Paul D. Thorn HHU Düsseldorf, DCLPS, DFG SPP 1516
High rational personal probability (0.5 < r < 1) is a necessary condition for rational belief. Degree of probability is not generally preserved when one aggregates propositions. 2
Leitgeb (2013, 2014) demonstrated the ‘formal possibility’ of relating rational personal probability to rational belief, in such a way that: (LT ) having a rational personal probability of at least r (0.5 < r < 1) is a necessary condition for rational belief, and (DC) rational belief sets are closed under deductive consequences. In light of Leitgeb’s result, I here endeavor to illustrate another problem with deductive closure. 3
Discounting inappropriate applications of (LT ) and some other extreme views, the combination of (LT ) and (DC) leads to violations of a highly plausible principle concerning rational belief, which I’ll call the “relevant factors principle”. Since (LT ) is obviously correct, we have good reason to think that rational belief sets are not closed under deductive consequences. Note that Leitgeb’s theory is not the primary or exclusive target of the argument. 4
The following factors are sufficient to determine whether a respective agent’s belief in a given proposition, , is rational: (I) the agent’s relevant evidence bearing on , (II) the process that generated the agent’s belief that , (III) the agent’s degree of doxastic cautiousness , as represented by a probability threshold, s, (IV) the features of the agent’s practical situation to which belief in are relevant, and (V) the evidential standards applicable to the agent’s belief that , deriving the social context in which the agent entertains . 5
It is intended that (I) through (V) outline a range of factors upon which facts about rational belief supervene. To be slightly more precise, I propose (for each proposition ) that: For any two possible agents who both believe , no difference in factors (I) through (V) implies no difference in the status of the respective beliefs, as rational or not.
I regard (DC) as expressing the following claim: For all possible agents, A, with unlimited deductive abilities, the set of propositions that it is rational for A to believe is closed under deductive consequences. I adopt the convention of saying that it is rational for an agent, A, to believe a proposition, , just in case there are grounds immediately available to A such that if A were to believe that and base her belief on those grounds, then A’s belief that would be rational. 7
One may consistently hold that having a rational personal probability of at least r (r < 1) is a necessary condition for rational belief, while also holding that a rational personal probability of one is a necessary condition for rational belief. In order to exclude the preceding possibility, I propose to treat the application of (LT ) in characterizing a theory of rational belief as appropriate just in case the theory admits cases where the rational personal probability for some proposition is r, and it is rational to believe that proposition. 8
For reductio, assume (LT ) and (DC). Consider two agents A 1 and A 2 , whose total evidence, and rational personal probability functions, PROB 1 and PROB 2 , exclusively concern two disjoint domains D 1 and D 2 , describable by the following propositional atoms: p 1 , …, p n for D 1 , and q 1 , …, q m for D 2 . Suppose that all of A 1 ’s and A 2 ’s beliefs are rational. Let 1 be the strongest proposition believed by A 1 , and 2 be the strongest proposition believed by A 2 . Assume that PROB 1 ( 1 ) = PROB 2 ( 2 ) = r, and r is the minimum ‘cautiousness threshold’ for A 1 and A 2 . 9
Suppose that A 1 and A 2 will proceed within the respective domains D 1 and D 2 via actions that are specific to each domain, and the result of performing various actions are immediate payoffs in units of utility. Suppose that both A 1 and A 2 engage in appropriate practical deliberations. Finally, suppose that contexts in which A 1 and A 2 entertain 1 and 2 are thoroughly asocial.
Now observe that it is possible to form a probability function, PROB 1 2 , which is defined over truth functional combinations of p 1 , …, p n , q 1 , …, q m , which: (i) agrees with PROB 1 regarding propositions that exclusively concern D 1 , (ii) agrees with PROB 2 regarding propositions that exclusively concern D 2 , and (iii) assigns probabilities to other propositions by treating propositions that exclusively concern D 1 as being probabilistically independent of propositions that exclusively concern D 2 . 11
Consider the sets of possible worlds with respect to D 1 and D 2 , respectively, which may be identified with propositions of the form ( )p 1 … ( )p n for D 1 , and ( )q 1 … ( )q m for D 2 . These possible worlds may be listed as: w P1 , …, w P2 n , and w Q1 , …, w Q2 m . The set of possible worlds with respect to the joint domain of D 1 and D 2 may be identified with the set propositions of the form: w P i w Q k . Let PROB 1 2 (w P i w Qk ) = PROB 1 (w P i ) PROB 2 (w Q k ), for all such combinations. 12
It is apparent that adopting the probability function PROB 1 2 would be a rational response (though perhaps not uniquely so) for an agent whose total evidence is the aggregate of A 1 ’s and A 2 ’s evidence (but see below). Consider an agent, A 1 2 , whose total evidence is the aggregate of A 1 ’s and A 2 ’s, and who rationally adopts PROB 1 2 .
Suppose that A 1 2 believes exactly the same propositions concerning domain D 1 as A 1 , forming them by type identical processes to the ones that produced A 1 ’s beliefs (and similarly for D 2 and A 2 ) Suppose that A 1 2 ’s degree of doxastic cautiousness is identical to that of A 1 and A 2 .
Suppose that the language of D 1 and D 2 concern mutually remote parts of A 1 2 environment, and this fact is transparent to A 1 2 . Suppose that the actions available to A 1 2 with respect to D 1 are identical to the ones available to A 1 , where A 1 2 ’s payoff for performing respective actions under various D 1 conditions are identical to the payoffs for A 1 (and similarly for D 2 and A 2 ).
Given the preceding, it would be appropriate (though perhaps not uniquely so) for A 1 2 to negotiate D 1 by engaging in deliberations that are identical to the ones employed by A 1 (and similarly for D 2 and A 2 ). Assume that A 1 2 proceeds in this manner. In that case, it’s clear that the features of A 1 2 ’s practical situation to which her belief that 1 is relevant are identical to the features of the A 1 ’s practical situation to which her belief that 1 is relevant (and similarly for A 2 and 2 ). As with A 1 and A 2 , assume that the context in which A 1 2 entertains 1 and 2 is thoroughly asocial.
There is no difference in factors (I) through (V) regarding A 1 2 ’s and A 1 ’s belief that 1 . So A 1 2 ’s belief that 1 is rational [by the relevant factors principle]. Similarly, A 1 2 ’s belief that 2 is rational. So it is rational for A 1 2 to believe 1 2 (by (DC)). But it is not rational for A 1 2 to believe 1 2 (by (LT ), since PROB 1 2 ( 1 2 ) = r 2 < r). Thus, by reductio, not (LT ) or not (DC). So not (DC), since (LT ).
(a) One can modify the above example so that PROB 1 and PROB 2 are both defined over the joint domain of D 1 and D 2 , assuming the rational permissibility of suspending belief regarding propositions about which one has no evidence, or the rational permissibility of imprecise personal probabilities. (b) One need not accept that PROB 1 2 is rational given the aggregate of A 1 ’s and A 2 ’s evidence. …
The End. Thanks for your attention.
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