Incompleteness and incomparability

Various arguments for (precise) probabilism assume that some relation or other is complete. Whether this is a preference over acts, or some “qualitative probability ordering”, the relation is assumed to hold one way or the other between any two elements of the domain. This hardly seems like it should be a principle of rationality, especially in cases of severe uncertainty. That is—to take the preference example—it is reasonable to have no preference in either direction. This is an importantly different attitude to being indifferent between the options. Mark Kaplan argues this point as follows:

اقتباس :: Both when you are indifferent between [ltr][size=18]A[/ltr] and [ltr]B[/ltr] and when you are undecided between [ltr]A[/ltr] and [ltr]B[/ltr] you can be said not to prefer either state of affairs to the other. Nonetheless, indifference and indecision are distinct. When you are indifferent between [ltr]A[/ltr] and [ltr]B[/ltr], your failure to prefer one to the other is born of a determination that they are equally preferable. When you are undecided, your failure to prefer one to the other is born of no such determination.(Kaplan 1996: 5)[/size]

There is a standard behaviourist response to the claim that incomparability and indifference should be distinguished. In short, the claim is that it is a distinction that cannot be inferred from actual agents’ choice behaviour. Ultimately, in a given choice situation you must choose one of the options. Which you choose can be interpreted as being (weakly) preferred. Joyce offers the following criticism of this appeal to behaviourism.

اقتباس :: There are just too many things worth saying that cannot be said within the confines of strict behaviorism… The basic difficulty here is that it is impossible to distinguish contexts in which an agent’s behavior really does reveal what she wants from contexts in which it does not without appealing to additional facts about her mental state… An even more serious shortcoming is behaviorism’s inability to make sense of rationalizing explanations of choice behavior. (Joyce 1999: 21)

On top of this, behaviourists cannot make sense of the fact that incomparable goods areinsensitive to small improvements. That is, if [ltr]A[/ltr] and [ltr]B[/ltr] are two goods that you have no preference between (for example, bets on propositions with imprecise probabilities) and if [ltr]A[size=13]+[/ltr] is a good slightly better than [ltr]A[/ltr], then it might still be incomparable with [ltr]B[/ltr]. This distinguishes incomparability from indifference, since indifference “ties” will be broken by small improvements. So the claim that there is no behavioural difference between indifference and incomparability is false.[/size]
Kaplan argues that not only is violating the completeness axiom permissible, it is, in fact, sometimes obligatory.

اقتباس :: [M]y reason for rejecting as falsely precise the demand that you adopt a … set of preferences [that satisfy the preference axioms] is not the usual one. It is not that this demand is not humanly satisfiable. For if that were all that was wrong, the demand might still play a useful role as a regulative ideal—an ideal which might then be legitimately invoked to get you to “solve” your decision problem as the orthodox Bayesian would have you do. My complaint about the orthodox Bayesian demand is rather that it imposes the wrong regulative ideal. For if you have [such a] set of preferences then you have a determinate assignment of [[ltr][size=18]p[/ltr]] to every hypothesis—and then you are not giving evidence its due. (Kaplan 1983: 571)[/size]

He notes that it is not the case that it is always unreasonable or impossible for you to have precise beliefs: in that case precision could serve as a regulative ideal. Precise probabilism does still serve as something of a regulative ideal, but it is the belief of an ideal agent in an idealised evidential position. Idealised evidential positions are approximated by cases where you have a coin of a known bias. Precise probabilists and advocates of IP both agree that precise probabilism is an idealisation, and a regulative ideal. However, they differ as to what kind of idealisation is involved. Precise probabilists think that what precludes us from having precise probabilistic beliefs is merely a lack of computational power and introspective capacity. Imprecise probabilists think that even agents ideal in this sense might (and possibly should) fail to have precise probabilistic beliefs when they are not in an ideal evidential position.
At least some of the axioms of preference are not normative constraints. We can now ask what can be proved in the absence of the “purely structural”—non-normative—axioms? This surely gives us a handle on what is really required of the structure of belief.
It seems permissible to fail to have a preference between two options. Or it seems reasonable to fail to consider either of two possibilities more likely than the other. And these failures to assent to certain judgements is not the same as considering the two elements under consideration to beon a par in any substantive sense. That said, precise probabilism is serving as a regulative ideal. That is, precision might still be an unattained (possibly unattainable) goal that informs agents as to how they might improve their credences. Completeness of preference is what the thoroughly informed agent ought to have. Without complete preference, standard representation theorems don’t work. However, for each completion of the incomplete preference ordering—for each complete ordering that extends the incomplete preference relation—the theorem follows. So if we consider the set of probability functions that are such that some completion of the incomplete preference is represented by that function, then we can consider this set to be representing the beliefs associated with the incomplete preference. We also get, for each completion, a utility function unique up to linear transformation. This, in essence, was Kaplan’s position (see Kaplan 1983; 1996).
Joyce (1999: 102–4) and Jeffrey (1984: 138–41) both make similar claims. A particularly detailed argument along these lines for comparative belief can be found in Hawthorne (2009). Indeed, this idea has a long and distinguished history that goes back at least as far as Koopman (1940). I.J Good (1962), Terrence Fine (1973) and Patrick Suppes (1974) all discussed ideas along these lines. Seidenfeld, Schervish, and Kadane (1995) give a representation theorem for preference that don’t satisfy completeness. (See Evren and Ok 2011; Pedersen 2014; and Chu and Halpern 2008, 2004; for very general representation theorems).

Incompleteness and incomparability

Incompleteness and incomparability :: تعاليق

Incompleteness and incomparability