Abstract
According to the fitting-attitude analysis of value (FA-analysis), to be valuable is to be a fitting object of a pro-attitude. In earlier publications, setting off from this format of analysis, I proposed a modelling of value relations which makes room for incommensurability in value. In this paper, I first recapitulate the value modelling and then move on to suggest adopting a structurally similar analysis of probability. Indeed, many probability theorists from Poisson onwards did adopt an analysis of this kind. This move allows to formally model probability and probability relations in essentially the same way as value and value relations. One of the advantages of the model is that we get a new account of Keynesian incommensurable probabilities, which goes beyond Keynes in distinguishing between different types of incommensurability. It also becomes possible to draw a clear distinction between incommensurability and vagueness (indeterminacy) in probability comparisons.
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Notes
These two forms of pluralism should not be confused, however, with a more familiar ‘substantive’ value pluralism. On that view, even if we keep the kind of proattitude and the ontological category of the value bearer fixed, there may be different and irreducible types of objects within this category that call for the attitude in question. In this sense one might be a pluralist about, say, political values, such as freedom, equality and justice. All these things fall within the same ontological category and all of them might be deemed to be independently desirable. Needless to say, this kind of substantive value pluralism is fully compatible with the FA-account. I am indebted to Richard Dietz and to an anonymous referee for asking me to clarify this issue.
Preference can be understood as a dyadic attitude or as a degree comparison between two monadic attitudes of favouring/disfavouring. On the latter approach, preference for \(x\) over \(y\) consists in favouring \(x\) to a higher degree than \(y\) (or disfavouring \(x\) to a lower degree than \(y\)). See Rabinowicz (2012) for an elaboration of this distinction.
However, instead of requiredness, Brentano appealed to the notion of correctness. That Brentano’s correctness is a normative cencept has been questioned by Danielson and Olson (2007). On their suggestion, correctness should instead be understood as analogous to truth. Does this mean that a proattitude is correct insofar as it is true to the value of its object? If so, then such a ‘representational’ interpretation of fittingness will make the FA-account circular. An alternative might be to treat correctness (representational fittingness) as a basically primitive notion. In this paper, I focus on the normative interpretation instead.
For some notions close to Chang’s parity, cf. Griffin’s (1986, p. 81, 96ff) “rough equality”, and Parfit’s (1984, p. 431) “rough comparability”.
Concerning my usage of “incommensurability”: In the ordinary parlance, “incommensurability” means that a common scale of measurement is missing for items under consideration. Which does not exclude that one of them might be better than the other. Thus, my usage differs the ordinary one. I would have used the term “incomparability” instead, were it not for the fact that, if Chang is right, two items might still in some sense be comparable in value, even if none of which is better than, worse than or equally as good as the other. I am indebted to an anonymous referee for pressing me on this point.
If it is objected that a discount does not make a trip better but only cheaper, then one might let \(x^{+}\) instead be the trip to Australia with an additional exhilarating bush-walk.
Cf. Rabinowicz (2008). I am indebted to an anonymous referee for pressing me on this issue.
Proof: In the interval model, if \(x^{+}\) is better than \(x\) and \(y^{+}\) is better than \(y\), (i) \(x^{+}_{min} > x^{max}\) and (ii) \(y^{+}_{min} > y^{max}\). There are two possible cases: (1) \(x^{max}=y^{max}\), or (2) \(y^{max}=x^{max}\). In case 1, (i) implies that (iii) \(x^{+}_{min} > y^{max}\), i.e., that \(\hbox {x}^{+}\) is better than y. In case 2, (ii) implies that (iv) \(y^{+}_{min} > x^{max}\), i.e., that \(y^{+}\) is better than \(y\) .
I am indebted to an anonymous referee for asking me to clarify this issue. A useful and lucid comparison between Sen and Levi can be found in Sen (2004).
Indeed, the model also makes clear what needs to be given up if one for example, like Temkin (2012), wants to reject the view that betterness must be transitive. This position is now translatable into the view that requirements on permissible preferences need to be correspondingly relaxed.
Instead of restricting the space of possibilities, one might instead suggest that this space should be expanded. Perhaps we should increase the number of possible preferential stances: For example, isn’t there such a relation as preferential parity, along with preference, dispreference, indifference, and a gap? Indeed, Chang herself seems to have thought so. In (2002b, p. 666), she wrote: “Perhaps most striking, the possibility of parity shows the basic assumption of standard decision and rational choice theory to be mistaken: preferring X to Y, preferring Y to X, and being indifferent between them do not span the conceptual space of choice attitudes one can have towards alternatives.” And, if one allows for preferential parity, why not split this stance into different subtypes, just as we have divided value parity into several subtypes, and why not in addition allow for several other possible preferential stances, which are analogues of the value relations we have identified? If we do this, however, then the number of possible value relations will dramatically increase. If the number of preferential stances is increased to 15 (which is the number of types of value relations in our taxonomy), the number of atomic types of value relations will go up to 2\(^{15}-1\). And for each new value relation we might then ask whether this relation does not have an analogue on the preferential level. If it does, however, then this leads to yet more value relations that need to be introduced; we are thus heading towards a veritable explosion of logical possibilities. (I am indebted to anonymous referees for raising these issues.)
The answer to this worry is to be sought in the analysis of preferential attitudes. We should be prepared to allow for preferential stances that we can distinguish, conceptually and phenomenologically, as different possible attitudes. We understand what value parity is, but is there such a thing as preferential parity? What would such an attitude consist in? And if preferential parity has subtypes, what do those subtypes consist in? The answers to these questions can be provided, I think, and some fine distinctions can be made, if one lets permissible preferential states of an agent, i.e. the different members of K, be non-empty sets of orderings rather than single orderings. Thus, for example, preferential parity between \(x\) and \(y\) would obtain in a preferential state containing both an ordering that ranked \(x\) above \(y\) and an ordering that ranked \(y\) above \(x\). But this process of increasing the complexity of the preferential base for taxonomy of value relations cannot be continued ad infinitum: phenomenology of possible preferential attitudes sets limits to purely technical constructions.
A biconditional like this might be accepted even by someone who isn’t prepared to view it as an analysis of probability, but rather takes it as an adequacy criterion for a satisfactory analysis of this notion. The modelling I develop in what follows is compatible with such a cautious position.
This account of probability lends itself, however, also to more ‘objective’ interpretations, on which the relativization to a body of evidence is replaced by relativization to time: Ought is then relativized either to the total evidence available at a time, or— in an even more objective vein—to everything that an ideal subject could know at the time in question (which would include full knowledge of the preceding history of the world and, possibly, also of the laws in accordance with which the world develops). On the latter version, probability would arguably coincide with objective chance. I am indebted to Christian List for pressing me on this point.
It is possible, in fact, to analyze objective chances in terms of FA-probabilities, on the lines of David Lewis’s well-known reduction of objective chances to rational credences given appropriately circumscribed states of knowledge or evidence. Cf. the preceding footnote.
But the terminology varies. In Galavotti (2011), ”the epistemic interpretation” is subdivided into two categories: the subjective and the logical interpretation. The adherents of the FA-approach are included among the representatives of the logical interpretation and the latter culminates with Carnap. For another useful recent historical account, see Zabell (2011).
This suggests that there was an earlier definition of “chance” which was like the definition that Poisson now wants to give for “probability”. But whose definition was it? Poisson’s or somebody else’s? We aren’t told.
It may be somewhat misleading to count Poisson as a precursor of the account of probability I am interested in. It seems that on his proposal an event is more probable on one kind of evidence rather than another if the former evidence gives us “more grounds to believe” it. Which is not quite the same as to say that it gives us grounds to believe it more, as the analysis I am interested in would have it. To put it differently: Stronger reasons to believe are not necessarily co-extensional with reasons to believe more strongly. For an analysis of “more probable than” in terms of stronger reasons to believe, cf. Skorupski (2010), Chap. 9.
He then, however, adds:
it would be unphilosophical to affirm that the strength of that expectation, viewed as an emotion of the mind, is capable of being referred to any numerical standard. [...] The rules which we employ in life-assurance, and in the other statistical applications of the theory of probabilities, are altogether independent of the mental phaenomena of expectation. They are founded upon the assumption that the future will bear a resemblance to the past; that under the same circumstances the same event will tend to recur with a definite numerical frequency; not upon any attempt to submit to calculation the strength of human hopes and fears. (p. 188)
It is unclear whether Boole here merely rejects the measurement of probability by measuring the strength of the actual psychological state of expectation, or rejects the previous suggestion that probability is rational expectation based on partial knowledge, “the reason we have to think that [an event] will occur”.
It is possible, though, that Jevons should not be counted among the adherents of the FA-approach. While he thinks the probability is normative with respect to belief, he is unwilling to define it in terms of what we ought to believe:
I prefer to dispense altogether with this obscure word belief, and to say that the theory of probability deals with quantity of knowledge ... An event is only probable when our knowledge of it is diluted with ignorance, and exact calculation is needed to discriminate how much we do and do not know. (ibid., p. 199).
In describing another example, Keynes is more categorical in his rejection of balancing. He suggests that in that example there is no admissible way of weighing different considerations against each other:
Consider three sets of experiments, each directed towards establishing a generalisation. The first set is more numerous; in the second set the irrelevant conditions have been more carefully varied; in the third case the generalisation in view is wider in scope than in the others. Which of these generalisations is on such evidence the most probable? There is, surely, no answer; there is neither equality nor inequality between them. We cannot always weigh the analogy against the induction, or the scope of the generalisation against the bulk of the evidence in support of it (ibid., p. 31).
For a nice succinct discussion of the advantages of this representation and for some further information on its historical roots in economics and statistics, see Weatherson (2002). Cf. also Weatherson (2007). An alternative, ‘linguistic’ representation of doxastic states will be briefly considered below.
Cf. Hawthorne (2009) for a linguistic representation of doxastic states and for a representation theorem connecting such representation with one in terms of sets of credence functions. However, it should be noted that Hawthorne’s regimented language is more restricted than the one I envisage; it is purely qualitative and thus does not introduce numerical values.
I am indebted to Christian List for suggesting this linguistic representation of doxastic states.
The mirroring of probability relations by credences cannot be complete, however: There are only four credential stances with regard to two propositions, but there are fifteen probability relations. If \(S\) satisfies conditions (i) and (ii) in Probability Mirror, then the credence gap in \(S\) mirrors probabilistic incommensurability, but it doesn’t distinguish between different atomic types of incommensurability. Even though some of those types can be reflected by different kinds of credence gaps, there are fewer kinds of credence gaps than there are atomic types of incommensurability.
Of course, if one wants to reserve the term “probability” for numerical assignments, as many theorists prefer to do, then “non-quantifiable probabilities” aren’t possible. Instead, one would need to have a special terminology for the concept we are talking about in this paper. Perhaps, we could refer to it as likelihood, as suggested by an anonymous referee. Propositions that lack numerical probabilities can still be more or less likely than other propositions. And while only some propositions have numerical probabilities, every proposition has a (qualitative) degree of likelihood, given by its equivalence class with respect to the relation of being equally likely. In this paper, however, the term “probability” is used in a broad sense and the distinction between probability and likelihood is not made.
In his (1921), Keynes to a large extent worked with what we might call conditional probabilities. He used expressions of the form \(a/h\), standing for the probability of a proposition \(a\) on the hypothesis that \(h\). As long as the condition \(h\) is positively probable (i.e., as long as for all \(S\) in K and all \(C\) in \(S\), \(C(h) > 0\)), conditional probabilities do not present any special problem for our model: We represent them by conditionalizing credence functions in all permissible states on positively probable conditions. But if we also want to allow conditional probabilities with conditions whose probability need not be positive (i.e., which receive value zero in at least some credence functions in some permissible doxastic states), the credence functions that are elements of doxastic states would have to be Popper functions instead. That is, they would need to be conditional functions that are defined even for the conditions that are assigned zero credence.
If one not only assumes that there is a unique permissible doxastic state, but also holds that this state is fully opinionated, then there is no longer any room for probabilistic incommensurabilities. Nor is there any room for propositions whose probablities are unmeasurable. Some of the post-Keynesian adherents of the FA-account of probability have chosen this uncompromising path. Perhaps the best example is Jeffreys (1931):
On a given set of data \(p\) we say that a proposition \(q\) has in relation to these data one and only one probability. If any person assigns a different probability, he is simply wrong [....] Personal differences in assigning probabilities in everyday life are not due to any ambiguity in the notion of probability itself, but to mental differences between individuals, to differences in the data available to them, and to differences in the amount of care taken to evaluate the probability. (p. 10)
While Jeffreys here postulates “one and only one probability” rather than one and only one permissible credence, he does seem to run the two together (which would be the right thing to do if only one degree of credence were permissible with regard to each proposition, given the evidence). This is confirmed by his (1939, p. 39), where he talks about “the unique reasonable degree of belief”.
There has recently been much discussion about peer disagreement. The central question in this debate is, as Van Wietmarschen (2013) puts it: “what should you believe about the disputed proposition if you have good reason to believe that an epistemic peer disagrees with you?” It is clear that the debate about peer disagreement has connections to debates about Uniqueness. I am indebted to Richard Dietz for pointing this out.
Needless to say, this is also the way vagueness can be injected into the intersection model of value relations. It night be indeterminate which preference orderings are permissible. Cf. Rabinowicz (2009b).
There might also be another route to vagueness: It might be argued that doxastic states aren’t sharply delimited. Thus, some permissible doxastic states might be ‘fuzzy’ sets of credence functions. However, to the extent that we are interested in the probability judgments and not in judgments about particular doxastic states, this possibility boils down to the possibility that the boundaries of class K are fuzzy. Yet another route to vagueness in probability judgments might have to do with their objects. If propositions aren’t sharply delimited entities, this vagueness will infect probability judgments that concern the propositions in question. In what follows, however, I disregard this and other potential sources of vagueness in probability assessments and focus on indeterminacies concerning the boundaries of the class of permissible doxastic states.
To illustrate, suppose that \(A\) and \(B\) are, respectively, propositions It will rain and It won’t rain in Keynes’s example, in which “the barometer is high, but the clouds are black” (Keynes 1921, p. 31). Now, consider a fair lottery with 1000 tickets, of which exactly one is going to win. Let \(B_{0}=B\) = It won’t rain, \(B_{1}\) = It won’t rain or ticket 1 will win, \(B_{2}\) = It won’t rain or one of the tickets 1 and 2 will win, .., \(B_{1000}\) = It won’t rain or one of the tickets 1, 2, ..., 1000 will win. We thus construct the relevant sequence by disjoining the original proposition \(B\) with propositions unrelated to \(B\) in such a way as to increase the probability of the disjunction by small steps all the way up to 1. Admittedly, it is a very artificial construction, but it should do for my purposes.
To illustrate, consider the sequence \(B_{0}=B\) = It won’t rain, \(B_{1}\) = It won’t rain and ticket 1 won’t win, \(B_{2}\) = It won’t rain and none of the tickets 1 and 2 will win, .., \(B_{1000}\) = It won’t rain and none of the tickets 1, 2, ..., 1000 will win.
But not vice versa, though. Allowing for probabilistic vagueness does not as such create a need for incommensurabilities. Introduction of incomensurabilities must be argued for independently.
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Acknowledgments
The ideas of this paper were first presented at a workshop on probability and vagueness at the University of Tokyo in March 2013. I am much indebted to the workshop organizers, Richard Dietz and Masaki Ichinose, and to the participants, in particular to Alan Hájek, Peter Pagin and Nick Smith, for their suggestions and encouragement. The paper was subsequently presented at several conferences: in Warsaw, Copenhagen, Glasgow, Venice, Chapel Hill and Rotterdam. I also lectured on the subject at the universities in York, Paris, Lund and LSE and at the Swedish Collegium for Advanced Study in Uppsala. I wish to thank the audiences at these various events for helpful comments. Special thanks are due to Olivier Roy, whose question at my talk at Munich Center for Mathematical Philosophy in 2012 spurred me pursue this project in the first place, and to Luc Bovens and Christian List, whose suggestions have been very helpful at the final stages of writing. Last but not least I want to thank the anonymous referees and the guest editor, Richard Dietz, for their incisive criticisms and constructive suggestions.
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Rabinowicz, W. From values to probabilities. Synthese 194, 3901–3929 (2017). https://doi.org/10.1007/s11229-015-0693-5
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DOI: https://doi.org/10.1007/s11229-015-0693-5