As we have seen, the appeal to decision theory plays a crucial role in what we take to be the strongest case for the possibility of radical interpretation. In this section, we look more closely at existing decision theories, and the representation theorems that have been proven for them, in in order to evaluate their suitability to play this role. Starting with Jeffrey’s decision theory and representation theorem—to which both Davidson and Lewis appeal—and moving on to further existing theories and theorems, we show that none of them is well-suited to play its designated role.Footnote 14 The trouble is that a radical interpreter who relies on the existing representation theorems faces a trilemma, since for all these theorems, at least one of the following is true:Footnote 15
Underdetermination. The theorems deliver a set of probability functions that disagree even in their rankings of the objects of belief, so a rational agent’s preferences are compatible with several mutually incompatible interpretations of her beliefs.
Inapplicability. The theorems impose constraints on preference that are both too normatively and psychologically demanding, and hence do not come close to being satisfied by any actual agent nor even by ideally rational agents.
Inaccessibility. The frameworks within which the theorems are proven contain objects the preference ranking of which the interpreter could not know on the basis of knowledge of the physical truths alone.
This trilemma arises as a direct result of the role decision theory must play for the interpretationist project to succeed. First, interpretationists take decision theory to provide a constitutive account of the attitudes, as specifying what it is for an agent to have beliefs, desires, or preferences at all. In order to fulfil this role, a decision theory must capture constraints on the attitudes that are minimal enough to be satisfied by all ordinary agents, with all of their foibles and imperfections. If a decision theory places overly demanding constraints on the attitudes, it fails to specify the necessary conditions for attitude possession, and gives rise to ‘false negatives’—genuine beliefs, desires or preferences that are misclassified as non-attitudes. Second, interpretationists hope that decision theoretic representation theorems provide a bridge from knowledge of PT to knowledge of S that satisfies circularity and deviance conditions. For a representation theorem to satisfy the circularity condition, it must take as ‘input’ purely physical information about an agent, and to satisfy the deviance condition, it must yield as ‘output’ only those probability and utility functions that accurately represent the agent’s beliefs and desires. On the input side, it must be possible for the radical interpreter to know whether an arbitrary agent satisfies the postulated constraints on the basis of knowledge of PT alone. On the output side, it must be possible for the radical interpreter to rule out all but the correct interpretation of the agent’s beliefs and desires. As we shall see in more detail in what follows, those existing decision theories and theorems that specify constraints that are suitably minimal, and plausibly take as input purely physical information about an agent, severely underdetermine the radical interpreter’s choice of interpretation, while those that limit underdetermination in their output do so by either postulating constraints that are too demanding to be applicable to ordinary agents, or whose inputs involve semantic or intentional information.
The representation theorem for Jeffrey’s theory, which Lewis (1974, p. 337) and Davidson (1985, 93ff; 1990, pp. 323–324) both assumed in their defence of the possibility of radical interpretation,Footnote 16 suffers from the underdetermination problem (as Jeffrey himself was well aware). The reason is essentially an issue brought up by Joyce, namely, that since Jeffrey does not assume ‘absurdly strong structural requirements on preference rankings’ (Joyce 1999, p. 197), preferences in Jeffrey’s theory only suffice to determine utility/probability products, but do not allow us to isolate the effect of the probability function from the effect of the utility function on the agent’s preferences (and choice behaviour). Indeed, all versions of expected utility theoryFootnote 17 assume that a rational person’s preferences between alternatives correspond to a product of the utilities and probabilities of the alternative’s outcomes.
More precisely, the problem is that even if an agent perfectly fits Jeffrey’s theory—that is, even if her preferences perfectly satisfy the Bolker–Jeffrey axioms—there will still be a (non-singleton) set P of probability functions, such that for each function p in P it is true that the agent’s preferences can be represented as maximizing expected utility (or ‘desirability’, to use Jeffrey’s term) relative to p (+some utility function u); but for any p, q in P such that p ≠ q, p and q will not agree on how to rank some propositions.Footnote 18 Hence, even if a radical interpreter knows all choices that Karla is disposed to make—assuming further that these dispositions match her preferences and that Karla’s preferences satisfy the Bolker–Jeffrey axioms—such knowledge does not suffice to determine, for many propositions X and Y, which of these propositions Karla believes more strongly.Footnote 19 In other words, preferences in Jeffrey’s theory do not entail a coherent comparative belief relation.
The result is an underdetermination of the interpretation of Karla that does not reflect any ordinary indeterminacy in her beliefs and desires. To see this, take any proposition X, that is neither a tautology nor a contradiction, and suppose that Karla does not have unbounded preferences (recall fn. 18). For any two probability functions, p and q, that represent Karla’s preferences, the possible size of the difference between p(X) and q(X) depends on the product p(X)u(X); the closer this is to 0, the less the difference will be. Now, there are, in Jeffrey’s framework, two fixed points that are common to any probability/utility representation: A tautology (which is assumed to be neutral in value) is assigned a utility of 0 and a probability of 1 according to all functions, while a contradiction is assigned a probability of 0 according to all functions.Footnote 20 Therefore, if the utility of X is close to the utility of a tautology, or the probability of X is close that of the contradiction, then the difference between p(X) and q(X) will be small. But if X has, say, a middling probability and is more desirable than the tautology, then the difference between p(X) and q(X) can be considerable. Most importantly, for any two logically independentFootnote 21 propositions X and Y, Karla’s attitudes to which satisfy the above constraints—i.e., the probability/utility product of each is not (close to) 0—it is possible to find two probability functions p and q that both represent her preferences, but for which, say, p(X) < p(Y) but q(Y) < q(X). In other words, if radical interpretation is based on a preference ordering that satisfies only the Bolker–Jeffrey axioms, then for any propositions X and Y, such that Karla is neither (almost) certain that both X and Y are false nor considers X and Y to be of (almost) neutral desirability, a radical interpreter will not be able to determine which of X and Y Karla believes to be more likely to be true.
To take an example, consider the following two propositions:
(X) It will snow in London in December 2020 and I will have a successful career.
(Y) It will snow in Copenhagen in December 2020 and I will have a successful career.
It is plausible that although Karla is much less convinced of the truth of both X and Y than the truth of a tautology, and while both X and Y are desirable to Karla, there is a fact of the matter as to whether Karla believes X or Y more strongly. But if there is such a fact, then Jeffrey’s theory leaves it underdetermined. Though Karla may in fact believe Y more strongly than X, a radical interpreter guided by Jeffrey’s theory cannot rule out the possibility that Karla believes X more strongly than Y. Thus, given the assumption of realism, and given that it is possible that Karla really does believe X more likely to be true than Y or vice-versa, there are semantic and intentional facts that the radical interpreter cannot come to know by appeal to Jeffrey’s theory. For Lewis, this means that radical interpretation simply cannot get off the ground.
For Davidson, it means that the data on which the Principle of Charity operates—the set of sentences that Karla holds true—is radically underdetermined. This is because, on any plausible view of how holding true is connected to subjective probabilities, an interpretation of Karla based on Davidson’s version of Jeffrey’s theory will, for many sentences s, result in one probability function according to which Karla holds s true, and another probability function according to which Karla does not hold s true. Since Davidson’s Principle of Charity tells the radical interpreter to assign meanings to the sentences of Karla’s language that maximize truth in the set of sentences she holds true, if it is underdetermined which sentences Karla holds true, it is underdetermined which interpretation is most charitable. It is unavoidably underdetermined which sentences Karla holds true because the set of propositions that get assigned probabilities and utilities in Jeffrey’s theory form an atomless Boolean algebra,Footnote 22 so we can be sure that wherever we set the threshold, as long as it is below 1, there will be some proposition (which might be a complex disjunction or conjunction) that gets a probability just above the threshold according to one representing probability function and a probability just below the threshold according to another. So, if Davidson is right in thinking that we can translate the propositions in Jeffrey’s framework into sentences, then Davidson’s version of radical interpretation will fail to determine which of several probability functions represents an agent’s beliefs.
It can be easily seen in a toy case how underdetermination in the representation of an agent’s comparative beliefs induces underdetermination in the interpretation of the agent’s language. Let p, q, be probability functions in P that represent Karla’s preferences by the lights of Davidson’s version of Jeffrey’s theory, and let s1, s2, be sentences in S: the set of sentences of Karla’s language whose probabilities are not close to a that of a contradiction, and whose utilities are not close to that of the tautology. Suppose that p(s1) = 0.6 and p(s2) = 0.4, while q(s1) = 0.4 and q(s2) = 0.6, and that the threshold for holding a sentence true is around 0.5 (though note that our point holds for any threshold below 1). Then, if p represents Karla’s degrees of belief, she holds s1 true but not s2, and if q represents her degrees of belief, she holds s2 true, but not s1. Now suppose that the radical interpreter sets out to maximize truth in the set of sentences Karla holds true. Relative to the stipulated choice of threshold, if he assumes p, he will arrive at an interpretation Ip that makes s1 true but not s2, whereas if he assumes q, he will arrive at an interpretation Iq that makes s2 true but not s1. Since Ip and Iq differ in their truth value assignments to s1 and s2, they are not equivalent. Thus, since it is underdetermined whether p or q represents Karla’s degrees of belief, it is underdetermined whether Ip or Iq is most charitable.
Furthermore, we have assumed for simplicity a threshold that determines how degrees of belief map onto sentences held true. Yet, not only is the choice of threshold underdetermined by the physical truths, it is underdetermined whether any fixed threshold determines the function from degrees of belief to sentences held true at all. For instance, the function from degrees of belief to sentences held true could vary with context, the agent’s desires, preferences or degrees of belief.Footnote 23 This only makes matters worse for the Davidsonian radical interpreter, since there are many sentences in S, many probability functions in P, and many functions from degrees of belief to sets of sentences the subject holds true (even allowing for vagueness), which combine to give a wide range of verdicts on which sentences Karla holds true. Such rampant underdetermination of which sentences Karla holds true in turn gives rise to rampant underdetermination of which interpretation of her is most charitable. Any way you slice it, radical interpretation based on Jeffrey’s theory does not satisfy the deviance condition.
Decision theorists have adopted three main strategies to avoid this radical indeterminacy in Jeffrey’s theory. Though these were not expressly formulated to address difficulties that arise for the interpretationist project—nor were they intended to apply to actual or ordinary agents—they are worth considering as potential solutions to those difficulties.Footnote 24 First, they have imposed further (both normative and structural) constraints on the agent’s preferences (e.g. Savage 1954);Footnote 25 second, they have enriched the set of things between which the agent is assumed to have preferences (e.g. Bradley 1998); third, they have postulated primitive epistemic facts, in particular, a comparative belief relation that cannot be inferred from the preference relation (e.g. Joyce 1999). All three strategies cause trouble for radical interpretation.Footnote 26
The problem with imposing further constraints on the agent’s preferences is that the stronger the constraints we impose, the less likely it becomes that any actual agent comes close to satisfying them, and radical interpretation of ordinary agents becomes impossible. So, although in theory, imposing some such constraints means that we can infer, from the agent’s preferences (or choice dispositions), a probability function and a utility function unique up to a choice of scale and starting point, we know that no actual agent will come sufficiently close to satisfying these constraints for radical interpretation to be possible.
Savage (1954) is the best-known advocate of the first strategy, that is, the strategy of deriving a unique probability function from a preference relation by imposing stronger constraints than Jeffrey does. A general problem with Savage’s strategy, from the perspective of radical interpretation, is that he delivers a representation theorem by imposing such strong structural constraints on an agent’s preferences that no actual (or even ideally rational) agent plausibly comes close to satisfying them. Moreover, even when the constraints are satisfied, they render radical interpretation impossible.
An important requirement of Savage’s is that the set A of acts that an agent is assumed to have preferences between contains all possible functions from the set S of states of the world to the set C of possible consequences. A particularly worrying implication of this requirement is the so-called constant act assumption: for any consequence c in C, there is some act a in A that delivers c in any possible state in S, and which is equally desirable as c. So, for Savage’s theorem to give hope for the possibility of radical interpretation, it must be possible to devise choice situations that reveal an agent’s attitudes to such ‘constant acts.’ But the problem is that many such constant acts will be impossible; both physically impossible and epistemically impossible according to the agent in question.
In recent work, Gaifman and Liu (2018) show that one can relax Savage’s constant action assumption by assuming that there are only two (non-equivalent) constant acts. This might suggest a way to render Savage’s theory of some use to the radical interpreter. However, while the weaker assumption is in many ways a great improvement, it is still disastrous from the perspective of radical interpretation. Suppose, for instance, that Karla takes there to be a greater than zero chance of a meteorite strike that kills all of humanity sometime in the near future. Even on the weakened constant act assumption, there will have to be some action in her preference ordering that delivers the same consequence in the state where the meteorite strikes as in, say, any state where humanity’s existence continues for another 200,000 years. But it is hard to see that such an action would be physically possible; let alone epistemically possible according to any sensible agent.
In the context of the interpretationist view that decision theory provides an account of the nature of belief and desire, the constant act assumption (even the weaker version) has the implausible implication that it is a necessary condition for an agent to have beliefs and desires at all that she has preferences with regard to acts that are physically impossible and/or epistemically impossible by her own lights. Moreover, even if this implausible hypothesis is true, the constant act assumption creates insuperable problems for a radical interpreter, who must devise choice situations that would reveal an agent’s attitudes to impossible acts. Yet, if an act is physically impossible, then there is no physically possible circumstance in which an agent could perform the act. This difficulty may seem less acute for the Davidsonian radical interpreter, who initially establishes Karla’s preferences over sentences, since it is physically possible to present Karla with choices between sentences that describe physically impossible circumstances. Nevertheless, if an act is epistemically impossible by Karla’s lights, she may have no preference with regard to the truth of a sentence describing that act. Either way, the true interpretation of Karla remains severely underdetermined by the evidence available to the interpreter, and radical interpretation is thwarted at the outset.
Indeed, Savage’s theory additionally faces the problem of inaccessibility. This is because S, C and A are sets that the modeler constructs in order to represent the agent, but do not necessarily correspond to how the agent herself conceptualizes states, consequences, and acts. This is clearly problematic from the perspective of radical interpretation since, for instance, the functions in A may not correspond to the acts that the agent herself takes to be available to her, and it is impossible to know, on the basis of PT, whether they do. This aspect of the agent’s mind is in effect inaccessible. All told, from the point of view of the radical interpreter, Savage’s theory is a non-starter.Footnote 27
Joyce (1999, pp. 138–145) suggests a very different way to recover a unique probability function from an extension of Jeffrey’s theory. The suggestion, which Jeffrey himself had mentioned in passing, is to add to Jeffrey’s framework a primitive comparative belief relation which, in addition to the preference relation, is taken to represent the attitudes of the agent. Crucially, the postulated comparative belief relation—which represents psychological states of being more confident in one proposition than another—cannot be derived from the preference relation (nor, in fact, from anything else).
It should be evident that, from the perspective of radical interpretation, this renders Joyce’s solution to the underdetermination problem a nonstarter. In fact, Joyce explicitly distances himself from any attempts to reduce degrees of belief to preference (see e.g. Joyce 1999, pp. 89–90). And, indeed, the comparative belief relation in Joyce’s framework cannot be known on the basis of ‘raw behaviour.’ Recall that the problem that Jeffrey’s original theory poses for radical interpretation is that even if an agent’s preferences satisfy the Bolker–Jeffrey axioms, her preferences at best entail a set P of pairwise inconsistent probability functions. Joyce’s result simply shows that if in addition to having a preference relation that satisfies the Bolker–Jeffrey axioms, an agent has a comparative belief relation that satisfies the right constraints, then the two relations together entail a unique probability function in P. But that means that all the radical interpreter can infer, from information about the agent’s preferences, is that if the person in question has some comparative belief relation with the right structure, then that comparative belief function will be represented by exactly one probability function in P. The problem is that the radical interpreter neither knows whether the agent in fact has such a relation, nor which probability function in P the relation—if there is one—corresponds to, on the basis of knowledge of the physical truths alone.Footnote 28 Once again, the underdetermination problem for the radical interpreter remains untouched.
Bradley (1998) proposes another type of solution to Jeffrey’s underdetermination problem, which involves enriching the domain of the preference relation. In particular, he shows that adding certain types of (non-truth functional) indicative conditionals to Jeffrey’s framework resolves the underdetermination.Footnote 29 As Bradley points out, the (quite standard) interpretation of the preference relation in Jeffrey’s framework as an attitude to news items makes it very plausible that agents have preferences between indicative conditionals: ‘One might prefer to learn, for instance, that if it’s chicken for dinner then a white wine will be served than to learn that if it’s beef, then beer will be served’ (Bradley 1998, p. 188).
However, the inclusion of indicative conditionals in Jeffrey’s decision theory undermines the use of the theory as a basis for radical interpretation.Footnote 30 For a radical interpreter to determine the agent’s preference between conditionals, there would have to be possible choice situations that would reveal such preferences. But as Bradley’s example illustrates, there are many (natural, common and simple) conditionals for which that would be impossible. For instance, although we can certainly construct choice situations that reveal an agent’s preference between chicken and white wine on one hand, and beef and beer on the other, it would be impossible to devise a choice situation that reveals the agent’s preference between ‘if chicken, then white wine’ and ‘if beef, then beer’. Thus, even with the assistance of Bradley’s extension of Jeffrey’s theory, the information available to the radical interpreter remains insufficient to determine the true interpretation of the agent.
It might seem that Davidson’s proposal to reformulate constraints on preference in terms of preferences over uninterpreted sentences fares better here. In this setting, perhaps all the radical interpreter would need to do to make use of Bradley’s extension would be to present Karla with suitable sentences of her language, and observe her preferences between them. However, if the indicative conditional is not a truth functional connective, the radical interpreter cannot use Davidson’s method to identify it, and thus cannot make use of Bradley’s extension at the point at which it is most needed. Recall that Davidson’s radical interpreter begins by identifying the Sheffer stroke in Karla’s language, and uses that to derive the remaining truth functional connectives; if the indicative conditional is not truth functional, the radical interpreter simply cannot use the Sheffer stroke method to identify it. At best, the Davidsonian radical interpreter might be able to interpret the indicative conditional in Karla’s language at the second stage, when he applies the Principle of Charity to maximize truth in the sentences Karla holds true. But if the indicative conditional can only be identified at this later stage, the radical interpreter cannot make use of Bradley’s extension to resolve the underdetermination inherent in Jeffrey’s theory, which is needed to determine which sentences the subject holds true—the very data on which the Principle of Charity is to be applied.
Perhaps it will be suggested that the radical interpreter could simply assume that the indicative conditional is truth-functional. He could then use the Sheffer stroke method to identify it and present the agent with choices between the relevant sentences containing it. However, such a move would render Bradley’s extension useless as far as deriving a unique probability function is concerned. If the interpreter wants to use Bradley’s method to ensure such uniqueness, he must know Karla’s preferences between sentences containing a non-truth functional indicative conditional. For, unless the interpreter assumes that the sentences in question have a non-truth functional structure—that is, without assuming that the agent’s background algebra is a ‘conditional algebra’—the interpreter is back in Jeffrey’s indeterminacy.