How to Do Things with an Infinite Regress: A Learning Theoretic Analysis of

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How to Do Things with an Infinite RegressA
Learning Theoretic Analysis of Normative
Naturalism
Kevin T. Kelly Department of Philosophy Carnegie
Mellon University kk3n_at_andrew.cmu.edu
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Two Methodological Paradigms
Confirmation
Learning
Partial support
Reliable convergence
Certainty
Entailment by evidence
Halting with the right answer
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Learning Theory
epistemically relevant worlds
K
correctness
H
hypothesis
M
1 0 1 0 ? 1 ? 1 0 ? ? ?
input stream
output stream
method
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Convergence
finite
M
With certainty
? ? 0 ? 1 0 1 0 halt!
finite
forever
M
In the limit
? ? 0 ? 1 0 1 0 1 1 1 1
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Reliability

• Guaranteed convergence to the right answer

1-sided
2-sided
worlds
output streams
input streams
H
K
M
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Underdetermination Unsolvabililty Complexity
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
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Example UniformitariansmMichael Ruse, The
Darwinian Revolution
Uniformitarianism (steady-state)
Degree of advancement
Stonesfield mammals (1814)
Catastrophism (progressive)
Degree of advancement
creation
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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.

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Underdetermination Unsolvabililty Complexity
AE
EA
Verifiable in the limit
Refutable in the limit
Uniformitarianism
Decidable in the limit
A
E
Verifiable with certainty
Refutable with certainty
Universal laws
Decidable with certainty
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Foundational Question

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

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The Familiar Options
. . .
Foundationalism No turtle has been found
Coherentism Everbodys doing it
Regress Orphan
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Learning Theoretic Analysis of Methodological
Regress
worlds
Input streams
Output streams
error
H
Presupposition method doesnt fail
M
success
error
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Regress of Methods
Same data to all
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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
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Worthless Regress

• M1 alternates mindlessly between acceptance and
  rejection.
• M2 always rejects a priori.

no!
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Pretense

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

• Poppers response to Duhems problem

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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
. . .
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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 2i1.
K
P2
P3
P1
. . .
Blue
Blue
Blue
Blue
Blue
Blue
P0
Green
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Observations

• Locality new methods added later in response to
  challenges.
• Collapse a single method could have succeeded
  in the same sense if the regress exists.
• Trade-off slower convergence for weaker
  presuppositions.
• Popperian frustration doesnt transcend the
  power of refutation with certainty, as required
  to deal with Duhems problem.

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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
Pn1

. . .
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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.
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Observations

• Transcendence nested regresses of verifiers
  exist for hypotheses that are only verifiable in
  the limit.

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The General Picture
AEA
EAE
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
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Naturalism 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.

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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.

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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
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Positive Reliability

• A method is positively negatively reliable iff
  it always succeeds when H is true false

Positive reliability
H
success
presupposition
M
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Positive Covering

• Ai Mi converges to acceptance.
• Bi Mi converges to rejection
• A regress positively negatively covers K iff
  the Ai Bi cover K.

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No Foundations or Direction
P0
Positive negative covering
P2
Decides in limit
Pn
. . .
Each pretends to Refute with certainty Decide
in the limit
Pn1
. . .
Positive negative reliability
H
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The Regress Problem
Confirmation What are the reasons for your
reasons?
Learning how can you learn whether you are
learning?
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Modified Example

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

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Reduction
H
2 retractions in worst case
Starts not rejecting
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Reliability
H
-H
H
M1 rejects M2 never rejects
M1 never rejects M2 never rejects
P1
M1 never rejects M2 rejects
M1 rejects M2 rejects
-P1
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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.

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Reliability
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Regress Tamed
method
regress
2 retractions starting with 1
Pretends to refute with certainty
Refutes with certainty
Complexity classification
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Finitary Reduction
P0
M
Set m count the rejections
0
data
0
output
. . .
1
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Infinitary Reduction
P0
M
0
data
?
output
. . .
k
1
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Reliability
P4
P3
P2
P1

P0
M1 accepts
M2 accepts
M3 accepts
M4 accepts
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Reliability
P4
P3
P2
P1

P0
M1 accepts
M2 rejects
M3 accepts
M4 accepts
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Infinitary Reduction
P0
Infinitely repetitive enumeration of regress
methods loaded with appropriate hypotheses
M
Increment if indicated method accepts.
0
f(0)
data
?
f(2)
. . .
k
1
f(3)
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Reliability
P4
P3
P2
P1

P0
M0 accepts inf. oft
M1 accepts inf. oft
M2 converges to rejection
M3 accepts inf. oft
M4 accepts inf. oft
Pointer eventually gets stuck at a copy of M2
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Converse Reduction
Given a limiting refuter
Construct this nested regress of pretending
verifiers
P0
. . .
M
Pi1 P0 v M converges to 0 no later than i1.
. . .
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Reliability
Suppose M accepts infinitely often
Then all the Pi are true.
P4
P3
P2
P1

P0
M1 accepts eventually
M2 accepts eventually
M3 accepts eventually
M4 accepts eventually
Because M accepts after each i.
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Reliability
Suppose M converges to rejection, say at stage
3
M rejects at stage 3
M rejects at stage 4
M accepts at stage 2
P4
P3
P2
P1

P0
M1 accepts eventually
M2 accepts eventually
M3 rejects forever.
M4 accepts eventually
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Infinitary Reduction
P0
P0
M
Increment if pointed method rejects.
0
e
0
. . .
k
1
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Infinitary Reduction
P0
Enumeration of pretending limiting verifiers
M
1
M1
Find first rejecting method up to k
M2
data
2-i
0
Mi
. . .
k
1
Mk
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Reliability

• Some method in the regress correctly converges to
  1 by positive covering.
• All methods that converge to 1 are correct by
  positive reliability.
• The earlier methods all converge by some time
  they pretend to decide in the limit.
• Thereafter, backward induction works.

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Six Reliability Concepts
One-sided
Two-sided
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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

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Empirical Conversion

• An empirical conversion is a method that produces
  conjectures solely on the basis of the
  conjectures of the given methods.

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Reduction and Equivalence

• Reduction B lt A iff
• There is an empirical conversion of an arbitrary
  regress achieving A into a regress achieving B.
• Methodological equivalence inter-reducibility.

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Simple Illustration

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

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Refinement Retractions
You are a fool not to invest in technology
Retractions
NASDAQ
0 1 1 0 ? 1 1 ? ? ? ? ?
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Retractions asComplexity Refinement
AE
EA
Verifiable in the limit
Refutable in the limit
Decidable in the limit
A E
E A
v
v
. . .
2 retractions starting with 0
2 retractions starting with 1
1 retraction starting with ?
A
E
1 retraction starting with 0
1 retraction starting with 1
0 retractions starting with ?
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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
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Limiting Verification Regresses
UI
P0
UI
Ever weaker presuppositions
P2
UI
Gradually verifies over UiPi
Pn
. . .
Each pretends to Verify in the limit
UI
Pn1
. . .
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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.

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