse than *X*. Thus, let (*h*_{X}, *l*_{X}) be the total welfare in *X*. An addition of the first kind means adding (*m*, *n*), *m* > 0, to (*h*_{X}, *l*_{X}), while an addition of the second kind means adding (*r*, *s*), *r* < 0. No matter what the values of *n* and *s* are, (*h*_{X} + *m*, *l*_{X} + *n*) is greater than (*h*_{X} + *r*, *l*_{X} + *s*).
General Non-Extreme Priority is also satisfied. In fact, the condition holds for any number *n *of lives added to population *X*. Let (*h*_{X}, *l*_{X}) be the total welfare in *X* and let welfare level **A** be represented by (*h*_{A}, *l*_{A}). Adding some life or lives with very high welfare and one life at level **A** to *X* yields (*h*_{X} + *h*_{A} + *m*, *l*_{X} + *l*_{A} + *r*), *m* > 0. Adding some life or lives with very low positive welfare and one life slightly above **A** to *X* yields (*h*_{X} + *h*_{A}, *l*_{X} + *l*_{A} + *u* + *s*), *u* > 0. Irrespective of the values of *r*, *s* and *u*, (*h*_{X} + *h*_{A} + *m*, *l*_{X} + *l*_{A} + *r*) is greater than (*h*_{X} + *h*_{A}, *l*_{X} + *l*_{A} + *u* + *s*).
It remains to consider Non-Elitism. Let level **A** be represented by (*h*_{A}, *l*_{A}), implying that level **B** is represented by (*h*_{A}, *l*_{A} - *r*), *r* > 0. Also, let level **C **be represented by (*h*_{C}, *l*_{C}), and let (*h*_{X}, *l*_{X}) be the total welfare in population *X*. Let *n* be the number of people in populations *B* and *A*∪*C*. According to our axiology, *B*∪*X* is at least as good as *A*∪*C*∪*X* iff (*nh*_{A} + *h*_{X}, *n*[*l*_{A} – *r*] + *l*_{X}) is at least as great as (*h*_{A} + [*n **–** *1]*h*_{C} + *h*_{X}, *l*_{A}_{ }+ [*n *– 1]*l*_{C} + *l*_{X}). Cancelling out the terms *h*_{X} and *l*_{X},* *this is equivalent to (*nh*_{A}, *n*[*l*_{A} – *r*]) being at least as great as (*h*_{A} + [*n* – 1]*h*_{C}, *l*_{A}_{ }+ [*n *-1]*l*_{C}). Since *h*_{A} > *h*_{C}, or *h*_{A} = *h*_{C} and *l*_{A}_{ }– *r* > *l*_{C}, this holds for any *n*.
Thus, if we assume a non-Archimedean theory of welfare, there are population axiologies which satisfy all of Arrhenius’s adequacy conditions. Moreover, such an axiology can, as in the case of non-Archimedean totalism, conform to the simple and intuitively appealing idea that a population is, other things equal, better than another iff it contains more welfare.
**6. The Plausibility of Non-Archimedean Theories of Welfare**
The non-Archimedean toy theory of welfare outlined in section 4 is rather simplistic. First, it assumes that all welfare levels are comparable. Second, the assumption that there are two kinds of welfare components, such that the smallest amount of the superior kind trumps any amount of the inferior kind is perhaps not very plausible.^{34} However, a non-Archimedean theory can be much more sophisticated than our toy model. First, it can allow for incomparability between welfare levels. Second, non-Archimedeanness need not be a simple matter of some welfare components being superior to others. Although the representation of welfare levels by ordered pairs of numbers may naturally suggest such a simple superiority view,^{35} this type of representation could be used also for other kinds of non-Archimedean theories. Conversely, non-Archimedean theories, including superiority views, can be mathematically represented in other ways.
On a more complex view, non-Archimedeanness may be a holistic effect, arising from the combination of different welfare components, none of which is in itself superior. To illustrate this possibility, let us assume an “objective list” theory of welfare, akin to the one suggested by Griffin. According to this theory, pleasure, knowledge, friendship, freedom, appreciation of beauty, development of one’s talents, purposeful activity, and so on, are positive welfare components. It seems plausible to claim that for a life to have very high welfare, it must contain all or most of these things, to some degree. A life containing just one or two types of welfare components will likely be of an impoverished kind, having at best moderately high welfare. Nevertheless, it appears that an impoverished life can always be slightly improved, by adding more welfare components of the type or types already included in the life. However, such additions can never result in a life with very high welfare. Any finite series of improvements that takes us from an impoverished life to a life with very high welfare must include at least one step at which a welfare component of a new type is added. Such a step, arguably, means an improvement that cannot be arbitrarily slight.
Obviously, much work is needed in order to furnish the details of such a holistic non-Archimedean theory of welfare. But the general idea has enough plausibility, I believe, to cast doubt on Archimedeanness. Consider a life, *p*_{1}, containing no positive or negative welfare components except one second of mild pleasure. Compared to *p*_{1}, a life, *p*_{2}, containing nothing but two seconds of the same kind of mild pleasure, seems slightly better. And so on. Thus, each such life, *p*_{i}, *i* = 1, 2, 3, … is a representative of a welfare level **X**_{i}, such that **X**_{i}_{+1} **≻** **X**_{i}, for all *i* > 1. Let us also suppose that the difference between adjacent levels **X**_{i}_{+1} and **X**_{i} is the same, no matter the value of *i*. That is, an added second of mild pleasure has constant marginal value. Further, let *p* *be the best life you can imagine, and let **X***** be its welfare level. Probably, *p** contains welfare components of many different kinds, interrelated in diverse ways. It is very plausible to claim that **X***** **≻** **X**_{i}, for any *i*. However, if Archimedeanness is true, the difference between **X*** and **X**_{1} is bridged by a finite number of differences, each roughly equal to the difference between **X**_{i}_{+1} and **X**_{i}. Hence, it follows that **X**_{j} **≿** **X***, for some positive integer *j*.
The assumption of constant marginal value may be questioned. Perhaps the difference between adjacent levels **X**_{i}_{+1} and **X**_{i} diminishes as *i *increases. However, the conclusion that **X**_{j} **≿** **X***, for some *j*, still follows from Archimedeanness, as long as there is, for any *i*, some *k* > *i*, such that the difference between **X**_{k} and **X**_{i} equals that between **X**_{2} and **X**_{1}.^{36} To deny that there is such a *k* is to claim that if a life contains a sufficient number of seconds of mild pleasure (and no other welfare components), *no* extra number of such seconds yields an improvement, in terms of welfare, equal to the improvement from one to two seconds of mild pleasure. This claim does not appear very plausible.
At the very least, I think we may conclude that Archimedeanness is not so evidently true that it can legitimately be presupposed, as a background assumption, in an impossibility theorem purporting to establish the non-existence of an acceptable population axiology.
**7. Arrhenius’s Other Impossibility Theorems**
The Crucial Assumption is a background assumption for each of Arrhenius’s six axiological impossibility theorems.^{37} If the Crucial Assumption is replaced by the weaker assumption that (**L**, **≿**) has a fine-grained S-discrete suborder, non-Archimedean totalism constitutes a counter-example also to Arrhenius’s first, fourth, and fifth impossibility theorems. (This is shown in the appendix.) His second and third theorems, on the other hand, include one adequacy condition that is violated by non-Archimedean totalism. This condition is as follows:
*Inequality Aversion: *For any triplet of welfare levels **A**, **B**, and **C**, **A **higher than **B**, and **B **higher than **C**, and for any population *A* with welfare **A**, there is a larger population *C* with welfare **C**, such that a perfectly equal population *B* of the same size as *A*∪*C* and with welfare **B**, is at least as good as *A*∪*C*, other things being equal.^{38}
Non-Archimedean totalism violates this condition if, for example, **A** is a very high welfare level, while **B** and **C** are very low positive levels. However, Inequality Aversion does not appear particularly compelling. Suppose that **A** is the highest welfare level you can imagine, while **B** and **C** are very low positive levels, **B** being merely slightly higher than **C**. The only difference between a **B**- and a **C**-life, let us assume, is that a **B**-life contains one extra second of mild pleasure. Inequality Aversion implies that if a population at level **C** is large enough, it is improved at least as much by giving everyone an extra second of mild pleasure, as by raising a great number of people to level **A**. This is surely contestable.
In all events, “Inequality Aversion” is a misnomer. If this condition should for some reason be accepted, this cannot essentially have to do with the badness of inequality. Standard, Archimedean total utilitarianism, according to which the total welfare in a population is measured on a real-valued ratio scale, satisfies Inequality Aversion. Since only the total sum of welfare matters for the value of a population, according to total utilitarianism, this theory is “inequality neutral”. It neither favours nor disfavours inequality in the distribution of welfare. Non-Archimedean totalism is inequality neutral in exactly the same way. This theory, too, ranks populations exclusively by their total sum of welfare. Hence, it is no less “inequality averse” than standard total utilitarianism.^{39}
Actually, Arrhenius acknowledges that Inequality Aversion is intuitively debatable.^{40} He nevertheless defends this adequacy condition, by arguing that it follows from the allegedly more compelling Non-Elitism condition.^{41} His proof of this implication relies, however, on the Crucial Assumption.^{42} Without this assumption, Non-Elitism does not imply Inequality Aversion, as is evident from the fact that non-Archimedean totalism satisfies Non-Elitism but not Inequality Aversion. The Crucial Assumption thus plays an important role also as regards the second and third theorems.
In addition to his axiological theorems, Arrhenius proves two normative impossibility theorems.^{43} Roughly, these theorems are designed to show that there are possible situations of choice, with populations as options, in which the agent cannot avoid choosing wrongly. The following condition is an adequacy condition in both of the normative theorems:
*Normative Inequality Aversion: *For any triplet of welfare levels **A**, **B**, and **C**, **A **higher than **B**, and **B **higher than **C**, and for any population *A* with welfare **A**, there is a larger population *C* with welfare **C**, such that if it is wrong in a certain situation to choose a perfectly equal population *B* of the same size as *A*∪*C* and with welfare **B**, then it is also wrong in the same situation to choose *A*∪*C*, other things being equal.^{44}
If non-Archimedean totalism is combined with the normative principle that one ought always to maximize welfare, the resulting theory violates Normative Inequality Aversion. However, this condition is hardly more compelling than Inequality Aversion. If **A** is a very high level, while **B** and **C** are very low positive levels, one can reasonably claim that one ought to raise a large number of people from level **C** to level **A**, rather than raising an even larger number of people from level **C** to level **B**. If so, choosing *B* is wrong, but choosing *A*∪*C* is not.
As in the case of its axiological cousin, moreover, Normative Inequality Aversion does not really seem to concern inequality. Standard total utilitarianism satisfies this condition, although it is no more inequality averse than non-Archimedean total utilitarianism.
**8. Conclusion**
Four of Arrhenius’s six axiological impossibility theorems presuppose that the order of welfare levels satisfies Archimedeanness. Without this assumption, there are counterexamples to the theorems, in the form of population axiologies satisfying all of Arrhenius’s adequacy conditions. The remaining two theorems also rely on Archimedeanness, albeit less directly. Many philosophers have made claims that are incompatible with Archimedeanness, and non-Archimedean theories of welfare do not seem implausible. Hence, Arrhenius’s theorems rest on controversial assumptions, and the prospects for finding an acceptable population axiology may not be quite as bleak as he argues.^{45}
This does not detract from the significance of Arrhenius’s work. By revealing a great number of inconsistencies or tensions among intuitively very plausible claims, his results considerably restrict the room for maneuver, in the search for a satisfactory population axiology.^{46}
**Appendix: Non-Archimedean Totalism and Arrhenius’s First to Fifth Theorems**
We shall verify that non-Archimedean totalism satisfies the adequacy conditions of Arrhenius’s first, fourth, and fifth theorems. The first theorem is as follows:
*The First Impossibility Theorem: *There is no population axiology which satisfies Egalitarian Dominance, Quality, and Quantity.
As compared to the sixth theorem, two of these adequacy conditions are new:
*Quality: *There is a perfectly equal population with very high positive welfare which is at least as good as any population with very low positive welfare, other things being equal.
*Quantity:* For any pair of positive welfare levels **A** and **B**, such that **B** is slightly lower than **A**, and for any number of lives *n*, there is a greater number of lives *m*, such that a population of *m* people at level **B** is at least as good as a population of *n* people at level **A**, other things being equal.
Non-Archimedean totalism obviously satisfies Quality. To verify Quantity, let levels **A** and **B **be represented by (*h*_{A}, *l*_{A}) and (*h*_{B}, *l*_{B}), respectively. **A** and **B** being positive, and **B** being slightly lower than **A** implies that *h*_{A} *=* *h*_{B} ≥ 0, and *l*_{A} > *l*_{B}. Further, if *h*_{A} *=* *h*_{B} = 0, then *l*_{B} > 0. Hence, *n* and *m* can be chosen so that *mh*_{B} > *nh*_{A}, or *mh*_{B }= *nh*_{A} and *ml*_{B} ≥ *n**l*_{A}, implying that the population at level **B **is at least as good as the one at level **A**.
Consider next the fourth theorem:
*The Fourth Impossibility Theorem:* There is no population axiology which satisfies Egalitarian Dominance, General Non-Extreme Priority, Non-Elitism, Weak Non-Sadism, and Quality Addition.
In this theorem, there is only one new condition:
*Quality Addition: *For any population *X*, there is a perfectly equal population with very high welfare, such that its addition to *X* is at least as good as the addition of any population with very low positive welfare to *X*, other things being equal.
To see that non-Archimedean totalism satisfies Quality Addition, let (*h*_{X}, *l*_{X}) be the total welfare in *X*. Adding a population with very high welfare means adding (*x*, *y*), *x* > 0, to (*h*_{X}, *l*_{X}), whereas adding a population with very low positive welfare means adding (0, *z*), *z* > 0. This implies that (*h*_{X} + *x*, *l*_{X} + *y*) is greater than (*h*_{X}, *l*_{X} + *z*), and hence that the former addition is better than the latter.
Let us now turn to the fifth theorem:
*The Fifth Impossibility Theorem:* There is no population axiology which satisfies Dominance Addition, Egalitarian Dominance, General Non-Extreme Priority, General Non-Elitism, and Weak Quality.
Three of these adequacy conditions are new:
*Dominance Addition:* An addition of lives with positive welfare and an increase in the welfare of the rest of the population does not make a population worse, other things being equal.
*General Non**-Elitism:* For any triplet of welfare levels **A**, **B**, and **C**, **A **slightly higher than **B**, and **B **higher than **C**, and for any one-life population *A* with welfare **A**, there is a population *C* with welfare **C**, and a population *B* of the same size as *A*∪*C* and with welfare **B**, such that for any population *X*, *B*∪*X* is at least as good as *A*∪*C*∪*X*, other things being equal.
*Weak Quality:* There is a perfectly equal population with very high welfare, a very negative welfare level, and a number of lives at this level, such that the high welfare population is at least as good as any population consisting of the lives with negative welfare and any number of lives with very low positive welfare, other things being equal.
That non-Archimedean totalism satisfies Dominance Addition is obvious. In section 5, we showed that it satisfies Non-Elitism. Since that proof is independent of the welfare levels in *X*, it also proves that General Non-Elitism is satisfied. To see that Weak Quality is satisfied, let (*h*_{A}, *l*_{A}), be the total welfare in a population *A *with very high welfare. It follows that *h*_{A} > 0. The total welfare in a population *B *of lives with very negative welfare is (*h*_{B}, *l*_{B}), *h*_{B} < 0. Adding a number of lives with very low positive welfare to *B*, yields a population *C* with* *a total welfare of (*h*_{B}* *+ 0, *l*_{B}* *+ *x*), *x* > 0. Since *h*_{A} > *h*_{B}, (*h*_{A}, *l*_{A}) is greater than (*h*_{B}* *+ 0, *l*_{B}* *+ *x*). Hence, *A* is better than *C*.
Let us finally check that non-Archimedean totalism satisfies all the adequacy conditions of Arrhenius’s second and third theorems, except Inequality Aversion. The second theorem is this:
*The Second Impossibility Theorem: *There is no population axiology which satisfies Dominance Addition, Egalitarian Dominance, Inequality Aversion, and Quality.
As already noted, it is obvious that non-Archimedean totalism satisfies these conditions, with the exception of Inequality Aversion.
It remains to consider the third theorem:
*The Third Impossibility Theorem: *There is no population axiology which satisfies Egalitarian Dominance, Inequality Aversion, Non-Extreme Priority, Non-Sadism, and Quality Addition.
Two of these adequacy conditions are new:
*Non-Extreme Priority: *There is a number *n *of lives such that for any population *X*, a population consisting of the *X*-lives, *n *lives with very high welfare, and one life with slightly negative welfare, is at least as good as a population consisting of the *X*-lives and *n* + 1 lives with very low positive welfare, other things being equal.
*Non-Sadism: *[For any population *X*] an addition of any number of people with positive welfare [to *X*] is at least as good as an addition of any number of lives with negative welfare [to *X*], other things being equal.
That non-Archimedean totalism satisfies Non-Sadism is evident. To verify Non-Extreme Priority, let (*h*_{X}, *l*_{X}) be the total welfare in *X*. Adding *n* lives with very high welfare to *X *means adding (*x*, *y*), *x* > 0, to (*h*_{X}, *l*_{X}), yielding at total welfare of (*h*_{X} + *x*, *l*_{X} + *y*). Then adding one life with slightly negative welfare yields (*h*_{X} + *x* + 0, *l*_{X} + *y* + *z*), *z* < 0. Instead adding *n* + 1 lives with very low positive welfare to *X* yields (*h*_{X} + 0, *l*_{X} + *u*), *u* > 0. Since *x* > 0, (*h*_{X} + *x* + 0, *l*_{X} + *y* + *z*) is greater than (*h*_{X} + 0, *l*_{X} + *u*), implying Non-Extreme Priority.
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