Updating for Externalists (S4) 1 J. Dmitri Gallow with Kripke Frames 2. 3. (S5) (B) Because conditionalization presupposes a conception of evidence on which, 2 with Experiments 4. 5. . . . rationalized certainty. 1.1 1. Conditionalization 1 Internalism & Externalism Internalism Externalism 1 Experience & Updating · Ruhr-Universität Bochum · July 20, 2017 Assume evidence is factive, and assume a Kripke semantics for E and T : Necessarily, if ϕ is your total time t evidence, then your time t evidence entails that ϕ is your total time t evidence. (a) � E ϕ � is true at w ifg � ϕ � is true at all worlds accessible from w . (b) � T ϕ � is true at w ifg � ϕ � is true at all and only worlds accessible from w . � ( T t ϕ → E t T t ϕ ) Tien, Internalism is equivalent to the S5 principle for E , Possibly, your total time t evidence is ϕ but your time t evidence does � ( ¬ E ϕ → E ¬ E ϕ ) not entail that ϕ is your total time t evidence. which is equivalent to the conjunction of the S4 and B principles for E , ◊ ( T t ϕ ∧ ¬ E t T t ϕ ) � ( E ϕ → EE ϕ ) If C is your current credence function, upon acquiring the total evidence � ( ¬ ϕ → E ¬ E ϕ ) e , you should be disposed to adopt a new credence function, C e , 1 such that, for every proposition ϕ , C e ( ϕ ) = C ( ϕ | e ) An experiment, E , is a set of propositions { ϕ 1 , ϕ 2 ,..., ϕ N } . if ϕ is evidence for you, then it is rational for you to be absolutely certain that You conduct the experiment E at time t ifg: ϕ , we should understand the operators E and T as so: (a) Your time t total evidence might be ϕ 1 . (a) � E ϕ � says that experience has rationalized certainty about ϕ . (b) Your time t total evidence might be ϕ 2 . (b) � T ϕ � says that ϕ is the strongest proposition about which experience has Tiroughout, ‘ C e ’ stands for the credence function which you should be disposed to adopt upon (c) Your time t total evidence might be ϕ N . acquiring the total evidence e .
Figure 1: An experiment which doesn’t form a partition. facie A partition is a set of propositions such that exactly one of the propositions in the set must be true. 3 Bonnie, and you’re going to sneak a peek under the cup closest to try to look under, nor do you know whether you’ll be successful. 10. Prima facie , in Sneak Peek, you might acquire any of the following total evi- dence propositions So, prima your which cup hides the ball. If you guess correctly, you win; if not, experiment is the non-partition 11. Internalist rejoinder: not be your total evidence full stop . (c) Once you include this additional epistemic evidence, the experiment will form a partition. 12. According to the internalist, what is your total evidence? 2 2 the cups and shuffmes the cups around. Tien, you attempt to guess While your back is turned, Bonnie places the ball under one of 8. 7. 2.1 Epistemic Evidence 9. As an argument for externalism, consider Sneak Peek. 6. Sneak Peek You and Bonnie are playing a game involving three cups and a ball. Given weak assumptions: 3 Bonnie wins. At t , your accomplice is going to attempt to distract you at t , if you can. However, you don’t know which cup will be closest to you at t , and therefore, you don’t know which cup you’ll ◃ Nothing at all ( ⊤ ) ◃ Tie ball is not beneath cup 1 ( ¬ 1 ) (d) It must be that: either your time t total evidence is ϕ 1 , or your time t ◃ Tie ball is not beneath cup 2 ( ¬ 2 ) total evidence is ϕ 2 , or ... , or your time t total evidence is ϕ N . ◃ Tie ball is not beneath cup 3 ( ¬ 3 ) Tiis defjnition entails: if you conduct the experiment E = { ϕ 1 , ϕ 2 ,..., ϕ N } ◃ Tie ball is beneath cup 1 ( 1 ) at time t , then { T t ϕ 1 , T t ϕ 2 ,..., T t ϕ N } is a partition. 2 ◃ Tie ball is beneath cup 2 ( 2 ) ◃ Tie ball is beneath cup 3 ( 3 ) However, for all the defjnition has to say, { ϕ 1 , ϕ 2 ,..., ϕ N } could fail to be a = E partition. (See fjgure 1.) {⊤ , ¬ 1 , ¬ 2 , ¬ 3 , 1 , 2 , 3 } (a) Internalism is equivalent to the claim that, necessarily, for all t , your time t experiment forms a partition. (a) While ¬ 1 might be your total evidence about the position of the ball , it will (b) Externalism is equivalent to the claim that, possibly, for some t , your time t experiment does not form a partition. (b) If you learn ¬ 1 , then you must also learn that you’ve learned ¬ 1 . (a) Tiey can’t say that it’s T ¬ 1 , so long as evidence is factive. (b) Tiey can’t say that it’s E ¬ 1 , since this is consistent with E ¬ 2 . (c) Tiey should say that your total evidence is T P ¬ 1 , where P is the partition P = { 1 , 2 , 3 } and � T P ϕ � is true at w ifg ϕ is the proposition ∪ p i p i ∈ P : In particular, that evidence is factive and that we can give a Kripke semantics for E and T . p ∩ { x | wR t x }̸ = ∅
3 6/42 3/42 3/42 0 6/42 0 0 0 0 0 0 6/42 cup 2. You know that, after a guess has been made, Bonnie always about the position of the ball. You guess that the ball is under Your accomplice does not distract Bonnie, so you learn nothing Sneak Peek (con’t) 15. Consider the following continuation of Sneak Peek. 3/42 3/42 3 Figure 3: future rational credence, and not your future credence. 4 Reflection (c) Shouldn’t you be able to reason as follows? “No matter what I learn, it will will be 1/2. will be 1/2. 16. Note: 1/3 3/42 1/3 1/3 2/42 2/42 2/42 0 3/42 Externalism, Conditionalization, & Reflection 0 3/42 6/42 3/42 0 0 3/42 2/42 2/42 2/42 3/42 1/3 0 1/3 0 3/42 0 0 6/42 0 0 1/3 Figure 2: An externalist pre-experimental credence distribution. 13. Tie internalist thinks that, in Sneak Peek, you are conducting the experiment 0 6/42 14. An externalist may think that you are conducting the experiment 3/42 1 2 3 1 2 3 TT P 1 T 1 TT P 2 T 2 TT P 3 T 3 TT P ¬ 1 T ¬ 1 TT P ¬ 2 T ¬ 2 TT P ¬ 3 T ¬ 3 TT P ⊤ T ⊤ An internalist pre-experimental credence distribution. (Here, P is the partition { 1 , 2 , 3 } , and � T P ϕ � says that ϕ is the strongest proposition learned about P .) E int = { T P ⊤ , T P ¬ 1 , T P ¬ 2 , T P ¬ 3 , T P 1 , T P 2 , T P 3 } (b) Suppose that your experiment is, as the externalist may think, E = {¬ 1 , ¬ 3 } . (see fjgure 2) (a) If you conditionalize on ¬ 1 , then your post-experimental credence that 2 E ext = {⊤ , ¬ 1 , ¬ 2 , ¬ 3 , 1 , 2 , 3 } (b) If you conditionalize on ¬ 3 , then your post-experimental credence that 2 (see fjgure 3) be rational for me to have credence 1/2 that 2 . So I should have credence 1/2 in 2 now.” 17. Tiis reasoning is endorsed by van Fraassen’s principle of reflection. 4 Your pre-experimental credence that ϕ should be equal to your expectation of your rational post-experimental credence that ϕ . reveals an empty cup. So you will either learn ¬ 1 or you will learn ! ∑ C ( ϕ ) = C e ( ϕ ) · C ( T e ) ¬ 3 . e ∈ E Tiis principle difgers from van Fraassen’s in that it advises you to defer (in expectation) to your (a) Suppose your prior credences are C ( 1 ) = C ( 2 ) = C ( 3 ) = 1 / 3 .
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