2. Preliminaries

Let $S$ be a non-empty, finite set of states with typical element $s$ . These states may be provisional, not very specific, subject to revision, and so forth. Let $O$ be a non-empty set of outcome or consequence elements represented by lowercase Latin letters (except $s$ ). Alternatives or options are functions $X: S \to O$ that associate states with outcomes. So ${X\left( s \right) \in O}$ is the outcome of act $X$ in state $s$ . Let ${\cal A}$ denote the set of all options or alternatives. Outcomes themselves can be embedded in the set of options by the usual technique of identifying an outcome $x$ with constant alternatives ${\mathfrak{c}_{\cal x}}$ : for all $s \in S,{\mathfrak{c}_{\cal x}}\left( s \right) = x$ .

Let ${\rm{\Sigma }}$ be an algebra of events over $S$ . Because $S$ is finite, we can just take ${\rm{\Sigma }} = {2^S}$ . Let $\dot\succsim\subseteq {\rm{\Sigma }} \times {\rm{\Sigma }}$ be a binary relation on ${\rm{\Sigma }}$ , which we will interpret as giving qualitative probability comparisons. The expression $E{\rm{\;\;}}\dot\succsim{\rm{\;\;}}F$ means that the event $E$ is at least as probable as the event $F$ . Given the appeals to pseudo-rationalizability and imprecise probabilities (IP) that follows, it is natural to assume that $\dot\succsim$ is a “partial likelihood relation” because such relations admit “multiprior” or IP representations.Footnote ³ That is, if $\dot\succsim$ is a partial likelihood relation, there exists a set $\mathbb{P}$ of probability functions on $\left( {S,{\rm{\Sigma }}} \right)$ such that, for all $E,F \in {\rm{\Sigma }}$ , $E{\rm{\;\;}}\dot\succsim{\rm{\;\;}}F$ if and only if $P\left( E \right) \ge P\left( F \right)$ for all $P \in \mathbb{P}$ . Boyer-Kassem alleges that “Peterson’s treatment of uncertainties lacks generality” (2017a, p. 2026). Partial likelihood relations are rather general and, unlike what Peterson assumes, allow that some events cannot be compared in terms of their likelihoods. A limiting case worth noting is the one in which $\dot\succsim$ is representable by the set of all probability functions on $\left( {S,{\rm{\Sigma }}} \right)$ . It is plausible, then, that all types of probabilistic uncertainty of concern in the literature on the precautionary principle can be seen as special cases of partial likelihood relations.Footnote ⁴

A menu is a non-empty, finite subset of ${\cal A}$ . Menus represent decision problems. A choice function is a set-valued function $C:{2^{\cal A}}\backslash \left\{ \emptyset \right\} \to {2^{\cal A}}\backslash \left\{ \emptyset \right\}$ such that $C\left( {\cal S} \right) \subseteq {\cal S}$ and $C\left( {\cal S} \right) \ne \emptyset $ for any menu ${\cal S} \subseteq {\cal A}$ .Footnote ⁵ Choice functions can be thought of as selecting the acceptable/admissible/choiceworthy options in a menu. Selecting the best alternatives according to some complete and transitive preference relation generates a particular choice function, but there are choice functions that cannot be reduced to binary comparisons. In general abstract choice theory, certain properties of choice functions are especially interesting and play important roles. Perhaps the most central such property is Sen’s property α.

(α) $${\cal S} \subseteq {\cal T} \Rightarrow {\cal S} \cap C\left( {\cal T} \right) \subseteq C\left( {\cal S} \right)$$

Property α—also known as contraction consistency, heritage, and Chernoff—requires that acceptable options remain acceptable when other items are removed from the menu. Sen’s property β requires that if two options, $X$ and $Y$ , are both acceptable in a menu, and $X$ remains acceptable when the menu is expanded to include additional options, then $Y$ must also remain acceptable.

(β) $${\cal S} \subseteq {\cal T},X,Y \in C\left( {\cal S} \right),{\rm{\;\;and\;\;}}X \in C\left( {\cal T} \right) \Rightarrow Y \in C\left( {\cal T} \right)$$

Together, properties α and β characterize those choice functions that are rationalizable by a weak order (complete and transitive) preference relation; that is, $C$ satisfies α and β if and only if there exists a complete and transitive relation $\succsim \subseteq {\cal A} \times {\cal A}$ such that, for all menus ${\cal S} \subseteq {\cal A}$ ,

$$C\left( {\cal S} \right) = \left\{ {X \in {\cal S}:X \ \succsim \ Y{\rm{\;for\ all\;\;}}Y \in {\cal S}} \right\}.$$

In other words, such choice functions can be regarded as choosing the best elements from a menu according to the relation $\succsim$ . In the presence of property α , property γ is strictly weaker than property β .

(γ) $$C\left( {\cal S} \right) \cap C\left( {\cal T} \right) \subseteq C\left( {{\cal S} \cup {\cal T}} \right)$$

Options that are acceptable in both menu ${\cal S}$ and menu ${\cal T}$ are acceptable in the menu ${\cal S} \cup {\cal T}$ . Together, property α and property γ characterize binary choice. A binary choice function is one for which there exists some binary relation—not necessarily a weak order—that rationalizes it.

Following a common convention in the more formal literature on the precautionary principle (e.g., Peterson, Reference Peterson2006; Boyer-Kassem, Reference Boyer-Kassem2017a; Stefánsson, Reference Stefánsson2019), we use letters $p,q, \ldots $ to denote catastrophic or fatal outcomes in $O$ (except for lowercase letters corresponding to the designation of an act: so ${\cal x},{{\cal x}_i}$ , etc. are not necessarily catastrophic when representing generic outcomes of an act $X$ ). I will not offer any substantive account of such outcomes here but will just flag that the ability to draw such a line in a substantive way makes strong measurability assumptions.Footnote ⁶

3. A generalization of Peterson’s first impossibility theorem

Peterson states impossibility results for two versions of the precautionary principle. I will state and prove generalizations of each. Peterson’s first impossibility result relies on a formulation of the precautionary principle given by his $PP\left( \alpha \right)$ . Informally, “If one act is more likely to give rise to a fatal outcome than another, then the latter should be preferred to the former; and if the two acts are equally likely to give rise to a fatal outcome, then they should be equi-preferred” (Reference Peterson2006, p. 597). In contrast to Peterson’s $PP\left( \alpha \right)$ , here, PP( $\alpha $ )_c is formulated for choice functions rather than binary relations, let alone a particular type of binary relation like a weak order. Say that some outcome ${\cal x}$ is not more choiceworthy than another outcome y when ${\mathfrak{c}_y}$ is acceptable in a choice between it and ${\mathfrak{c}_{\cal x}}$ .

Let X be an alternative such that there is at least one outcome that is not more choice worthy than p. Then,
(1) if the likelihood of an outcome that is not more choiceworthy than p is greater for X than for Y, X is not acceptable in the menu {X, Y}; (2) if an outcome that is not more choiceworthy than p is as likely for X as for Y, then neither X nor Y is ruled out.	(PP( $\alpha )_c$ )

Informally, PP( $\alpha )_c$ says that if one act is more likely to lead to a catastrophic outcome than another, then the former is unacceptable. If the two acts are equally likely to yield a catastrophic outcome, then neither is uniquely acceptable in a binary choice between them. (More formal statements of the various assumptions included in the theorems are provided in the Appendix.) Even if proponents of precautionary reasoning intend a stronger principle that legislates choice even when the sorts of likelihood comparisons made in PP( $\alpha)_c $ are unavailable, it seems plausible to think such a principle would imply something very much like PP( $\alpha)_c $ in those special cases in which such likelihood comparisons are available.Footnote ⁷ Our assumptions allow that some (other) outcomes may not be comparable in terms of likelihood and that some acts (including $X$ and $Y$ themselves in the statement of PP( $\alpha)_c $ may not be comparable in terms of a binary preference relation.

Next, Cov_c is a choice-theoretic analogue of Peterson’s covariance condition intended to capture the idea that “everything else being equal, the less likely a bad outcome is to occur, the better” (Reference Peterson2006, p. 597).

( $\text Cov_c $ )

Effectively, alternative $X^{\rm{\prime}}$ is just like $X$ , only more likely to yield ${{\cal x}_j}$ than ${{\cal x}_i}$ . Because ${\mathfrak{c}_{{{\cal x}_j}}}$ is uniquely acceptable in the choice between it and ${\mathfrak{c}_{{{\cal x}_i}}}$ , according to Cov_c , $X^{\rm{\prime}}$ is uniquely acceptable in a binary choice between it and the original alternative $X$ .

We can now state a rather substantial generalization of Peterson’s first impossibility result.

It is worth stressing that theorem 1 requires neither the dominance nor the ordering assumptions of Peterson’s impossibility result for his analogue of PP( $\alpha $ )_c (2006, Theorem 1, p. 600). So, in addition to generalizing Peterson’s $PP\left( \alpha \right)$ and covariance conditions, Theorem 1 generalizes Peterson’s result by dropping two further assumptions completely. Among other things, this helps us to isolate the source of inconsistency.

4. Incommensurability

In general, talk of (unique) admissibility cannot be construed as coded talk of an alternative being better than others in a menu. The latter description assumes a binary relation that determines choiceworthiness. But only in special cases—namely, when properties α and β are both satisfied—does choiceworthiness reduce to binary comparisons.

Peterson makes the standard assumption that preferences are given by a weak order.Footnote ⁸ This implies that any two alternatives $X$ and $Y$ can be compared, with $X$ at least as good as $Y$ or $Y$ at least as good as $X$ . Neither PP( $\alpha)_c $ , 2, nor, consequently, theorem 1 appeals to a weak order. In particular, the impossibility reported there does not rely on any general commensurability assumptions. The theorem does not even assume that the choice function satisfies the central internal coherence constraint property $\alpha $ , only that $C$ is, in fact, a choice function. So the prospect of incommensurability providing a context for coherent application of the precautionary principle as formulated in PP( $\alpha $ )_c seems dim by theorem 1. Nevertheless, incommensurability is a central issue in debates about the precautionary principle, and it is important to explain how relaxing the assumption of a weak order desirability ranking in generalizing Peterson’s impossibility results allows for forms of incommensurability. I will illustrate this with the property of path independence, although I stress that the generalizations do not assume even this much.

In order to respond to Boyer-Kassem’s criticism—which we return to in section 5—that many and some of the most important applications of the precautionary principle are to cases lacking determinate desirability comparisons, cases in which certain outcomes are incommensurable, Peterson considers extending the scope of his observations so that it applies to incommensurable options. He then suggests a choice rule for handling such options: “If $X$ and $Y$ are incomparable, and the agent is rationally permitted to choose $X$ , then it is also rationally permitted to choose $Y$ ” (Peterson, Reference Peterson2017, p. 2037, with options capitalized for consistent notation). Peterson concedes that “[t]his new condition is controversial (because some think it may make us vulnerable to money pump arguments) […] I am not claiming that this is the correct analysis of incomparability, but the example suggests that if we knew how incomparable values should be linked to rational choice it would also be possible to construct a corresponding impossibility theorem” (Peterson, Reference Peterson2017, p. 2037). It is about this point that Boyer-Kassem expresses skepticism that the program sketched can be carried out in detail because various assumptions, including the Archimedean and weak order preference conditions, require modification.

That the precautionary principle is inconsistent with some choice rule or other is not by itself a challenge to the precautionary principle. For a plausible challenge, it is important either to identify a plausible decision rule for incommensurability or to show that no choice rule at all supports the coherent articulation of precaution along the lines Peterson pursues. The results reported in sections 3 and 5 pursue the second strategy. In this section, however, I want to consider not the rule that Peterson tentatively suggests but path-independent choice. This is a natural and nontrivial generalization of the standard assumption of a weak order that has been used in thinking about incommensurability, as I explain later in the article.Footnote ⁹ Of course, adding the assumption of path independence to my generalizations of Peterson’s impossibility results for the precautionary principle will not help skirt the limitations reported there, but my point is that the leaner observations recorded here apply to path-independent choice functions, among others. As a result, simply appealing to incommensurability—which can be naturally modeled for path-independent choice functions—is no automatic out.

Intuitively, path independence says that the order of consideration or presentation of options in a menu does not affect the final choice.

(PI) $$C\left( {{\cal S} \cup {\cal T}} \right) = C\left( {C\left( {\cal S} \right) \cup C\left( {\cal T} \right)} \right)$$

In a sense, path-independent choice functions allow harder choice problems ( $S \cup T$ ) to be decomposed into easier problems ( $S$ and $T$ ). That choice from a menu is independent of the way options are divided up for consideration is an important feature of standard weak order rationalizability that path independence retains (cf. Danilov and Koshevoy, Reference Danilov and Koshevoy2005). A consequence of path independence, then, is that certain forms of manipulation are excluded. A decision maker’s choice from a menu cannot be manipulated by presenting options in a different order.Footnote ¹⁰ Aizerman and Malishevski show that such choice functions admit an interesting representation. If $C$ is a path-independent/pseudo-rationalizable choice function, then there is a set ${\{ {\succsim_i}\} _{i \in I}}$ of weak orders such that, for any menu ${\cal S}$ ,

(1) $${C\left( {\cal S}\right) = \bigcup\limits_{i \in I} {max_S} \ \succsim_i}.$$

In other words, rather than selecting the optimal elements in a menu with respect to a single preference relation, a path-independent choice function selects those elements that are optimal according to at least one preference relation in the set ${\{ {\succsim_i}\} _{i \in I}}$ . One interpretation of the Aizerman and Malishevski decomposition is that there is indeterminacy in preference when $\left| I \right| \gt1$ . For example, it could be the case that, according to one permissible way of evaluating things, $X{ \succ _i}Y$ , and according to another, $Y{ \succ _j}X$ .Footnote ¹¹

So, if $C$ is path independent, then there exists a set of weak orders ${\{ {\succsim_i}\} _{i \in I}}$ such that, for any menu ${\cal S}, C\left( {\cal S}\right) = \bigcup\limits_{i \in I} {max_S} \ \succsim_i$ . In words, $C$ selects those alternatives in ${\cal S}$ that are maximal according to some relation ${\succsim_i}$ in the set ${\{ {\succsim_i}\} _{i \in I}}$ . The elements of ${\{ {\succsim_i}\} _{i \in I}}$ can be interpreted as rival but permissible assessments of the alternatives in terms of desirability. Such sets arise in cases of indeterminacy or vagueness in desirability assessments. And at least in some such cases, the rankings may correspond to various permissible ways of trading off certain fundamental valuations of the alternatives, certain ways of compromising between different values. There are a number of ways to define categorical desirability from a set of desirability orderings. The simplest may be to just take the intersection of the orderings: $\succsim = {\bigcap}_{i \in I}\,\succsim_i$ (e.g., Sen, Reference Sen2004, p. 672). This proposal allows for the possibility that $X \succ Y$ even if $X\,\succsim_i Y$ for some $i \in I$ . That is, it is possible that strict categorical desirability comparisons hold that are not unanimously shared among the ${\succsim_i}$ . Another proposal is for categorical desirability to consist of the unanimously held strict desirability comparisons and the unanimously held indifferences (cf. Sen, Reference Sen2004, p. 674). Levi proposes a more complicated account according to which categorical desirability consists of unanimous strict desirability, unanimous indifference, and unanimous weak desirability (Reference Levi, Basu and Kanbur2008). In all three of these proposals, if $X{ \succ _i}Y$ and $Y{ \succ _j}X$ for some $i,j \in I$ , then $X$ and $Y$ are categorically incommensurable.

Like Peterson’s rule, path independence is phrased in choice-theoretic rather than relation-theoretic terms—although, as we have seen, there is a set-based relation-theoretic representation via pseudo-rationalizability. Path independence, however, does not vindicate Peterson’s rule for incommensurable options.

Example 1. Let ${\cal A} = \left\{ {X,Y,Z} \right\}$ . Let $C$ be a path-independent choice function on ${\cal A}$ that is rationalized by $\left\{ {{\succsim_1},{\succsim_2}} \right\}$ , defined as follows:

$$X{ \succ _1}Y{ \succ _1}Z$$

$$Z{ \succ _2}Y{ \succ _2}X$$

Here, and $Y$ are categorically incommensurable. Relative to ${ \succ _1}$ , alternative $X$ is more desirable; according to ${ \succ _2}$ , the opposite comparison obtains. Peterson’s suggested rule would imply that $Y$ is acceptable in the menu $\left\{ {X,Y,Z} \right\}$ because $X$ is. But this is not the case for $C$ : $C\left( {\left\{ {X,Y,Z} \right\}} \right) = \left\{ {X,Z} \right\}$ . The alternative $Y$ is not maximal according to any permissible evaluation of the options in that menu.

The fundamental idea is that $X \in C\left( {\left\{ {X,Y} \right\}} \right)$ does not imply that, for some categorical desirability weak order $\succsim$ , $X \,\succsim\, Y $ . It could be that $X$ is at least as desirable as $Y$ , but it could also be the case that $X$ is strictly more desirable than $Y$ , or even—and this is the key point—that $X$ and $Y$ are incommensurable. In example 1, $X$ and $Y$ are categorically incomparable; it is indeterminate whether $X$ is more desirable than $Y$ or $Y$ is more desirable than $X$ .

Path independence is one important way of thinking about incommensurability. I will return briefly to it in section 7. My point here is not to argue that path independence is definitely mandatory for rational choice but to illustrate how theorems 1 and 2—which assume only a choice function and not path independence—extend the limitations for the precautionary principle to forms of incommensurability. Arguably, any plausible theory of decision making is committed to the bare assumption of choice function.

5. A generalization of Peterson’s second impossibility result

Peterson’s second theorem is the more “refined and powerful one,” according to Boyer–Kassem (Reference Boyer-Kassem2017a, p. 2028). The formalization of the precautionary principle in this theorem is the weakest and most general that Peterson considers, “so weak that it cannot reasonably be refuted by any advocate of the precautionary principle” (Reference Boyer-Kassem2017, p. 599). Still, the theorem is not fatal for the precautionary principle, Boyer-Kassem claims, because it assumes general commensurability in desirability. And he finds Peterson’s subsequent reply to that worry unconvincing: “When answering the second part of my objection, Peterson suggests an escape route: change the scope of the theorem so that it applies to incommensurable outcomes. I doubt this can be done—not only does the Archimedean condition need a reformulation, but also the total order condition” (Boyer-Kassem Reference Boyer-Kassem2017b, p. 2040). I have explained how incommensurability is consistent with the extremely minimal assumption of a choice function. My task now is to generalize the relevant precautionary principle and Archimedean condition accordingly.

Peterson’s informal statement of his weakest formulation of the precautionary principle is this: “If one act is more likely to give rise to a fatal outcome than another, then the latter should be preferred to the former, given that (i) both fatal outcomes are equally undesirable, and (ii) not negligibly unlikely, and (iii) the nonpreferred act is sufficiently more likely to lead to a fatal outcome than the preferred one” (Reference Peterson2006, p. 599). In his formalization of this property, Peterson assumes that the relevant fatal/catastrophic outcome for both acts is equi-preferred to the best fatal outcome (namely, $p$ ) (Reference Peterson2006, p. 601). Clause (ii) implies that we are outside the context of the potential application of the de minimis principle, according to which sufficiently improbable outcomes can be ignored or treated very differently in deliberation (Peterson, Reference Peterson2002; Lundgren and Stefánsson, Reference Lundgren and Orri Stefánsson2020). The choice-theoretic version replaces assumptions about preference and desirability with assumptions about acceptability (and again, we do not assume completeness of the partial likelihood relation).

Let Y be an alternative that is more likely to result in a catastrophic outcome than alternative X. Y is not acceptable in the menu {X,Y} if
(1) there is exactly one outcome of each of X and Y—x i and y j , respectively—that is catastrophic, and neither catastrophic outcome is more choice worthy than the other; (2) neither x i nor y j is negligibly unlikely, and (3) Y is sufficiently more likely to lead to y j than X is to lead to x i .	( $\text{PP}(\delta)_c $ )

PP(δ)_c generalizes Peterson’s constraint because it does not require that both catastrophic outcomes are equally undesirable, although the property is implied by that special case. It requires instead that neither catastrophic outcome is more choiceworthy than the other. PP(δ)_c only governs cases of acts that have a single catastrophic outcome. This is suggested by Peterson’s more formal articulation of this version of the principle in his appendix. As with PP(α)_c, this special case is plausibly implied by any stronger precautionary principle that governs additional cases, such as when acts have multiple possible catastrophic outcomes. Unlike Peterson’s version, PP(δ)_c does not assume that the relevant catastrophic outcomes are indifferent to $p$ , although again, clause 1 is implied by that special case as well. We also do not assume, as Peterson does, that $X$ is strictly preferred to $Y$ , only that $Y$ is not acceptable in the binary choice between it and $X$ .

The only other assumption of Peterson’s that is needed to state a choice-theoretic generalization of his second impossibility theorem is a version of what he calls an Archimedean condition. In spirit, Peterson’s Archimedean condition is similar to the Archimedean or continuity assumptions associated with von Neumann and Morgenstern expected utility theory. It captures a sense in which trade-offs should be considered in choice. As Peterson puts it, “advocates of the precautionary principle must be willing to admit that, to some extent, both the likelihood and the desirability of an outcome matter” (Reference Peterson2006, p. 599). His informal statement of this assumption is as follows: “If the relative likelihood of a nonfatal outcome is increased in relation to a strictly better nonfatal outcome, then there is some (nonnegligible) decrease of the relative likelihood of a fatal outcome that counterbalances this precisely” (Reference Peterson2006, p. 599). As I explain in section 6, it is the assumption of a trade-off Archimedean condition on which many objectors focus. My informal statement of the choice-theoretic version of this condition again generalizes assumptions about preference or desirability to choiceworthiness and tracks Peterson’s formal version.

There are atleast two alternatives $X,Y \in {\cal A}$ such that $Y$ is acceptable in the menu $\left\{ {X,Y} \right\}$ , but
(1) $X\;{\rm{and}}\;Y\;{\rm{have\;the\;same\;set\;of\;possible\;outcomes}},{\rm{\;}}$ (2) ${\rm{no\;outcome\;for\;either\;alternative\;is\;negligibly\;unlikely}},$ (3) ${\rm{there\;is\;exactly\;one\;catastrophic\;outcome\;}}{x_i},{\rm{and\;}}$ (4) $Y\;{\rm{is\;sufficiently\;more\;likely\;than\;}}X\;{\rm{to\;lead\;to\;}}{x_i}.$	(Archc)

Arch_c is considerably weaker than Peterson’s Archimedean condition on a few counts. First, by using choice functions, we do not require indifference between the acts $X$ and $Y$ , only that $Y$ is acceptable in the menu $\left\{ {X,Y} \right\}$ . This latter requirement is consistent with indifference between $X$ and $Y$ , with $Y$ being strictly preferred to $X$ , and also with $X$ and $Y$ being incommensurable according to a categorical desirability relation.Footnote ¹² Second, Peterson’s condition quantifies over all alternatives. By contrast, Arch_c only asserts the existence of a pair of options in ${\cal A}$ meeting the stated assumptions.Footnote ¹³ Third, Arch_c does not commit to a particular means—such as how many likelihood relations between outcomes are modified—by which $Y$ attains its admissibility in the choice between it and $X$ . Arch_c is so weak that it does not really resemble an Archimedean condition, that label being retained for continuity with the literature on Peterson’s results. The interest in working with a much weaker condition is not limited to the fact that the associated mathematical results are stronger; the main point is that the assumption is more difficult to deny. If a proponent of precaution wishes to deny Arch_c , the scope of the precautionary principle is more significantly restricted still. Its applicability demands that not even two alternatives as are mentioned in Arch_c exist, that not even this much trade-off reasoning is allowed. The existence of such a pair of alternatives is, of course, consistent with the incommensurability of the options $X$ and $Y$ and with extremely widespread incommensurability in general.

We can now state the generalization of Peterson’s second impossibility theorem.

Boyer-Kassem suggests that there is already a conflict between Peterson’s most general statement of the precautionary principle and his Archimedean condition without invoking the additional assumptions of Peterson’s theorem (Reference Boyer-Kassem2017a, p. 2031). Theorem 2 verifies that this is true even when those assumptions are stated choice-theoretically. As with theorem 1, theorem 2 generalizes Peterson’s corresponding result, not only by substantially weakening the precautionary and Archimedean assumptions but also by dropping two other assumptions—those of a weak order desirability ranking and a certain dominance condition—altogether.

6. Responses to some reservations about Peterson’s results

About Peterson’s impossibility theorems, Sprenger writes, “The source of the problem is the intuition that both the probability and desirability/potential harm of an outcome matter and that they can, to some extent, be traded off against each other. This view is deeply entrenched in most accounts of rational decision making” (Reference Sprenger2012, pp. 883–884). It is the Archimedean condition that encodes some of this deeply entrenched intuition. The Arch_c formulation of Peterson’s Archimedean condition and theorem 2 allow us to offer responses to three objections that have been voiced in the literature. Although I will not argue that Peterson is definitely correct on all of the issues involved, I think the results presented in this article show that some objections focus on inessential features of Peterson’s observations and thereby fail to be satisfactory responses to some of the concerns that they raise.

First, Boyer-Kassem objects to the commensurability of catastrophic and noncatastrophic outcomes assumed by Peterson’s original formulation of the Archimedean condition. He writes, “if one accepts PP (in Peterson’s sense), one is committed to this view of incommensurability between fatal and nonfatal outcomes. Now, the problem is that the Archimedean condition is saying exactly the opposite: by stating that a change in the likelihood of nonfatal outcomes can be compensated by a change in the likelihood of fatal outcomes, it assumes that the desirability of fatal and nonfatal outcomes can be compared—even if one change of likelihood has to be much smaller than the other—and thus that fatal and nonfatal outcomes are commensurable” (Reference Boyer-Kassem2017a, p. 2031). The source of the conflict, Boyer-Kassem seems to be suggesting, is the commensurability between certain types of outcomes—namely, fatal and nonfatal—that is illicitly assumed by the Archimedean condition. As it is stated here, however, Arch_c is formulated in terms of a choice function rather than a binary desirability relation. Given our (lack of) assumptions about $C$ , we cannot infer commensurability from acceptability. Furthermore, Arch_c only makes an assumption about a single pair of alternatives. So, even if Arch_c were making an assumption about the commensurability of fatal and nonfatal outcomes, the scope of the commensurability assumed would be very minimal.

These same points regarding Arch_c can be used to address a second, related objection that Boyer-Kassem raises: “My second criticism against the Archimedean condition is that it assumes a value commensurability between outcomes in general […] the Archimedean condition assumes that all outcomes can be compared, so that changes in the likelihood of some outcomes can be compensated by changes in the likelihood of some other outcomes. This gives another reason to reject the Archimedean condition” (Reference Boyer-Kassem2017a, p. 2031). Here, the worry concerns the assumption of commensurability “in general” rather than just between fatal and nonfatal outcomes. But to repeat, Arch_c does not assume general value commensurability, certainly not that all outcomes can be compared. For one thing, the assumptions in place on choice functions in theorem 2 do not secure comparability of alternatives (or all constant alternatives/outcomes) by a binary desirability relation. For another, Arch_c makes a claim only about a single pair of alternatives.

A third objection to interpreting Peterson’s results as trouble for the precautionary principle comes from Steel (Reference Steel2015). Steel’s objection, like Boyer-Kassem’s, focuses on the Archimedean condition: “For what increase of credibility of catastrophe relative to poor would precisely offset the advantage accruing from the increase of the credibility of excellent relative to good? I submit that there is no non-arbitrary way to answer such a question” (Reference Steel2015, p. 42). Steel is referring here to Peterson’s informal statement of his Archimedean condition. The “precise offsetting” shows up in Peterson’s more formal statement of the condition as a claim about indifference between two alternatives: an initial alternative and one that results from it by increasing the likelihoods of both a catastrophic outcome and an excellent outcome. But Arch_c and t 2 help us to see that the focus on “precisely offsetting” is something of a red herring because, as the theorem establishes, an assumption of indifference is inessential to the derivation of a contradiction.

I do not anticipate that the approach presented here will have left Peterson’s critics and advocates of the precautionary principle more generally bereft of replies. I am not even attempting to reply to all criticisms that have been voiced about Peterson’s interpretation of his results, focusing in this section on some concerns raised about the Archimedean condition. Perhaps some of the foregoing objections can be repurposed to articulate new objections to the choice-theoretic assumptions presented here, even if, as they’re stated, they fall short. Alternatively, one might object to the foregoing formulations of the precautionary principle on different grounds.Footnote ¹⁴ I hope, however, that this approach can help to structure further debate and clarify which issues are really at stake.

7. An alternative approach to the precautionary principle

In essence, the impossibility we encounter in theorem 2 is a conflict between trade-off reasoning, in the form of Arch_c, on the one hand, and precautionary reasoning, in the form of PP(δ)_c, on the other. One possibility worth exploring is that the problem arises because both types of reasoning are applied simultaneously. A different approach, which I will now consider, would be to apply them lexicographically.Footnote ¹⁵

The most developed account of weighing trade-offs is, of course, expected utility theory. Standard expected utility theory, however, might—and has been thought to—fall short when it comes to deep uncertainty and value incommensurability. Certainly, many fans of the precautionary principle think so. States of uncertainty are restricted to numerically determinate probability judgments. And a basic starting assumption in the classical formulations of expected utility is that desirability weakly orders the alternatives so that there is no incommensurability. Generalizations of expected utility theory have been developed that drop both of the evidently restrictive assumptions of completeness and numerically determinate probabilities. A natural and very general extension of the standard theory allows for sets of probabilities and sets of utilities rather than the assumptions of a single probability function and a single utility function for a decision maker.Footnote ¹⁶ Before, in motivating partial likelihood relations, I mentioned how two events may fail to be comparable in terms of likelihood. Similar remarks apply when we come to the expected utilities of options. Assume for the moment that desirability is a weak order and has a utility representation. If $\mathbb{P}$ is a set of probabilities representing $\dot\succsim$ , it could be the case that, relative to one $P \in \mathbb{P}$ , $E{U_P}\left( X \right) \gt E{U_P}\left( Y \right)$ , whereas relative to another $P^{\rm{\prime}} \in \mathbb{P}$ , $E{U_{P^{\rm{\prime}}}}\left( Y \right) \gt E{U_{P^{\rm{\prime}}}}\left( X \right)$ . In such cases, one might think, and some decision theories imply, that considerations of expected utility secure no categorical preference between $X$ and $Y$ .

A natural generalization of the injunction to maximize expected utility when probabilities and utilities may not be determinate is $E$ -admissibility, propounded prominently by Levi (e.g., Reference Levi1980). Where $\mathbb{P}$ is a set of probabilities and $\mathbb{U}$ a set of utilities, the $E$ -admissible options are those that maximize expected utility with respect to some $P \in \mathbb{P}$ and some $U \in \mathbb{U}$ . Let’s suppose that utility is determinate for ease of exposition. In Levi’s interpretation, elements of $\mathbb{P}$ are permissible probability assessments, so $E$ -admissibility restricts choice to those options that are best according to some permissible way of evaluating risk. Put differently, and abstracting from reference to states for simplicity, the $E$ -admissible options are given by the set

$$\left\{ {X \in {\cal S}:\exists P \in \mathbb{P} \forall Y \in {\cal S}\;\;E{U_P}\left( X \right) \ge E{U_P}\left( Y \right)} \right\}.$$

Note that a choice function induced by $E$ -admissibility is pseudo-rationalizable, with the set of expected utility rankings as the rationalizing set of weak orders. $E$ -admissibility reduces to expected utility maximization when $\mathbb{P}$ is a singleton (given our assumption that utility is determinate also). $E$ -admissibility has been considered by some to be excessively permissive (for a recent example of such a critique, see, e.g., Mogensen and Thorstad, Reference Mogensen and Thorstad2022). In response to this concern, Levi and others have considered a certain combination of $E$ -admissibility with some other rule applied lexicographically as a second-tier criterion for narrowing the set of admissible options. The most prominent such two-tiered rule is especially interesting for my purposes in the present section, but presenting it requires a bit more setup.

In earlier work, Hansson suggests a model of precautionary reasoning differing from the ones we have been considering so far: “The maximin rule can be used as a formal version of the precautionary principle” (Reference Hansson1997, p. 293). Each alternative is associated with a security level, that alternative’s worst outcome for any possible state. Maximin selects those alternatives that have maximal security levels, thereby maximizing the minimum. As with other proposed formulations of the precautionary principle, the extent to which maximin ignores trade-offs has been the subject of criticism Reference Luce and Raiffa1957, pp. 279-280). Some find that even as a formulation of the precautionary principle, the conditions of applicability of maximin are either too restrictive or that the rule disregards information outside of those conditions. Bartha and DesRoches think identifying the precautionary principle with maximin makes it too discontinuous with expected utility theory: “we reject the Maximin interpretation of PP because such an identification makes it impossible to clarify the relationship between PP and ordinary expected utility maximization. Maximin operates in the framework of decisions under ignorance; standard decision theory applies to decisions under risk. There is no element in common. [ $ \ldots $ ] our goal is to show that PP is more closely related to standard decision theory than the Maximin interpretation allows” (Reference Bartha and Tyler DesRoches2021, p. 8709). On its own, maximin may well be too conservative, wasteful of valuable information, subject to convincing counterexamples, and so forth. But, following an important strand of research in decision theory and pace Bartha and DesRoches, I want to consider a role for maximin in a unified setting that allows for both ignorance and risk.

In the setting of imprecise probabilities, the manifestation of the conservative, maximin approach to decision making is sometimes called ${\rm{\Gamma }}$ -maximin (Gärdenfors and Sahlin, Reference Gärdenfors and Sahlin1982; Gilboa and Schmeidler, Reference Gilboa and Schmeidler1989). Suppose that $\mathbb{P}$ is a set of probability distributions on some common measurable space. According to ${\rm{\Gamma }}$ -maximin, we should restrict choice to those options with the greatest minimal expected utility. Continuing with the simplifying assumption that utility is determinate, the choice set is given by

$$\left\{ {X \in {\cal S}:\mathop {{\rm{inf}}}\limits_{P \in \mathbb{P}} E{U_P}\left( X \right) \ge \mathop {{\rm{inf}}}\limits_{P \in \mathbb{P}} E{U_P}\left( Y \right){\rm{\;\;for\ all\;\;}}Y \in {\cal S}} \right\}.$$

Under complete uncertainty—when no probability distributions are excluded from $\mathbb{P}$ —maximin and ${\rm{\Gamma }}$ -maximin coincide (e.g., Berger, Reference Berger1985, p. 216). ${\rm{\Gamma }}$ -maximin is a significant generalization, and it clarifies at least one way of understanding the relationship between the precautionary principle and ordinary expected utility maximization. When $\mathbb{P}$ is a singleton (and utility is determinate), ${\rm{\Gamma }}$ -maximin and expected utility maximization coincide. Perhaps ${\rm{\Gamma }}$ -maximin could serve as the sort of bridge Bartha and DesRoches are looking for, connecting maximin and (even substantial generalizations of) standard decision theory.Footnote ¹⁷

I’ll call the rule that first eliminates all options that are not $E$ -admissible and then applies ${\rm{\Gamma }}$ -maximin to the surviving options $E + {\rm{\Gamma }}$ . This two-tiered or lexicographic rule has been studied in the literature on IP decision theory (e.g., Levi, Reference Levi1986; Seidenfeld et al., Reference Seidenfeld, Schervish and Kadane2012). As with ${\rm{\Gamma }}$ -maximin and $E$ -admissibility, $E + {\rm{\Gamma }}$ reduces to expected utility maximization when $\mathbb{P}$ is a singleton. To the extent that $E$ -admissibility is a generalized form of trade-off reasoning and ${\rm{\Gamma }}$ -maximin captures some of the precautionary impulse, $E + {\rm{\Gamma }}$ is one way of reconciling these general reasoning styles. Whether $E$ -admissibility satisfies Arch_c is more or less a matter of the richness of the set of outcomes: provided two such alternatives as described in the four clauses exist, $Y$ will be acceptable in a choice between it and $X$ so long as it maximizes expected utility with respect to some $P \in \mathbb{P}$ . In some cases, ${\rm{\Gamma }}$ -maxmin will satisfy PP(δ)_c. The key clause is the third: $Y$ is sufficiently more likely to lead to its catastrophic outcome than $X$ is to lead to its catastrophic outcome. If the greater likelihood of $Y$ ’s catastrophic outcome makes it such that $X$ ’s minimal expected utility (across $P \in \mathbb{P}$ ) is greater than $Y$ ’s minimal expected utility, then PP(δ)_c is satisfied: $Y$ is not acceptable in a choice between it and $X$ . To repeat, for $E + {\rm{\Gamma }}$ , both $X$ and $Y$ must maximize expected utility with respect to some probability assessment that has not been excluded (that is, some $P \in \mathbb{P}$ ).

Two further comments on ${\rm{\Gamma }}$ -maximin and PP(δ)_c are in order. One way of securing a tighter link between the two is to use ${\rm{\Gamma }}$ -maximin to flesh out the content of $Y$ ’s being “sufficiently more likely” to lead to its possible catastrophic outcome than $X$ is to lead to its possible catastrophic outcome. If we define “sufficiently more likely” here as the likelihood of Y’s catastrophic outcome is greater than that of X’s and is such that X’s minimal expected utility is greater than Y’s minimal expected utility, the satisfaction of PP(δ)_c is secured. In the context of $E + {\rm{\Gamma }}$ , PP(δ)_c would be secured only at the second tier, for the restricted application to $E$ -admissible options. Alternatively, rather than appealing to ${\rm{\Gamma }}$ -maximin as a second-tier criterion, we could simply impose ${\rm PP({\delta )_c}}$ or a suitable strengthening of it as a tie-breaking rule after first restricting choice to $E$ -admissible options.

In some recent publications, Bartha and coauthors explore an alternative lexicographic model of precautionary reasoning (Reference Bartha and Tyler DesRoches2017; Reference Bartha and Tyler DesRoches2021; Reference Bengio, Russell, Musk, Wozniak, Mostaque and Yang2023). Their approach requires avoiding catastrophe first and then maximizing expected utility—a sort of reversal of the order of operations when compared to $E + {\rm{\Gamma }}$ . In favor of $E + {\rm{\Gamma }}$ , one might point out that choiceworthy options are forced to pass trade-off analysis with respect to at least some feasible probability assessment, which, as proponents and critics alike point out, can be quite a weak requirement in the presence of deep uncertainty. We might think of $E + {\rm{\Gamma }}$ as making precautionary reasoning palatable to (broad-minded) expected utility partisans. Strong partisans of precautionary reasoning, on the other hand, might find Bartha et al.’s lexical approach more congenial to their initial inclinations by not subordinating precaution to trade-off reasoning but, instead, advancing the converse subordination. Others may find the lexicographic approach objectionable in general, reasoning that vast advantages in security are worth some sacrifice in trade-off superiority and, similarly, that vast advantages on the trade-off ledger surely license at least some sacrifice in security level.Footnote ¹⁸ Whether such a view can be precisely and coherently articulated remains to be seen, as far as I am aware.Footnote ¹⁹ At any rate, there are multiple routes for further exploring a reconciliation of trade-off and precautionary reasoning.