Learning & Value Change (c) I have one to ofger. . 5. Tie point of today’s talk: J. Dmitri Gallow idential norms which are not explained in terms of the single-minded pursuit of accuracy. (b) Alternatives are needed. 2 epistemic value’ and ‘opinions which are epistemic-value-dominated are Bayesianism 6. where 7. Tie Bayesian account of rational learning: you should be a probabilistic con- ditionalizer. Probabilism Conditionalization irrational’ (a) Existing accuracy-fjrst approaches to rational learning presuppose ev- (b) Consequentialist deontic norms like ‘it is rational to maximize expected 2. 1 Introduction 1. Daniel’s beliefs are irrational but true; Melissa’s beliefs are rational but false. temic good; together with So rational belief is not true belief. But isn’t there still some connection be- tween rationality and truth? 1 A tempting thought: Daniel’s beliefs were true, but they were likely to be liefs to be less accurate than other beliefs he could have adopted instead. (a) Tie axiological claim that (properly measured) accuracy is the sole epis- false. 4. Tie accuracy-fjrst project is to derive all evidential norms from: (a) If the accuracy-fjrster is right, then Daniel should have expected his be- Tie accuracy-fjrster attempts to vindicate this tempting thought. 3. Modality & Method Workshop · Center for Formal Epistemology, CMU · June 10, 2017 At a time t , your opinions are representable with a credal state < W , A , c t > , (a) W = { w 1 , w 2 , . . . w N } is a fjnite set of doxastically possible worlds; (b) A ⊆ ℘ ( W ) is a set of propositions; and (c) c t : A → [ 0, 1 ] is your time t credence function which represents the stength of your belief in all propositions in A . At all times t , c t should be a probability function. Tiere should be some ur-prior credence function c such that, for all times t and all A , E ∈ A such that E could be your total evidence at t , c t , E ( A ) = c ( A | E ) (a) ‘ c t , E ’ is the credence function you are disposed to adopt, at t , upon re- ceiving the total evidence E .
3 P4. Rationality requires you to think that your own credences See, e.g. , Oddie (1997), Joyce (2009), and Pettigrew (2011). Epistemic V dence function. tion, even without receiving any evidence P3. It is impermissible to change your credences without re- ceiving evidence. C1. So, epistemic value must be proper (b) Tie second appeals to immodesty as a rational requirement: dence function. are epistemically better than any other credences you could (a) Tie fjrst appeals to epistemic conservativism: have held instead. C1. So, epistemic value must be proper ery non-probabilistic credence is accuracy dominated by some probabilistic credence, and no probabilistic credence function is accuracy dominated. (a) Tius, the above arguments, if successful, vindicate the rational norm Probabilism in terms of accuracy and accuracy alone. 4 Conditionalization & Accuracy 4.1 Take 1 1 2 10. Why Propriety? Two arguments. 1 are choiceworthy to the extent that they maximize expected (epistemic) alue 8. Propriety 9. value. (b) Tiis is the consequentialist deontic norm which says (epistemic) acts P1. For any probability p , there is some evidence you could have that would make it permissible to have p as your cre- ‘ V ( c , w ) ’ is the epistemic value of holding the credence function c at world P2. If another credence function c is at least as valuable as your own, then it is permissible to adopt c as your credence func- w . (a) Accuracy-fjrst epistemology claims that V ( c , w ) is entirely a function of the accuracy of c at w . ‘ V c ( c ∗ ) ’ is how epistemically valuable the credence function c ∗ is, from the standpoint of the credence function c . P1. For any probability p , there is some evidence you could (a) Leitgeb & Pettigrew (2010): if your credence is a probability, p , then, have that would make it permissible to have p as your cre- for all c , V p ( c ) ! ∑ = V ( c , w ) · p ( w ) w ∈ W 11. Predd et al. (2009) show that, if V is a proper measure of accuracy, then ev- An epistemic value function V is proper ifg, for every probability function p and every credence function c � p , V p ( c ) < V p ( p ) 12. Leitgeb & Pettigrew (2010): if p is your (probabilistic) credence, then you should be disposed, upon learning E , to adopt a new credence which
3 ii. (b) Tie elimination of worlds at stage 1 either: 13. Tiis norm, together with the following theorem, epistemic value. the remaining worlds to pick a posterior which maximizes expected arg max Stage 2: you use your prior (no longer probabilistic) credences over i. relies upon an evidential norm like “do not treat a world as epis- nal learning. imize expected epistemic value. Why should we accept this norm? 15. Note: P5 does not follow from, and in fact confmicts with, the norm to max- P7. Tieorem 1. P6. Epistemic value is (properly measured) accuracy. new credence which maximizes expected epistemic value i. temically possible if it is incompatible with your evidence”; or (Conditionalization) sistent with your evidence . def defensible ( cf . Levinstein (2012)). def (2010) show that curacy measure you use. In the case of the quadratic, Leitgeb & Pettigrew 17. Tie solution to this maximization problem depends upon which proper ac- maximizes expected epistemic value amongst those credence functions con- ii. Take 2 4.2 irrational. climate change is a hoax perpetrated by the Chinese afuer a snowfall is of accuracy; in the second, we must deny that becoming certain that (c) In the fjrst case, we’ve failed to reduce all evidential norms to the pursuit is treated as an brute and not rationally evaluable fact. 14. Tiis afgords the following argument for Conditionalization: maximizes expected epistemic value in all possibilities consistent with E . ! ∑ = arg max p ( w ) · V ( c , w ) p E c w ∈ E Tieorem 1 (Generalized from Leitgeb & Pettigrew (2010)) . If V is proper, then, for any probabilistic p and any E , ∑ p ( w ) · V ( c , w ) = p ( − | E ) c w ∈ E 16. Leitgeb & Pettigrew (2010): if p is your (probabilistic) credence, then ! you should be disposed, upon learning E , to adopt a new credence which entails that, if V is a proper accuracy measure, then p E = p ( − | E ) ! ∑ = arg max p ( w ) · V ( c , w ) p E ( ⋆ ) P5. Upon learning that E , you should be disposed to adopt a c : c ( E )= 1 w ∈ E c ( ¬ E )= 0 in all possibilities consistent with E . Tieorem 2 (Leitgeb & Pettigrew (2010)) . If V = Q , then the solution to C2. Upon learning that E , you should be disposed to condi- the maximization problem in ( ⋆ ) is tionalize on E . = p ( AE ) + || AE || p ( A || E ) || E || · ( 1 − p ( E )) 18. Updating your degrees of belief from p to p ( − || E ) is not epistemically (a) Leitgeb & Pettigrew appear to presuppose a 2-stage theory of ratio- Stage 1: upon learning E , you eliminate all ¬ E worlds from W . 19. Levinstein: we should keep the norm ( ⋆ ), but instead of the quadratic ac- curacy measure Q , we should use the logarithmic L ′ , where L ′ ( c , w ) = ln [ c ( w )]
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