Abstract
Normative notions are often explained in terms of reasons, which (allegedly) can be weighted and combined, for instance in order to know what one ought to do. But what is their weight? How do they combine? This chapter applies measurement theory to these questions. I argue that normative reasons cannot be consistently weighted and aggregated for purely formal, rather than substantial, reasons and that this is a prima facie novel, non ad hoc argument for normative particularism.
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- 1.
At first approximation, it is often said that normative reasons are considerations in favor of doing something. Some take reasons to be primitive (or fundamental) for other normative notions (in the vicinity of this view are Scanlon 2014; Parfit 2011; Dancy 2004). Raz (2016) takes reasons to be those things our rational capacities respond to; Kearns and Star (2009, 2013, forthcoming) take reasons to be evidence one ought to act in a certain way. Recently, Dietrich and List (2013) showed how decision theory can be recast in reason-theoretic terms. There are other stances, but such substantive issues are orthogonal to scope of this paper.
- 2.
Both Pollock (1995), pp. 101–2 and Schroeder (2007), Chap. 7 hold a close relative of this view. Pollock takes complex reasons as independent from the simple reasons that compose them, so that the weight (priority) of the compound is unrelated to the weight (priority) of the simpler reasons. Schroeder assigns weight just to sets of reasons, not to their members. Dietrich and List (2017) defines a weighing relation between sets of elements as well, in their formal apparatus.
- 3.
- 4.
Some prefer to think about pro tanto and pro toto, a distinction that is usually thought to go back to Ross (1930). Pro tanto reasons are considerations which can be outweighted, whereas pro toto reasons are all-things-considered, “final” considerations. This distinction is applied—sometimes with slightly different terminology—across the board to principles, duties, and oughts. Sometimes one finds prima facie reasons for pro tanto reasons. This terminology might be a bit misleading insofar as one thinks of them as considerations which are reasons on their face, but might turn out not to be. The intended use of prima facie reasons is instead for considerations which are reasons but are pro toto only prima facie. I hesitate to drag pro toto reasons in the discussion, because further assumptions would be needed, for instance about whether the “sum” of all relevant undefeated reasons in favor is a different entity, plausibly the pro toto reason, or not.
- 5.
This picture is naive on many different levels, some obvious, some less so. Just for one underappreciated issue, Gerts (2016a, b) argues that there is a robust distinction between justifying weight and requiring weight. My arguments abstract away from this and other substantive issues to focus on certain formal ones.
- 6.
This distinction is explicitly drawn in Lord and Maguire (2016a), perhaps for the first time.
- 7.
Such characterizations consider certain properties of automorphisms, i.e. isomorphisms from qualitative structures to themselves. See for instance Krantz et al. (1971), Roberts (1979). If one doesn’t limit oneself to real numbers, the number of scales increases dramatically. For instance Cameron (1989) showed that there is an infinite number of scales in the rational numbers.
- 8.
A classical atomic mereological structure (i.e. an atomic Boolean algebra without bottom) on reasons is used by Brown (2014) who, however, assumes (without justification) that the weight measure of reasons is a ratio scale, and therefore additive. Having a more sensible mereological structure, for example, opens the possibility of having a fine-grained logic of reasons, for instance based on the ideas of Fine (forthcoming) or the hyperintensional deontic logic developed in (Author). Having a mereological structure, moreover, lets one deal more perspicuously with Moore-style holistic questions, namely whether smaller and smaller parts, if any, have value (strength, weight) independently from their wholes. It’s not clear, however, that an additional binary operation can account for modifiers. Intuitively, as we think of reasons aggregation as a sort of addition-like operation, we could think of modifiers as a multiplication-like operation. However, there are several different arguments against this intuitive similarity. First, while aggregation is an operation on reasons, modifiers need not be reasons: modification, then, cannot be an operation exclusively on reasons. Second, if we think of aggregation as an addition-like operation, and modification as a multiplication-like operation, we might find tempting to try to define modification in terms of repeated aggregation—which of course can be done only by imposing certain conditions on structures. Third, with a multiplicative operator, it would be natural to introduce distinguished elements (think of 0, 1) in the structure, and it is not clear that reason structures would support them. Finally, Bader (2016), Sect. 4 shows that (conditions and) modifiers cannot be integrated in the reasons themselves, which seems in line with our considerations. We leave this problem open for the time being, as we make do with simpler structures. A multiplicative operation could however be useful to model undercuttering defeaters, once we add a 0 element in the structure.
- 9.
Artemov (2015) introduces an ordering on the terms constituting the reasons in a justification logic framework. Setting aside the fact he is interested in probabilistic evidence, his framework is inadequate in a normative context for several reasons: first, being a probability function, even if the underlying measure is not countably additive and defined on a \(\sigma -\)algebra, it will still be finitely additive and defined on an algebraic structure which is far too rich for a reason structure; second (and more importantly) the terms have a lattice-theoretic structure, i.e. are endowed with a partial order (and other additional properties); again, a partial ordering is inadequate to model the importance, strength or weight of the normative reasons, as they cannot be antisymmetric: if two reasons have the same weight (strength, importance) they are not necessarily the same reason; third, we might want also to assign negative values, not just positive values to reasons for not doing something (the former a probability measure cannot do, the latter Artemov can do).
- 10.
Horty (2012) (and see also Horty and Nair forthcoming)—who writes about normative reasons but from a slightly different vantage point—employs a strict partial order (i.e. an irreflexive, transitive, (and therefore) asymmetric (if \(a < b\) then not \(b < a\)) relation. However, there’s a one-one correspondence between strict partial orders and partial orders. For the right direction, just consider the reflexive closure given by \( x \le y\) if \( x < y\) or \(x = y\), and we have the corresponding partial order; for the left direction, just consider the reflexive kernel. Thus the critical remarks of the previous footnote apply.
- 11.
For an introduction to the incommensurable/incomparable distinction (in the case of value) , cf. at least Chang (1997) and Hsieh (2008). These distinctions, however sketchy, do not seem exhaustive. Just consider bad reasons (as compared to good reasons). Are they still reasons? Are they incommensurable? Settling this issue is outside the scope of this paper.
- 12.
An example of a common non-associative (and non-commutative) operation is vector product. Weighted averages are non-associative as well.
- 13.
A case which might be construed as an example is discussed by Kearns (2016), p. 179: “John is in pain. This is a reason for me to give him painkillers. John is also in severe pain. This is also a reason for me to give him painkillers. Still, these reasons do not combine their strengths so that they together make a stronger case for my giving John painkillers than each does independently.” In this case it is not clear whether there is the same reason counted twice, or pain and severe pain are two distinct reasons. Clearly Kearns’s is not a formal discussion, so he does not say whether he considers aggregation undefined in these cases, thus making the operation partial, or whether—since one reason is included in the other—their aggregation just results in the weight of the weightier one. If this were the case for all reasons, the ordering would rather be a partial ordering, and the aggregation operation, with some additional conditions, would just be the lattice-theoretic join, with the consequence that the reasons structure would be much more well behaved. Elsewhere, Kearns (p. 186) suggests that evidence aggregates just when it is independent (in the epistemic case, we might look at conditional probabilities). Now, given Kearns’s view of normative reasons as evidential, one might hypothesize that normative reasons do not aggregate when they aren’t independent. A reviewer notes that, conceiving of practical reasons as evidential, one could get a measure of the strength of a reason p to do q as the strength of evidence p provides for the claim that one ought to do q. On these premises, one would get a measure out of evidential reasons quite easily, but only on the condition that the underlying reason structure is quite algebraically rich, something like a \(sigma-\)algebra, whose adequacy for practical reasons I criticized in fn. 9 (main concerns are the existence of a complement for every reasons, and the set theoretic (i.e. associative, distributive) interpretations of aggregation (and something like fission/intersection)). However, such an approach, while evidently fine, is contrary to the spirit of this paper, whose idea is to start with premises which are as neutral and as general as possible and see whether on formal grounds they point towards a specific, substantive theory. With regard to idempotency, quite interestingly (Fogal 2016) mentions a couple of examples of this phenomenon he calls out as double counting. While he does not specifically discuss the issue of reason aggregation in this context, he takes this phenomenon as an indicator of intracontextual variation. Since a more formal account is lacking, I can’t say how close his ideas are to those explored here; however, his account seems at least compatible with the view that reasons cannot be consistently aggregated. To be fair, it must be noted that Fogal is in general quite skeptical about the reasons-first approach—so this particular point seems in line with his broader take on the issue.
- 14.
An objection is the following: one cannot aggregate a reason with itself, but just with a qualitative identical (but numerically distinct) reason: in this case the whole idempotency discussion would become moot. One one hand, a metaphysical discussion about qualitative identity and numerical distinctness is well beyond the concerns of this paper and I think orthogonal to the issues it raises; on the other hand, for the idempotency discussion to go through it is enough just to consider every element of R as an equivalence class of all qualitatively identical reasons, or a representative member of such a class. After all, if they are qualitatively identical they surely share all properties relevant to their normative importance, ordering, and aggregation behavior.
- 15.
An immediate thought would be to disallow for idempotency of the aggregation operation, or impose some other ad hoc modifications. But I don’t see any independent reason to do so.
- 16.
Why, one may ask, the algebra of propositions, where set-theoretic union is idempotent, is indeed representable with a real-valued function? The algebra of propositions has stronger structural properties than our reason structures. Since there are real-valued representations of non-Archimedean, or at any rate non-associative idempotent structures (see Krantz et al. 1971, pp. 293–7), the most natural question is therefore whether there are slightly stronger conditions, replacing Archimedeaness, which would make the existence of a real-valued representation possible. Such conditions are usually “solvability” conditions, and are not readily adapted to reason structures.
- 17.
Cf. Narens (1985), Skala (1975). In particular Skala proves that an ordered commutative group can be embedded into an ultrapower of the (naturally ordered) additive group of the reals, which is non-Archimedean, that is, has a non-standard representation. However, an ordered commutative group is readily seen to be much stronger than plausible reasons structures: even setting aside other group axioms, associativity cannot always be assumed, and monotonicity is almost always considered unacceptable in practical reasoning. Moreover, since we will argue that there is no representation at all, we neglect here the very interesting characterizations of scales in terms of automorphisms, homogeneous and point-uniqueness features, and completions of gappy structures with ideal elements. I’d like to mention that an established and more well-known case of non-representability is that of lexicographical orders. Lexicographical orders give also a nice intuitive idea of why a nonstandard representation is needed, given that there is an “infinite” distance between some elements, as it were. Jack Woods (p.c.) suggests that, if the scale of reasons were non-standard, one could explain facts which are indeed reasons, but make an infinitesimal contribution. I don’t know whether there are other interesting connections to reason structures as they are presented here.
- 18.
It is well known that so-called impossibility theorems are not “absolute”, but “relative” to very specific models, usually isomorphic to the real numbers. This is the case especially with results from utility theory and social decision theory. But this needs not be the case. As Rizza (2016) shows, difference in modeling (for instance by requiring a representation into non standard natural or real numbers) turns impossibility theorems to (relative) possibility results.
- 19.
More precisely, let \((X, \mathcal {A}, \mu )\) be a measurable space. A function \(\mu \) is additive if:
-
1.
for all \(A \in \mathcal {A}, \mu (A) \ge 0\)
-
2.
\(\mu (A) \le \infty \)
-
3.
\(\mu (\bigcup ^{\infty }_{i = 1} A_i) = \sum ^{\infty }_{i = 1} \mu (A_i)\), for every i, j, s.t. when \(i \ne j, A_i \cap A_j \ne \emptyset \)
-
1.
- 20.
Were x, y not disjoint, we would need to consider submodularity and supermodularity instead. Nothing of philosophical significance turns on this. See also Torra et al. (2014).
- 21.
There seems to be some degree of similarity to cases where two pieces of defeasible epistemic evidence count separately in favor of something, but jointly against it: “People in the Smith family in village V tend to be either sober church-goers or heavily drinking atheists. Hearing that John, a person from village V, goes to church three times a week is evidence (reason to believe) that he might be a Smith (evidence 1). Likewise, hearing that John drinks heavily three times a week is evidence that John might be a Smith (evidence 2). However, evidence 1 and 2 together are reason to believe that John is not a Smith (because the Smiths are heavy drinkers or church-goers, but not both)”. However, weight and defeasibility should not be confused, at least in the case of normative reasons. I thank Eve Kitsik for bringing my attention to these cases and for the example.
- 22.
With regard to this particular example, a reviewer pointed out that that it is a bit weird that a reason involves someone being humiliated, but rather someone getting their comeuppance. This would then presuppose an enabler, and sorting out the conditionality and individuating correctly the basic reasons would then make the conflict go away. While the problem in this particular example may be one of description, one can come up with several, very basic examples that behave in non-additive ways: The fact that it rains is a reason not to go for a run. The fact that it is hot is a reason not to go for a run. (The fact that it rains) and (the fact that it is hot) is a reason to go for a run. If also in this case one thinks that the issue is merely one of description, then the burden of proof has now shifted, for we need a systematic account of how to proceed in these instances.
- 23.
Bader (2016), Sect. 6 is able to treat some cases additively, and some cases non-additively (pp. 53–4); although the his proposed account is quite different from mine, in that his involves factorizations and separability conditions (for which see Oddie 2001a, b), additive cases result from comparisons of members of the same subspaces of the Cartesian product of all factors; non-additive cases result from considering members from different subspaces. Nair (2016) proposes that independent reasons aggregate additively, where independency is cashed out via the (classical, in moral philosophy) distinction between derivative and non-derivative reasons, where reasons are independent just when they are non-derivative. However, it is not clear how to represent these ideas in a proper measurement-theoretic framework like the one employed in this paper. Beside requiring the aggregation operation be partial, i.e. undefined in certain cases, a plausible idea (mentioned in a previous footnote) would be to have a second ordering imposed on reasons, namely, a partial ordering, standing for a parthood relation, and allowing just for the aggregation of atomic reasons, i.e. reasons with no proper parts, which would ensure a certain independency. Nair’s derivative/non-derivative distinction is not, however, based on ideas having to do with part/whole. I take this just as a first approximation.
- 24.
Although non sufficiently.
- 25.
See Nagel (1970), Kagan (1988) and in particular Berker (2007). A couple of comments. First, it’s not clear why Berker requires uniqueness, a strong assumption which is not needed at all (there are possibly infinite scales for temperature, say, but with the same structure—we just consider what they have in common). Existence (of at least one scale) is enough. Second, some particularists and Berker (2007) (Sect 3 and fn 18), beside non-additivity, require a finiteness condition. It is not clear whether this means that (i) no representation is possible; (ii) the aggregation function would need (not) be just finitely, rather than countably, additive; or, even more strongly (iii) that such a function is not finitely expressible. Claim (i) is the argument I put forward in Sects. 7.3 and 7.4; claim (ii) seems hardly a relevant point (for normative matters); claim (iii) would need additional justification, for infinitary (or higher-order) languages are regularly used when pursuing (moral) metaphysics.
- 26.
This interpretation seems to support Lord’s and Maguire’s thought that “one strategy for avoiding weighted notions is to accept deep normative dilemmas (Lord and Maguire 2016a, p. 5)”.
- 27.
Issues about supervenience and grounding arguments against moral or normative reductive arguments come to mind. On this topic see Bader (2015a).
- 28.
For recent work on the topic, cf. Bader (2015b); on reason weighing and the role of context, cf. Philips (1987); for the question of holism and moral particularism, cf. McNaughton and Rawling (2008). Other impossibility theorems in the theory of individual versus collective judgment aggregation are reviewed in List (2012), List and Dietrich (forthcoming).
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Faroldi, F.L.G. (2019). The Scale of Normative Reasons. In: Hyperintensionality and Normativity. Springer, Cham. https://doi.org/10.1007/978-3-030-03487-0_7
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