Abstract
Thomas Kroedel has recently proposed an interesting Pareto-style condition on permissible belief. Despite the condition’s initial plausibility, this paper aims at providing a counterexample to it. The example is based on the view that a proper condition on permissible belief should not give permission to believe a proposition that undermines one’s belief system or whose epistemic standing decreases in the light of one’s de facto beliefs.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
As Kroedel (2017, fn.11) has noted, his condition has some similarity to dominance principles known from decision and game theory. One could argue that from a game-theoretic perspective, his condition would be closer to the spirit of a Pareto principle if it stated that the permission to believe a number of proposition persists if at least one proposition’s epistemic standing improves while the epistemic standing of no other proposition deteriorates. However, nothing essential hinges on this.
Challenging Eder’s condition is not Kroedel’s main motivation. His aim is to defend his well-known permissibility solution to Kyburg’s (1961) famous lottery paradox as presented in Kroedel (2012). For objections against the permissibility account see Littlejohn (2012) with a reply in Kroedel (2013a) and Littlejohn (2013) with a reply in Kroedel (2013b). Another objection has been proposed by Huber (2014).
In fact, Kroedel’s way of representing the lotteries is slightly different but equivalent to how they are represented here: he assumes three tickets and a blank where the blank being drawn represents the event that all three tickets lose which corresponds to the proposition \(p\wedge q\wedge r\). Analogously, the propositions \(p\wedge q\), \(p\wedge r\) and \(q\wedge r\) can be represented by introducing further types of tickets. For simplicity and without loss of generality, however, we will omit this detour. Instead of lotteries, one can also think of the discussed cases very naturally as situations where one is playing a slot machine: p, q and r are the propositions that the first, second or third reel does not land on the relevant fruit, \(p\wedge q\), \(p\wedge r\) and \(q\wedge r\) are the propositions that the respective pairs of reels miss the relevant fruit and \(p\wedge q\wedge r\) is the proposition that all reels miss it.
One remark on the involved probabilities: unlike Kroedel, I do not make the assumption that certain Boolean combinations of propositions are impossible and should therefore receive zero probabilities. The relevant propositions are \(\lnot p\wedge \lnot q\wedge \lnot r\), \(\lnot p\wedge \lnot q\wedge r\), \(\lnot p\wedge q\wedge \lnot r\) and \(p\wedge \lnot q\wedge \lnot r\). In a lottery situation, it would be quite natural to assume that each of them is impossible because at most one ticket can win. However, since Kroedel’s lotteries are supposed to mimic situations analogous to Makinson’s (1965) preface paradox, this assumption should be relaxed: all combinations except \(\lnot p\wedge \lnot q\wedge \lnot r\) should be considered possible since the preface paradox only assumes at least one of the preface propositions is false, i.e. at least one of the propositions p, q or r is true. I will further relax this assumption and allow all preface propositions to be true and hence allow the combination \(\lnot p\wedge \lnot q\wedge \lnot r\) to receive non-zero probability. This will make our calculations slightly more difficult but we can still provide the lower and upper bounds for the relevant probabilities. More important, however, it will make the argument more general because it refers a larger class of probability functions.
An anonymous reviewer has pointed out that it could be seen as problematic to evaluate the probability of a proposition in the light of an agent’s de facto beliefs. A promising alternative, according to the reviewer, would be Timothy Williamson’s (2000) proposal that probability should only be taken conditional an agent’s knowledge. While I agree that there is much potential for a discussion about this issue, I am afraid that addressing it in a footnote would not do justice to it. Hence, it is left for future research. Nevertheless, I am very grateful to two reviewers for a number of stimulating remarks on this aspect.
References
Eder, A.-M. A. (2015). No match point for the permissibility account. Erkenntnis, 80(3), 657–673.
Huber, F. (2014). What is the permissibility solution a solution of? A question for Kroedel. Logos and Episteme, 5(3), 333–342.
Kroedel, T. (2012). The lottery paradox, epistemic justification and permissibility. Analysis, 72(1), 57–60.
Kroedel, T. (2013a). The permissibility solution to the lottery paradox—Reply to Littlejohn. Logos and Episteme, 4(1), 103–111.
Kroedel, T. (2013b). Why epistemic permissions don’t agglomerate—Another reply to Littlejohn. Logos and Episteme, 4(4), 451–455.
Kroedel, T. (2017). The lottery, the preface, and conditions on permissible belief. Erkenntnis. https://doi.org/10.1007/s10670-017-9911-5.
Kyburg, H. E. (1961). Probability and the logic of rational belief. Middletown: Wesleyan University Press.
Littlejohn, C. (2012). Lotteries, probabilities, and permissions. Logos and Episteme, 3(3), 509–14.
Littlejohn, C. (2013). Don’t know, don’t believe: Reply to Kroedel. Logos and Episteme, 4(2), 231–38.
Makinson, D. C. (1965). The paradox of the preface. Analysis, 25(6), 205–207.
Pollock, J. L. (1986). The paradox of the preface. Philosophy of Science, 53(2), 246–258.
Sen, A. (1970). The impossibility of a Paretian liberal. Journal of Political Economy, 78(1), 152–157.
Williamson, T. (2000). Knowledge and its Limits. Oxford: Oxford University Press.
Acknowledgements
I would like to thank (in alphabetical order) Roman Heil, Thomas Kroedel, Patricia Rich and Moritz Schulz for helpful comments. Special thanks to Thomas Kroedel for taking the time to discuss some of my questions in detail and for encouraging me to submit this short paper. I am also indebted to three anonymous referees whose remarks helped me to improve this paper. This work was funded by Grant SCHU 3080/3-1 to Moritz Schulz from the DFG as part of the Emmy-Noether-Group Knowledge and Decision.
Author information
Authors and Affiliations
Corresponding author
Appendix
Appendix
- Case 1 :
-
Let \(P(p)=P(q)=P(r)=75/100\) and \(P(p\wedge q\wedge r)=25/100\). According to the Boole–Fréchet inequalities, it holds that \(25/100\le P(p\wedge q), P(p\wedge r), P(q\wedge r)\le 75/100\). Using the upper bound yields \(P(r|p\wedge q)\ge 25/100\times 100/75\), using the lower bound yields \(P(r|p)\ge 25/100\times 100/75\) and \(P(r|q)\ge 25/100\times 100/75\). Analogously, \(P(p|r)\ge 25/100\times 100/75\), \(P(q|r)\ge 25/100\times 100/75\) and \(P(p\wedge q|r)=25/100\times 100/75\).
- Case 2 :
-
Let \(P(p)=P(q)=77/100\), \(P(r)=76/100\) and \(P(p\wedge q\wedge r)=30/100\). According to the Boole–Fréchet inequalities, it holds that \(30/100\le P(p\wedge q)\le 77/100\) and \(30/100\le P(p\wedge r), P(q\wedge r)\le 76/100\). Using the upper bound yields \(P(r|p\wedge q)\ge 30/100\times 100/77\), using the lower bound yields \(P(r|p)\ge 30/100\times 100/76\) and \(P(r|q)\ge 30/100\times 100/76\). Analogously, \(P(p|r)\ge 30/100\times 100/76\), \(P(q|r)\ge 30/100\times 100/76\) and \(P(p\wedge q|r)=30/100\times 100/76\).
- Case 3 :
-
Let \(P(p)=P(q)=P(r)=75/100\), \(P(p\wedge q\wedge r)=27/64\) and \(P(p\wedge q)=P(p\wedge r)=P(q\wedge r)=9/16\). This yields \(P(r|p)=P(r|q)=P(r|p\wedge q)=9/16\times 100/75\). Analogously, \(P(p|r)=P(q|r)=9/16\times 100/75\) and \(P(p\wedge q|r)=27/64\times 100/75\).
Rights and permissions
About this article
Cite this article
Koscholke, J. On the Pareto Condition on Permissible Belief. Erkenn 84, 1183–1188 (2019). https://doi.org/10.1007/s10670-018-0003-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10670-018-0003-y