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Lexical views without abrupt breaks

by Magnus Vinding. First published in 2020.

Summary

This essay aims to illustrate a formal result in value theory with a few concrete examples. These examples demonstrate some specific ways in which one can assign lexical disvalue to extreme suffering without being vulnerable to some of the objections commonly raised against lexical views.

Introduction

The perhaps most common objection against lexical views is some variation of the sequence argument.

One version of this argument runs roughly as follows: for any value entity claimed to be lexically worse than some given bad, we can construct a sequence of intermediate bads in which adjacent elements do not differ greatly in badness. Therefore, the argument goes, each bad in the sequence can plausibly be outweighed by a sufficiently large amount of the slightly milder bad. And so by continually exchanging greater bads for sufficiently large amounts of lesser bads, it seems that we for any given amount of the worst bad in our sequence can find a large amount of the mildest bad that is worse.

The formal result presented below shows that this is not a valid argument against lexical views in general. Specifically, the last step in the argument does not follow, even if we grant the preceding premise.

The formal result

Arrhenius and Rabinowicz (2015, p. 232) define and prove the following result in terms of weak and strong superiority, yet we shall here rephrase their result in terms of inferiority (the definitions and the proof of the following result are equivalent for superiority and inferiority).

Define ‘strong inferiority’ and ‘weak inferiority’ as follows:

Strong Inferiority: An object e is strongly inferior to an object e′ if and only if e is worse than any number of e′-objects.

Weak Inferiority: An object e is weakly inferior to an object e′ if and only if for some number m, m e-objects are worse than any number of e′-objects.

In other words, strong inferiority implies that any amount of A is worse than any amount of B, whereas weak inferiority implies that some amount of A is worse than any amount of B.

Suppose that we have a domain of objects ordered by the relation “is at least as bad as”, and assume that this relation is complete (i.e. all objects in the domain are ordered by this relation) and transitive (cf. Arrhenius & Rabinowicz, 2015, p. 232).

One can then prove the following observation (Arrhenius & Rabinowicz, 2015, p. 234):

Consider any two objects e and e′ such that e is worse than e′. If e is weakly inferior to e′, without being strongly inferior to it, then the domain must contain a finite decreasing sequence of objects in which the last element is strongly inferior to the first one, but no element is strongly inferior to its immediate successor.

A purported implication

It is often claimed that lexical views require us to accept some single step where strong inferiority kicks in. That is, somewhere in an ordered sequence of value entities — e1, e2, … , ek — there must exist a single step such that one instance of en+1 is worse than any amount of en.

The result presented above shows that this is not true. One can hold that the final object in a given sequence is strongly inferior to the first object without accepting strong inferiority between any adjacent elements in the sequence.

To be specific: say that 3 e-objects are worse than any amount of e′-objects, but a single e-object is not (i.e. we have weak inferiority that kicks in at m = 3 e-objects). We can now construct a sequence in which there is no strong inferiority kicking in at any single step:

1 e′-object, 1 e-object, 2 e-objects, 3 e-objects

We have strong inferiority between the first and the final element in the sequence, but not between adjacent elements. This sequence is also a counterexample to the sequence argument outlined in the introduction: each bad can be outweighed by a sufficient amount of the preceding bad, yet no amount of the first bad can outweigh the last bad in the sequence.

Concrete examples

It is not obvious how the formal result outlined above could be relevant in the real world, as it does not, in itself, point to any substantive views in value theory. For that, we need to replace the abstract symbols (“e′-objects” and “e-objects”) with some concrete, plausible value entities.

Example 1: m seconds of extreme suffering

Philosopher Jamie Mayerfeld defends a view according to which a certain intensity and duration of extreme suffering has lexical priority over any amount of mild suffering (Mayerfeld, 1999, pp. 179-180).

To make a direct translation from the formal result above, we may say that the e′-object is a single second of the experience of stubbing one’s toe hard, while the e-object is a single second of intense, torturous suffering. We thus get a view according to which a certain duration of torment is worse than any duration of toe stubbing, yet where there is no strong inferiority between these distinct value entities.

Note that one can defend different versions of this view. For example, one may hold that the weak inferiority only kicks in for a certain duration of torment experienced by the same person (and one may further maintain that this duration of torment must occur in a single streak), or one can hold that the weak inferiority obtains regardless of who and where the torment is experienced.

Example 2: m extremely bad lives

A similar view can be defended at the level of populations. For example, one may think that extremely bad lives are weakly, but not strongly inferior to lives that are, by comparison, just mildly bad. One way to formalize such a view may be to say that the disvalue of mildly bad lives levels off asymptotically, whereas the disvalue of extremely bad lives does not (Arrhenius & Rabinowicz, 2015, sec. 12.7). This would mean that for a certain number of extremely bad lives, no sum of mildly bad lives could be worse.

Such a view is more commonly defended in the context of miserable lives versus happy lives, and suffering versus happiness — i.e. the value of such purported goods levels off asymptotically, yet the disvalue of such bads does not diminish as their numbers increase (Hurka, 1983; Gloor, 2016; Vinding, 2020, sec. 6.2).

Example 3: m torture methods combined

Say we have distinct, but equivalently bad torture methods, T, while t is a toe stub. One may hold that a large enough amount of toe stubs can be worse than any given duration of torture with a single torture method, yet that the unbearable suffering resulting from the simultaneous use of, say, three different torture methods is worse than any amount of toe stubs.

Example 4: m experiential subcomponents of suffering

Say that p is mild pain while C is an experiential component of intense suffering — e.g. the experience of an intensely painful headache, or intense despair. One may hold that C is weakly, but not strongly superior to p in a way that is similar to example 3.

That is, one may hold that a sufficient duration of mild pain can be worse than any given duration of a single intense experiential component of suffering, while a certain number of such intense components experienced simultaneously is worse than any duration of mild pain, e.g. because such a combination of intensely bad feelings gives rise to unbearable suffering.

References

Arrhenius, G. & Rabinowicz, W. (2015). Value Superiority. In Hirose, I. & Olson, J. (eds.), The Oxford Handbook of Value Theory. Oxford University Press.

Hurka, T. (1983). Value and Population Size. Ethics, 93, pp. 496-507.

Gloor, L. (2016/2019). The Case for Suffering-Focused Ethics. Ungated

Mayerfeld, J. (1999). Suffering and Moral Responsibility. Oxford University Press.

Vinding, M. (2020). Suffering-Focused Ethics: Defense and Implications. Ratio Ethica. Ungated