How to Do Things with an Infinite Regress

How to Do Things with an Infinite Regress: A Learning Theoretic Analysis of “Normative Naturalism” Kevin T. Kelly Department of Philosophy Carnegie Mellon University kk 3 n@andrew. cmu. edu

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

Learning Theory epistemically relevant worlds K correctness H M input stream method hypothesis 1010? 1? 10? ? ? output stream

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

Reliability output streams input streams worlds = Guaranteed convergence to the right answer H K 1 -sided 2 -sided Verification Refutation Decision Converge to 1 Don’t converge to 0 Converge to 1 Don’t converge to 1 Converge to 0 M

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

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

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

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 Decidable with certainty Universal laws

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

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

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

Regress of Methods H Same data to all M 1 P 1 M 2 P 2 M 3 P 3 M 4

No Free Lunch Principle 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. … l The Regress achievement Single-method achievement Scale of underdetermination

Worthless Regress l l M 1 alternates mindlessly between acceptance and rejection. M 2 always rejects a priori. H M 1 P 1 M 2 “no!”

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

Nested Refutation Regresses Ev er I U P 0 M 1 P 1 we I U ak er P 2 pr es I U M 2 P 0 . . . os iti on s I U Each pretends to Decide with certainty Refute with certainty Pn M up p Mk+1 Pn + 1 Refutes with certainty over Ui. Pi . . .

Example P 0 Regress of deciders: “ 2 more = forever” M 1 Halt at stage 3. Output 0 iff blue occurs. K Blue P 3 Blue Blue Mi Halt at stage 2 i + 1. Output 0 iff blue occurs at 2 i or 2 i+1. P 2 P 1 P 0 Pi Green . . .

Infinite Verification Regresses Ev U P 0 I er M 1 we ak P 1 er I U Pn up M po sit io ns I U … es . Mk+1 Pn+1 . . . Refute in the limit . . Each pretends to Verify with certainty P 2 pr U I M 2 P 0 Refutes in the limit over Ui. Pi

Example: Uniformitariansm Michael Ruse, The Darwinian Revolution Regress of 2 -retractors equivalent to a single limiting refuter: P 0 = uniformitarianism Halt with acceptance when first schedule is violated. M 1 Keep rejecting until then. Pi = P 0 is true if the first i schedules are all false. Pi-1 Mi Accept before the ith schedule is refuted. Reject when the ith schedule is refuted. Accept and halt when the i+1 th schedule is refuted.

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

Naturalism Logicized l l l Unlimited Fallibilism: every method has its presupposition checked against experience. No free lunch: captures objective 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.

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.

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

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

Modified Example Same as before l But now M 1 pretends to refute H with certainty. l H M 1 P 1 M 2

Reduction H Reject when just one rejects M Accept otherwise M 1 ? ? ? 1 0 0 0 M 2 ? 1 1 1 0 0 M 1 1 0 0 0 1 1 Starts not rejecting 2 retractions in worst case

Reliability H M Reject when just one rejects Accept otherwise H -H P 1 M 1 never rejects M 2 never rejects M 1 rejects M 2 never rejects -P 1 M 1 rejects M 2 rejects M 1 never rejects M 2 rejects

Converse Reduction M decides H with at most 3 retractions starting with acceptance. l Choose: l – P 1 = “M retracts at most once” – M 1 accepts until M uses one retraction and rejects thereafter. – M 2 accepts until M retracts twice and rejects thereafter. l Both methods pretend to refute.

Reliability Retractions used by M 0 1 2 H true false true M 1 never rejects M 2 never rejects P 1 true rejects false

Regress Tamed regress H H M 1 Pretends to refute with certainty method P 1 M 2 2 retractions starting with 1 Refutes with certainty Complexity classification

Six Reliability Concepts Two-sided One-sided Decision Verification Refutation Certain Halt with correct answer Halt with “yes” iff true Halt with “no” iff false Limiting Converge to correct answer Converge to “yes” iff true Converge to “no” iff false

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

Empirical Conversion l An empirical conversion is a method that produces conjectures solely on the basis of the conjectures of the given methods. H M 1 M P 1 M 2 P 2 M 3

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

Simple Illustration P 1 is the presupposition under which M 1 refutes H with certainty. l M 2 refutes P 1 with certainty. l H M 1 P 1 M 2

Refinement: Retractions You are a fool not to invest in technology Retractions NASDAQ 0110? 11? ? ?

AE EA Verifiable in the limit Refutable in the limit Decidable in the limit A A v. E . . . E 2 retractions starting with 0 v 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 ? Retractions as Complexity Refinement

Finite Regresses P 0 Pretends : n 1 retractions starting with c 1 P 0 M 1 Pretends : n 2 retractions starting with c 2 M P 1 M 2 Sum all the retractions. Start with 1 if an even number of the regress methods start with 0. P 2 . . Pn . n 2 retractions starting with c 2 Mk+1 H

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

Example Conversion to single refuting method P 0 1 if all the methods in the regress currently say 1. M 0 otherwise Blue Blue P 0 K P 3 Blue P 2 P 1 Green . . .