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Are there infinite welfare differences among living things?


Suppose, as biocentrists do, that even microorganisms have a good of their own--that is, some objective form of welfare. Still, human welfare is vastly greater and more valuable. If it were infinitely greater, individualistic biocentrism would be pointless. But consideration of the facts of evolutionary history and of the conceptual relations between infinity and incommensurability reveals that there are no infinite welfare differences among living things. It follows, in particular, that there is some very large number of bacteria whose aggregate overall welfare is not less than the welfare of a human being.


Biocentrism, incomparability, incommensurability, infinity, sentience, welfare

Organic life, we are told, has developed gradually from the protozoon
to the philosopher, and this development, we are assured, is
indubitably an advance. Unfortunately it is the philosopher, not the
protozoon, who gives us this assurance, and we can have no security
that the impartial outsider would agree with the philosopher's
self-complacent assumption.
                                            Bertrand Russell (1963: 81)

Russell poses a puzzle. On the one hand, it must be more than a hubristic delusion to think that evolution from the protozoan to the philosopher is an advance, not only in complexity, but in value. On the other, we who make that judgment are far from impartial. How can we grasp this development objectively?


The question matters to biocentrists, who hold that all living beings have objective, non-anthropogenic values that ought to figure into our moral decisions. Paul W. Taylor thought the answer lay in what he called the 'biocentric outlook'. One who achieves this outlook, he thought, thereby attains a realisation of 'species impartiality'--the notion that 'all living things, human and nonhuman alike, have the same inherent worth' (1986: 267). Taylor maintains, as Russell suggests, that humanity's apparent superiority in value is an anthropocentric illusion. Affirming, then, that humans are not inherently more valuable than other living things, and assuming also, as is uncontroversial, that they are not inherently less valuable, Taylor infers that all living things are of equal inherent worth (1981: 83).

But even if there is such a thing as inherent worth (1) and humans have neither more nor less of it than other living things, equality does not follow. For it is possible that the inherent worth of humans is neither greater than nor less than nor equal to, but rather incomparable with, that of other living things. Incomparability is difference without superiority--and hence also without inferiority. It has nothing to do with our inability to assess difference. Value x is incomparable to value y if and only if x is in fact neither greater than nor less than nor equal to y. (2) What we know (or think we know) is irrelevant. Incomparability exists when, as mathematicians say, values are merely partially, rather than linearly, ordered. The possibility of value incomparability invalidates Taylor's argument.

The notion of equality among living things has, moreover, not a whiff of empirical plausibility. Greater knowledge and objectivity lead not to a realisation of universal biocentric equality, but to appreciation of near-universal difference. A tarantula and a tamarisk tree are not in any interesting sense different-but-equal. They are just different--full stop. Assuming that each has some sort of objective, non-anthropogenic value, those values, too, may well be just different--each neither greater than, nor less than, nor equal to the other.

That some pairs of organisms have incomparable values does not, however, imply that all do. A partial order is not a lack of order. One can still maintain, with most contemporary biocentrists and animal ethicists, that there are gradations of non-anthropogenic value--that, for example, sentient beings generally have more of it than non-sentient beings and that some sentient beings have more of it than others (Nolt, 2013, 2015: 178-80).


But in what does this allegedly objective non-anthropogenic value consist? The central assumption of all biocentrists is that each living being--be it bacterium, betony or bison--can fare well or poorly. Any organism may, for example, be whole or injured, well or poorly nourished, healthy or diseased. How well or ill it fares is clearly a matter of degree.

That some organisms fare better than others is, moreover, an objective fact. John O'Neill (1992) defines two senses in which an object may have a property objectively: a weak sense in which it can have that property in the absence of evaluating agents (typically humans) and a strong sense in which its having that property can be characterised without reference to evaluating agents. Nonhuman organisms have welfare properties objectively in both senses. They fared well or ill before humans evolved and would have done so even if we had never evolved (see Nolt, 2004; 2009). Organisms also have value in less objective senses. But that need not concern us here.

Given the diversity of living things and of their aims, it is unreasonable to expect that the objective welfare of each is comparable with the objective welfare of all others. It is therefore likely that their welfare values are only partially ordered.

From the mere fact that living things have objective welfare, nothing follows about their moral significance (Nolt, 2006; O'Neill, 1992). Anthropocentrists regard welfare, however conceived, as morally significant only in humans. Sentientists deem it morally significant only in sentient animals. Biocentric individualists regard it as morally significant even in beings as humble as bacteria. Among biocentric individualists, some (who typically are consequentialists) hold that its moral significance is proportional to its degree: the greater the welfare (either positive or negative) the more significant.

Yet discussion of objective welfare in environmental ethics has focused almost exclusively on its moral significance, or lack thereof. This paper, however, concerns objective welfare itself. The title question is axiological, not ethical: 'Are there infinite welfare differences among living things?'--not 'Are there infinite differences in moral significance among living things?' (3) Yet, as was just noted, for some biocentrists, these questions are essentially equivalent.

The objective welfare of living things varies, of course, over their lifetimes. A life may sometimes go well, sometimes poorly; but since I am painting here with a broad brush, I will ignore variations within the lifetime of an organism and consider only the lifetime total. So henceforth, when I speak of the welfare of an organism, I mean its lifetime welfare. This is its average momentary welfare over its lifetime, multiplied by the length of its life. (4) Lifetime welfare thus varies with lifespan. Given two organisms with the same constant positive level of welfare, for example, if one lives longer than the other, then it has the greater lifetime welfare. Conversely given two organisms with equal lifetimes, if one has greater average welfare, it has the higher lifetime welfare.


Robin Attfield poses an important problem for biocentrists regarding the comparative welfare of sentient and non-sentient organisms. He frames the problem in terms of moral significance; but since, like many biocentric consequentialists, he regards moral significance as proportional to welfare, we may justly interpret his remarks as applying to welfare as well:

If plants (or bacteria) have any more-than-negligible moral
significance, then in their millions their interests must sometimes
outweigh those of individual humans or other sentient beings...
(Attfield, 1983: 154).

Attfield finds this consequence objectionable and seeks to avoid it by hypothesising that plants or bacteria may

... have moral standing and yet have an almost infinitesimal moral
significance, so that even large aggregations of them did not outweigh
the significance of sentient beings in cases of conflict. It could be
that their moral significance only makes a difference when all other
claims and considerations are equal (or non-existent) (Attfield, 1983:
154, italics added).

The implications of Attfield's response depend crucially, as we shall see, on the meaning of the phrase 'almost infinitesimal'.

A positive value x is infinitesimal relative to a positive value y if and only if x is so much smaller that any finite number of x-units is always less than y --that is, if only if y is infinitely greater than x. Correlatively, positive value y is infinite relative to positive value x if and only if x is infinitesimal relative to y. (5) Now either there is some finite number of bacteria, say--perhaps a number greater than the number of actual bacteria on Earth--whose aggregate welfare is not less than that of a human, or there isn't. If there is such a number, then the welfare of a bacterium, however tiny, is not infinitesimal relative to the welfare of a human. (6) If there is no such number--that is, if the aggregate welfare of any finite number of bacteria is less than that of the human--then the welfare of one single bacterium is infinitesimal relative to that of the human. In that case, moreover--assuming, as Attfield (1983) implicitly does, that degrees of moral significance, and hence of welfare, are linearly ordered--this bacterium's welfare is also infinitesimal relative to the miniscule but non-infinitesimal welfare of the bacterium of the previous case. It follows that either a bacterium's welfare is infinitesimal relative to that of a human or it is infinitely greater than any value that is infinitesimal relative to that of the human. There can be no 'almost' about it.

Some thinkers seem to have countenanced genuinely infinite differences. Better Socrates dissatisfied than a pig satisfied, said John Stuart Mill--and some interpreters think that he meant infinitely better. (7) No number of piggly lives, he suggested, could contain as much happiness as the life of a single Socrates. Mill felt confident in his answer because he held that wise men (among whom he counted himself) have experienced pleasures and pains both like those of the pig and like those of Socrates. But who, even among the wise, has experienced piggly pleasures as pig? More to the point, who, even among the wise, could experience a literally infinite difference in pleasure?

Perhaps, then, Mill was wrong about philosophers and pigs. But surely, we may think, the lifetime welfare of a Socrates with its rich fund of intellectual experiences is infinitely greater than the lifetime welfare any number of bacteria, which presumably experience nothing at all. If so, then the lifetime welfare of a single bacterium is genuinely infinitesimal relative to the lifetime welfare of Socrates.

Attfield's worry then takes hold: the welfare of non-sentient beings could in itself have no practical importance if it were infinitesimal relative to ours; for in that case any finite amount of welfare for even a single human would outweigh the welfare not only of any non-sentient being, but also of any aggregate of them. This is a direct consequence of two indisputable facts: (1) there have only been finitely many living things in Earth's evolutionary history and (2) a finite sum of values infinitesimal relative to a given value x must itself be infinitesimal relative to x. Hence if the welfare of non-sentient entities were infinitesimal relative to ours, their welfare could affect moral decisions only if those decisions made no more than an infinitesimal difference to sentient animals--a difference so small as to be wholly indiscernible. In practice, therefore, biocentric individualism would be indistinguishable from sentientism.


But caution is needed here. As Attfield's use of the phrase 'almost infinitesimal' suggests, it is easy to underestimate the chasm between the genuinely infinitesimal and the miniscule-but-finite. A value that is infinitesimal relative to familiar values, though logically conceivable, would be infinitely smaller than any value we could ever perceive or measure. Though mathematically distinct from zero, it would be so small that it could never be distinguished from zero in any practical way. Yet the objective welfare even of an individual bacterium is detectable, and its various parameters are measurable, if only with technological aid.

There is, moreover, an independent reason to reject infinite welfare differences: the evolutionary tree is finite. In particular, there is only a finite (though very large) number of reproductive steps from the earliest living organisms to human beings. Hence, since a finite series of finite differences cannot add up to an infinite difference, there can be no infinite welfare differences between humans and any of their ancestors--unless an infinite leap in welfare occurred at one or more of the reproductive steps.

Could there have been such an infinite leap? It is conceivable. There is no special reason, of course, to suppose that it occurred with the birth of the first member of the species Homo sapiens. But it might have happened later, with the cultural development of human autonomy and self-directed lives--or earlier, with the emergence of sentience. Maybe there were several such leaps. Maybe, due to some sort of vagueness, one or more of these leaps was 'smeared out' over many generations. In any case, if there is an infinite welfare gap between humans and the earliest forms of life, then such a leap must have occurred somewhere.

For purposes of exposition, it will be easiest if we focus on a single hypothetical instance. What we learn from that instance can then be generalized to others. Attfield is not alone in suggesting that the appearance of sentience was peculiarly significant. (8) Suppose, then, for the sake of argument, that an infinite leap in welfare occurred with the emergence of the first sentient organism in a particular lineage. How could that have happened?


Sentience, the ability to enjoy or suffer, is among the most fundamental forms of phenomenal consciousness. To sentient beings like us its fluctuations are uniquely salient. It is therefore plausible that there are two kinds of welfare--one that non-sentient beings possess and one that they lack.

Let's call the first kind of welfare biotic. Biotic welfare is, roughly, physical health and its constituents--adequate nourishment, efficient respiration, bodily integrity, and the like. Biocentrists describe the biotic welfare of an organism in various ways: achievement of the normative goals encoded in its genetic set (Rolston, 1988: 98-101), its autopoietic functioning (Nolt, 2009; 2015, sec. 6.4.2), the development or fulfilment of the full range of its capacities (Attfield, 1999: 39; 2014a: 44), satisfaction of its biopreferences (biofunctional goals) (Agar, 2001: 94), etc. For the purposes of this paper, we need not choose among these closely related characterisations. One thing on which all agree is that capacity for biotic welfare increases at least roughly with the complexity of the organism. I assume that as well.

Sentient animals have some degree of biotic welfare just as non-sentient beings do. But they also have a second kind of welfare, which we may call experiential. In its positive forms, experiential welfare is usually understood as happiness, pleasure, satisfaction of desires, or the like. But it has negative forms as well: misery, pain, frustration of desires, or the like. If the positive and negative forms are in balance or absent, experiential welfare may be neutral or zero.

Like biotic welfare, experiential welfare is both objective and non-anthropogenic. Though we cannot always tell what a sentient animal is experiencing, that it has positive or negative experiences is a fact wholly independent of us. Sentient animals enjoyed the warmth of sunshine and suffered the misery of illness, for example, long before Homo sapiens appeared on Earth.

Hedonists and preference axiologists, including many traditional neoclassical economists, (9) maintain that experiential welfare is the only ultimate or intrinsic value, though they admit that biotic welfare may be instrumental to its production. But biocentrists find non-anthropocentric moral significance in both.

Since sentient animals are capable of both kinds of welfare, they also have some configuration of overall welfare. Their overall welfare is high to the extent that their biotic and experiential welfare are both high--roughly, that is, to the extent that they are physically fit and feeling fine. It is lower to the extent that either is less. The overall welfare of non-sentient beings lacks the experiential component and hence is just their biotic welfare.


With the advent of sentience, the experiential component of overall welfare rises (let's assume) from zero to some positive value. Isn't that an infinite increase?

Almost certainly not. Consider an analogy. Suppose that overall wealth consists of both money and property and that I have a car but no money. Now you give me a penny. It would be pointless and misleading to say that relative to what it was (namely, zero) my monetary wealth has increased infinitely; for, relative to zero, all positive quantities, even those that are infinitesimal relative to familiar quantities, are likewise 'infinite'. Relative to any discriminate standard--euros or dollars, for example--the increase in my monetary wealth is finite and slight, as is the increase in my overall wealth. Likewise, the inference that with the evolution of sentience experiential welfare must have increased infinitely merely because it rose from zero to some positive value is invalid; for, relative to any discriminate standard of experiential welfare, that increase may well have been (and probably was) finite and slight. The point is that a value can properly be considered finite or infinite only relative to some positive, though typically conventional, standard--not relative to zero.

Our aim, moreover, is to compare the overall welfare of sentient and non-sentient living things. This is composed of both biotic and experiential welfare. Our question is whether the overall welfare of early sentient animals might somehow have been infinitely greater than the overall welfare (which is just the biotic welfare) of their close non-sentient ancestors. And we have seen that to answer that question, we need some positive standard of overall welfare. But the logic of any such standard depends upon whether biotic and experiential welfare are commensurable.


Two positive kinds of value are commensurable in the sense relevant here if and only if some positive amount of one exceeds some positive amount of the other. Thus, for example, if biotic welfare were physical health and positive experiential welfare were pleasure, biotic and experiential welfare would be commensurable if some number of units of pleasure were objectively greater than some number of units of physical health. It is not obvious that this even makes sense, and I am not claiming that it does. But perhaps we can imagine advances in neuroscience that enable us to reduce degrees of both health and pleasure to common physical parameters in terms of which we could compare them.

If the two forms of welfare are not commensurable, then they are incommensurable. Incommensurability is incomparability with a vengeance. Two positive values are incommensurable if and only if no amount of either exceeds any amount of the other--in other words, if any amount of one is incomparable with (neither greater than nor less than nor equal to) any amount of the other. Assuming again, for the sake of illustration, that biotic welfare is physical health and positive experiential welfare is pleasure, then biotic and experiential welfare would be incommensurable if no amount of pleasure were either greater than or equal to or less than any amount of health.

Incommensurability is no stranger to discussions of objective welfare. The widely cited Human Development Index (HDI), for example, is a measure of overall human welfare constructed from three components that are, at least arguably, incommensurable: longevity, educational attainment and wealth. It is reasonable to suppose that no amount of wealth in dollars, for example, is objectively either greater than or equal to or less than any number of years of longevity--even though people may sometimes be willing to trade one for the other. Nevertheless, the index combines scores on each of the three parameters into a single crude numerical measure of something like human welfare. Likewise, even if biotic and experiential welfare are incommensurable, we could still combine them into a single index of overall welfare. There is no non-arbitrary way of doing this. Every method involves conventional stipulations. Some induce spurious comparability. But it can be done.

On the face of it, biotic and experiential welfare are incommensurable--and this indeed is my view. But for completeness' sake, we should consider what would follow if they were commensurable.

Assume, then, that biotic and experiential welfare are commensurable, and imagine a pair of wholly non-sentient parents whose offspring is to some degree sentient. The parents have certain levels of biotic welfare. Their offspring, which inherits its genes from them, must be similar to them physiologically, even allowing for mutations. Hence its level of biotic welfare can hardly be infinitely greater than theirs. Any infinite overall welfare increase, then, must be due to the additional experiential welfare.

One consideration speaks immediately against this: experiential welfare may be negative. If pain, for example, were the dominant early form of sentience, then the advent of sentience would have produced, not a gain (much less an infinite gain), but a decline in overall welfare. There have even been philosophers--among them some Buddhists, Arthur Schopenhauer and David Benatar (2006)--who have suggested that sentient life is so fraught with suffering that lifetime experiential welfare is never positive. Whatever one thinks of such views, the possibility of negative experiential welfare weighs against the idea that there was an infinite overall welfare increase in the initial step from a non-sentient organism to a sentient one.

Setting that worry aside, however, let's assume for the sake of argument that there was an infinite overall welfare increase and that this was due to the additional experiential welfare. The experiential welfare of the offspring must therefore be infinitely greater than the biotic welfare of the parent(s). We have granted that this is conceptually possible, since we are assuming for the sake of argument that biotic and experiential welfare are commensurable in the sense defined above. Surely, however, the biotic welfare of the parents was finitely assessable by some set of physical parameters. Hence whatever parameters define the experiential welfare of the offspring must be comparable with those physical parameters but infinitely greater. Given the genetic and physical similarity of parent and offspring, that is not empirically credible.

But perhaps sentience took not just one reproductive step but many generations to emerge. Still, the number of generations from any organism to any of its descendants is finite. Emergence of an infinite gap in any finite number of generations is no more credible than its emergence in one. Moreover, once experiential welfare appears--to however slight a degree--any variation in it from generation to generation would almost certainly be finite. Given that biotic and experiential welfare are commensurable, there is no room in the entire evolutionary sequence for an infinite gap.


Consider, then, the more plausible of the two alternatives: that experiential and biotic welfare are incommensurable--that is, no amount of one is comparable with any amount of the other. Trivially, then, experiential welfare cannot exceed biotic welfare. But could the overall welfare of the sentient offspring, which is a combination of both biotic and experiential welfare, somehow be infinitely greater than the overall welfare of the non-sentient parents, which is purely biotic?

We are now entering conceptual territory wherein the notions of incommensurability and infinity commingle. Though generally unfamiliar, the terrain is not unintelligible. The remainder of this section presents a brief orientation in the form of visual aid and crucial definitions.

We may depict the incommensurability of biotic and experiential welfare by drawing them as two orthogonal scales, like the axes of two-dimensional Cartesian coordinates (just so, in three dimensions we might depict the three parameters of the HDI; more generally, with n incommensurable values we need an n-dimensional space.) The orthogonality of the scales represents the fact that no amount of value on either of them registers at all on the other. We position the coordinates so that their zero points coincide and stipulate that the horizontal axis represents biotic welfare and the vertical axis experiential welfare, as in Fig. 1.

In a more adequate representation, each dimension of welfare might be decomposable into further incommensurable parameters, so that there are more dimensions to worry about, but let's ignore that complication. There may also be other kinds of objective welfare associated, for example, with the autonomy of persons, or person-like animals, so that still more dimensions may be needed for a more complete representation of overall welfare. Again, to keep things simple, we ignore these complications, none of which would materially affect the argument. (10)

I hold that there are no negative biotic values, and therefore no points to the left of 0 on the horizontal axis, in which case zero (possibly representing death) is as low as it is possible to go on the biotic welfare axis (Nolt, 2015: 176). But we can leave that question undecided here. There clearly are, however, as was noted above, negative experiential values. These are represented by points below 0 on the vertical axis.

The overall welfare of any non-sentient being is represented by some point on the horizontal axis. There is no vertical component, since that being has no experiential welfare. The overall welfare of a sentient animal, by contrast, has both biotic and experiential components; it is represented by a point that is not on either axis, such as point x. Unlike points representing biotic welfare alone (horizontal axis) or experiential welfare alone (vertical axis), those representing overall welfare are not linearly ordered.

Each point on the diagram represents an objective welfare value, understood as an ordered pair consisting of a biotic value and an experiential value. The diagram presupposes the plausible, though substantive, assumption that welfare values can be added and therefore can be multiplied by positive integers. Thus 3u, for example, is u+u+u, which is three times greater than u. We may think of 3u as a value equal to the sum of the welfare of three individuals, each of which has welfare u. Of course u itself is an ordered pair <[u.sub.1],[u.sub.2]>, where u1 is a biotic value and [u.sub.2] is an experiential value. So 3u = <[u.sub.1],[u.sub.2]> +<[u.sub.1],[u.sub.2]> + <[u.sub.1],[u.sub.2]> = <[u.sub.1]+[u.sub.1]+[u.sub.1],[u.sub.2]+[u.sub.2]+[u.sub.2]> = <3[u.sub.1],3[u.sub.2]>.

In general, objective welfare values that are analysed into two or more components would be represented as n-tuples in an n-dimensional space. If values y and z each have n components, so that y = <[y.sub.1],....[y.sub.n]> and z = <[z.sub.1],....[z.sub.n]>, then z is greater than y if and only if for each integer i such that 1[less than or equal to]i[less than or equal to]n, [z.sub.i] is greater than or equal to yi and for at least one such i, [z.sub.i] is strictly greater than [y.sub.i].

In Figure 1, for example, values greater than x are the ones greater in both dimensions or greater in one and equal in the other; they are represented by points above and/or to the right of the given value. Values less than a given value are those less in both dimensions or less in one and equal in the other; they are represented by points below and/or to the left of the given value. So, in Figure 1, 0, u, v, 2v and 3v are less than x. A point coinciding with x represents overall welfare equal to x. Other points (those to the left and above or to the right and below the given value) represent amounts of overall welfare that are incomparable with it. Thus, among the points marked in Figure 1, 2u and 4v are incomparable with x.

It is easy to see that no positive value on either axis is greater, less than, or equal to any positive value on the other--in other words, that any two positive values on different axes are incomparable. Formally, values y = <[y.sub.1],....[y.sub.n]> and z = <[z.sub.i],....[z.sub.n]> are incomparable if and only if for some integers i and j such that 1[less than or equal to]i[less than or equal to]n and 1[less than or equal to]j[less than or equal to]n, [y.sub.i] is less than [z.sub.i] and [z.sub.j] is less than [y.sub.j].

One can also see in the case of value v, for example, that multiplying any positive value on the horizontal axis by any value produces a value that is incomparable with every positive value on the vertical axis and hence incommensurable with any value on the vertical axis.

As was noted above, value z is infinite relative to value y if and only if for any integer n, ny is less than z. Values infinite relative to u or x are so large that their representative points lie at an infinite distance from 0 in the upper right quadrant and so cannot be depicted in our diagram, no matter how much we expand it.

Value z is infinitesimal relative to value y if and only if for any integer n, nz is less than y. Positive values that are infinitesimal with respect to u or x are so small that they lie infinitely close to zero in the upper right quadrant or on the axes that border it. Both u and x are greater than any positive infinitesimal, but v is greater than a positive infinitesimal only if that infinitesimal lies on the horizontal axis.


Now our question, to return to it, is: Assuming that biotic and experiential welfare are incommensurable, could a sentient being's overall welfare be infinitely greater than the merely biotic welfare of its parent(s)?

To assess overall welfare, we need, as was noted above, some standard of overall welfare that, like the HDI, is serviceable despite the incommensurability of its components, which, in this case, are biotic and experiential welfare. There are many ways of defining such a standard. All are conventional and to some degree arbitrary, but to our question all yield the same answer.

For the sake of illustration, suppose we adopt as our standard the overall lifetime welfare of the average bullfrog. (I assume that bullfrogs have some degree of both biotic and experiential welfare.) The example is fanciful, of course. We don't know enough about animal experience to make any but the crudest comparisons with such a standard. But suppose that we could, perhaps via advances in neuroscience, do much better. We might determine, for example, that the overall lifetime welfare of a certain pig was greater than a thousand bullfrog units. Given such a standard, we could use it to make meaningful comparisons of the overall welfares of living things, sentient or not.

Such a standard would be represented in Figure 1 by a value like u. So imagine that u is one bullfrog unit, 2u is two bullfrog units, and so on. For any values y and z, let's say that y delimits z if there is some integer n such that ny is greater than z. Then in Figure 1, u delimits x, since 3u is greater than x. Similarly, x delimits u, since 1x (that is, x itself) is greater than u. It is, in fact, obvious that u delimits any value depicted in the diagram and almost as obvious that u does not delimit any value whose biotic or welfare component is infinite. Because welfare values are not linearly ordered, they do not in general have an exact bullfrog value. But because u delimits all the finite ones, for any finite welfare value w there is some integer n such that w is less than nu.

Now our question, once again, is whether a sentient being's overall welfare could be infinitely greater than the merely biotic welfare of its parent(s). In terms of the diagram, the non-sentient parents' overall welfare is represented by a point that, like v, picks out some positive amount of purely biotic value. The sentient offspring's overall welfare, if positive, is represented by a point like x that picks out some positive degree of welfare with both biotic and experiential components. Thus our question becomes: could such a point (call it s, for 'sentient') be infinite relative to a point like v?

No--and the reason is interesting. Even if, implausibly, we represent s's experiential component as positive and infinite--for example, we place s at an infinite distance directly above x--still s would not be infinite relative event to the purely biotic welfare of a value like v, for, as Figure 1 indicates, 4v would not be less than s, but rather incomparable with it. Likewise, s would not be infinite in bullfrog units; for, as the diagram reveals, taking u as one bullfrog unit, s would not be greater than three bullfrog units. The infinity of s's experiential component thus would avail nothing, for we are reasoning under the assumption that biotic welfare and experiential welfare are incommensurable, so that no amount of experiential welfare (not even an infinite amount) can be greater than any positive amount of biotic welfare.

This, of course, is just an example, but it illustrates a general principle: assuming that biotic and experiential welfare are incommensurable, the overall welfare s of a sentient animal is infinite relative to the purely biotic welfare v of a non-sentient being only if s is infinite relative to v in biotic welfare. (11) But, as was noted above, surely there is no infinite increase in biotic welfare with the advent of sentience. Therefore, relative to any overall welfare standard that, like u, delimits and is not infinite relative to the biotic welfare of the parent(s), the overall welfare of the offspring cannot be infinite.

Earlier we derived the same conclusion from the assumption that biotic and experiential welfare are commensurable. Hence in no case could overall welfare have increased infinitely relative to such a standard.

The reproductive step to sentience, moreover, was just an example. We might suppose that an infinite leap in overall welfare took place somewhere else in the evolutionary sequence. But similar reasoning would apply in any such case. Hence, generalising, we may conclude that at no reproductive step (or, as was noted earlier, finite series of reproductive steps) in the entire evolutionary tree is there an infinite increase in overall welfare.

Though we have been considering an infinite welfare increase from parents to offspring (primarily because we began with the human lineage, in which welfare presumably increased over evolutionary time), for the essentially same reasons there can be no infinite welfare decrease from parents to offspring. Hence, as a final generalisation, we may infer that:

For any overall welfare standard u and any parent/offspring pair, the
offspring has overall welfare infinite relative to u if and only if the
parents do; and, similarly, the offspring has overall welfare
infinitesimal relative to u if and only if the parents do.

From this principle, a brisk and direct argument leads to the conclusion that there are no infinite overall welfare differences among living things.


To prove that conclusion, we must attend to the structure of the evolutionary tree. Two assumptions are crucial:

1. The number of reproductive steps between an organism and any of its ancestors is finite, and

2. Any two living organisms have a common ancestor.

The first is undeniable. The second has been doubted, but the doubts are speculative. In any case, it is certainly true of all the organisms with which we are familiar.

The argument, then, is as follows. Choose any standard u of overall welfare that is reasonable in the sense that it meets these two conditions:

A. u delimits the overall welfare of every living thing. (Certainly we can construct such a u, though if there are infinite welfare differences, then u must be infinite relative to the overall welfares of some organisms.)

B. There is a least one living thing a relative to whose overall welfare u is not infinite. (This requirement is merely a matter of proper scaling. Any standard that was infinite relative to the overall welfare of every living thing would be uselessly indiscriminate.)

Now pick any living thing b. By assumption 2, a and b have a common ancestor g. By assumption 1, there are finitely many reproductive steps from g to a and from g to b.

Since we have made u large enough to delimit the overall welfare of each living thing (condition A), no living thing has overall welfare that is infinite relative to u. If there are infinite welfare differences among living things, then, the welfares of some living things must be infinitesimal relative to u. It was, however, shown above that for any parent/offspring pair, the offspring has overall welfare infinitesimal relative to u if and only if the parent does. By the transitivity of the biconditional, then, both a and b have overall welfare infinitesimal relative to u if and only if g does. And again by the transitivity of the biconditional, a has overall welfare infinitesimal relative to u if and only if b does. Hence since (by condition B) a's overall welfare is not infinitesimal relative to u, it follows that b's overall welfare is not infinitesimal relative to u. But b was an arbitrarily selected living thing. Therefore, no living thing has overall welfare infinitesimal relative to u. Hence u delimits but is not infinite relative to the welfare every living thing. Thus relative to u--that is, relative to any standard of overall welfare that meets the reasonable minimal conditions A and B, there are no infinite welfare differences among living things.


It follows--however discomforting--that the overall welfare of a bacterium is not infinitesimal relative to the overall welfare of a human being. It is indeed miniscule; and if (as is most plausible) biotic and experiential welfare are incommensurable, then there is no number of bacteria whose aggregate welfare exceeds the overall welfare of a healthy human. But there is some very large number of bacteria whose aggregate welfare is not less than--though it may be incomparable with--your overall welfare and mine.

How does the incomparability or even incommensurability of some pairs of overall welfare values affect the possibility of consequentialist decision-making? Not severely, according to Hsieh (2007) and Attfield (2014b: 49-52), and I have offered additional reasons for optimism (Nolt, 2015, secs. 6.4.3 and 7.3.2). But vindication of that optimism is a task for another context.


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Department of Philosophy University of Tennessee, Knoxville 818 McClung Tower Knoxville, TN 37996-0480, USA


(1.) In (Nolt, 2009: 142-147) I argue that there is not--assuming that it is supposed to be something more than just the welfare of (or goodness for) those organisms.

(2.) The term is used variously by philosophers, but this is a standard mathematical definition. See, for example Trotter, 1992: 3.

(3.) I have a broader biocentric ethic in mind, but there is no room to explicate it here. For a sketch, see sections 6.4-6.9 of (Nolt, 2015).

(4.) The mathematically inclined may prefer to think of it as the definite integral over time, from the beginning to the end of an organism's life, of its momentary welfare.

(5.) Some readers may balk at these definitions of infinitesimal and infinite quantities--and for good historical reason. They are essentially the definitions used by Newton and Leibniz in their development of the calculus. Berkeley, however, subsequently showed that their conceptions were riddled with paradox, thus generating a widespread conviction that such infinite and infinitesimal quantities are unintelligible. Much later, Abraham Robinson (1966) showed how to make sense of these classical definitions of infinite and infinitesimal quantities in a rigorous, non-paradoxical formal context. I am implicitly relying here on Robinson's analysis. For a brief and accessible treatment of it, see Henle and Kleinberg (2003).

(6.) I make here the natural assumption that the total objective welfare of a multiplicity of organisms is just the sum of their individual welfares.

(7.) According to Elinor Mason (2011), for example, 'Mill argues that there are higher and lower pleasures, and that the higher pleasures (pleasures of the intellect as opposed to the body) are superior, in that higher pleasures can outweigh lower pleasures regardless of the quantity of the latter'. Thus, given a lower pleasure of value x and a higher pleasure of value y, any finite sum of x's must be less than y--which is to say that higher pleasures are infinite relative to lower pleasures.

(8.) It is for this reason that many animal ethicists, most notably Peter Singer (1990: 6-8), restrict moral consideration to sentient animals. More directly relevant is Trevor Hedberg's comment in a forthcoming review of Nolt, 2015 for Ethics, Policy and Environment that: There is at least one candidate for a step at which an infinite difference in value could arise: when a sentient offspring was produced by non-sentient parents. While the first sentient organisms probably did not have much sentience, they still possessed a morally relevant capacity that their predecessors lacked entirely. Might this step in the reproductive chain ground an infinite difference in value? I am not sure, but it is a reasonable thought

(9.) The latter hold that value is determined by preference, which is supposed to be experiential, though they typically assess it by behavioural indicators of willingness to pay.

(10.) It is also possible that there is no analysis of objective welfare into a finite number of linear scales. A generalised version of argument developed below works in that case as well, even if objective welfare values are only partially ordered. But to show that requires a more abstract approach, which is beyond the scope of this paper.

(11.) It might seem that s should be infinite relative to v whenever s = <[s.sub.1],[s.sub.2]>, v = <[v.sub.1],0>, and [s.sub.1], [s.sub.2] and [v.sub.1] are positive. But unless the biotic welfare [s.sub.1] of s is infinite relative to the biotic welfare [v.sub.1] of v, there is some integer n such that [s.sub.1] is less than [nv.sub.1]. Hence nv is not less than s, so that s is not infinite relative to v.
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Author:Nolt, John
Publication:Environmental Values
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Date:Feb 1, 2017
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