Incomparable Values
Analysis, Axiomatics, and Applications
John Nolt
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PART I
Preliminaries 1
1 I ncomparability, Axiologies, Axiomatics, and Model Theory 3
1.1 Incomparability 3
1.2 S mall Improvement Argument for Incomparability 4
1.3 Incomparability from Multiple Scales 5
1.4 T he Compulsion to Linearize 8
1.5 Axiologies 10
1.6 Axiomatic Method 15
1.7 Model Theory 17
1.8 T he Eclipse of Axiomatics 20
1.9 P rocess and Pitfalls 24
2 L ogic, Comparison, Hasse Diagrams, and Set Theory 27
2 .1 L ogical Conventions, Terminology, and Notation 27
2.2 Axioms Regarding “Is Worse Than” (A1–A2) 29
2.3 C omparability and Incomparability 32
2.4 C ompression Convention 35
2.5 Hasse Diagrams 36
2.6 S et theory 39
2.7 S ets and Bounds 42
PART II Basic Formal Axiology 49
3 A rithmetical Value Structures 51
3 .1 Arithmetical Axioms (A3–A7) 51
3.2 Theorems Concerning Addition 54
3.3 Theorems Concerning Zero 56
3.4 Theorems Concerning Negatives 57
3.5 Hasse Diagrams for Addition and Negatives 59
3.6 Hasse Diagrams as Models of the Axioms 62
3.7 Variation in Hasse Diagrams 64
3.8 Some Fallacies 65
3.9 Subtraction 67
3.10 Addition and Subtraction Tables 70
4 Cartesian Models 74
4 .1 C artesian Coordinate Systems and Cartesian Models 74
4.2 C artesian Models: The Geometric Interpretation 75
4.3 C artesian Models: The Algebraic Interpretation 78
4.4 S ubtraction, Minima and Maxima 83
4.5 C artesian Models vs. Hasse Diagrams 85
4.6 E mergence and Incomparability with Zero 86
4.7 S ome Model-Theoretic Concepts 88
5 I nteger Multiplication, Delimitation, and Commensurability 93
5 .1 Multiplication by Non-negative Integers 93
5.2 T he Tubular Model 99
5.3 Non-tubularity Axiom (A8) 103
5.4 Q uantification over Integers 108
5.5 Delimitation 109
5.6 C ommensurability and Incommensurability 110
5.7 Applying Cartesian Models 113
6 I ncomparability Obscured 117
6.1 Value Indices 117
6.2 Unit Scales 119
6.3 C hang’s Concept of Parity 123
6.4 Is Incomparability Vagueness? 129
6.5 Values, Value-Bearers, and Abstraction 134
6.6 Acknowledging Incomparability 138
7 Aggregating Welfare over Populations 140
7.1 S ituations and Populations 140
7.2 A ssumptions of Population Welfare Axiology 141
7.3 Notation and Two Special Axioms 142
7.4 T heorems on Welfare Aggregation 145
7.5 Inferred Values as Bounds 148
7.6 Intersituational Comparison 151
7.7 Some Fallacies 154
7.8 The Alleged Impossibility of Interpersonal Comparison 155
7.9 The Pareto Principle 157
7.10 D ifferent-Individual Comparisons 159
7.11 Incomparability with Zero and the Value of a Life 164
8 Evaluating and Comparing Situations 167
8.1 Situational Value and Situational Comparison 167
8.2 The Principle of Welfare-functionality 169
8.3 P rinciples of Inclusion, Exclusion, Equality, and Isovaluence 171
8.4 “Repugnant” Conclusions 181
8.5 An Ironic Conclusion 184
8.6 Nonidentity and Individual -affecting Principles 187
8.7 A n Individual affecting Principle for Nonlinear Value Structures 192
8.8 A Consistency Theorem 194
8.9 The Average Principle and the Mere Addition Paradox 198
8.10 Welfare Equality 201
8.11 Maximin 204
8.12 Hybrid Principles 205
8.13 All -Things-Considered Betterness and Transitivity 206
8.14 Q uality and Quantity 208
8.15 Parfit’s Ultimate View 211
9 T he Deontic Logic of Decision
9.1 Maximizing Choice Principle 220
9.2 Structure of the Outcome Set 222
9.3 P ropositional Deontic Logic 225
9.4 S emantic Postulates 229
9.5 D eontic Theorems on the Outcome Set 233
9.6 D eontic Decision Models 237
9.7 T he Consistency of the Logic 242
10 Ethical Rational Decisions
10.1 I mplications of Max 247
10.2 Unexcelled Choice Principle (Unex) 250
10.3 S atisficing 256
viii Contents
10.4 Moving Diagonal, Optimization, and Sufficientarian Choice 263
10.5 Adequacy Indices 267
10.6 Equilibration and Indeterminacy 270
10.7 Strict Satisficing 275
10.8 Hyperbolic Satisficing 280
10.9 Expected Value 290
10.10 Multicriteria Decision Analysis 292
10.11 D ecision Under Ignorance 297
10.12 Stepping Back 300
PART IV
11 Absolute Value and Concept Generalization
11.1 Absolute values in Cartesian models 3 06
11.2 L east upper bound axiom (A9) 3 08
11.3 P roperties of the absolute value operation 312
11.4 G eneralized delimitation and commensurability 315
11.5 Absolute difference 318
11.6 C oorthanticity and Antiorthanticity 322
11.7 Orthogonality 328
11.8 Reflections 333
12 B ounds of Finite Value Sets 337
12.1 L east Upper Bounds of Finite, Nonempty Value Sets 337
12.2 Adding Least Upper Bounds 343
12.3 Value Set Negatives and Greatest Lower Bounds 347
12.4 Positive and Negative Parts of Values 353
12.5 Adding and Subtracting Value Sets 358
12.6 Integer Multiplication of Value Sets 363
12.7 R ectangles, Boxes, and Triangle Equalities 368
12.8 O rthanticity in Detail 374
13 Value Analysis 385
13.1 Subvalues 385
13.2 Unions and Intersections of Values 392
13.3 S ubvalue sets 397
13.4 E lements and Element Sets 399
13.5 F inite Analyzability 403
13.6 T he Dimensions of Finitely Analyzable Values 407
14.1 Relative Infinity for Positive Values 411
14.2 Infinity and Infinitesimals in Mathematics 411
14.3 Relative Infinity in Axiology 413
14.4 The Full Concept of Relative Infinity 415
14.5 Relative Finitude 421
14.6 Finite and Infinite Commensurability 423
14.7 Nonstandard Models 426
14.8 Other Sources of Infinity 430
14.9 The Archimedean Principle 433
14.10 T he Non-infinitude of Objective Welfare 436
1 I ncomparability, Axiologies, Axiomatics, and Model Theory
1.1 I ncomparability
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 Comparison of a simple linear order with a simple merely partial 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 . << << << gfed cb a
The diagram at right represents a partially ordered collection of seven values. Some of these are comparable. Thus: , ,and . < << < < ca gf da db
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 than d. But since m + is better than 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 disparate value criteria. The familiar phrase “that’s like comparing apples and oranges” expresses this idea. Presumably, it means that an apple and an orange are so different that neither is better (in some overall sense) than the other, nor are they equal in value. Suppose for the sake of illustration that this is true. And suppose too, to keep the following example simple,
t hat each apple is indistinguishable from every other apple, and likewise for oranges. Then each type of fruit determines a linear unit scale: numbers of oranges or numbers of apples. Ten apples, let’s assume, are collectively twice as good as five apples (and the same is true for oranges), but the value of ten oranges is neither greater than nor less than, nor equal to, the value of five apples. Nevertheless—and this point proves crucial later—the two scales share a common zero value. Zero apples have the same value as zero oranges—none at all.
Now consider four possible situations in which a person (call her Alice) has different numbers of apples and oranges, as shown in Table 1.1.
Table 1.1 Four possible situations in which Alice has various numbers of apples and oranges.
The value that Alice possesses in these situations can be represented either as pairs of numbers—with, say the number of apples first and the number of oranges second, as in Table 1.1—or (since the two scales share a common zero value) as points on a two-dimensional diagram, as in Figure 1.2.
Here, each point represents the total value that Alice possesses in one of the four situations. That total value is comprised of two incomparable values: those of apples and oranges.
Figure 1.2 T he data of Table 1.1 plotted in a two-dimensional graph.
We can read the comparison relations worse-than, better-than, equal-to and incomparable-with from the graph. Values better than a given value are the ones better in both dimensions (apples and oranges) or better in one and equal in the other; they are represented by points above and/or to the right of the given value. Values worse than a given value are the ones worse in both dimensions or worse in one and equal in the other; they are represented by points below and/or to the left of the given value. The only value equal to a value is itself. Thus the values incomparable with a given value are those above it and to its left or below it and to its right. Using the symbol ‘ ’ to mean “is incomparable with,” the graph thus represents all of the following relationships among the labeled values:
Translating these formulas into a Hasse diagram, we obtain Figure 1.3.
Figure 1.3 depicts the comparison relations expressed in the 15 formulas listed prior to it, though it omits the numerical information contained in Figure 1.2. Where partially ordered values are not associated with precise quantities, Hasse diagrams are generally the best way to represent them; but we can also use Hasse diagrams even if, as with apples and oranges, we have exact numbers.
Not all incomparability results from combining linearly ordered scales. In the case of the music and the dinner, for example, determination to compare the pleasures of the two may motivate us to attempt to analyze each into multiple components. Dinners might be evaluated by the tastes,
Figure 1.3 T he data of Table 1.1 represented in a Hasse diagram.
s mells, and visual presentation of the food, the ambience, the conviviality of the company, etc. Live music might be evaluated by the quality of the emotions it stimulates, the originality or virtuosity of the musicians, and so on. But these attempts at analysis may be crude at best.
For sufficiently robust value structures, full analysis into multiple linearly ordered scales is sometimes possible. Chapter 13 explains how. But such an analysis is not always possible. The component values may be merely qualitative and non-numerical. And the values that the separate criteria aim to assess may themselves be merely partially ordered.
1.4 T he Compulsion to Linearize
Though many value structures are merely partially ordered, people often feel compelled to rank values linearly. Consider, for example, an essay contest in which essays are rated on a variety of criteria—writing style, originality, clarity, significance of ideas, and so on. The judges may be required to rank whole essays linearly, even though one essay may be better than another on some of the criteria and worse than the other on others. Ties may be permissible, but to declare two essays incomparable is hardly ever allowed.
To avoid incomparability and yet keep their rankings consistent, the judges may assign a numerical score for each criterion and then add the scores to obtain a single numerical score. Or, they may first multiply the scores by numerical weighting factors to indicate their importance and then add. (There are various ways to do this, but all are to some extent arbitrary; see Sections 6.1 and 10.6–10.8.) Since numbers are linearly ordered, the overall scores (raw or weighted) are as well, but they incorporate this arbitrariness.
Although the judges may be compelled to linearize by social constraints or their own discomfort, they are not compelled by logic. There is nothing inconsistent in refusing to assign numerical scores and instead expressing the value of each essay as an array of scores, one for each criterion, as we did for apples and oranges. These arrays will not always be linearly ordered, but why suppose they should be?
The compulsion to linearize has produced notable lapses in logic. Over the course of this book, we’ll examine many such fallacies. The simplest amounts merely to overlooking the possibility of incomparability. I call it the trichotomy fallacy.
In The Case for Animal Rights, for example, Tom Regan argues that all animals with certain capacities, whether human or non-human, have equal inherent value. He calls such animals “subjects of a life.” Regan holds that it is morally pernicious to admit superiority or inferiority of inherent value among subjects of a life, for at least two reasons: (1) historically, that has led to “chattel slavery, rigid caste systems, and gross disparities in the quality of life available to citizens of the same state,” and (2) differential inherent value would be unjust, since no individual is responsible for the nature,
gifts, and talents that she or he was born with (Regan 2004, 234). Let’s grant all this for the sake of argument. From these premises—that subjects of a life have inherent value and that none has more or less of it than any other—Regan infers that all subjects of a life have equal inherent value. But that doesn’t follow. Regan has committed the trichotomy fallacy.
The same fallacy recurs in Respect for Nature, Paul W. Taylor’s seminal work in biocentric ethics. Like Regan, Taylor argues for a doctrine of moral equality, but in this case for all living things. Traditionally, Taylor says, ethical theory has taken the superior ethical value of humans for granted, rooting it in capacities such as “rationality, aesthetic creativity, individual autonomy, and free will” that we alone are alleged to possess. Surely these capacities are valuable. But, Taylor asks, valuable to whom and for what? To us, for the sake of our interests, he answers—but not in any absolute sense. Likewise, the capacities of each organism are valuable to it for the sake of its interests (Taylor 1986, 130). Thus, he concludes, our inherent worth is no greater than that of any other living thing. Again, let’s grant these premises for the sake of argument. Now Taylor, like Regan, infers a principle of moral equality—that non-humans are equal in inherent worth to humans:
Rejecting the notion of human superiority entails its positive counterpart: the doctrine of species impartiality. One who accepts that doctrine regards all living things as possessing inherent worth—the same inherent worth, since no species has been shown to be either “higher” or “lower” than any other.
(Taylor 1981, 83)
There are two mistakes here. One is a fallacy of ignorance: from the premise that no species has been shown to be higher or lower in inherent worth than any other, it does not follow that no species is higher or lower in inherent worth than any other. But what is for our purposes the more interesting error lies in the inference from no species is higher or lower in inherent worth than any other to all species have the same inherent worth.
Taylor, too, has committed the trichotomy fallacy. A very different instance of the trichotomy fallacy occurs in Gottfried Leibniz’ notorious argument that ours is the best of all possible worlds. The crux of the argument is this:
If this world is not the best of all possible worlds, then either • God was not powerful enough to bring about a better world; or
• God did not know how this world would develop after his creation of it (i.e., God lacked foreknowledge); or
• God did not wish this world to be the best; or
• God did not create the world; or
• t here were no other possible worlds from which God could choose.
None of these five conditions hold.
Therefore, this world is the best of all possible worlds.2
The inference is logically valid, but the initial premise overlooks a sixth possibility: that possible worlds are so ordered by the betterness relation that none has maximal value. (For a simple finite example of this, look back at the diagram at right in Figure 1.1) The trichotomy fallacy is thus implicit in that initial premise. Given the variety of possible worlds, which Leibniz took to be infinite in number, the non-existence of a maximum is quite conceivable—perhaps almost inevitable—whatever the criteria of betterness are.
1.5 A xiologies
Because we are investigating values, a brief survey of their varieties is in order. Theories of aesthetic, prudential, or ethical values are called axiologies. Axiologies are of two types: relational and non-relational. Relational axiologies conceive values as values for beings for which things can be bad or good, better or worse. These beings may be people, but they may also be non-human animals or organisms, or even collectives of certain kinds—for example, institutions or ecosystems. A relational axiology holds that for every value there must be an answer to the question “value for whom (or what)?” On a relational view, a situation or event may be better or worse for you or me, perhaps even for ocelots or oaks, but it can’t be better or worse simpliciter
Non - relational axiologies hold that some things (conditions, events, situations, etc.) have value independently of their relations to valuers. Thus, for example, on certain Platonic conceptions, the value of a thing is constituted by its perfection (degree of Being), but perfection need not be valuable for any valuer. The value of perfection is therefore not relational but absolute. Similarly, some Kantians regard the values of persons as non-relational.
Many of the results of this book are applicable to both relational and non-relational values, but some—especially those of Chapters 7 and 8, which deal specifically with the welfare of individuals and populations— presume relational axiologies. Of these, there are three main types: hedonism, preference theories, and objective welfare theories.
The most venerable of relational axiologies is hedonism —the idea that goodness is pleasure or happiness and badness is pain or suffering. Hedonistic axiologies are relational, since suffering is in each case the suffering of an individual, and so is bad for that individual in a way that it is not bad for others. (Sometimes hedonists or their opponents imagine pleasures and pains that are impersonal and disembodied, but nevertheless good or bad, though not good or bad for anything. But these philosophical fictions need not concern us here.)
Classical hedonistic utilitarians, most notably Jeremy Bentham, propose to aggregate degrees of pleasure, which are positive, and degrees of suffering or pain, which are negative, into a single value that they regard as the degree of overall happiness or utility, either for an individual or for a population. Situations with much pleasure and little suffering have positive utility. Those with much suffering and little pleasure have negative utility. Those in which pleasure and suffering are either absent or precisely balanced have zero utility. Bentham and his followers judge the goodness or badness of an action by the total utility it produces for those whom it affects.
It is often assumed that hedonistic utility can be measured on a linear scale. But if raw pleasures, pains, joys, and sufferings themselves are what is ranked, their sheer diversity renders that assumption incredible. Which, for example, is worse: to live for months in mortal terror during a bloody military siege or to live for a similar time with bone cancer? Even those who have experienced both may not be able to say. Mortal terror may be just too different from penetrating but localized bodily pain to permit ranking.
The difficulties are compounded if we try to aggregate an individual’s joys and sufferings to obtain an overall assessment. Actually, we can’t do it at all. The best we can do is ask the individuals, assuming they are verbal, for a subjective assessment of their overall happiness. But subjective assessments are sensitive to many variables, including who is asking and how the question is asked.
Difficulties are compounded further if we try to aggregate the hedonistic utilities of many individuals—and still further if we try to compare or aggregate hedonistic utility across species (Nolt 2013). Many of these difficulties stem from our ignorance. It is hard to obtain reliable estimates. But when there are huge qualitative differences in subjective experience, so that the person herself cannot determine which is better or worse, then there may be genuine incomparability and hedonistic utilities may be merely partially ordered. Still, difficulties of assessment may obscure this.
Something closer to an objective linear order might be obtained by considering individuals’ reactions to pain and pleasure. We might, for example, treat hedonistic value behaviorally as the degree to which pain and suffering are disabling or aversive, and pleasure and happiness are
enabling or rewarding, so that it becomes objectively assessable. Or, more in keeping with the subjectivity of hedonism, we might assess hedonistic value as the degree to which an individual likes or dislikes an experience. Hedonism then becomes what Derek Parfit has called preferencehedonism (Parfit 1984, 493–4).
Preference-hedonism, however, is unstable. Having admitted the goodness of satisfying desires to feel pleasure or avoid pain, we have little reason not to admit the goodness of satisfying other sorts of desires. The idea that goodness is the satisfaction of any sort of preference is the purest and crudest form of preference axiology. Preference-axiological utility, then, is the degree to which overall preference satisfaction exceeds preference frustration. (When preference frustration is greater than preference satisfaction, that utility is negative.)
Whereas hedonism simply stipulates that goodness is pleasure or happiness, and nothing else, what preference axiology counts as good for an individual depends on that individual’s preferences. On this view, we determine what is good for ourselves by choosing our values and life projects. Rather than pleasure, people may, for example, choose to pursue power, money, fame, heroism, creativity, wisdom, or piety—any of which may or may not be pleasurable.
Preference axiology is foundational to neoclassical economics and widely assumed in policy-making. But preference - satisfaction per se particularly, as understood in these disciplines—is neither a precondition nor a guarantee of welfare, happiness, or goodness. People who are thoughtful, informed, and well-intentioned do tend to prefer what is good for them and others; but thoughtless, misguided, ignorant, or malevolent people tend not to. Sometimes, what is best for us is not to get what we most desire—as, for example, when we cling to toxic relationships. Sometimes, as Buddhists teach, we benefit most not by satisfying preferences but by relinquishing them.
Yet we are constantly barraged by attempts to shape our preferences and instill more of them. Advertising and propaganda—which are lucrative, powerful, and pervasive—exist precisely to create and manipulate preferences, often by appealing deceptively to narcissism, lust, anger, envy, guilt, fear, or greed. Satisfaction of such preferences is often harmful.
Satisfaction of our native preferences can be harmful too, as Leo Tolstoy reminds us in Anna Karenina. Anna and Vronsky have been having an affair, and she has left her husband for him:
Vronsky meanwhile, in spite of the complete fulfillment of what he had so long desired, was not completely happy. He soon felt that the realization of his longing gave him only one grain of the mountain of bliss he had anticipated. That realization showed him the eternal error men make by imagining that happiness consists in the
gratification of their wishes. … Soon he felt rising in his soul a desire for desires—boredom. Involuntarily he began to snatch at every passing caprice, mistaking it for a desire and a purpose.
(Tolstoy 1992, 548)
Clearly, preference satisfaction per se is not definitive of any plausible conception of welfare, happiness, or goodness.
What about the satisfaction of rational preferences, those based on reasoned and informed deliberation? Defining welfare as the satisfaction of rational preferences might indeed yield a more adequate utility theory. But this is fantasy. Any actual deliberation may err, and actual humans are only intermittently and ephemerally rational. Speculations about the outcomes of rationally disinterested deliberations can therefore amount to little more than educated guesses as to what would turn out to be best. But if that is the game, then preferences are largely irrelevant. Some prior notion of goodness is doing the axiological work.
More promising for ethics are objective welfare axiologies. They hold that goodness, positive utility, or welfare is constituted not—or at least not wholly—by subjective states such as pleasure or satisfaction of desire, but by conditions that are objectively good. Health, for example, is good for everyone, including even those rare individuals who don’t desire it. Objective welfare axiologies that posit more than one objective good are often called objective list theories. The objectivity of these goods is typically rooted in the biological needs and/or social nature of human beings. Brad Hooker, for example, has advocated an objective list theory according to which human welfare consists at minimum of “friendship, achievement, important knowledge, and pleasure” (Hooker 2008, 130). Health, as I have noted, is another such good. Other plausible candidates include wealth sufficient to needs, longevity, productive employment, security, and freedom from oppression. By including pleasure on his list, Hooker also assimilates hedonistic utility into an objective list theory.
But objective list conceptions of welfare are perhaps the least likely of all to generate linear value structures. This is due largely to their multidimensionality. Multidimensional values, like the values of apples and oranges, tend to arrange themselves nonlinearly. Objective list conceptions are, moreover, potentially quite broad. Since all living things are objectively capable of welfare—at minimum of health—they need not be restricted to humans, or even to sentient beings (Nolt 2009; 2021, sec. 5). And broad axiologies, too, tend to generate nonlinear value structures. Such axiologies are now, however, unavoidable. We cannot deal ethically and rationally with climate change, for example, or the impending threat of mass extinction, without juggling a multiplicity of objective values. The differences between my approach to an ethical decision (C hapter 10) and those that still take subjective preference as the paradigm of value
stem largely from that realization. Even so, all the relational axiologies discussed here—pleasure and the absence of pain, preference satisfaction and non-frustration, and objective welfare—can be enhanced by a rigorous and accurate understanding of incomparable values.
When an axiology is conjoined with a choice principle that specifies how choices or actions should be shaped by their good and bad consequences, the result is a consequentialist ethic. The classic choice principle is: do what yields the greatest utility. But, among merely partially ordered utilities there need not be a greatest. Choice then requires some modification of the classic principle. Chapter 10 considers the alternatives.
Historically, the possibility of incomparability was little noticed. When noticed, it was generally seen as an objection to consequentialism. Writing of consequentialist decision-making in environmental ethics, John O’Neill, Alan Holland, and Andrew Light (2008), for example, note that choices frequently require trade-offs—sacrifices of one sort of good for another. But, they caution,
… some values may not be convertible into other currencies of value … A second problem is that the standard interpretation of the [trade-off] metaphor suggests … that we are trying to produce a state of affairs with the greatest amount of value, and we do this by comparing the gains and losses of different dimensions of value, trading these off until we arrive at a result that produces the greatest gains in values over losses in values.
(79)
In the end, they conclude, “there may simply be no ‘best’ choice. … Reason cannot ensure for us in advance that conflicts of values can be resolved and tragic choices avoided” (86).
True. But, O’Neill, Holland, and Light take incomparability as a reason to reject consequentialist decision-making and replace it with a “procedural account of rationality” according to which “what matters is the development of deliberative institutions that allow citizens to form preferences through reasoned dialogue, not the refinement of ways of measuring given preferences and aggregating them to arrive at a putative ‘optimal’ outcome” (85). This rejection of consequentialism—which they caricature as preference utilitarianism with a maximizing choice principle—is premature. Trade-offs are inevitable on any moral theory, and citizen deliberation can hardly ensure the resolution of value conflicts or the avoidance of tragic choices. If incomparability renders the aim of promoting the common good hopeless, what principles will guide the citizens in their deliberations? Before giving up on consequentialism, we ought at least to examine the extent to which its decision procedures can accommodate merely partially ordered values. Chapters 9 and 10 attempt that.
1.6 A xiomatic Method
A secondary aim of this book is to show that formal axiomatics and model theory can advance axiology. Regrettably, contemporary philosophers have largely abandoned these methods. The explanation is in part historical.
The axiomatic method reached its classical zenith in the work of the Greek geometer Euclid. His magnum opus is known simply as Elements. In Western and near Eastern culture, it served as both the standard geometry text and the highest exemplar of clear, indubitable, rational thought for two millennia.
In Elements, Euclid deduces a wide swath of geometry, together with some number theory and algebra, from a small set of axioms and postulates.3 Axioms were, throughout most of the Western tradition, taken to be principles so lucid and self-evident as to be indubitable, so that any proposition validly deduced from them would also be undeniable. Ideally, they were also to be rich in implications, so that a large, substantial, and sophisticated body of knowledge could be deduced from them. Philosophers hoped to use the axiomatic method to expunge doubt from wide swaths of knowledge.
The method had other attractions as well. It provided a rational way of organizing what was known, clarifying fundamental concepts, and precisely articulating relations among them; and it fostered deep, broad, and precise conceptual analysis.
Principles deduced from a set of axioms are called theorems. Of these, Elements contains hundreds—some of such power and sophistication that it can seem miraculous (as it did to me when I was in high school) that so intricate an edifice of knowledge could be built upon such a slender foundation. The miracle, it turns out, is partly illusory—but more of that shortly.
Euclid’s Elements captured the imaginations not only of mathematicians but of philosophers. One of the beauties of the axiomatic method is that once a theorem is proved, it is tied back to the axioms by a series, sometimes quite lengthy, of inexorable logical steps. Intuitive leaps are unnecessary. Each step is recorded in the proof of the theorem and of the prior theorems on which it rests. Anyone who doubts the theorem can trace its derivation back to the axioms, which are supposed to be self-evident. In his biography of the philosopher Thomas Hobbes, John Aubrey recounts Hobbes’ first encounter with this procedure:
He was 40 years old before he looked on Geometry; which happened accidentally. Being in a Gentleman’s Library, Euclid’s Elements, lay open, and ‘twas the 47 El. libri I. He read the proposition. By G— sayd he (he would now and then sweare an emphaticall Oath by way
of emphasis) this is impossible! So he reads the Demonstration of it, which referred him back to such a Proposition; which proposition he read. That Proposition referred him back to another, which he also read … that at last he was demonstratively convinced of that trueth. This made him in love with Geometry.4
Thus Hobbes overcame his initial skepticism of the Pythagorean theorem. Some venerable philosophers of the Western tradition—most notably Rene Descartes (in the Meditations) and Baruch Spinoza (in his Ethics)— made extensive use of the axiomatic method. Often they botched it. The method is, of course, only as reliable as the axioms from which it starts and the inferences by which it proceeds. Philosophers made plenty of bad inferences, but some of their most conspicuous failures lay in the axioms they chose. This central premise from Descartes’ Third Meditation is a case in point: there must be at least as much reality in the total efficient cause as in its effect” (Descartes 1960, 39). Descartes regarded this abstruse relic of scholastic metaphysics as indubitable by “the light of nature” and hence worthy of axiomatic status.
Natural philosophers (who are nowadays called natural scientists) sometimes did better. Newton formulated his three laws of motion as axioms for celestial mechanics. Using astronomical observations and the inferential techniques of the infinitesimal calculus—which he and, independently, Leibniz had invented—he was able to explain and predict the motions of the tides and planets.
Still, all was not well. George Berkeley showed that because the inferential methods of Newton and Leibniz lacked rigor, they could, despite their astonishing successes, readily be made to yield absurdities. Mathematicians then labored through much of the nineteenth century to clarify the relevant concepts and develop a consistent and rigorous calculus.
From this work emerged the formalist movement, whose aim was to eliminate the need for intuition that had so often led to error by breaking down all mathematical reasoning into simple steps of recognizably valid form. Its methods revealed previously unnoticed gaps in Euclid’s reasoning—which is why I said a few paragraphs back that Euclid’s miracle had been partly illusory. In his Grundlagen der Geometrie, published in 1899, the consummate mathematician David Hilbert reworked geometry into a sound and logically rigorous axiomatic system—at the cost of burgeoning technicality.
Others sought to axiomatize other areas of mathematics.5 Toward that end, Gottlob Frege developed the predicate calculus, the first fully formal logical system powerful enough to express sophisticated mathematical reasoning. With it he aimed to derive much of mathematics from axioms of pure logic, and thus to prove that mathematics is logic and nothing
besides. A major step in this effort was to have been his Grundgesetze der Arithmetik, published in two volumes (1893 and 1903).
To achieve the conceptual power necessary for this task, Frege adopted an axiom according to which (in contemporary terms) every predicate defines a set. In 1902, Bertrand Russell showed that this axiom is self-contradictory, thus refuting Frege’s work. Russell’s proof was shockingly elementary. He simply observed that if every predicate defines a set, then the predicate “is not a member of itself” defines the set of things that are not members of themselves. Call this set S. S cannot be a member of itself, since the only sets it contains are those that are not members of themselves; but since S contains all sets that are not members of themselves, it has to be a member of itself. This contradiction refutes the claim that every predicate defines a set.
Russell himself, together with Alfred North Whitehead, subsequently sought to complete Frege’s project of deriving all known mathematics, by using a different—and presumably consistent—set of axioms. Between 1910 and 1913 they published this work in the monumental three-volume Principia Mathematica. Principia provided the means to deduce all the mathematics that was known at the time, but it did not realize Frege’s hope of reducing mathematics to logic, because not all of its axioms were purely logical. Still, it was a triumph for the formal axiomatic method. The foundations of mathematics seemed secure at last.
1.7 Model Theory
In geometry, formal methods achieved another triumph: proof of the consistency of non-Euclidean geometries that described previously unimaginable curved spaces. This discovery vastly expanded human knowledge of what is geometrically possible, but at first it seemed little more than a mathematical curiosity. Euclidean geometry was, so far as anyone knew, sufficient to describe spatial relations in the physical universe. Then, again in the first decade of the twentieth century, Hermann Minkowski, reflecting on the work of Albert Einstein, realized that the universe conforms to a weird and exquisite four- dimensional geometry that encompasses time as well as space and is in fact non-Euclidean.
The question that spurred the development of non-Euclidean geometry was whether one of Euclid’s axioms could be derived from the others. That axiom is equivalent to this apparently obvious statement about points and lines in a plane:
Parallel Postulate: If L is a line and P is a point not on L, then there is one and only one line through P that does not intersect L.
Figure 1.4 suggests the idea:
