1. Introduction
Parfit defends the Lexical View as a way to avoid the Repugnant Conclusion.Footnote 1 This view involves requiring a case of weak superiority to be true of the general betterness relation. Parfit acknowledges the objection to weak superiority, that it forces us to accept an implausible discontinuity in value between qualitatively very similar objects. But, he argues, this objection depends on an assumption of ‘evaluative precision’. Once we give up this assumption and allow for ‘imprecise equality’, the objection does not apply.
The present article is concerned with evaluation of this strategy. In section 2, I introduce Parfit's arguments. In sections 3–4, I develop a formal framework, which I then use in sections 5–6 to study the properties of the weak superiority relation under conditions not allowing for and allowing for imprecise equality, respectively. I use the results from these sections to evaluate Parfit's arguments in sections 7–8. Then, in section 9, I confront the most plausible version of the Imprecise Lexical View with Arrhenius’ Sixth Impossibility Theorem. Section 10 summarizes the results.
2. Introduction to Parfit
This section will introduce Parfit's arguments and thereby point out where the formal analysis will set in. Parfit's main ambition in population ethics is to avoid
The Repugnant Conclusion: Compared with the existence of many people who all have some very high quality of life, there is some much larger number of people whose existence would be better, even though these people would all have lives that were barely worth living.Footnote 2
To this end, Parfit proposes
The Lexical View: If many people exist who all have some high quality of life, that would be better than if there existed any number of people whose lives, though worth living, would be, in certain ways, much less good.Footnote 3
The Lexical View represents an instance of the kind of relation whose properties will be studied in the next sections. As for domain, Parfit in his arguments considers a sequence of possible types of lives a, b, c, …, z where each life is better than its successor, but any two neighboring lives are qualitatively very similar.
Parfit calls his sequence a ‘continuum’; however, he merely means that the difference in value between neighboring lives is ‘small’. But it matters whether the sequence is discrete, or it is truly continuous (dense). It will be a task to reframe the arguments in a framework that allows for denseness.
As I understand the proposal, a sufficiently large population of people with high, but not necessarily very high, quality of life may be lexically better than any number of people with much less good lives, but lives still worth living, that is, better than barely worth living. When the loss of quality of life reaches a certain size, it can no longer be compensated by quantity.
One possibility regarding the Repugnant Conclusion Footnote 4 would be to appeal to what Parfit calls
Diminishing Value Views: Though it would always be in itself better if there existed any extra person whose life would be worth living, the goodness of there being more such people would steadily diminish, and would have some upper limit.Footnote 5
But he rejects this kind of view. Instead, he appeals to
The Simple View: Anyone's existence is in itself good if this person's life is worth living. Such goodness has non-diminishing value, so if there were more such people, the combined goodness of their existence would have no upper limit.Footnote 6
Clearly, Parfit must mean ‘non-diminishing marginal value’. However, the concept of ‘non-diminishing marginal value’ alludes to numerical representation of value, the possibility of which Parfit otherwise denies. Thus, it will be a challenge to give it a meaning without assuming such numerical representation.
Parfit defends the Lexical View in at least two different ways. The first consists in meeting a serious objection to lexical views in general, namely that if in a sequence of objects of decreasing value, the first is lexically better than the last, the range of intermediate things must involve at least one sharp discontinuity between neighboring objects, and this is implausible. Parfit's strategy is to meet this objection by giving up an underlying assumption of evaluative precision; and the relevant arguments are mainly to be found in an unpublished manuscript.Footnote 7
Clearly, if the Lexical View is true, the Repugnant Conclusion is false. The second type of defense consists in addressing seemingly plausible premises that would force us to accept the Repugnant Conclusion and thereby make the Lexical View untenable. In this regard, Parfit is particularly concerned with so-called Continuum Arguments. A Continuum Argument is based on a premise of the following form:
[C]ompared with the existence of many people who would all have lives that were equally worth living, there are some much larger numbers of people whose existence would be better, though these people would all have lives that would be slightly less worth living.Footnote 8
Recursive application of this premise combined with transitivity leads to the Repugnant Conclusion, Parfit claims. Also, in the case of the Continuum Argument, Parfit sees the remedy in giving up an underlying assumption of evaluative precision.
By evaluative precision, Parfit appears to imply that value differences can always be measured on a real-valued cardinal scale. He makes at least two independent claims as to why this is not the case. The first claim is that equality cannot always be precise: ‘when neither of two things is better than the other, we may need to know whether this relation is precise’. It need not be, and Parfit adopts a definition of imprecise equality from John Broome.Footnote 9 The second claim is that, because Parfit assumes Non-diminishing Marginal Value, he seems to commit himself to the possibility that the value of populations has no upper limit. If lexical superiority is supposed to dominate such populations, Parfit is forced to reject the Archimedean Property.Footnote 10 These claims will be explained in the formal section.
Corresponding to the two types of defense outlined above, two relevant lines of thought can be identified in the literature. One is roughly concerned with the question that if we have a sequence of objects of decreasing value, what are the implications of claiming that the first is lexically better than the last?Footnote 11 With one minor exception to be mentioned below, the implications of introducing imprecise equality have not yet been examined in this context.
The other line of thought is the literature on Continuum Arguments. As initiated by Larry Temkin and Stuart Rachels (under the name of ‘spectrum arguments’), these arguments were understood as a challenge for the transitivity of the betterness relation. However, Parfit understands them rather as arguments for repugnant-like conclusions that refute lexical views. From this perspective, the question becomes: what are necessary and sufficient conditions for blocking the premises of a Continuum Argument in order to allow for a case of lexical betterness? In this case, there is an answer in the literature involving imprecise inequality, but it is confined to the case of a discrete sequence.Footnote 12
Although the two strategies and the two lines of thought are distinct, they have important overlaps and it is beneficial to address them together.
3. The framework
I shall adapt the framework, which was developed by Arrhenius and Rabinowicz, and later used by Jensen,Footnote 13 but which in the present context needs a minor revision. The abstract framework allows me to present results in a general form which is independent of any specific interpretation.
Consider a set of objects L weakly ordered by a binary relation Q.Footnote 14 Two objects are of the same type, if they are qualitatively identical with respect to Q; in that case, they belong to the same equivalence class. Call the set of types or equivalence classes P and let Q be the simple order on P induced by Q, which is a complete, transitive and antisymmetric relation.Footnote 15 If two objects are equally good, a weak order allows that they may be different objects; but in a simple order they must be identical (the same object). I shall be concerned with this domain P of types in this article (indicated by the use of bold), even though I shall often write ‘object’ for simplicity. But all definitions could also apply directly to the members of the set L.
All the literature mentioned merely considers discrete finite sequences of objects of decreasing (or increasing) value. However, the more general case of continuous sequences of objects shall not be excluded from the outset. ‘Continuity’ shall be cashed out in terms of the sequence in question being an order-dense subset of the set of atomic objects P:
Definition 1: A subset S of a set of atomic object types P simply ordered by a binary relation Q is order-dense in P, if and only if, for all x and y in P such that xQy, there exists z in S, such that xQz and zQy.
The definition implies for all types in P that, if xQy, then there is always a type z in the subset S, which lies between x and y in terms of Q.
Definition 2: A subset S of a set of atomic object types P simply ordered by a binary relation Q is discrete, if and only if, it is not order-dense in P.
On the set P is defined a concatenation operation, designated with ‘◦’, by which two types can be conjoined into a new object, which also belongs to P. Call such new objects in P formed by concatenation compound objects, and call the subset of the original atomic types P A. The concatenation procedure forms a whole (new object) out of the original types. But this could just mean that the objects are put side by side and considered together. Thus, a population will be defined as an object concatenated out of the lives of its individual members.
Concatenation is assumed to be associative; and P is assumed to be closed under concatenation.Footnote 16Self-concatenation is to concatenate an object x with an object of the same type as itself any number of times m, which object is designated mx.
Handfield and Rabinowicz use a comparable framework,Footnote 17 which however is limited to what I call self-concatenation only. In fact, they do not define any concatenation procedure; instead they talk about quantities of objects, assuming that, for any object, ‘any quantity is possible’. However, the set of quantities they sketch has its members indexed by natural numbers, and their examples only use natural numbers, so in practice their framework is comparable with self-concatenation as I define it.
Between pairs of types in the set P, I assume the existence of the following asymmetricFootnote 18 and transitive relation, ‘– is better than –’. The betterness relation evaluates types according to their overall betterness. On the set of atomic objects P A, I assume the betterness relation to be induced by Q: for all types x and y, x is at least as good as y (see Definition 5 below), if and only if xQy.
In this article, I shall only be concerned with cases where all members of P are positively valuable, and in particular, I shall assume that concatenation is value-increasing, i.e. for all x and y, x◦y is better than x. But the framework could allow for negatively valuable objects.
On the domain corresponding to P, Arrhenius and Rabinowicz defined a complete and transitive weak betterness relation, ‘– is at least as good as –’; and Jensen followed them.Footnote 19 My choice here of ‘– is better than –’ as primitive constitutes the revision of their framework. The two approaches can easily be transformed into each other. However, for the purpose of introducing evaluative imprecision, the present framework is intuitively easier to grasp, and it is more in line with Parfit's presentation.
We need to distinguish between:
Definition 3: xis precisely equally as good as y, if and only if x is not better than y and y is not better than x, and for any third object z, z is better than x, if and only if z is better than y, and x is better than z, if and only if y is better than z.
Definition 4: xis imprecisely equally as good as y, if and only if, x is not better than y and y is not better than x, and there exists a third object z, such that z is better than x, and z is not better than y, or x is better than z, and y is not better than z.Footnote 20
The latter part of Definition 3 ensures that ‘– is precisely equally as good as –’ is an equivalence relation (reflexive, symmetric and transitive),Footnote 21 whereas the latter part of Definition 4 indicates that ‘– is imprecise equally as good as –’ is not, because it is not necessarily transitive.Footnote 22 If imprecise equality is allowed, value differences cannot be measured precisely on a real-valued scale, because no real-valued numerical representation of betterness will then be possible, be it cardinal or ordinal.Footnote 23
Hence, one aspect of evaluative precision can be cashed out by:
Condition 1: For all x and y, exactly one of the following holds: x is better than y, y is better than x, x is precisely equally as good as y.
Note that since Q is assumed to be a weak order, it follows that Condition 1 must hold for the betterness relation on the set of atomic objects P A.
Allowing for evaluative imprecision involves that Condition 1 is rejected and replaced by:
Condition 2: For all x and y in P but not in P A, exactly one of the following holds: x is better than y, y is better than x, x and y are precisely equally good, x and y are imprecisely equally good.
Let me add the following definition, which holds under both Condition 1 and Condition 2:
Definition 5: x is at least as good as y, if and only if x is better than y or x is precisely equally as good as y.
Definition 5 ensures that weak betterness is transitive.Footnote 24 I shall be concerned with the properties of lexical betterness:Footnote 25
Definition 6: An object x is lexically better than [or weakly superior to] an object y, if and only if, for some positive integer m and all positive integers n, mx is better than ny.
Definition 7: An object x is strictly lexically better than [or superior to] an object y, if and only if, for all positive integers n, x is better than ny.
4. Non-diminishing marginal value
The literature has showed that it makes a lot of difference for lexical betterness whether for the inferior object y, when it is concatenated to itself any number of times n, the marginal contribution diminishes and eventually converges to zero, such that there will be an upper limit for the value of the object ny, or self-concatenation of y has constant marginal value and thus never reaches an upper limit.Footnote 26
However, the concepts of ‘diminishing’ and ‘non-diminishing marginal value’ allude to numerical representation of value on a cardinal scale, of which the possibility cannot be assumed under the condition of evaluative imprecision. So how can these terms be given a meaning without assuming such numerical representation?
I shall ignore the possibility that self-concatenation of an object could have increasing marginal value and simply assume that it has either constant marginal value or diminishing marginal value. I suggest ‘constant marginal value’ can be captured by the concept of a standard sequence.
A standard sequence defines value contributions having non-zero, equal spacing in some intended numerical representation.Footnote 27 Thus, the existence of standard sequences must be a necessary condition for constant marginal value, even if a numerical representation, for other reasons, happens to be impossible.
Definition 8: A sequence x, 2x, 3x, … constitutes a standard sequence based on the atomic object x, if and only if there exists some other object s (the standard), such that for all integers n ≥ 0, (n+1)x is precisely equally as good as nx◦s:
x and s are precisely equally good
2x and x◦s are precisely equally good
3x and 2x◦s are precisely equally good
…
A standard sequence can be either finite or infinite.
The problem is that it does not seem possible to construct a standard sequence based on an atomic object in the domain described above. If s and x are precisely equally good, they belong to the same type. If that is the case, both s and x may diminish in marginal value as they each are concatenated with a sequence nx of the same type. The standard should not be susceptible to diminishing marginal value, so it needs to be of a different type.
Therefore, consider:
Condition 3: Non-diminishing Marginal Value for a sequence x, 2x, 3x, … based on the atomic object x: For all atomic objects y of a type different from x, if x is better than y, then, for all n ≥ 1, x◦nx is better than y◦nx.
and
Condition 4: Non-increasing Marginal Value for the sequence x, 2x, 3x, … based on the atomic object x: For all atomic objects y of a type different from x, if y is better than x, then, for all n ≥ 1, y◦nx is better than x◦nx.
These conditions make it possible to define a standard sequence in a domain where Definition 8 does not apply, i.e. where there is no atomic object s which is both (1) of a type different from the atomic object x and (2) precisely equally as good as x:
Definition 9: The sequence x, 2x, 3x, … based on the atomic object x is a standard sequence, if it fulfills Condition 3 and Condition 4.
I shall later make use of the following condition:
Condition 5: Non-diminishing Marginal Value for the sequence tx, 2(tx), 3(tx), … based on the compound object tx: For all atomic objects y of a type different from x, if tx is better than y, then, for all n ≥ 1, tx◦nx is better than y◦nx.
However, note:
Observation 1: Given that self-concatenation is value increasing, Condition 5 is implied by Condition 3.
Proof: See Appendix 2.
In the literature, the possibility of diminishing marginal value (with an upper bound) is contrasted with the following independence condition:Footnote 28
Condition 6: Strong Independence: For all atomic or compound objects x, y and z, if x is at least as good as y, then x◦z is at least as good as y◦z.
This condition is a principal necessary condition for value additivity; it applies to all triples of objects and is thus much stronger than Condition 3, which is merely concerned with self-concatenation. But note that Condition 5 is implied by it. Another independence condition is:
Condition 7: Weak Independence: For all atomic objects x and y and all atomic or compound objects z, if x is at least as good as y, then x◦z is at least as good as y◦z.
Conditions 3 and 4 are implied by Weak Independence.
Definition 8 and Definition 9 both allow that a standard sequence has no upper limit. But lexical betterness implies that some standard sequence has an upper limit. Consider:
Definition 10: A standard sequence x, 2x, 3x, …, nx based on the atomic object x is strictly bounded if there exists an object y (atomic or compound) of another type, such that for all n, y is better than nx.
Consider next:
Condition 8: Archimedean Property: Every strictly bounded standard sequence is finite.
This condition is necessary for representing betterness with a real-valued cardinal scale since such a scale would have to satisfy the Archimedean Property of real numbers: for any positive number x, however small, and any number y, however large, there is some integer n, such that ny is at least as great as x.Footnote 29
Under Condition 3, lexical betterness violates the Archimedean Property. ‘x is lexical better than y’ implies that no number of ys can bridge the value of some number of xs; but if self-concatenation of any type fulfills Non-diminishing Marginal Value and the Archimedean Property, any upper bound should be reachable in a finite number of self-concatenations. Therefore, under these conditions, Condition 8 needs to be rejected and this provides another reason for the impossibility of a real-valued cardinal scale.Footnote 30
I shall now present some results that follow under Condition 1. These results pose various problems for the Lexical View. Later, I shall evaluate the extent to which these problems can be met by rejecting the evaluative precision involved in Condition 1 by replacing it with Condition 2.
5. Results under precision
Consider first the negation of lexical betterness. If x is not lexically better than y then, for all m, there is some n, such that mx is not better than ny. Under Condition 1, this means that for all m, there is some n, such that ny is as least as good as mx; this means further (given that self-concatenation is value increasing) that for all m, there is some n’ = n + 1, such that n’y is better than mx.
Definition 11: x is exchangeable with y if, for all m, there is some n, such that ny is better than mx.
Thus, since under Condition 1, exchangeability and lexical betterness are negations of each other, the following holds:
Observation 2: Under Condition 1, for all x and y, either x is lexically better than y, or x is exchangeable with y.
One problem that has been raised for lexical betterness is that if, for some sequence b, …, q, b is lexically better than q, then one member of the sequence must be lexically better than its successor, and this seems to be an implausible discontinuity in value.
Arrhenius and Rabinowicz proved:
Observation 3: Suppose that at-least-as-good-as is a complete and transitive relation on the domain. Then, in any finite sequence of objects in which the first element is lexically better than the last element, there exists at least one element that is lexically better than its immediate successor.Footnote 31
However, it is possible to obtain a considerably more general result:
Observation 4: Consider a (possibly order-dense) set of objects b, …, q of decreasing value, where b is lexically better than q. Assume that ‘– is better than –’ is transitive and that Condition 1 holds. Let x belong to the set {x | b is better than x and x is better than q}. For any x: Assume that b is not lexically better than x, i.e. is exchangeable with x. Then x is lexically better than q.
Proof: See Appendix 2.
Corollary to Observation 4: For any x: Assume that x is not lexically better than q, i.e., is exchangeable with q. Then b is lexically better than x.
The corollary follows by Modus Tollens.
This demonstrates that a case of lexical betterness has much stronger implications than those indicated by Observation 3. Consider first the case, where b is exchangeable with all x. Then not only b but all q’s predecessors in the sequence after b are lexically better than q. Alternatively, if all x are exchangeable with q, then b is lexically better than all its successors down to and including q.
If the sequence is discrete, Observation 4 makes the claim that one element will be lexically better than its immediate successor more precise, namely that either will q’s nearest predecessor be lexically better than q, or b will be lexically better than its immediate successor.
If the sequence is an order-dense subset of P A, then there will be a discontinuity in value between an object arbitrarily close to q and q itself, or between b and an object arbitrarily close to b. These sharp discontinuities between qualitatively close objects appear implausible.
Another problem that has been raised for lexical betterness is a so-called Continuum Argument. Footnote 32 This argument is believed to rule out lexical betterness. Consider
Premise (A) In the discrete sequence a, …, z, each object is exchangeable with its successor.
The following holds:
Observation 5: If, for some discrete sequence a, …, z, Premise (A) holds, and transitivity holds for betterness, then a is exchangeable with z.
Proof: See Appendix 2.
Corollary to Observation 5: If betterness is transitive, then exchangeability is transitive.
Proof: Follows directly from the proof of Observation 5.Footnote 33
We know from Observation 4 that if the conclusion of the Continuum Argument is false, such that a is lexically better than z, then either object y immediately preceding z must be lexically better than z; or a must be lexically better than its immediate successor b. In other words, in order to block the Continuum Argument, one case of lexical betterness between neighbors (at either end of the sequence) is necessary.
Suppose, on the other hand, that some object in the sequence, say p, is not exchangeable with its successor, q. Then the Continuum Argument is blocked at this point. Although a may be exchangeable with p, a is not necessarily exchangeable with q, which means that a may be lexically better than q. Hence, the Continuum Argument is blocked precisely at the point where an object for the first time in the sequence is not exchangeable with its successor. Of course, if lexical betterness between neighboring objects is implausible anyway, this way to block the Continuum Argument seems dubious.
However, the Continuum Argument only works if the sequence a, …, z is both finite and discrete. If the sequence a, …, z is an order-dense subset of P A, then the conclusion of the Continuum Argument does not follow. Consider the following restatement of Premise (A) to allow for an order-dense subset of P A:Footnote 34
Premise (B): For all x in the set {x | x is better than z}, there is some y, such that x is both better than and exchangeable with y.
In this case, we might never reach all the way down to z. The fact that, starting from a, we can keep on finding some x which is progressively worse, and which can be exchanged with some y that is still worse than this x, such that, by transitivity, each new y can be exchanged for a, is compatible with a being lexically better than z, and more generally, with some b in the set being lexically better than some q in the set (worse than b); repeated application of Premise (B) and transitivity might only get us arbitrarily close to q but never actually make us reach it.
Premise (A) is only plausible if the members of the discrete sequence are sufficiently close. But there is no principled way to demonstrate, for any actual sequence, that the members are close. And given denseness, there is even no such thing as neighboring objects; between any two objects in the sequence, there will be infinitely many objects.
However, even if Premise (B) is compatible with the b being lexically better than q, this blocking of the Continuum Argument does involve a serious discontinuity in value, as we saw above.
Finally, I need to present another observation under Condition 1, or rather, merely the weaker assumption that Condition 1 applies for cases of lexical betterness:
Condition α: if x is lexically better than y, then Condition 1 applies for all mx and ny, i.e., for all m and n: either mx is better than ny, or ny is better than mx, or mx and ny are precisely equally good.
Neither Observation 3 nor Observation 4 assumes Non-diminishing Marginal Value. But it does make a difference to assume Non-diminishing Marginal Value:Footnote 35
Observation 6: Assume x is better than y. Condition 3 and Condition 5 (Non-diminishing Marginal Value) and Condition α implies that, if x is lexically better than y, then x is also strictly lexically better than y, i.e., if for some m and all n, mx is better than ny, then x is better than ny.
Proof: See Appendix 2.
This means that, given Non-diminishing Marginal Value, lexical betterness collapses into strict lexical betterness, that is, if no ny can be at least as good as mx, no ny can even be at least as good as x.
When objects come in natural units, e.g., lives, it is clear what a drop from some number m to 1 involves. But what about objects that do not come in natural units? Handfield and Rabinowicz,Footnote 36 for instance, deal with types of harms, which do not seem to come in natural units. However, a type of harm seems to be determined by its intensity; and these harms are assumed to be quantifiable by their duration. If they are, these harms in fact satisfy the condition of Non-diminishing Marginal Value. Take a unit of time as small as you want. Observation 6 implies that if some duration of one type of harm is better than any duration of another, then even a duration of the first type of harm as short as the smallest unit of time is better than any duration of the other type.
In all cases, if lexical betterness involves an implausible discontinuity in value in the first place, this becomes a much more implausible discontinuity in value if we assume Non-diminishing marginal Value.
What does Constant Marginal Value imply for the Continuum Argument? Premise (B) allows for cases of lexical betterness with an upper limit. But suppose we accept Conditions 4 and 5 together with the Archimedean Property (Condition 8). Consider an infinite standard sequence q, 2q, 3q, …, nq according to Definition 9, where n is any integer, and let b be an object which is better than q. If b were lexically better than q, then the standard sequence q, 2q, 3q, …, nq would be strictly bounded; but since it is infinite, b being lexically better than q would violate the Archimedean Property. Hence, the Archimedean Property excludes any case of lexical betterness under Constant Marginal Value. This follows directly from the assumptions and does not depend on any Continuum Argument.
Suppose next that we accept Constant Marginal Value (Conditions 3 and 5) but reject the Archimedean Property (Condition 8). This will allow for cases of lexical betterness, since a strictly bounded standard sequence does not have to be finite. However, because Observation 6 also holds under these assumptions, any instance of lexical betterness will involve a much stronger discontinuity in value than need be the case without Constant Marginal Value.
6. Imprecise equality
I shall now examine how allowing for imprecise equality, i.e., exchanging Condition 1 with Condition 2, affects the results of the previous section. Under Condition 2, lexical betterness and exchangeability are not simply negations of each other; negating one leads to a weaker version of the other.
Definition 12: An object x is minimally exchangeable with (not lexically better than) an object y, if and only if, for all m, there is n, such that mx is not better than ny.
Definition 13: An object x is minimally lexically better than Footnote 37 (not exchangeable with) an object y, if and only if there is m, such that for all n, ny is not better than mx.
Minimal exchangeability is thus implied by exchangeability (Definition 10) but not vice versa; and minimal lexical betterness is implied by lexical betterness (Definition 5), but not vice versa. Let me add a definition of a weaker version of strict lexical betterness (Definition 6), which is implied by it, but not vice versa:
Definition 14: An object x is strictly minimally lexically better than an object y, if and only if, for all positive integers n, ny is not better than x.
Handfield and Rabinowicz introduce the concept of radical imprecise equality:Footnote 38
Definition 15: An object x is radically imprecisely equal to an object y, if and only if there is m’, such that for all m ≥ m’, there is n m, such that for all n ≥ n m, mx is imprecisely equal to ny.
It is easy to show
Observation 7: Given that concatenation is value increasing, x is both minimally lexically better than and minimally exchangeable with y is equivalent with x is radically imprecisely equal to y.
Proof: See Appendix 2.
As explained by Handfield and Rabinowicz,Footnote 39 radical imprecise equality is quite different from the behavior of imprecise equality in the small-improvement type of cases described by Chang.Footnote 40 The point of these examples is that, in cases where none of the objects is better than other, small improvements on either side may not change that; hence imprecise rather than precise equality is true. But it seems implicit in these examples that large improvements on either side would make one object better than the other. However, the imprecise equality between objects involved in radical imprecise equality is persistent despite improvements on either side, however large.
Handfield and Rabinowicz prove the following result, which, given Observation 7, follows immediately from Definitions 11 and 12:Footnote 41
Observation 8: For all x and y, one of the following holds: x is lexically better than y, x is exchangeable with y, x is radically imprecisely equal to y (i.e., x is both minimally lexically better than and minimally exchangeable with y).
It follows that someone who wants to deny that some object x is exchangeable with another object y, without confirming that x is lexically better than y (or vice versa), by way of appealing to imprecise equality, is committed to claim that these objects are radically imprecisely equal. This is a stronger claim than may have been envisaged.
Now consider what radical imprecise equality may imply for Observation 4 above. The crucial point was step (4) in the proof. Now, under Condition 2, there is the possibility that if, for all m, there is some n, such that nx is not better than mb, this could imply that nx could be imprecisely equally as good as mb. This fact is used in the proof of Observation 9:
Observation 9: Consider a (possibly order-dense) set of objects b, …, q of decreasing value, where b is lexically better than q. Assume that ‘– is better than –’ is transitive and that Condition 2 holds. Let x belong to the set {x | b is better than x and x is better than q}.
(9.1) For any x: Assume that b is not lexically better than x. Then x is minimally lexically better than q.
(9.2) For any x: Assume that x is not lexically better than q. Then b is minimally lexically better than x.
Proof: See Appendix 2
Corollary 1 to Observation 9.1 and 9.2: Assume that x is minimally lexically better than but not lexically better than q. Then x is minimally exchangeable with q. Assume that x is minimally exchangeable with but not exchangeable with q. Then x is minimally lexically better than q.
Proof: Follows from Observation 8.
Corollary 2 to Observation 9.1: For any x: Assume that x is not minimally lexically better than (i.e., exchangeable with) q. Then b is lexically better than x.
Corollary 2 to Observation 9.2: For any x: Assume that b is not minimally lexically better than (i.e., exchangeable with) x. Then x is lexically better than q.
Proof: These corollaries follow by Modus Tollens.
Observation 9 applies under Condition 1 as well as Condition 2. In the former case, ‘– is not lexically better than –’ means ‘– is exchangeable with –’, and Observation 9 simply reduces to Observation 4, as is also clear from the two Corollaries 2. In the latter case, ‘– is not lexically better than –’ may mean ‘– is minimally exchangeable, but not exchangeable with –’, implying radical imprecise equality.
What are the implications of introducing radical imprecise equality in a sequence of objects of decreasing value? It might be worth spelling out the possibility of a full zone of radical imprecise equality between some endpoints b and q:
Corollary 3 to Observation 9.1: Assume that all x are radically imprecisely equal to q. Then b is radically imprecisely equal to all x.
Corollary 3 to Observation 9.2: Assume that b is radically imprecisely equal to all x. Then all x are radically imprecisely equal to q.
Next, suppose that, after a zone {x | b < x ≤ m} where b is exchangeable with x, there is another zone {x | m < x < q}, such that b is not exchangeable with x, but radically imprecisely equal to x. Then it follows that all x in the zone are radically imprecisely equal to q. Since b is exchangeable with m, it also follows that m is lexically better than q. But then we can apply Corollary 3 with m as first object instead of b with the result that m is radically imprecisely equal to all members x of the zone, and all members x of the zone are radically imprecisely equal to q.
Alternatively, we might have a zone {x | b < x < m}, where b is lexically better than m, m and all its successors are lexically better than q, b is radically imprecisely equal to all x in the zone, and all x in the zone are radically imprecisely equal to m.
What has been obtained as compared with Observation 4? Instead of the single borderline of lexical betterness, a zone of radical imprecise equality has been inserted. At the borderlines at either end of this zone, radical imprecise equality appears, and the size of the zone determines the minimal interval across which lexical betterness holds.
This can all be said to soften the sharp borderline of lexical betterness, but the question is how much. One aspect is the size of the zone of radical imprecise equality. The smaller it is, the less the discontinuity appears to be softened. But even if it is large, the fact that the objects at either end of the zone are radically imprecisely equal to all members of the zone in between seems to dissolve the difference in value between the members of the zone. This seems to indicate that the discontinuity in value because the first end point is lexically better than the second to a large extent remains independent of the size of the zone between them.
Another aspect is the situation at either borderline of the zone – between two pairs of neighboring objects in the case of a finite, discrete sequence, or between two pairs of arbitrarily close objects in the case of a dense sequence. Here, the single case of lexical betterness under Condition 1 is transformed into two cases of minimal lexical betterness, each of which still represents some discontinuity of value as compared with exchangeability.
All in all, even if some softening has been obtained, it can be questioned how much it amounts to. However, I believe further evaluation depends on analysis of the actual interpretation of the domain.
Observation 9 also confirms, and makes more precise, a point made by Handfield and Rabinowicz, that in order to block the Continuum Argument by the help of imprecise equality between some pair(s) of neighboring objects, one case of radical imprecise inequality between a pair of neighbors in the (finite, discrete) sequence is incompatible with a being lexically better than z – at least two cases are needed.Footnote 42
From Observation 9, we learn that if b is lexically better than q, then there needs to be a zone of radical imprecise equality consisting of at least one object somewhere in between. Could we not just claim that, say, p is imprecisely equal to it successor q? No, because it follows from Observation 9 that if b is exchangeable with p, then p is lexically better than q. Contrary, if b is radically imprecisely equal to p, then p’s predecessor o must be imprecisely equal to p, because if it were not, o would be exchangeable with p, and then since b (by assumption) is exchangeable with o, by transitivity it would also be exchangeable with p. Hence, two cases of radical imprecise equality must exist, one at either border of the zone consisting of p.
Finally, I shall prove an analogue of Observation 6 for minimal lexical betterness:
Observation 11: Assume x is better than y. Condition 3 and Condition 5 (Non-diminishing Marginal Value) implies that, if x is minimally lexically better than y, then x is also strictly minimally lexically better than y, i.e., if for some m and all n, ny is not better than mx, then ny is not better than x.
Proof: See Appendix 2
What I said above concerning Observation 6 about units applies here as well.
Consider a (possibly order-dense) set of objects b, …, q of decreasing value, where b is lexically better than q, and assume that both Condition 2 and Non-diminishing Marginal Value holds. Observation 6 implied that b is strictly lexically better than q, and that fact will persist in spite of Condition 2 and in spite of Observation 11, as long as imprecise equality cannot hold between any mb and nq, that is, as long as Condition α holds.
But Observation 11 does imply that any instance of x being radically imprecisely equal to y will imply that x is strictly minimally lexically better than y, meaning that for all m ≥ 1 and all but very small n, mx is imprecisely equal to ny. Thus, the difference in value between members of the zone of radical imprecise equality as compared with either endpoint of a case of lexical betterness will be further erased; and at either end of the zone, strict minimal lexical betterness will be true representing a more serious discontinuity in value.
7. Implications for Parfit's lexical view of the results under precision
Here are a few remarks on the relevant interpretation of the formal framework. In his arguments, Parfit considers a sequence of possible types of lives a, …, z where each life is better than its successor, but any two neighboring lives are qualitatively very similar. Thus, we have as the domain L the set of possible lives. Each life is characterized by a certain quality of life (level of welfare), and the set of lives are weakly ordered by their quality of life (the relation Q). Two lives are of the same type if and only if they have the same quality of life. Hence, the domain becomes the set of types P A, the set of lives with varying qualities.
A population is simply a compound formed by concatenation of lives of various types or – at the limit – just one life. Since Parfit's examples always assume that a population consists of some number of people all with the same quality of life, a population simply becomes a type of life, concatenated with itself some number of times.
Parfit acknowledges that, under Condition 1, if b is lexically better than q, then there is at least one life within this sequence which is lexically better than its successor, but he claims that it only follows if we assume evaluative precision. He could merely point out that Observation 3 assumes completeness and thus relies on precision in terms of a relation of precise equality. Instead, he sets up his own argument for the conclusion that, under Condition 1, p is lexically better than q and shows how this argument can be met by denying the assumption of precision, that is, by denying Condition 1.Footnote 43
However, Parfit's argument has more implications than he himself exploits, since it can be used to prove Observation 4, which demonstrates that one instance of lexical betterness, such as b being lexically better than q, draws with it many more instances of lexical betterness than merely p being lexically better than q. Parfit's argument can further be used to prove Observation 9, which similarly demonstrates far more details about the consequences of introducing imprecise equality than he himself draws out.
Concerning the Continuum Argument, Parfit also seems to acknowledge that, under Condition 1, rejection of Premise (A) at one step is implausible, because it would involve an instance of lexical betterness between neighboring lives in a discrete sequence. However, he does not consider the possibility of denseness at all, which would make the Continuum Argument futile, but still leave a discontinuity at the limit.
And importantly, he appears to be unaware of the implications of assuming Non-Diminishing Marginal Value spelled out in Observation 6.Footnote 44
8. Implications for Parfit's lexical view of giving up precision
The greatest challenge for Parfit's proposal comes from Observation 6. The condition of Non-diminishing Marginal Value, which Parfit confirms, implies that lexical betterness collapses into the much stronger strict lexical betterness. This would imply that even one life of the superior quality of life is better than any number of lives of the inferior quality of life. I believe Parfit would consider it a reductio of lexical betterness, if it collapses to strict lexical betterness. This means that Observation 6 can be understood as an impossibility theorem, which makes the condition of Non-diminishing Marginal Value (Conditions 3 and 5) incompatible with the Lexical View.
Parfit does not address this problem. Neither is it addressed through the introduction of imprecise equality. Admittedly, the proof of Observation 6 makes use of Condition α, which is incompatible with Condition 2 in cases of lexical betterness. But the Imprecise Lexical View involves by definition at least one case of lexical betterness, which could not be true, if radical imprecise equality could hold between the lives in question. Non-diminishing Marginal Value further amplifies the problems relating to the introduction of imprecise equality, as Observation 11 demonstrates.
This raises the question of whether the Non-diminishing Marginal Value condition should be abandoned, if the Imprecise Lexical View is otherwise defensible. I myself share Parfit's intuition that Non-diminishing Marginal Value is a very plausible requirement in population ethics. But given the theoretical benefits of giving it up, should the Imprecise Lexical View without it end up being a viable view, it might be worth revisiting.
Interestingly, Parfit does not make use of Non-diminishing Marginal Value in his arguments about imprecision. Thus, the following discussion of the consequences of introducing (radical) imprecise equality does not assume Non-diminishing Marginal Value. Consider the implications for the Imprecise Lexical View of Observation 9 for a sequence of lives of descending quality of life (dense or discrete) b, …, q. Observation 9 implies that there may be an interval, say from m to q, where m is lexically better than q, and a zone in between, all the members of which will be radically imprecisely equal to both m and q.
One problem is that within the zone, differences between options tend to get blurred. In comparison with populations based on m or q, for any sufficiently large population size based on lives in the open interval in between m and q, there is no difference in value in choosing one population over another. There is a dilemma visible here. If the distance between m and q is short, this would appear to result in an implausible discontinuity between them after all. If the distance between m and q is long, it appears implausible that it should make almost no difference which option in between we choose. However, compared with a reasonable balance between these two horns, no zone of imprecise equality or even no case of lexical betterness might be even more implausible.
The next thing to consider is discontinuity between close objects. At each border of the zone of radical imprecise equality, there will be an instance of radical imprecise equality between close objects. In the following I mean ‘close’ to cover both the case of discreteness (in which case the objects are neighbors) and the case of denseness (in which case ‘close’ means ‘arbitrarily close’). Is not radical imprecise equality between close objects still an implausible discontinuity?
I grant that lexical betterness between close objects is implausible and, consequently, that minimal exchangeability is more plausible.Footnote 45 But why is it that a very small loss of quality of life could not be outweighed by some much larger gain in quantity, that is, why is minimal lexical betterness plausible? What is special about the differences between close objects at the borders of the zone, where minimal lexical betterness is true, as compared with any other pair of close objects outside the zone where it apparently is not true?
To point at the mere possibility of imprecise equality does not provide a justification, since we need to know why it holds exactly here. However, in the context of the Continuum Argument, Parfit actually considers minimal lexical betterness between any neighboring lives plausible, as he conjoins minimal exchangeability with minimal lexical betterness for all neighboring pairs, at least if they are sufficiently close:Footnote 46
Compared with the existence of some number of people, it would not always be worse if instead there existed many more people who would have a slightly lower quality of life. But we should deny that such truths would be precise. We should then claim that no slight loss in quality of life would either be outweighed by, or outweigh, any such gain in quantity.
But why could these truths (about exchangeability) not be precise? Moreover, the problem with the proposal is that it has odd consequences. Suppose b is radically imprecisely equal to c, c is radically imprecisely equal to d, and so on further down to m, but that b is lexically better than d, and c is lexically better than e, and so on down to m (in case the sequence is dense, allow a difference between close objects and less close objects). Then it follows from Observation 9 that c is radically imprecisely equal to q, d is lexically better than q, e is radically imprecise equal to q, and so on, which appears very difficult to justify.
In my view, a better justification of the two necessary instances of minimal lexical betterness between close objects assumes that the overall case of lexical betterness (e.g., m is lexically better than q) has an independent, non-arbitrary justification. But then it is necessary to accept that thereby a border is established in the sequence, from where no loss of quality of life below the value of some number of ms can ever be outweighed, regardless of quantity; this border is most plausibly followed by a zone of radical imprecise equality, which necessarily involves another instance of minimal lexical betterness between close objects at the lower end.
Thus, I suggest that the most plausible version of the Imprecise Lexical View is one that establishes a lower bound m for the exchange of quality with quantity. Below m, there is a zone, such that m and all lives better than m are radically imprecisely equal to all members of the zone, but lexically better than the life at the lower end of the zone q, and all lives worse than q; and all members of the zone are radically imprecisely equal to q, and all lives worse than q. Call this version of the Imprecise Lexical View (without Non-diminishing Marginal Value) the Lower Bound for Exchangeability View.
The zone of radical imprecise equality will reduce the arbitrariness of picking out a sharp limit, but it seems unavoidable that the zone will have sharp borderlines, of which it may be difficult to justify the non-arbitrariness. In the last resort, it is a matter of judging which theory of population is the least implausible in comparison.
9. Violation of Arrhenius’ non-elitism condition?
There is reason to consider another question. If the Lower Bound for Exchangeability View is to have any future in population ethics, it must not only be able to avoid the Repugnant Conclusion but do so without implying other counterintuitive conclusions. In effect, the hardest challenge in this regard comes from the Sixth Impossibility Theorem.Footnote 47
It is well known that lexical views violate a condition of Inequality Aversion:Footnote 48 For any triplet of lives x, y, z of descending value, there is a positive integer n, such that for all numbers m > n, if we have nx, mz and (m+n)y, then (m+n)y is at least as good as nx◦mz. The increase from z to y and the obtained equality can compensate for the loss from x to y. But this is denied by the Lower Bound for Exchangeability View, if x is lexically better or at least minimally lexically better than y. I demonstrate in Appendix 3 how the Lower Bound for Exchangeability View violates the more sophisticated Condition β, which is implied by the Non-Elitism Condition of the Sixth Impossibility Theorem.Footnote 49
However, the Non-Elitism Condition,Footnote 50 even though it implies Condition β, cannot itself be rejected for the same reasons, according to Arrhenius.Footnote 51 It is not entirely obvious that the Lower Bound for Exchangeability View violates the Non-Elitism Condition, but for logical reasons, it must. I demonstrate how in Appendix 3.
Roughly, the Non-Elitism Condition claims that, in a discrete domain, a small decrease of quality of life in one life from a to b can always be compensated by a larger increase from z to b for a sufficiently high number of lives. This condition brings out a further dimension of discontinuity of lexical betterness. Assume Condition 1 and suppose that a is lexically better than its successor b. This sets an upper bound ma for any number of bs. But for (m-1)a, there is some number n such that nb is better than (m-1)a. If z is exchangeable with b, then by transitivity, there is a number n’ such that n’z is better than (m-1)a. In that case, the value decrease from ma to (m-1)a◦b is so large, because we get below the limit of ma, that it cannot be compensated. The fact that a could be radically imprecisely equal to b is not of any help here. See Appendix 3 for details.
Arrhenius' theorem presupposes a discrete (sub)set of P A . If there is an order dense subset of the set of lives P A, then we can also form a discrete fine-grained subset P A 1; and if some condition is intuitively plausible of an order dense subset of P A, it is also plausible of a discrete subset of P A1, Arrhenius says.Footnote 52 But as we saw with Premise (A) of the Continuum Argument, an argument often starts from what seem intuitively plausible conditions of a discrete subset of P A 1, which may not be true on an order dense subset of P A. The assumption that there exists a discrete domain, which fulfills certain conditions, is certainly not innocent if denseness is true.Footnote 53 The space does not allow an investigation of how Arrhenius’ theorem is dependent on the assumption of discreteness, but I believe there is room for some suspicion, not least when it comes to lexical views.
Setting this point aside and accepting discreteness, the question is to what extent the Lower Bound for Exchangeability View can be considered independently plausible. The violation of Condition β involves a disregard for the value of equality among (some of the) lexically dominated lives. This is an unavoidable feature of any Lexical View. Surely, there are strong intuitions about the value of equality. But there might also be strong intuitions about lexical betterness, particularly, I suggest, about a lower bound for exchangeability. The question is, then, which intuition is more credible. Given that the Lower Bound for Exchangeability View may represent a way to avoid the Repugnant Conclusion, the overall weighing may be in its favor.Footnote 54
Consider next the following claim:
Roughly, the intuition which the Non-Elitism Condition tries to capture is that there is at least some very small decrease in the welfare of one of the best off lives which can be compensated for by an increase in welfare for at least some number of worst off people.Footnote 55
The question is to what extent this intuition relies on an assumption that the comparison in all cases is independent of who else lives. Since the Lower Bound for Exchangeability View rejects Non-diminishing Marginal Value, it also rejects Weak Independence (Condition 7). But this raises the possibility that a comparison could be influenced by who else lives, which is at stake when the Lower Bound for Exchangeability View violates the Non-Elitism Condition. And again, the overall weighing could be in favor of the Lower Bound for Exchangeability View.
10. Summary of results
A formal framework has been developed to study the properties of lexical betterness, in general as well as regards Parfit's Imprecise Lexical View in particular. In contrast to comparable frameworks from the literature, (1) the present framework allows not only for a domain of a discrete finite sequence, but also a dense one; and (2) a condition of Non-diminishing Marginal Value is formulated without assuming numerical representation by a real-valued cardinal scale of value.
Under conditions that do not allow for imprecise equality (i.e. conditions of full comparability), a result which goes beyond existing literature is proved: If in some sequence of objects (dense or discrete) the first is lexically better than the last (and assuming some regularity), then either all members of the sequence (except the last) are lexically better than the last, or the first is lexically better than all its successors including the last. Moreover, it was proved that the condition of Non-diminishing Marginal Value implies that lexical betterness collapses into strict lexical betterness. In passing, it was observed (1) that the Continuum Argument only holds for a discrete sequence, but not for a dense one; and (2) that Non-diminishing Marginal Value excludes any case of lexical betterness, unless the Archimedean Property is rejected.
The analysis of conditions allowing for imprecise equality builds on Handfield and Rabinowicz's notion of radical imprecise equality, which is shown to be equivalent to the conjunction of the negation of lexical betterness and the negation of exchangeability, and on their result that for all objects x and y, either x is lexically better than y, or x is exchangeable with y, or x is radically imprecisely equal to y (i.e. x is both minimally lexically better than and minimally exchangeable with y).
Allowing for imprecise equality, the following general result is obtained: In some sequence of objects (dense or discrete) where the first is lexically better than the last, there may be a zone of radical imprecise equality before the last object or after the first, or possibly a zone occupying all of the space between the first and the last object, such that the object before the zone is lexically better than the last. This also confirms, and makes more precise, a point made by Handfield and Rabinowicz, namely that, in order to block the Continuum Argument by the help of imprecise equality between some pair(s) of neighboring objects, at least two cases are needed, namely one at either end of the zone.
Moreover, it is shown that Non-diminishing Marginal Value implies that minimal lexical betterness collapses into strict minimal lexical betterness.
Applied on Parfit's Imprecise Lexical View, it is argued that the most defensible version is what I call the Lower Bound for Exchangeability View, which still does involve some implausible discontinuities. Also, it violates Arrhenius’ Non-Elitism Condition. However, the logical space for alternatives in the area is quite limited, so in the end the Lower Bound for Exchangeability View might turn out to be the least implausible option.Footnote 56
Supplementary material
The supplementary material for this article can be found at https://doi.org/10.1017/S0953820819000517.
