kevin t. kelly department of philosophy carnegie mellon university kk3n@andrew.cmu.edu
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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
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