Kevin T. Kelly Department of Philosophy Carnegie Mellon University kk3n@andrew.cmu.edu

1. How to Do Things with anInfinite Regress:A Learning Theoretic Analysis of “Normative Naturalism” Kevin T. Kelly Department of Philosophy Carnegie Mellon University kk3n@andrew.cmu.edu

2. Two Methodological Paradigms Confirmation Learning Partial support Reliable convergence Certainty Entailment by evidence Halting with the right answer

3. Learning Theory epistemically relevant worlds K correctness H hypothesis M 1 0 1 0 ? 1 ? 1 0 ? ? ? input stream output stream method

4. Convergence finite With certainty: M ? ? 0 ? 1 0 1 0 halt! finite forever In the limit: M ? ? 0 ? 1 0 1 0 1 1 1 1…

5. Reliability = Guaranteed convergence to the right answer 1-sided 2-sided worlds output streams input streams H K M

6. Underdetermination =Unsolvabililty = Complexity AE EA Verifiable in the limit Refutable in the limit Decidable in the limit E A Verifiable with certainty Refutable with certainty Decidable with certainty

7. Example: UniformitariansmMichael Ruse, The Darwinian Revolution Uniformitarianism (steady-state) Degree of advancement Stonesfield mammals (1814) Catastrophism (progressive): Degree of advancement creation

8. Example: UniformitariansmMichael Ruse, The Darwinian Revolution • Refutable in the limit: • Say “yes” when current schedule is refuted. • Return to “no” after a schedule survives for a while • Not verifiable in the limit: • Data support a schedule until we say no. • Nature refutes the schedule thereafter.

9. Underdetermination =Unsolvabililty = Complexity AE EA Verifiable in the limit Refutable in the limit Uniformitarianism Decidable in the limit E A Verifiable with certainty Refutable with certainty Universal laws Decidable with certainty

10. Foundational Question • Every method is reliable only under empirical background conditions. • How do we find out whether they are true?

11. The Familiar Options . . . Foundationalism No turtle has been found Coherentism Everbody’s doing it Regress Orphan

12. Learning Theoretic Analysis of Methodological Regress worlds Input streams Output streams error Presupposition = “method doesn’t fail” H M success error

13. P1 P2 P3 H M1 M2 M3 M4 Regress of Methods Same data to all

14. No Free Lunch Principle • The instrumental value of a regress is no greater than the best single-method performance that could be recovered from it without looking at the data directly. … Regress achievement Single-method achievement Scale of underdetermination

15. P1 H M1 M2 Worthless Regress • M1 alternates mindlessly between acceptance and rejection. • M2 always rejects a priori. “no!”

16. Pretense • Mpretends to refute H with certainty iff M never retracts a rejection. • Popper’s response to Duhem’s problem

17. P0 P1 M M1 M2 Mk+1 Nested Refutation Regresses UI P0 UI Ever weaker presuppositions P2 UI Refutes with certainty over UiPi Pn . . . Each pretends to Decide with certainty Refute with certainty UI Pn+1 . . .

18. P0 Pi Mi M1 Example Regress of deciders: “2 more = forever” Halt at stage 3. Output 0 iff blue occurs. Halt at stage 2i + 1. Output 0 iff blue occurs at 2i or 2i+1. K P2 P3 P1 . . . Blue Blue Blue Blue Blue Blue P0 Green

19. P0 P1 M M1 M2 Mk+1 Infinite Verification Regresses UI P0 UI Ever weaker presuppositions P2 UI Refutes in the limit over UiPi Pn . . . Each pretends to Verify with certainty Refute in the limit UI Pn+1 … . . .

20. Pi-1 M1 Mi Example: UniformitariansmMichael Ruse, The Darwinian Revolution Regress of 2-retractors equivalent to a single limiting refuter: P0 = uniformitarianism Pi = P0 is true if the first i schedules are all false.

21. EAE AEA The General Picture Gradual refutability Gradual verifiability AE EA Verifiable in the limit Refutable in the limit Decidable in the limit A E Verifiable with certainty Refutable with certainty Decidable with certainty

22. Naturalism Logicized • Unlimited Fallibilism: every method has its presupposition checked against experience. • No free lunch: capturesobjective power of empirical regresses. • Truth-directed: finding the right answer is the only goal. • Feasibility: reductions are computable, so analysis applies to computable regresses. • Historicism: dovetails with a logical viewpoint on paradigms and articulations.

23. References • The Logic of Reliable Inquiry. Oxford, 1996. • “Naturalism Logicized”, in After Popper, Kuhn and Feyerabend , Nola and Sankey eds., Kluwer, 2000. • “The Logic of Success”, forthcoming BJPS, December 2000.

24. Traditional Thinking No matter how far we extend the [infinite] branch [of justification], the last element is still a belief that is mediately justified if at all. Thus as far as this structure goes, wherever we stop adding elements, we … have not shown that the conditions for the mediate justification of the original belief are satisfied. William Alston, 1976

25. The Regress Problem Confirmation: What are the reasons for your reasons? Learning: how can you learn whether you are learning?

26. P1 H M1 M2 Modified Example • Same as before • But now M1 pretends to refute H with certainty.

27. M Reduction H Starts not rejecting 2 retractions in worst case

28. M Reliability H H -H P1 M1 never rejects M2 never rejects M1 rejects M2 never rejects -P1 M1 rejects M2 rejects M1 never rejects M2 rejects

29. Converse Reduction • M decides H with at most 3 retractions starting with acceptance. • Choose: • P1 = “M retracts at most once” • M1 accepts until M uses one retraction and rejects thereafter. • M2 accepts until M retracts twice and rejects thereafter. • Both methods pretend to refute.

30. Reliability

31. P1 H H M1 M2 M1 Regress Tamed method regress 2 retractions starting with 1 Pretends to refute with certainty Refutes with certainty Complexity classification

32. Six Reliability Concepts Two-sided One-sided

33. Table of Opposites • Confirmation • Coherence • State • Local • Internal • No logic of discovery • Computability is extraneous • Weight • Explication of practice • Learning theory • Reliability • Process • Global • External • Procedure paramount • Computability is similar • Complexity • Performance analysis

34. P1 P2 H M1 M2 M3 M Empirical Conversion • An empirical conversion is a method that produces conjectures solely on the basis of the conjectures of the given methods.

35. Reduction and Equivalence • Reduction: B<A iff There is an empirical conversion of an arbitrary regress achieving A into a regress achieving B. • Methodological equivalence = inter-reducibility.

36. P1 H M1 M2 Simple Illustration • P1 is the presupposition under which M1 refutes H with certainty. • M2 refutes P1 with certainty.

37. Refinement: Retractions You are a fool not to invest in technology Retractions NASDAQ 0 1 1 0 ? 1 1 ? ? ? ? ?

38. AE EA Retractions asComplexity Refinement Verifiable in the limit Refutable in the limit Decidable in the limit E A A E v v . . . 2 retractions starting with 0 2 retractions starting with 1 1 retraction starting with ? E A 1 retraction starting with 0 1 retraction starting with 1 0 retractions starting with ?

39. P0 P1 M M1 M2 Mk+1 Finite Regresses P0 Pretends : n1 retractions starting with c1 Sum all the retractions. Start with 1 if an even number of the regress methods start with 0. Pretends : n2 retractions starting with c2 P2 Pn . . . n2 retractions starting with c2 H

40. Logic of Historicism • Global historical perspective • Articulation : paradigm :: simple : complex • No time at which a paradigm must be rejected. • Eventually one paradigm wins. • Fixed “rules of rationality” may preclude otherwise achievable success.

41. M Example Conversion to single refuting method P0 K P2 P3 P1 . . . Blue Blue Blue Blue Blue Blue P0 Green