How to Do Things with anHow to Do Things with an Infinite Regress:Infinite Regress:

A Learning Theoretic Analysis of A Learning Theoretic Analysis of “Normative Naturalism”“Normative Naturalism”

Kevin T. KellyDepartment of Philosophy

Carnegie Mellon University

kk3n@andrew.cmu.edu

Two Methodological Paradigms Two Methodological Paradigms

Entailment by evidence Halting with the right answer

Certainty

Confirmation Learning

Partial support Reliable convergence

Learning TheoryLearning Theory

M

epistemically relevant worlds

method

hypothesis

input stream1 0 1 0 ? 1 ? 1 0 ? ? ?

correctness

output stream

H

K

ConvergenceConvergence

MIn the limit: ? ? 0 ? 1 0 1 0 1 1 1 1…

finite forever

MWith certainty: ? ? 0 ? 1 0 1 0 halt!

finite

= Guaranteed convergence to the right answer

MKH

Verification Refutation Decision

Converge to 1Don’t

converge to 0Converge to 1

Don’t converge to 1

Converge to 0 Converge to 0

ReliabilityReliability

1-sided 2-sided

wor

lds

inpu

t stre

ams

outp

ut s

tream

s

Underdetermination =Underdetermination = Unsolvabililty = Unsolvabililty =

ComplexityComplexity

Verifiable inthe limit

Refutable in the limit

Decidablewith certainty

Decidable inthe limit

Verifiablewith certainty

Refutablewith certainty

AE EA

AE

Example: UniformitariansmExample: UniformitariansmMichael Ruse, Michael Ruse, The Darwinian RevolutionThe Darwinian Revolution

Uniformitarianism (steady-state)

Catastrophism (progressive):

Deg

ree

ofad

vanc

emen

t

Stonesfield mammals (1814)

creation

Deg

ree

ofad

vanc

emen

t

Refutable in the limit: – Say “yes” when current schedule is refuted.– Return to “no” after a schedule survives for a

whileNot verifiable in the limit:

– Data support a schedule until we say no.– Nature refutes the schedule thereafter.

Example: UniformitariansmExample: UniformitariansmMichael Ruse, Michael Ruse, The Darwinian RevolutionThe Darwinian Revolution

Underdetermination =Underdetermination = Unsolvabililty = Unsolvabililty =

ComplexityComplexity

Verifiable inthe limit

Refutable in the limit

Decidablewith certainty

Decidable inthe limit

Verifiablewith certainty

Refutablewith certainty

AE EA

AE

Uniformitarianism

Universal laws

Foundational QuestionFoundational Question

Every method is reliable only under empirical background conditions.

How do we find out whether they are true?

The Familiar OptionsThe Familiar Options. . .

FoundationalismNo turtle has been found

CoherentismEverbody’s doing it

RegressOrphan

Learning Theoretic Analysis of Learning Theoretic Analysis of Methodological RegressMethodological Regress

MH

successPresupposition =

“method doesn’t fail”

wor

lds

Inpu

t stre

ams

Out

put s

tream

s

error

error

RegressRegress of Methods of Methods

Same data to allM1

H

M2

P1

M3

P2

M4

P3

No Free Lunch PrincipleNo Free Lunch PrincipleThe 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

Scale of underdetermination

Single-method achievement

Worthless RegressWorthless Regress

M1 alternates mindlessly between acceptance and rejection.

M2 always rejects a priori.

M1

H

M2

P1

“no!”

PretensePretense

M pretends to refute H with certainty iff M never retracts a rejection.

•Popper’s response to Duhem’s problem

Nested RefutationNested Refutation Regresses Regresses

M1

P0

M2

P1

Mk+1

P2

. . .

. . .

Pn

Pn + 1

M

P0

UI

UI

UIRefutes with

certaintyover UiPi

Each pretends to Decide with certaintyRefute with certainty

Ever weaker presuppositions

UI

ExampleExample

Blu

e

Blu

e

Blu

e

Blu

e

Blu

e

Blu

e

P0

P1P2 P3

. . .

Green

M1

P0

Halt at stage 3.Output 0 iff blue occurs.

Mi

Pi

Halt at stage 2i + 1.Output 0 iff blue occurs at 2i or 2i+1.

Regress of deciders:“2 more = forever”

K

Infinite Verification RegressesInfinite Verification Regresses

M1

P0

M2

P1

Mk+1

P2

. . .

. . .

Pn

Pn+1

M

P0

UI

UI

UIRefutes

in the limitover UiPi

Each pretends to Verify with certainty

Refute in the limit

Ever weaker presuppositions

UI

…

Example: UniformitariansmExample: UniformitariansmMichael Ruse, Michael Ruse, The Darwinian RevolutionThe Darwinian Revolution

M1

P0 = uniformitarianism

Mi

Pi-1

Regress of 2-retractors equivalent to a single limiting refuter:

Halt with acceptance when first schedule is violated.

Keep rejecting until then.

Accept before the ith schedule is refuted.

Reject when the ith schedule is refuted.

Accept and halt when the i+1th schedule is refuted.

Pi = P0 is true if the first i schedules are all false.

Verifiable inthe limit

Refutable in the limit

Decidable withcertainty

Decidable inthe limit

Verifiable withcertainty

Refutable withcertainty

AE

Gradualrefutability

Gradual verifiability

AE EA The The General General PicturePicture

AEAEAE

Naturalism LogicizedNaturalism Logicized 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.

ReferencesReferences• 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 ThinkingTraditional ThinkingNo 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 ProblemThe Regress Problem

Confirmation: What are the reasons for your reasons?

Learning: how can you learn whether you are learning?

Modified ExampleModified Example

Same as before But now M1 pretends to refute H with certainty.

M1

H

M2

P1

ReductionReduction

MReject when just one rejects

Accept otherwise

H

M1 ? ? ? 1 0 0 0 0 0

M2 ? 1 1 1 1 1 1 0 0

M 1 1 1 1 0 0 0 1 1

2 retractions in worst caseStarts not rejecting

ReliabilityReliability

M1 never rejects

M2 rejects

M1 rejects

M2 rejects-P1

M1 rejects

M2 never rejects

M1 never rejects

M2 never rejectsP1

-HH

MReject when just one rejects

Accept otherwise

H

Converse ReductionConverse 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.

ReliabilityReliability

Retractions used by M

0 1 2

H true false true

M1 never rejects rejects rejects

M2 never rejects never rejects rejects

P1 true true false

Regress TamedRegress Tamed

M1

H

M2

P1

Refutes with certainty

M1

H

2 retractionsstarting with 1

Complexityclassification

Pretends torefute with

certainty

regress method

Six Six Reliability ConceptsReliability Concepts

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

One-sidedTwo-sided

Table of Opposites 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

Empirical ConversionEmpirical Conversion An empirical conversion is a method that

produces conjectures solely on the basis of the conjectures of the given methods.

M1

H

M2

P1

M3

P2

M

Reduction andReduction and Equivalence Equivalence

Reduction: B < A iffThere is an empirical conversion of an arbitrary regress

achieving A into a regress achieving B.Methodological equivalence = inter-reducibility.

Simple IllustrationSimple Illustration

P1 is the presupposition under which M1 refutes H with certainty.

M2 refutes P1 with certainty.

M1

H

M2

P1

Refinement: RetractionsRefinement: Retractions

You are a fool not to invest in technology

0 1 1 0 ? 1 1 ? ? ? ? ?NASDAQ

Retractions

2 retractionsstarting with 0

2 retractionsstarting with 1

0 retractionsstarting with ?

1 retractionstarting with ?

1 retractionstarting with 0

1 retractionstarting with 1

A E E A

AEv

v

Verifiable inthe limit

Refutable in the limit

Decidable inthe limit

AE EA

. . .

Retractions Retractions asas

Complexity Complexity RefinementRefinement

Finite RegressesFinite Regresses

M1

P0

M2

P1

Mk+1

P2

. . .Pn

M

P0

Pretends : n1 retractionsstarting with c1

Pretends : n2 retractionsstarting with c2

n2 retractions

starting with c2

Sum all the retractions.Start with 1 if an even number of the regressmethods start with 0.

H

Logic of HistoricismLogic of Historicism

Global historical perspectiveArticulation : paradigm :: simple : complexNo time at which a paradigm must be

rejected. Eventually one paradigm wins.Fixed “rules of rationality” may preclude

otherwise achievable success.

ExampleExample

M1 if all the methods in the regress currently say 1 .

0 otherwise

P0

Conversion to single refuting methodB

lue

Blu

e

Blu

e

Blu

e

Blu

e

Blu

e

P0

P1P2 P3

. . .

Green

K