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04 April 2023
A Correction to this paper has been published: https://doi.org/10.1007/s11098-023-01943-5
Notes
As Easwaran explains ((Easwaran, 2016), Appendix C), everything that can be done in this framework with three qualitative attitudes (B, D, S) can be done (without loss of generality) with only two attitudes (B, \(\overline{B}\)).
It is worth noting that this notion of coherence is considerably weaker than traditional coherence norms for belief. As we will see below, not only is it compatible with inconsistent belief sets — it is even weaker than Lockean norms for belief.
Staffel discusses various ways of measuring “degree of incoherence” (or “degree of irrationality”) of an agent’s credence function c. Any of these could be plugged-in to \(\epsilon\)-Representability in order to yield a precisification of the concept. In my discussion of the examples in this section, I will plump for one precisification, but others are possible.
This also gives us another (qualitative) way to measure the “degree of irrationality” of an agent with (credence function c and) belief set \(\mathbf {B}\). We can say that they are (or their belief set \(\mathbf {B}\) is) “1 belief away from probabilistic/Lockean representability (at a threshold compatible with their epistemic utilities).”
Alternatively, from a qualitative perspective, could say that our agent is \(\epsilon =1\) full belief from being probabilistically representable. For the purposes of these comments, I will stick mainly to quantitative assessments of “approximate coherence.”
One might wonder at this stage — why not replace (ii) and (iii) with (ii*) treating the agent as if she assigns \(c(\sim p) \ge t\) for some Lockean threshold c and (iii*) treating the agent as if she is Pseudo-Jeffrey conditionalizing on \(\sim p\)? Pseudo-Conditionalization can be thought of as moving to the closest probability function such that \(c(\sim p) = 1\). Pseudo-Jeffrey conditionalization can be thought of as moving to the closest probability function such that \(c(\sim p) \ge t\) (Diaconis & Zabell, 1982). While this would solve the problem with (ii), it would not solve the problem with (iii). Both varieties of pseudo-updating involve a “renormalization” process which can be modeled (in mechanical terms) as computing the solution of a computationally complex constrained optimization problem.
I’ve changed the examples slightly. Here, I use “X” and “Y” rather than “A” and “B,” and our Table 2 is a bit more detailed than Staffel’s Table (on page 194), since we’re including the entire algebra of propositions. This is because we will be assessing the “degrees of incoherence” of the results of the various proposals, and it seems that this should be measured with respect to the entire algebra of propositions. Otherwise, our functions c, PC(Y), \(GR_{0.9}\) and SR(Y) in Table 2 are the same as \(c_2\), PC(B), \(GR(>0.9)\), and SR(B) in Staffel’s table. Finally, I have added an additional column for another strategy [\(PC(p_{0.9})\)], which I introduce below.
For some additional perspective on the examples, it is also worth noting that the distance in SED from PC(Y) to SR(Y) is 0.943; the distance in SED from SR(Y) to \(GR_{0.9}\) is 0.778; and the distance in SED between PC(Y) to \(GR_{0.9}\) is 0.107.
It is true that conditionalization is — as a matter of general computational complexity qua algorithm — far more complex than either (GR) or (SR). But, in particular cases, it is a bit less clear (to me) precisely how we ought to gauge the trade-off between (actual) degree-of-incoherence and (actual) computational cost.
Savage Savage (1954) would not have seen things this way. For him, preferences (and other qualitative attitudes like comparative confidence) were fundamental, and the numerical probability and utility functions in his representation theorem were merely theoretically useful measurement-theoretic posits.
Dr. Truthlove would not see things this way. For him, full beliefs are fundamental, and the numerical probability and utility functions in his representation theorem are merely theoretically useful measurement-theoretic posits.
de Finetti (1951) would not have seen things this way. For him, comparative confidence relations were fundamental, and the numerical probability functions in Scott’s representation theorem would have been seen by him merely as theoretically useful measurement-theoretic posits.
This part is a bit tricky, since it not totally clear how an agent with incoherent credences should evaluate the subjective value of gambles. Although, see Hedden (2013) for a defense of sticking with the standard EU approach of judging “actions by taking the sum of the values of each possible outcome of that action, weighted by one’s credence that the action will result in that outcome.”
References
Clarke, R. (2013). Belief Is Credence One (in Context)
de Finetti, B. (1951). La ‘logica del plausible’ secondo la concezione di Polya
Diaconis, P., and Zabell, S. (1982). Updating subjective probability
Dorst, K. (2019). Lockeans Maximize Expected Accuracy
Easwaran, K. (2016). Dr. Truthlove or: How I Learned to Stop Worrying and Love Bayesian Probability
Easwaran, K and B. Fitelson (2015). Accuracy, Coherence, and Evidence
Foley,R. (2009). Beliefs, degrees of belief, and the Lockean thesis
Harsanyi, J. (1985). Acceptance of Empirical Statements: A Bayesian Theory Without Cognitive Utilities
Hedden, B. (2013). Incoherence without exploitability
Kraft, C., Pratt, J., and Seidenberg, A. (1959). Intuitive probability on finite sets
Lewis, D. (1974). Radical Interpretation
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Staffel, J. (2020). Unsettled Thoughts
Acknowledgements
It is such a pleasure and a privilege to be able to contribute to a symposium in honor of Julia Staffel’s Unsettled Thoughts. I have such fond memories of Julia’s visit to UC–Berkeley in the summer of 2009 (she was then a graduate student at USC) to work on her project on “degrees of incoherence.” It was a novel and exciting project, and that was a fun summer. Over the course of the next decade, the project blossomed into a masterful treatise. Congratulations Julia and her collaborators on a remarkable achievement! And, many thanks to Julia for her helpful comments on previous versions of these remarks
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Fitelson, B. Remarks on staffel on full belief. Philos Stud 180, 385–393 (2023). https://doi.org/10.1007/s11098-022-01836-z
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DOI: https://doi.org/10.1007/s11098-022-01836-z