Updating for Externalists (S4) 1 J. Dmitri Gallow with Kripke - - PowerPoint PPT Presentation

Updating for Externalists

• J. Dmitri Gallow

Experience & Updating · Ruhr-Universität Bochum · July 20, 2017

1 Internalism & Externalism Internalism Necessarily, if ϕ is your total time t evidence, then your time t evidence entails that ϕ is your total time t evidence. (Tt ϕ → Et Tt ϕ) Externalism Possibly, your total time t evidence is ϕ but your time t evidence does not entail that ϕ is your total time t evidence. ◊(Tt ϕ ∧ ¬Et Tt ϕ) Conditionalization If C is your current credence function, upon acquiring the total evidence e, you should be disposed to adopt a new credence function, Ce,1 such that, for every proposition ϕ, Ce(ϕ) = C (ϕ | e) 1. Because conditionalization presupposes a conception of evidence on which, if ϕ is evidence for you, then it is rational for you to be absolutely certain that ϕ, we should understand the operators E and T as so: (a) Eϕ says that experience has rationalized certainty about ϕ. (b) Tϕ says that ϕ is the strongest proposition about which experience has rationalized certainty.

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Tiroughout, ‘Ce’ stands for the credence function which you should be disposed to adopt upon acquiring the total evidence e.

1.1 with Kripke Frames 2. Assume evidence is factive, and assume a Kripke semantics for E and T: (a) Eϕ is true at w ifg ϕ is true at all worlds accessible from w. (b) Tϕ is true at w ifg ϕ is true at all and only worlds accessible from w. 3. Tien, Internalism is equivalent to the S5 principle for E, (¬Eϕ → E¬Eϕ) (S5) which is equivalent to the conjunction of the S4 and B principles for E, (Eϕ → EEϕ) (S4) (¬ϕ → E¬Eϕ) (B) 2 with Experiments 4. An experiment, E, is a set of propositions {ϕ1,ϕ2,...,ϕN }. 5. You conduct the experiment E at time t ifg: (a) Your time t total evidence might be ϕ1. (b) Your time t total evidence might be ϕ2. . . . (c) Your time t total evidence might be ϕN . 1

Figure 1: An experiment which doesn’t form a partition. (d) It must be that: either your time t total evidence is ϕ1, or your time t total evidence is ϕ2, or ..., or your time t total evidence is ϕN . 6. Tiis defjnition entails: if you conduct the experiment E = {ϕ1,ϕ2,...,ϕN } at time t , then {Tt ϕ1,Tt ϕ2,...,Tt ϕN } is a partition.2 7. However, for all the defjnition has to say, {ϕ1,ϕ2,...,ϕN } could fail to be a

• partition. (See fjgure 1.)

8. Given weak assumptions:3 (a) Internalism is equivalent to the claim that, necessarily, for all t , your time t experiment forms a partition. (b) Externalism is equivalent to the claim that, possibly, for some t , your time t experiment does not form a partition. 2.1 Epistemic Evidence 9. As an argument for externalism, consider Sneak Peek. Sneak Peek You and Bonnie are playing a game involving three cups and a ball. While your back is turned, Bonnie places the ball under one of the cups and shuffmes the cups around. Tien, you attempt to guess which cup hides the ball. If you guess correctly, you win; if not,

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A partition is a set of propositions such that exactly one of the propositions in the set must be true.

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In particular, that evidence is factive and that we can give a Kripke semantics for E and T.

Bonnie wins. At t , your accomplice is going to attempt to distract Bonnie, and you’re going to sneak a peek under the cup closest to 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 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 ◃ Nothing at all (⊤) ◃ Tie ball is not beneath cup 1 (¬1) ◃ Tie ball is not beneath cup 2 (¬2) ◃ Tie ball is not beneath cup 3 (¬3) ◃ Tie ball is beneath cup 1 (1) ◃ Tie ball is beneath cup 2 (2) ◃ Tie ball is beneath cup 3 (3) So, prima facie your experiment is the non-partition E = {⊤,¬1,¬2,¬3,1,2,3}

• 11. Internalist rejoinder:

(a) While ¬1 might be your total evidence about the position of the ball, it will not be your total evidence full stop. (b) If you learn ¬1, then you must also learn that you’ve learned ¬1. (c) Once you include this additional epistemic evidence, the experiment will form a partition.

• 12. According to the internalist, what is your total evidence?

(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 TP¬1, where P is the partition P = {1,2,3} and TPϕ is true at w ifg ϕ is the proposition ∪

pi ∈P : p ∩ {x|wRt x}̸=∅

pi 2

1 2 3 T1 6/42 T2 6/42 T3 6/42 T¬1 3/42 3/42 T¬2 3/42 3/42 T¬3 3/42 3/42 T⊤ 2/42 2/42 2/42 1/3 1/3 1/3 Figure 2: An externalist pre-experimental credence distribution.

• 13. Tie internalist thinks that, in Sneak Peek, you are conducting the experiment

Eint = {TP⊤,TP¬1,TP¬2,TP¬3,TP1,TP2,TP3} (see fjgure 2)

• 14. An externalist may think that you are conducting the experiment

Eext = {⊤,¬1,¬2,¬3,1,2,3} (see fjgure 3) 3 Externalism, Conditionalization, & Reflection

• 15. Consider the following continuation of Sneak Peek.

Sneak Peek (con’t) Your accomplice does not distract Bonnie, so you learn nothing about the position of the ball. You guess that the ball is under cup 2. You know that, after a guess has been made, Bonnie always reveals an empty cup. So you will either learn ¬1 or you will learn ¬3. (a) Suppose your prior credences are C (1) = C (2) = C (3) = 1/3. 1 2 3 TTP1 6/42 TTP2 6/42 TTP3 6/42 TTP¬1 3/42 3/42 TTP¬2 3/42 3/42 TTP¬3 3/42 3/42 TTP⊤ 2/42 2/42 2/42 1/3 1/3 1/3 Figure 3: An internalist pre-experimental credence distribution. (Here, P is the partition {1,2,3}, and TPϕ says that ϕ is the strongest proposition learned about P.) (b) Suppose that your experiment is, as the externalist may think, E = {¬1,¬3}.

• 16. Note:

(a) If you conditionalize on ¬1, then your post-experimental credence that 2 will be 1/2. (b) If you conditionalize on ¬3, then your post-experimental credence that 2 will be 1/2. (c) Shouldn’t you be able to reason as follows? “No matter what I learn, it will 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

Reflection Your pre-experimental credence that ϕ should be equal to your expectation of your rational post-experimental credence that ϕ. C (ϕ)

!

= ∑

e∈E

Ce(ϕ) · C (Te)

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Tiis principle difgers from van Fraassen’s in that it advises you to defer (in expectation) to your future rational credence, and not your future credence.

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• 18. Tie only way to satisfy both conditionalization and reflection in the ex-

periment E = {¬1,¬3} is to either be certain that 2 or to be certain that ¬2. (a) Suppose your prior credence that 2 is x ̸= 0,1. (b) Tien, C (2 | ¬1) > x and C (2 | ¬3) > x. (c) So, if you satisfy conditionalization, then your pre-experimental cre- dence that 2 is not your expectation of your rational post-experimental credence that 2, and you violate reflection.

• 19. So, if the externalist thinks that there are experiments like {¬1,¬3}, then they

face a choice between conditionalization and reflection. 4 Reflection, Deliberate Self-Delusion, & Intentionally Biased Inquiry

• 20. Salow (forthcoming): reflection has an important role to play in preventing

rational agents from engaging in deliberate self-delusion.

• 21. Suppose you can design the experiment E = {¬1,¬3}, and there’s some propo-

sition p which you’d like to believe. (a) Tell a confjdant who knows the truth about whether p to put the ball beneath cup 2 ifg p. (b) Conduct the experiment. (c) You’ll end up more confjdent that p no matter what. (d) Rinse & Repeat to become as confjdent that p as you wish.

• 22. Such a procedure is not rational inquiry; it is deliberate self-delusion. No

sensible epistemology will say otherwise. So we should accept the principle No Self-Delusion A rational agent may not design an experiment which they know in advance will leave them more confjdent of some proposition than they were before they designed the experiment.

• 23. What Salow has shown is that the externalist who accepts No Self-Delusion

must reject Conditionalization. (a) No Self-Delusion is non-negotiable. So the externalist must not be a conditionalizer.

• 24. Should the externalist accept reflection? Salow argues ‘yes’.

(a) Tie reason for accepting No Self-Delusion should also persuade us to accept the more general No Biased Inquiry No Biased Inquiry A rational agent may not design an experiment which they ex- pect to leave them more confjdent of some proposition than they were before they designed the experiment. (b) Tiat is, it should be that, for all ϕ, your expectation of Ce(ϕ) − C (ϕ) is zero ∑

e∈E

[Ce(ϕ) − C (ϕ)] · C (Te) = 0 But this holds ifg you satisfy the principle of Reflection.

• 25. So: the reasons which persuaded us to accept No Self-Delusion should also

persuade us to accept the principle of Reflection.

• 26. So: the externalist needs an update rule which satisfjes the principle of Re-

flection in non-partitional experiments. 5 An Externalist Update

• 27. Hild (1998b) and Schoenfield (forthcoming) advise the externalist to re-

ject conditionalization and to instead embrace epistemic conditional- ization. Epistemic Conditionalization If C is your pre-experimental credence function, then upon receiv- ing the total evidence e, you should be disposed to adopt a new credence function, Ce, such that, for every proposition ϕ, Ce(ϕ) = C (ϕ | Te)

• 28. Epistemic Conditionalization secures the principle of Reflection.

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(a) If E = {e1, e2,..., eN } is your experiment, then {Te1,Te2,...,TeN } must be a partition. (b) Tien, by the law of total probability, C (ϕ) = ∑

ei ∈E

C (ϕ | Tei) · C (Tei) = ∑

ei ∈E

Cei (ϕ) · C (Tei)

• 29. However, Epistemic conditionalization is of no help for the externalist,

since it entails internalism. (a) Recall: Eϕ, says that experience rationalizes certainty about ϕ. (b) So, if experience rationalizes conditionalizing on Tϕ, then Tϕ is evi- dence, ETϕ. (c) According to Epistemic Conditionalization, experience rationalizes conditionalizing on Tϕ whenever it is true. (d) So, (Tϕ → ETϕ)

• 30. Gallow (2014) advises rejecting conditionalization in certain non-

partitional experiments—specifjcally, when experimental results are theory- dependent. (a) In a paradigmatic case, the most you’ll be in a position to believe with certainty is that, if t is the true background theory, then the experimental result is e (b) So, your total evidence will be t → e (c) Tien, Gallow (2014) prescribes to update with holistic conditional- ization Holistic Conditionalization If C is your pre-experimental credence function, then upon learn- ing t → e, you should adopt a new credence function, Ct →e, such that, for every proposition ϕ, Ct →e(ϕ) = C (ϕ | t ∩ e) · C (t ) + C (ϕ | ¬t ) · C (¬t )

• 31. However, Holistic Conditionalization is not general enough to serve the

externalist’s needs. 1 2 3 T¬1 1/6 2/6 T¬3 2/6 1/6 1/3 1/3 1/3

(a) A pre-experimental credence distribution

1 2 3 T¬1 1/6 4/6 T¬3 1/6 1/3 2/3

(b) Tie result of updating on ¬1 with excondi

Figure 4: Updating with excondi (a) In cases like Sneak Peek, there need not be any theory-dependence. (b) And when there is no theory-dependence, Holistic conditionaliza- tion reduces to conditionalization. (c) Moreover, Holistic Conditionalization has problems. In particular, it does not allow the background theory t to be (dis)confjrmed.

• 32. An new update rule for the externalist:

Externalist Conditionalization When conducting the experiment E, upon receiving the evidence e, you should be disposed to adopt a new credence function, Ce, such that, for every ϕ, Ce(ϕ) = ∑

a∈A[E]

C (ϕ | a) · C (a | Te) (a) ‘A[E]’ is the set of atoms of the experiment E (b) Tie atoms of the experiment E are all the non-empty conjunctions of (negations of) the e ∈ E. 5

• 33. Externalist Conditionalization entails the principle of Reflection.

∑

e∈E

Ce(ϕ) · C (Te) = ∑

e∈E

∑

a∈A[E]

C (ϕ | a) · C (a | Te) · C (Te) = ∑

a∈A[E]

C (ϕ | a) ∑

e∈E

C (a | Te) · C (Te) = ∑

a∈A[E]

C (ϕ | a) · C (a) = C (ϕ)

• 34. When E is a partition, excondi reduces to condi

(a) If E is a partition, then, for each e ∈ E, e =||= Te. (b) And, if E is a partition, then A[E] = E. So: Ce(ϕ) = ∑

a∈A[E]

C (ϕ | a) · C (a | Te) = ∑

e∗∈E

C (ϕ | e∗) · C (e∗ | Te) = C (ϕ | e) · C (e | Te) = C (ϕ | e)

• 35. Externalist conditionalization will also agree with conditionalization

if your experiment is epistemically neutral, where Epistemic Neutrality A non-partitional experiment E, in which you will learn something about the partition P, is epistemically neutral ifg, for all p ∈ P, and all e such that e ∈ E, C (p | Te) = C (p | e) (a) E.g., the experiment shown in fjgure 2 is epistemically neutral.

• 36. In the paradigm cases where holistic conditionalization was meant to ap-

ply, externalist conditionalization agrees with holistic conditional- ization. (a) Suppose that your experiment is E = {t → e1, t → e2,..., t → eN } where {e1, e2,..., eN } is a partition. (b) And suppose that, for each ei, C (t | Tei) = C (t ). (c) Tien, the atoms of your experiment will be {¬t , t ∩e1, t ∩e2,..., t ∩eN }, and Ct →ei (ϕ) = C (ϕ | t ∩ ei) · C (t ∩ ei | Tei) + C (ϕ | ¬t ) · C (¬t | Tei) = C (ϕ | t ∩ ei) · C (t | Tei) + C (ϕ | ¬t ) · C (¬t | Tei) = C (ϕ | t ∩ ei) · C (t ) + C (ϕ | ¬t ) · C (¬t ) i. Note also: if C (t | Te) ̸= C (t ), then excondi will allow the back- ground theory to be (dis)confjrmed. 6 In Summation

• 37. If you are an externalist, you must choose between reflection and condi-

tionalization

• 38. Salow (forthcoming) showed us that you should not choose conditional-

ization, and that there is good reason to accept reflection.

• 39. My proposed alternative, excondi,

(a) will always satisfy the principle of reflection (b) reduces to conditionalization in partitioning experiments and when- ever the experiment is epistemically neutral. (c) reduces to holistic conditionalization in paradigm cases of theory- dependence, and solves that rule’s troubles with confjrming background theories. References Bronfman, Aaron. 2014. “Conditionalization and not Knowing that One Knows.” Erkenntnis, vol. 79 (4): 871–892. 6

Elga, Adam. 2013. “Tie puzzle of the unmarked clock and the new rational refmec- tion principle.” Philosophical Studies, vol. 164: 127–139. Gallow, J. Dmitri. 2014. “How to Learn from Tieory-Dependent Evidence;

• r Commutativity and Holism: A Solution for Conditionalizers.” Tie British

Journal for the Philosophy of Science, vol. 65 (3): 493–519. [5] Hild, Matthias. 1998a. “Auto-Epistemology and Updating.” Philosophical Studies,

• vol. 92: 321–361.

—. 1998b. “Tie Coherence Argument Against Conditionalization.” Synthese, vol. 115: 229–258. [4] Salow, Bernhard. forthcoming. “Tie Externalist’s Guide to Fishing for Compli- ments.” Mind. [4], [6] Schoenfield, Miriam. forthcoming. “Conditionalization does not (in general) Maximize Expected Accuracy.” Mind. [4] Stalnaker, Robert C. 2009. “On Hawthorne and Magidor on Assertion, Con- text, and Epistemic Accessibility.” Mind, vol. 118 (470): 399–409. van Fraassen, Bas C. 1984. “Belief and the Will.” Tie Journal of Philosophy,

• vol. 81 (5): 235–256. [3]

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