Confjrmation Tieory 2 What we want from a theory of confjrmation: (a) A qualitative account of confjrmation. Pittsburgh Summer Program 1 (b) A quantitative measure of confjrmation. i. (c) We’d like our theory of confjrmation to be both formal and intersubjective . i. ii. You can’t always get what you want inductive inference is non-monotonic (or defeasible). 5. Hempel (1945a,b): any theory of confjrmation which satisfjes these two plausi- ble principles will say that every proposition confjrms every other proposition. Entailments Confirm (ec) Consequence Condition (cc) 6. Hempel: even if we weaken these principles like this, Laws are Confirmed by Their Instances 4. i. 1 2. 1 Confirmation & Disconfirmation 1. 3. i. While deductive inference is monotonic (or indefeasible), Center for the Philosophy of Science, University of Pittsburgh · July 7, 2017 Formal : we can say whether E confjrms H by looking only at the Sometimes, a piece of evidence, E , gives reason to believe a hypothesis, H . syntax, or logical form, of E and H . When this is so, say that E confjrms H . Intersubjective : we can all agree about whether E confjrms H . Other times, a piece of evidence, E , gives reason to dis believe a hypothesis, H . When this is so, say that E disconfjrms H . Just because we have some evidence, E , which confjrms H , this doesn’t mean that we should think H is true. (a) Confjrmation is a matter of degree. E could confjrm H by giving a slight but not conclusive reason to believe H . (b) Evidence for H could be defeated. We could have some evidence E which If H entails E , then E confjrms H . confjrms H while having a total body of evidence which disconfjrms H . If E confjrms H , then E confjrms anything which H entails. If P deductively entails C , then P & Q also deductively entails ◃ C (a) Take any two propositions A and B . (b) A & B entails A . So, by ec, A confjrms A & B . ◃ If E confjrms H , it doesn’t follow that E & F confjrms H (c) A confjrms A & B (above) and A & B entails B . So, by cc, A confjrms B For any H , E : does E confjrm H ? A law statement of the form “All F s are G s” is confjrmed by an F For any H , E : to what degree does E confjrm H ? G .
Equivalence Condition 3.1 . . . All unobserved emeralds are grue 8. counterinduction. So a theory of confjrmation must go beyond logical form. 3 Confirmation & Probability Probability Tie fjrst observed emerald is grue 9. which also has the following properties: diagram or a probabilistic truth-table . 11. We introduce the following defjnition : def if and only if 3.2 From Probability to Confirmation 13. One popular confjrmation measure: Tie second observed emerald is grue A purely formal theory of confjrmation cannot distinguish induction from All unobserved emeralds are green has not been observed before 2018 and is blue. nearly everything will end up confjrming any universal law statement. Take, for a toy example, the law statement “All ravens are black”. (a) “All ravens are black” is equivalent to “All non-black things are non-ravens” (b) By Laws are Confirmed by Their Instances, a green leaf confjrms the hypothesis that all non-black things are non-ravens (c) By (a), (b), and Equivalence Condition, a green leaf confjrms the hy- pothesis that all ravens are black. 7. 2 (b) Tien, there is no syntactic, formal difgerence between this inductive infer- Tie second observed emerald is green . . ence (which is a strong inductive inference): Tie fjrst observed emerald is green . If E confjrms H , then E confjrms anything which is equivalent to H . A probability function , Pr , is any function from a set of propositions, � , to the unit interval, [ 0 , 1 ] Pr : � → [ 0 , 1 ] Ax1. If the proposition ⊤ is necessarily true, then Pr ( ⊤ ) = 1 . Ax2. If the propositions A and B are inconsistent, then Pr ( A ∨ B ) = Pr ( A ) + Pr ( B ) . Goodman (1955): in order to say whether “All F s are G s” is confjrmed by an F G , we must know something about what ‘ F ’ and ‘ G ’ mean . 10. If Pr is a probability function, then we may represent it with a muddy Venn (a) Say that a thing is grue ifg it has been observed before 2018 and is green or = Pr ( A & B ) Pr ( A | B ) , if defjned Pr ( B ) 12. We may say that the propositions A and B are independent (according to Pr ) Tie n th observed emerald is green Pr ( A & B ) = Pr ( A ) · Pr ( B ) and this inductive inference (which is a counter-inductive inference): Given a probability function Pr , we may construct a confjrmation measure C , (a) C ( H , E ) gives the degree to which the evidence E confjrms the hypothesis Tie n th observed emerald is grue H . D ( H , E ) = Pr ( H | E ) − Pr ( H )
(a) Tiere are other possibilities— e.g. , i. some rational agent. 18. Tie Bayesian then endorses the following norms of rationality: Probabilism the axioms of probability. Conditionalization It is a requirement of rationality that, upon acquiring the total ev- (a) Terminology: ii. 4 20. A pragmatic justifjcation of Bayesianism: (a) A pragmatic justifjcation of probabilism: If your degrees of belief don’t satisfy the axioms of probability, then you could be sold a combination of bets which is guaranteed to lose you money come what may . (Ramsey 1931) (b) A pragmatic justifjcation of conditionalization: If you stand to learn than conditionalization, then you could be reliably sold a series of bets Bayesian Confirmation Theory 3 15. It is a consequence of the defjnition of conditional probability that: 14. All of these measures will agree about the following: (a) If Pr ( A ) = 1 , then the agent thinks that A is certainly true. (b) If Pr ( A ) = 0 , then the agent thinks that A is certainly false. � Pr ( H | E ) � R ( H , E ) = log (c) If Pr ( A ) = 1 / 2 , then the agent is as confjdent that A is true as they are that Pr ( H ) � Pr ( E | H ) A is false. � L ( H , E ) = log Pr ( E | ¬ H ) It is a requirement of rationality that your degrees of belief Pr satisfy (a) If Pr ( H | E ) > Pr ( H ) , then E confjrms H (b) If Pr ( H | E ) < Pr ( H ) , then E disconfjrms H (c) If Pr ( H | E ) = Pr ( H ) , then E neither confjrms nor disconfjrms H idence E , you are disposed to adopt a new credence function Pr E which is your old credence function conditionalized on E . Tiat is, for all H , Pr ( H | E ) = Pr ( E | H ) · Pr ( H ) Pr E ( H ) = Pr ( H | E ) = Pr ( E | H ) Pr ( E ) · Pr ( H ) Pr ( E ) 16. So, we may say: E confjrms H if and only if Pr ( H | E ) > Pr ( H ) Pr is the agent’s prior credence function. Pr E is the agent’s posterior credence function. Pr ( E | H ) · Pr ( H ) > Pr ( H ) Pr ( E ) 19. Tie Bayesian theory of confjrmation says that E confjrms H ifg Pr ( E | H ) > Pr ( E ) Pr E ( H ) > Pr ( H ) Tiat is: E confjrms H if and only if H did a good job predicting E . And E disconfjrms H ifg Pr E ( H ) < Pr ( H ) (a) In order to do ‘a good job’ predicting E , H doesn’t have to make E likely. (b) Also, in order to do a good job predicting E , it is not enough for H to make E likely. (c) To do a good job predicting E , H must make E more likely than its negation, ¬ H . whether E , and you are disposed to revise your beliefs in any way other 17. Tie Bayesian interprets Pr as providing the degrees of belief , or the credences , of which are guaranteed to lose you money no matter what. (Teller 1976)
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