Abstract
Christine Korsgaard sees normative generalizations as provisionally universal, in the sense that exceptions to them have reasons for being exceptions and that they could in principle be revised into more specific and precise absolutely universal rules. Do exceptions to normative generalizations have such explanations? Can such generalizations always be revised into or replaced by absolutely universal rules? The answer depends on the structure of practical space, and, specifically, the degree to which normative relations are definable. Distinguishing degrees of definability in practical space opens up paths for generalist and particularist conceptions of normativity.
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Notes
A small sample:
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“Any set of circumstances with the characteristics described by the reasons offered [counts as similar]... a reason in one case is a reason in all cases—or else it is not a reason at all” (Singer, 1961,1971, 28; see also 138).
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“The universality of a reason judgment is a formal consequence of the fact that taking something to be a reason for acting is not a mere pro-attitude toward some action, but rather a judgment that takes certain considerations as sufficient grounds for its conclusion” (Scanlon, 1998 74; see also 367, 371–372)
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“To say that necessarily all finite rational agents have such [basic moral] reasons for action commits one to the claim that such reasons are universal in scope” (Timmons, 2006, 188).
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Baier (1958), Harman (1975), Raz (1975a, b, 1999), Dancy (1981, 2004), Morreau (1997), Horty (1997, 2012), Little (2000, 2001), Lance and Little (2004, 2006, 2007), Kukla and Lance (2009), Alvarez (2010), and Bonevac (2016, 2018) have all developed generalist theories of reasons incorporating defeaters and other elements of nonmonotonicity.
I have phrased this noncommittally—“might be thought to presuppose”—because I want to focus on the conditions under which it would be reasonable to interpret such principles as provisionally universal. (Dancy, 2000, 2004) argues against the claim that reasons presuppose principles, general or universal, as inconsistent with holism about reasons. Under what conditions is it appropriate to hope even for general principles underlying reasons? That is an important and subtle question dividing generalists and particularists that I do not attempt to answer here. It has important implications for finite expressibility, as I explain further below.
If absolute universality requires a universalized biconditional, then we also need All Bs are (A and C)s.
The reasons themselves may or may not be articulable or finitely expressible. If we think of reasons as facts (Raz 2009) or as considerations in favor of something (Dancy, 2004; Scanlon, 1998, 2014), for example, it is not obvious that the reasons must be finitely expressible, for neither the set of considerations nor the considerations themselves need be so expressible. If we think of reasons as linked to explanations, as answers to why-questions (Skow, 2016), in contrast, they seem to be inherently articulable. Principles, in any case, can be articulated. Thanks to an anonymous referee for bring this issue to my attention.
Wiggins takes this to be Aristotle’s view. In terms to be developed below, he may be interpreting Aristotle as holding that practical space is rough but not primitive; some normative relations are partially definable, making it possible to formulate some absolutely universal rules such as Murder is always wrong (see Nicomachean Ethics II, 6, 1107a9–17), while others are not. But Aristotle treats these rules as absolutely universal because the subject terms are already thick, i.e., partly normative; they “have names that already imply badness” (1107a10). So, Aristotle may think that practical space is primitive after all.
It is common in the literature to link this to a principle of strong supervenience: the distribution of normative properties and relations depends solely on the distribution of non-normative properties and relations. That can be misleading, however, in several respects. First, the supervenience in question must be global and transworld; see below. Second, the distribution of properties and relations at issue must not depend on rigid designation; it cannot include properties like being Socrates or standing between Zeno and Callimachus. Third, one way to understand particularism is that it replaces the Generalization principle with a supervenience principle, arguing not only that the latter does not imply the former but also that the Generalization principle is false. Finally, the distinctions to come have a normative/non-normative interpretation, which is my focus here, but are broader, applying to any distinctions between kinds of properties and relations. For a detailed discussion of the relation of supervenience to generality and universality, see Dancy 1981, 373–375, 380–382.
The extent of that region depends on our theory of counterfactual conditionals and other tools for hypothetical reasoning; see, for example, (Williamson, 2020). But whatever analysis we choose or heuristics we adopt, we must be able to consider an uncountable number of hypothetical scenarios.
It will make no difference hereafter whether we think of a model as consisting of worlds and entities of various kinds within those worlds or conceive it as consisting of an uncountable domain of world-bound counterparts or object-world pairs, with interpretations of normative and non-normative predicates on that domain. That decision can be left to our theory of hypothetical reasoning.
To be precise: Let M be a model for L, with D as its domain. Say that a formula \(A (\vec {x})\) of L defines an n-ary relation \(R \subseteq D^n\) in M iff \(R = \{ <\vec {a}> : M \vDash A (\vec {a}) \}\). R is L-definable in M iff there is a formula of L defining it in M, and L-definable simpliciter iff there is a formula of L that defines it in all models of L. (Hereafter, where there is no danger of confusion, I sometimes drop reference to language L.) Tarski’s theorem, saying that arithmetical truth is not arithmetically definable, asserts that there is no formula of the language of arithmetic true of all and only the Gödel numbers of true sentences of arithmetic. A claim that a normative term such as better than is indefinable in L analogously amounts to the assertion that there is no formula of L true, in each model, of all and only pairs of things in which the first is better than the second. Since there are nondenumerably many relations on a nondenumerable domain, but only denumerably many relations definable in L, this is not unusual. Indefinability is the norm; definability is the exception.
Audi assumes that All A are B if they are also C and All (A and C) are B are equivalent.
This fact differentiates definability from the stronger notion of recursive enumerability. A recursively enumerable set need not have a recursively enumerable complement.
If the domain is finite, and every element of the domain is definable, then every set is definable; Korsgaard is vindicated. If the domain is countable and every element is definable, then every set has a definable cover. Because the subsets might be finite, however, this merits little interest; it is compatible with some sets being primitive in the sense, defined below, of having no infinite definable subsets. The Uncountability assumption is thus crucial to the results of this paper. It guarantees that there are indefinable members of the domain; that there are sets that lack definable covers; and that analyzability does not imply having a definable cover.
I’m not being honest, said all by itself, seems to generate an analogue of the liar paradox, for example. If I’m being honest in saying that, I’m not being honest; and, if I’m not being honest in saying it, I am being honest.
See Tzouvaras forthcoming. Tzouvaras, by analogy with recursion theory, calls sets with something similar to this property totally non-immune. That terminology risks confusing concepts concerning definability with related but non-equivalent concepts in recursion theory. (A set is immune iff it is infinite and has no infinite recursively enumerable subsets (Dekker, 1953).) It also employs a double negative that seems to obscure the nature of the concept in question. Totally non-immune sets are not strictly speaking totally non-immune, even in his sense, for any infinite set can be divided into two disjoint immune sets. I should note that Tzouvaras’s results appear to conflict with mine in this paper. But the conflict is only apparent; without the Infinity assumption, and with individual constants denoting each object in the domain in the language, the conceptual landscape (of arithmetic, for example, which is his concern) is quite different.
Say that R has a definable cover. Then R is a union of infinite definable sets \(R_1, ..., R_i, ....\). For any infinite definable A, then, \(R \cap A = (R_1 \cap A) \cup ... \cup (R_i \cap A) \cup ....\), where each nonempty \(R_i \cap A\) is infinite and definable. So, R is analyzable. (Note the importance to this argument of the Infinity assumption, that in practical space all nonempty sets are infinite. Without that assumption, the claim would not hold; some or all the \(R_i \cap A\) might be finite.) The converse does not hold. That any intersection of R with an infinite definable set has an infinite definable subset does not suffice to show that those intersections constitute a definable cover of R.
Every relation that has a definable cover is definable if the underlying topology generated by the class of definable sets is compact. The other converses do not hold even then. To see that they do not hold in general, it suffices to note that, for a given L, there are at most \(\aleph _0\) definable sets, and at most \(2^{\aleph _0}\) sets with definable covers. There is no upper bound to the number of analyzable or partially definable sets.
Are there other interesting concepts in the neighborhood? What about notions apparently stronger than analyzability, such as the property that its intersection with any infinite definable set is infinite and definable, or has a definable cover? Since the universal set is infinite and definable, these collapse into definability and having a definable cover, respectively. What about notions apparently weaker than analyzability, such as having an intersection with an infinite definable set that is infinite and definable, has a definable cover, or is partially definable? They all collapse into partial definability. Say R has an infinite definable subset A. Then, since \(A = R \cap A\), R has an intersection with an infinite definable set that is definable and, a fortiori, has a definable cover and is partially definable. More interesting are related candidates, such as the property that its intersection with any set that is partially definable (or analyzable, or has a definable cover) is partially definable (or analyzable, or has a definable cover). I leave their investigation to future work (Table 1).
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Bonevac, D. Provisional Universality. Erkenn (2022). https://doi.org/10.1007/s10670-022-00641-8
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DOI: https://doi.org/10.1007/s10670-022-00641-8