Incomparable Values

Analysis, Axiomatics and Applications

John Nolt

• Publisher: Routledge
• Year: 2022
• ISBN:
  9781000515305
• Edition: 1
• Topic: Philosophie
• Subtopic: Ethik & Moralphilosophie

People tend to rank values of all kinds linearly from good to bad, but there is little reason to think that this is reasonable or correct. This book argues, to the contrary, that values are often partially ordered and hence frequently incomparable.

Proceeding logically from a small set of axioms, John Nolt examines the great variety of partially ordered value structures, exposing fallacies that arise from overlooking them. He reveals various ways in which incomparability is obscured: using linear indices to summarize partially ordered data, relying on an inadequately defined concept of parity, or conflating incomparability with vagueness. Incomparability can enrich and clarify a range of topics including the paradoxes of Derek Parfit, rational decision theory, and the infinite values of theology. Finally, Nolt shows how to generalize many of the concepts introduced earlier, explores the intricate depths of certain noteworthy partially ordered value structures, and argues for the finitude of value.

Incomparable Values will be of interest to scholars and advanced students working in ethics, value theory, rational decision theory, and logic.

Part I Preliminaries

1 Incomparability, Axiologies, Axiomatics, and Model Theory

DOI: 10.4324/9781003098324-2

1.1 Incomparability

Incomparability is value difference without inferiority, and thus without superiority. Two values are incomparable if and only if neither is greater than or less than or equal to the other.¹ Any quantities some of which may be greater or less than others—weights, for example, or temperatures, or even pure numbers—may be thought of as values. But the values discussed here are degrees of goodness or badness. For such values, “greater than” means “better than” and “less than” means “worse than.”

We are accustomed to thinking of values as linearly ordered. In a linear order, each value is comparable with—that is, better than, worse than, or equal to—every other. Given any two values x and y, there are only three possibilities: x is worse than y, x and y are equal, or x is better than y. When all pairs of values in a given set of values are so related, that set obeys the law of the three possibilities—the law of trichotomy. Its members can be lined up, each with all worse values below it and all better values above it.

Where members of a set of values are incomparable, that set is merely partially ordered. (Because, by an awkward but firmly established convention, linearly ordered sets count as partially ordered sets, the term “merely” is needed to distinguish partially ordered sets that are not linearly ordered.) Mere partial orders are “wilder” and more various than linear orders. Yet a partial order is not a lack of order.

Figure 1.1 depicts an example of each type. Representations such as this are known as Hasse diagrams (Trotter 1992, 5). Mathematicians use them to depict finite partial orderings of anything, but we’ll always interpret the dots as values. Where two dots are connected by a line or an unbroken descending series of lines, the higher of the two represents a better value. (Horizontal lines are not allowed.) Dots not connected by a line or descending series of lines represent incomparable values. The diagram at left in Figure 1.1 depicts a set of seven linearly ordered values. Each value is comparable with all the others. Thus, using the symbol “<” for “is less than” or “is worse than,” this diagram signifies that

$$g<f<e<d<c<b<a.$$

Figure 1.1 Comparison of a simple linear order with a simple merely partial order.

The diagram at right represents a partially ordered collection of seven values. Some of these are comparable. Thus:

$$\begin{array}{l}c<a,\\ g<f<d<a,\text{and}\\ d<b.\end{array}$$

But a and b are incomparable. Value c is incomparable with every value except a and itself (all values are, of course, equal to themselves). Value e is incomparable with every other value.

That two values are incomparable doesn’t mean merely that we don’t know which is better than or equal to the other. It means that in fact neither is better than or equal to the other. Such arrangements are, if unfamiliar, perfectly conceivable. There is no a priori reason why all values should be linearly ordered. Indeed, since linear orderings are special cases of partial orderings, they ought to be rare. It would be a kind of miracle if all the value structures that mattered to us were linear. Or, if that thought seems too Platonistic (metaphysically realistic), think of it this way: if we have somehow constructed values only in linear orders, still we can construct them otherwise.

Incomparable values are not incomparable in every sense of the term, but only in the specific sense that neither is better than nor equal to the other. Often they can be compared indirectly. They may, for example, be bounded above or below by other values, and these bounding values might provide information about their relationship. (In the diagram at right in Figure 1.1 for example, c and d are bounded above by a, and a and b are bounded below by d.) Some partially ordered values can be added to or subtracted from one another. Some can be multiplied by integers. Some have negatives. Some can be decomposed into subvalues. Such operations facilitate many kinds of indirect comparison.

1.2 Small Improvement Argument for Incomparability

Despite their relative unfamiliarity, there is nothing esoteric about incomparable values. We meet them in everyday life. Consider the different kinds of pleasure that various experiences—say, attending an outstanding music performance and enjoying an outstanding dinner—may produce. Call the pleasure of the music m and that of the dinner d. Now, suppose that m and d are good in such different ways that neither is better than the other. Must they, then, be equal? Evidently not, for imagine the pleasure m⁺ of the music if it had been just marginally better. Values m⁺ and d are still so different that m⁺ is not better d. But since m⁺ is better m, it follows that m and d are not equal. Therefore d is neither better than nor worse than nor equal to m.

This line of reasoning is known as the small improvement argument (Chang 1997, 23–6; 2002a, section 1). Once you grasp the pattern, it’s easy to spot in many contexts. If you find the example of the concert and dinner unconvincing (perhaps food gives you much more pleasure than music, or vice versa), consider other kinds of pleasure. If you find pleasure too subjective, then consider degrees of health and wealth. It is difficult to deny that some degrees of health are neither better nor worse than some degrees of wealth and that a small improvement in one of them doesn’t alter this. It follows that those degrees of health and wealth are not equal either. Hence they are incomparable.

That conclusion can be challenged. One can concede that none of the following claims is true yet maintain that all three are so vague that none of them is false either:

The pleasure of the dinner is greater than that of the concert.

The pleasure of the dinner is less than that of the concert.

The pleasure of the dinner is equal to that of the concert.

If so, then the statement that the pleasure of the dinner is incomparable with that of the concert is likewise neither true nor false—and hence not true. We may thus seek refuge from incomparability in vagueness.

Vagueness is indeed commonplace. It pervades natural language like a fog. The three claims just mentioned are all vague to some degree. Yet it is not obvious that they are so vague as to preclude the truth or falsity of sentences that contain them. On the contrary; the incomparability of the two pleasures is discernible through the fog. If you can’t discern it yet, bear with me. Eventually, in Section 4.5, incomparability will appear in clear air, without the slightest wisp of vagueness.

1.3 Incomparability from Multiple Scales

Incomparability often results from the simultaneous use of dispa...

Table of contents

• Cover
• Half Title
• Series Page
• Title Page
• Copyright Page
• Table of Contents
• Preface
• Part I Preliminaries
• Part II Basic Formal Axiology
• Part III Ethical Decision Theory
• Part IV Bounds
• Acknowledgments
• References
• Index