Brentano's Metaethics Jonas Olson Brentano's main metaethical writings are two: The Origin of Our Knowledge of Right and Wrong, which is based on a 1889 lecture before the Vienna Law Society (Brentano 1889/1969), and The Foundation and Construction of Ethics, based on Brentano's lectures on practical philosophy from 1876--‐94 (Brentano 1952/1973b). Early on in Origin, Brentano presents his metaethical stance succinctly: Is there such a thing as a moral truth taught by nature itself and independent of ecclesiastical, political, and every other kind of social authority? Is there a moral law that is natural in the sense of being universally and incontestably valid-valid for men at all places and all times, indeed valid for any being that thinks and feels-and are we capable of knowing that there is such a law? ... My own answer is emphatically affirmative. (1969: 6) Brentano's metaethical stance thus appears thoroughly realist and objectivist. However, his theory has several features that one does not expect to find in a theory that holds that moral truths are objective and universal. Most notably, Brentano holds that moral judgements conceptually presuppose emotions, and that emotions are a precondition for moral knowledge. He denies that there is a separate realm of values and norms, independent of the emotive attitudes of love and hate, and he also denies that evaluative properties and facts are among the contents of acts of valuing. These features of Brentano's view all tie in with notorious difficulties for realist objectivist views in metaethics, chief among which are accounting for (i) the nature of moral truths and how we come to know them, and (ii) moral judgements' connection to attitudes that motivate action. This chapter explains Brentano's metaethical theory, and how it purports to deal with such difficulties. §1 considers Brentano's account of moral judgement, moral truth, and moral knowledge. §2 considers Brentano's metaethics from the perspective of the modern metaethical debate. 2 1. Brentano on Moral Judgement, Moral Truth, and Moral Knowledge 1.1. Three kinds of intentional acts In order to understand Brentano's views about moral judgements and in particular the way in which they presuppose emotions, we first need to consider briefly a pivotal ingredient in Brentano's philosophy, namely the threefold classification of intentional acts (see CHAP. 9).1 First, there are what Brentano calls 'Vorstellungen', often translated in English as 'ideas' or 'presentations', such as my thought of the fiords in New Zealand. Second, there are judgements ('Urteile'). Every judgement presupposes a presentation in that the content of the presentation is what the judgement is about. But the judgement is an additional intentional relation that consists either in an act of affirmation ('Anerkennen') or rejection ('Verwerfung') of what is presented (see CHAP. 10). Third, there are emotions, or conative attitudes ('Gemütsbewegungen'). The most fundamental are love and hate.2 Like judgements, emotions also presuppose presentations, in that they are directed at the contents of presentations (see CHAP. 11). 1.2. Valuing and Valuing Correctly Brentano's ethics is teleological. He thinks of ethics as the practical discipline that studies the 'highest end' and the means of attaining it (1973b: 7, 79, 121--‐2): it is right to choose, or strive, for the best option, or end, of those available, and wrong not to do so (see CHAP. 23). Consequently, evaluative judgements, and in particular judgements about intrinsic goodness, badness, and indifference, are central to moral judgement and moral theorizing. So what is it to judge something valuable? In order to understand this we need to start with valuing. To value something positively is to love it, and to value something negatively is to hate it. To value something intrinsically is to love it or hate it for its own sake or as an end. Clearly, however, some things that are valued are not valuable. To see the difference between what is valued and what is valuable we need to consider the Brentanian notion of correctness (Richtigkeit). Unlike presentations, judgements and emotions can be correct or incorrect. For a thing to be positively valuable is for it to be such that loving it is correct; for a thing to be 3 negatively valuable is for it to be such that hating it is correct (1969: 18). If it is correct to love (hate) a thing, it is incorrect to hate (love) that thing. For a thing to be indifferent is for it to be such that neither love nor hate is the correct response to it. For a thing to be better than another is for it to be such that preferring it to the other thing is correct (1969: 26).3 For example, to say that pleasure is intrinsically good is to say that it is correct to love pleasure as an end; to say that pain is intrinsically bad is to say that it is correct to hate pain as an end; to say that pleasure is intrinsically better than pain is to say that it is correct to prefer as an end pleasure to pain. To say that the number 2 is indifferent is to say that it is correct neither to love nor to hate the number 2. To value something is not yet to judge that thing valuable, and judging a thing valuable is distinct from valuing it. Valuing belongs in the third category of basic kinds of intentional acts-that of emotions-and emotions are not judgements. Still, Brentano holds that evaluative judgements presuppose emotions in that evaluative judgements are primarily judgements about emotions. For example, to judge that pleasure is intrinsically good is to judge that loving pleasure as an end is correct. Brentano holds that emotions are a precondition for moral and evaluative knowledge in that in loving something, e.g., pleasure, and experiencing that love as correct, I immediately see that pleasure is the kind of thing that can be loved with a love that is correct, and so I can reject loving pleasure with a love that is not correct. All of this raises a host of questions. For example, how can my experience of loving pleasure with a love that is correct reveal to me a moral truth (that pleasure is among the highest ends) and thereby provide me with moral knowledge? And most fundamentally, what is it for an emotion to be correct? 1.3. Correct Emotions Brentano explains correctness in emotions by analogy with correctness in judgements. For a judgement to be correct is for it to concord with a judgement made by someone who judges with self--‐evidence (Evidenz). Self--‐evident judgements are guaranteed to be correct, and they are based either on 'inner perception' or on presentations of objects that are rejected apodictically (1966b: 130; 1969: 19--‐20). An example of an evident judgement of the former kind is my judgement that I now have a visual experience of blue. It may or may not be true that my visual experience of blue corresponds to 4 something in non--‐mental reality, but when I have the experience of sensing blue I can go on immediately to affirm with self--‐evidence that I am now sensing blue. An example of an evident judgement of the second kind is the judgement that three is equal to two plus one. Here we consider the presentation of a combination of three with two plus one 'by means of a negative copula' (1969: 112), and we reject this combination apodictically. That is, we deny apodictically that 3 ≠ 2+1. By reflecting on this judgement, Brentano says, we obtain the concept of impossibility (in this case we reject apodictically that three is not equal to two plus one, and by reflecting on this judgement we conclude that it is impossible that three is not equal to two plus one) (1969: 112). Judgements that are based on self--‐evidence in either of these two ways are experienced as correct (als richtig charakterisierten). It is important to see that this is not a matter of an experience of a high degree of conviction (1969: 20). Many judgements are held with great confidence, although they are neither evident nor correct. How, then, can I come to recognise an experience of correctness? Brentano holds that the experience of correctness becomes manifest when I compare a self--‐evident judgement with a contrary judgement (e.g., affirmation versus rejection of my present visual experience of blue, or rejection versus affirmation of 3 ≠ 2+1). I then see that the judgements differ in a crucial respect: the original judgement is experienced as being correct, while the contrary judgement is not (1969: 20; Chisholm 1986: 49--‐50). Once I have grasped the experiential feature of correctness through this kind of comparative manoeuvre, I am able to recognise it in other judgements too. I shall not here consider whether Brentano is right that there is an experiential feature of correctness in our judgements and that we can become aware of it by comparing different kinds of judgements. Let us note instead that truth, according to Brentano, reduces to correctness: a true judgement is simply a correct judgement (see CHAP. 21). As we have seen, all evident judgements are correct, but not all correct judgements are evident. Judgements that are not evident are blind judgements. Blind judgements can still be correct; they are so if they concord with judgements that would be made by someone whose judgements are based on self--‐evidence. As with judgements in general, so with moral and evaluative judgements: For example, for the judgement that knowledge is intrinsically good to be true is for the judgement that it is correct to love knowledge for its own sake to be correct. 5 At this point one may feel that we are still in the dark about what a correct emotion is. We have been told that for a judgement to be correct is for it to concord with an evident judgement, but there seems to be no such thing as a "self--‐evident emotion", so the analogy between correct judgement and correct emotion may seem to fail. However, Brentano insists that there is an analogue of self--‐evidence in the emotional domain (1969: 22), for just as some judgements are experienced as correct, some emotions are experienced as correct. Just as correctness of judgements can become experientially manifest by comparing these judgements to conflicting judgements (i.e., judgements that are not correct), correctness of emotions can become experientially manifest by comparing conflicting emotions. For example, when we love for its own sake an experience of pleasure or a piece of knowledge, we may compare this love with an imagined attitude of hating for its own sake that experience of pleasure or that piece of knowledge. We then see that the love has something that the hatred lacks: the love is experienced as correct and the imagined hatred is not. Just like correctness of judgements, correctness of emotions is thus an experiential feature with a distinctive phenomenology, such that 'we know it when we have it'. And just as in the case of judgements, we can come to know that we have it by comparing experiences of correct emotions to experiences of emotions that are not correct. Ultimately, then, it seems that we have to rest content with the conclusion that correctness of judgements and emotions is not susceptible to analysis or constitutive explanation; in order to understand what correctness is, one has to experience it. 1.4. Moral Knowledge But how do I go from love of a particular experience of pleasure, or particular piece of knowledge, to love of pleasure or knowledge quite generally, i.e., to love of other instances of the same kind? That is, how do I go from valuing (for its own sake) a particular experience of pleasure or a particular piece of knowledge, to valuing (for their own sake) other instances of pleasure or knowledge? The answer is that in loving for its own sake an experience of pleasure or a piece of knowledge, I consider and love it as such. On the basis of loving for its own sake a particular experience of pleasure or a particular piece of knowledge I can universalize my love, so that it takes as its object pleasure or knowledge quite generally. If I experience as correct my loving for its own sake a particular experience of pleasure, or a particular piece of knowledge, what I 6 experience as correct is loving for its own sake a particular experience of pleasure, or piece of knowledge, as such. On that basis I can realize immediately that anything relevantly similar, i.e. any other relevantly similar experience of pleasure or relevantly similar piece of knowledge, is such that loving it for its own sake is correct.4 Via this kind of universalisation manoeuvre, I can come to experience as correct loving for its own sake pleasure or knowledge quite generally (see, e.g., Brentano 1969: 111--‐13). This is meant to be analogous to the way in which I can go from the judgement about a thing to a universal judgement about a class of things, e.g., from the rejection of a round square to the rejection of all round squares. When I reject round squares, I consider as such a square that is round, and this presentation causes me to reject all round squares. I perceive the causal link between the presentation of a round square and my rejection of round squares, which I experience as correct. This apodictic judgement constitutes my a priori insight that there are no round squares.5 Similarly, when I judge that every experience of pleasure is good, I start by considering a particular experience of pleasure, which is such that it is not correct to love it as such. This presentation causes me to reject experiences of pleasure that are not correct to love as such (i.e., insofar as they are experiences of pleasure). I perceive the causal link between the presentation of an experience of pleasure that is not correct to love as such and my rejection of experiences of pleasure that are not correct to love insofar as they are experiences of pleasure.6 This constitutes my insight that the nature of pleasure is such that any experience of pleasure, insofar as it is an experience of pleasure, is correct to love (for its own sake). Knowledge about what one ought to do depends ultimately on knowledge about what is good,7 which, as we have seen, relies on insights into what can be correctly loved for its own sake. Since there is a plurality of (kinds of) things that are such that they can be loved with a love that is correct, knowledge about what to do also relies on insights about what (kinds of) things can be correctly preferred to other (kinds of) things. This is because, as noted in §1.2 above, Brentano's normative ethics is teleological, holding that we ought always to bring about the best available outcome. To that extent, Brentano takes the good to be prior to the (morally) right. But it is important to bear in mind that one distinctive feature of his view is that the correct is conceptually prior to the good. 7 This last feature makes Brentano's view immune to the so--‐called 'wrong kind of reason problem,' which has prompted extensive discussions in connection with recent attempts to reduce the evaluative to the normative (see Rabinowicz and Ronnow--‐ Rasmussen 2004). The leading idea behind such attempts is that for a thing to be good is for it to be such that there are reasons to respond to it with a favourable attitude (Scanlon 1998). This idea has considerable intuitive appeal, but a major difficulty is that it is easy to imagine situations in which there are reasons to favour a thing although the thing in question is not good. Consider a powerful and evil demon who threatens to punish us with severe pain unless we favour him for his own sake. Intuitively, there is in such a dire circumstance reason to favour the demon for its own sake, but it is not at all intuitive that the demon is thereby good. Brentano's view is not vulnerable to this problem since on his view, for a thing to be good is not for it to be such that there is reason favour it, but to be such that loving it is (would be) correct. It may well be that there is conclusive reason to love the evil demon for his own sake, i.e., it may well be that we ought to love him for his own sake, because doing so shields us from severe pain. For Brentano, this means that loving the love of the demon for his own sake would be correct, because loving him for his own sake would prevent something that is correct to hate for its own sake (in this case, experiences of severe pain). However, loving the demon for his own sake cannot be correct.8 It is a natural complaint that Brentano's account sheds no light, as it merely replaces one presumed primitive (value) with another (correctness). But note first that if Brentano is right that we have an 'inner' access to an experiential feature of correctness of emotions, independently of our intuitive judgements about value, his account of value in terms of correct emotions does shed light after all. This is admittedly a very big 'if', however. Note, secondly, that regardless of whether Brentano's view of correctness as an experiential feature with a distinctive phenomenology is defensible, his account establishes in a neat way a necessary connection between values and correct emotion. 2. Brentano's Metaethics and Modern Metaethics 8 We saw in the beginning of this article that while Brentano's metaethical stance is avowedly realist and objectivist, it has some features that readers familiar with modern metaethical debates would not expect to find in views based on such a stance. In this section we shall explore further how Brentano's view is best categorized and understood, using distinctions and taxonomy of modern metaethics. 2.1. Rationalism or Sentimentalism? The first part of Brentano's The Foundation and Construction of Ethics is a critical survey of what other philosophers before him, in particular British and German rationalists and sentimentalists of the eighteenth and nineteenth centuries, said about how we acquire moral knowledge. The rationalists maintained that moral knowledge is attainable on purely rational grounds. Samuel Clarke and Richard Price, for example, argued that by means of rational reflection alone we can come to know necessary moral truths. Brentano rejected synthetic a priori principles in ethics and elsewhere (1973b: 73) and he found a lot to agree with in Hume's sentimentalist account and in his critique of rationalist views like Clarke's and Price's. Brentano took Hume's own position to be that moral judgements are feelings, but he complained that Hume's arguments did not in fact support that conclusion. What they supported was something weaker, namely that feelings are necessary preconditions for moral judgements (1973b: 50). According to Brentano, Hume overlooked the phenomenon of correct feelings. If I experience the love of an object as correct, I can affirm immediately, on that basis, that the love of the object is correct. In such a case, the judgement is a 'cognition' (Erkenntnis), i.e., a correct judgement (1973b: 68). My judgement that my feeling of love or hate is correct might of course be incorrect, in which case it is not a cognition. In sum, Brentano's view incorporates both rationalist and sentimentalist elements. We have seen that feelings are a precondition for moral judgements and that moral judgements are judgements about the correctness of feelings. We saw in §1.4 how my love or hate of a particular object can be universalised, by focusing on the object as such, to other particular objects of that kind. 2.2. Cognitivism or Non--cognitivism? Brentano's metaethical theory concerns first and foremost the psychology of valuing and of moral judgement. As we have seen, Brentano is clearly a cognitivist about evaluative 9 and moral judgement; such judgements are judgements about the correctness of feelings, and judgements are cognitive acts. There is, however, one aspect of his position that seems congenial to contemporary expressivism. This has to do with expressivist treatments of disagreement. When two people disagree about whether, say, knowledge is good, standard cognitivist accounts explain what is going on in terms of the two people taking the same kind of cognitive attitude-belief-to different propositions- that knowledge is good and that knowledge is not good, respectively. Expressivists cannot do that. They have to say instead something to the effect that the persons' utterances of 'Knowledge is good' and 'Knowledge is not good' express different attitudes to the same object, and that this amounts to disagreement. This is structurally similar to how Brentano thinks of disagreement. Recall that the judgement that there are pink rats consist in an affirmation of pink rats, while the judgement that there are no pink rats consists in a rejection of pink rats. Similarly, to judge that knowledge is good is to affirm correct love of knowledge and to judge that knowledge is not good is to reject correct love of knowledge. These cases amount to disagreement because simultaneous affirmation and rejection of the same thing are inconsistent, i.e., they cannot both be correct. As expressivist Allan Gibbard puts it, 'Fregeans' start out with negation and explain disagreement in terms of believing the negation of what the other person believes. 'Non--‐Fregeans,' like Brentano and many contemporary expressivists, start out instead with different kinds of mental attitudes that are inconsistent with one another, in terms of which they explain disagreement (Gibbard 2012: 273--‐4).9 2.3. Realism or Anti--realism? What counts as realism is a notoriously disputed issue, but we saw at the outset that Brentano's metaethical stance seems thoroughly realist in that he takes moral truth to be 'universally and incontestably valid' and independent of human conventions. We have seen, however, that on Brentano's view, facts about what is valuable, right, wrong, etc., are not independent of our attitudes, for to be valuable, right, wrong, etc., reduce to being something that can be correctly loved or correctly hated. One might worry that this dependence of evaluative and moral facts on our attitudes is in tension with realism. But this worry can be put to rest. For recall that evaluative and moral facts are dependent on, indeed reducible to, correct attitudes, and not simply to our actual 10 attitudes, or to attitudes we would have under non--‐normatively specifiable ideal circumstances.10 Brentano's account is thus not a version of a standard response--‐ dependence account. Brentano does hold that it is correct to love attitudes that are correct (e.g., love of knowledge and hatred of pain), but that is a substantive evaluative claim and not an analysis or what it is for an attitude to be correct.11 It is also noteworthy that Brentano holds that the property of goodness is not part of the content of acts of valuing. For example, to value knowledge positively is simply to love knowledge. And the object of such love is simply knowledge considered as such. One might think that this is in tension with a thoroughgoing realism, but it is not clear why it should be. According to Brentano, for an object to be good (bad) is for it to be such that it can be loved (hated) with a love (hatred) that is correct, and this merely commits him to the view that goodness and badness are not conceptually primitive. It is not clear why that should affect the account's claim to being a thoroughgoing version of realism. 2.4. Naturalism or Non--naturalism? As we have seen, it is a distinctive feature of Brentano's account that correctness of attitudes is the primitive normative notion, in terms of which goodness, rightness, wrongness, ought, etc., are analysed. Since correctness is a primitive notion it cannot be analysed in other terms, but that leaves open the question of whether it is a natural or a non--‐natural property. It is well known that G. E. Moore found much to favour in Brentano's philosophy,12 and Brentano's views seem akin the kind of foundationalist intuitionism that is traditionally associated with non--‐naturalism. It is difficult to draw a clear--‐cut distinction between natural and non--‐natural properties, but the distinction is still very much in use and it is therefore worth investigating where Brentano's position fits in. According to one influential proposal, which stems from Moore, natural properties are all and only the kinds of properties that form the subject matter of the natural sciences, including psychology (Moore 1903b: 92).13 On this view, correctness seems to be natural property. For recall that we have said that correctness is an experiential feature of judgements and emotions. As such, it seems to be a kind of property that belongs to the subject matter of psychology. 11 According to another influential view, related to Moore's, non--‐natural properties are causally inert and epistemically accessible to us only via a priori reflection. But this is not the case with Brentanian correctness. For again, the correctness of a judgement or emotion can be experienced as features of judgements and emotions and such experiences can cause apodictic judgements. Correctness is epistemically accessible to us only in experiences of judgements and emotions, and not via mere a priori reflection. 2.5. Internalism or Externalism? On Brentano's view, there seems to be a necessary connection between valuing something and being motivated, for to value something is to love or hate it, and love and hatred seem clearly to be motivationally efficacious attitudes. But what about the connection between judgements that something is good or bad and motivation? Here Brentano's view is more nuanced than to simply affirm or deny that there is a necessary connection between evaluative judgements and motivation. My judgement that something is good may be based on my love of that (kind of) thing and my experiencing that love as correct. It is only in such cases that I can be said to possess knowledge that the thing in question is good. However, it is also possible to judge that something is good, i.e., such that it can be loved with a love that is correct, on other bases, e.g., testimony. In such cases my judgement may or may not be correct, but it is not experienced as correct. As we saw in §1.3 above, such a judgement is blind and does not amount to knowledge. We can thus say that Brentano's position is a qualified version of internalism, according to which there is a necessary connection between knowing that something is good and being motivated to pursue it.14 References Chisholm, R. M. 1986. Brentano and Intrinsic Value. Cambridge: Cambridge University Press. Chrudzimski, A. and B. Smith. 2004. "Brentano's Ontology: From Conceptualism to Reism," in The Cambridge Companion to Brentano, edited by D. Jaquette. Cambridge: Cambridge University Press: 197--‐219. Danielsson, S. and J. Olson. 2007. 'Brentano and the Buck--‐Passers,' Mind 116: 511--‐22. 12 Gibbard, A. 2012. Meaning and Normativity. Oxford: Oxford University Press. Moore, G. E. 1903a. Review of The Origin of Our Knowledge of Right and Wrong, International Journal of Ethics 14: 115--‐23. Moore, G. E. 1903b. Principia Ethica, rev. ed., edited by T. Baldwin. Cambridge: Cambridge University Press, 1993. Rabinowicz, W. and T. Ronnow--‐Rasmussen 2004. 'The Strike of the Demon: On Fitting Pro--‐Attitudes and Value,' Ethics 114: 391--‐423. Scanlon, T. 1998. What We Owe to Each Other. Cambridge MA: MIT Press. Schroeder, M. 2008. Being For: Evaluating the Semantic Program of Expressivism. Oxford: Oxford University Press. 1 The ontology of intentional objects is a notoriously difficult issue on which Brentano's views changed over time. However, the issue has no immediate bearing on the discussion of Brentano's metaethics, pursued in this chapter. Suffice it to say here that Brentano at no point accepted a Fregean view according to which the contents of intentional acts are abstract propositions. Brentano rather thought of the contents of intentional acts as things and their properties, and in his reistic period, towards the end of his life, simply as things. For a general discussion of Brentano's ontological views, see Chrudzimski and Smith (2004). 2 The terms 'love' and 'hate' should be interpreted broadly. R. M. Chisholm suggests that 'pro--‐emotion' and 'anti--‐emotion' may be less misleading terms (Chisholm 1986: 18). For Brentano's own account of love and hate, see, e.g., 1969: 142--‐60. 3 Brentano conceives of preferences not as choice dispositions, but as emotions with a phenomenological nature. He says that to prefer A to B is to love A more than B (or to hate B more than A) (1969: 143), but he also says that 'more' does not signify greater intensity of love or hate (1969: 26). Brentano seems to think of preference as an emotive phenomenon of comparative acts of loving or hating (e.g., loving A more than B), as opposed to simple acts of loving or hating (e.g., loving A) (1969: 143). 4 Here one might of course wonder what exactly is meant by 'relevantly similar'? Is an experience of sadistic pleasure relevantly similar to an experience of non--‐sadistic pleasure? What about an undeserved experience of pleasure and a deserved experience of pleasure? What about pieces of trivial knowledge (such as the number of hairs on my head) and pieces of non--‐trivial knowledge (such as whether there is extra--‐terrestrial life)? Different possibilities are open here, but Brentano's view was that any experience of pleasure (or piece of knowledge) is good insofar as it is pleasure (or knowledge). For example, sadistic pleasure (pleasure in the bad) is good, i.e., it is correct to love it, insofar as it is pleasure, but bad, i.e., it is correct to hate it, insofar as it is pleasure in the bad. See, e.g., Brentano 1973b: 196. 5 As we saw above, Brentano also held that reflection on apodictic judgements is the basis of our modal judgements. See Chisholm (1986: 42--‐5, 49--‐51) for a helpful discussion. 13 6 The italics are meant to forestall scope ambiguities. 7 According to Brentano, '[e]thics must first determine which ends are rightly to be striven for as the highest ones (1973b: 79), and '[o]nly the end that exceeds all others in value can be the correct ultimate end, i.e., it must be the best, but only in so far as it is attainable' (1973b: 122). 8 See Danielsson and Olson (2007) for discussion. 9 Whether this is a good explanation of disagreement is a matter of debate in contemporary metaethics. Mark Schroeder argues that in explaining disagreement in this way, expressivists help themselves to everything that needs to be explained (Schroeder 2008: ch. 3). Gibbard (2012: appendix 2) responds to Schroeder. 10 According to Brentano, 'there is no guarantee that every good thing will arouse in us an emotion that is experienced as being correct' (Brentano 1969: 24). 11 See Danielsson and Olson 2007: 518--‐20. 12 See Moore's review of The Origin of Our Knowledge of Right and Wrong (Moore 1903a). 13 The implicit suggestion is that all other kinds of properties are non--‐natural. This is unsatisfactory, since it implies that supernatural properties are also non--‐natural, and that does not sit well with Moore's own views. 14 An earlier version of this chapter was presented at a seminar at Stockholm University. I thank the participants, in particular Per Algander, Krister Bykvist, Per Martin--‐Löf, and Frans Svensson, for very useful discussions. I am also very grateful to the editor of this volume for his helpful suggestions. Work for this chapter was supported by a generous grant from Riksbankens Jubileumsfond (Grant no. 1432305).