The Way Not Taken or How to Do Things with an Infinite Regress

1
The Way Not TakenorHow to Do Things with an
Infinite Regress
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Epistemic Naturalism

• The efficacy of science is a scientific question
• Giere, Laudan, etc.

3
Impasse
Coherentism
. . .
. . .
. . .
Foundationalism
regress
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Two Methodological Paradigms
Certainty
Entailment by evidence perfect support
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Two Methodological Paradigms
Problem of induction
Certainty
Entailment by evidence perfect support
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Two Methodological Paradigms
Problem of induction
Confirmation
Partial support
Certainty
Entailment by evidence perfect support
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Two Methodological Paradigms
Problem of induction
Eureka!
Confirmation
M
H
data
Partial support
method
Certainty
Entailment by evidence
Halting with the right answer
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Two Methodological Paradigms
Problem of induction
Confirmation
Learning
Partial support
Convergent procedure
Eureka!
Certainty
Entailment by evidence
Halting with the right answer
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Two Methodological Paradigms
Problem of induction
Confirmation
Learning
Partial support
Convergent procedure
Certainty
Entailment by evidence
Halting with the right answer
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Two Methodological Paradigms
Problem of induction
Confirmation
Learning
Partial support
Convergent procedure
Certainty
Entailment by evidence
Halting with the right answer
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Two Methodological Paradigms
Strategic perspective External Process
paramount Computation is special case
Short-run perspective Internal No logic of
discovery Computation is extraneous
Problem of induction
Confirmation
Learning
Partial support
Convergent procedure
Certainty
Entailment by evidence
Halting with the right answer
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Curious Similarities

• Normative Naturalism
• Historicist approach
• Larry Laudan (1996) Beyond Positivism and
  Realism. Boulder Westview.

• Formal Learning Theory
• Logical/Computational approach
• Kevin Kelly (1996) The Logic of Reliable
  Inquiry. New York Oxford.

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Laudans Normative Naturalism

• Laudans reconstruction of nihilism
• Methodological rules must be universal.
• But no such method is apparent in history.
• Laudans Response
• Methodological oughts are hypothetical.
• Aims and beliefs about means can differ.
• So diversity /gt normative nihilism

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Comments

• Straw man
• such violations are not accidental events. On
  the contrary, we see that they are necessary for
  progress. (Against Method, p. 23).
• Better critique of Feyerabend
• A few simple failures /gt impossible.
• Other sources of variation
• Efficient algorithms are problem-specific.

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Laudans Normative Naturalism

• So empirical methodology is empirical.
• Circle?
• Laudans response Methodological
  foundationalism.

(R3)
(R4)
justifies
(R2)
justifies
(R1)
uncontroversial method
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Laudans Foundational Rule (R1)

• If
• actions of a particular sort, m, have
  consistently promoted certain cognitive ends e,
  in the past, and
• rival actions, n, have failed to do so,
• then
• assume that future actions following the rule
• if your aim is e, you ought to do m
• are more likely to promote those ends than
  actions based on the rule
• if your aim is e, you ought to do n.

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Comments

• Gratuitous historicism
• Why not computer simulation and algorithm
  analysis?
• Doesnt work for non-verifiable goals (Robert
  Nola)
• Truth,
• Empirical adequacy,
• problem-solving effectiveNESS (Laudan).
• Transitivity question
• Each method in the chain promotes a goal.
• What goal does the CHAIN of methods promote?

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Laudans Normative Naturalism

• Progress
• Not verifiable, so cannot serve as a warranting
  goal.
• Should be explained as the result of repeated
  achievement of other, observable goals.

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Comment

• Simpler to drop the radical empiricism and adopt
  progress as an immediate goal.

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Laudans Normative Naturalism

• Concern unrestrained goal-relativism.
• Axiology
• Goals must be achievable.
• Goals must be reflected in scientific history.

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Summary

• Agreement
• Hypothetical norms survive historicist critique.
• Ought implies can.
• Focus on progress.
• Critique
• Over-emphasis on empirical correlation.
• Ignores structure of method and problem.
• Progress should be a direct aim.
• Transitivity issue The empirical regress says
  what each method achieves. What does the whole
  regress achieve?

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Learning Theory Primer
Putnam Weinstein Gold Freiwald Barzdin Blum Case S
mith Sharma Daley Osherson Etc.
epistemically relevant worlds
K
correctness
H
hypothesis
M
1 0 1 0 ? 1 ? 1 0 ? ? ?
data stream
conjecture 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
Verification Refutation Decision
Converge to 1 Dont converge to 0 Converge to 1
Dont converge to 1 Converge to 0 Converge to 0
worlds
conjectures
data
H
K
M
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Some Reliability 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-sided
Two-sided
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Underdetermination Degrees of Unsolvability
Verifiable in the limit
Refutable in the limit
Decidable in the limit
Verifiable with certainty
Refutable with certainty
Decidable with certainty
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Underdetermination Degrees of Unsolvability

• Catastrophism
• The coin is unfair
• Computability

• Uniformitarianism
• The coin is fair
• Uncomputability

Verifiable in the limit
Refutable in the limit
Decidable in the limit
Verifiable with certainty
Refutable with certainty

• Phenomenal laws

• Observable existence

Decidable with certainty

• It will rain tomorrow

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Underdetermination Complexity
AE
EA

• The coin is fair
• Computability
• Catastrophism

• The coin is unfair
• Uncomputability
• Uniformitarianism

Verifiable in the limit
Refutable in the limit
Decidable in the limit
A
E
Verifiable with certainty
Refutable with certainty

• Universal laws

• Existence claims

Decidable with certainty
clopen/recursive

• It will rain tomorrow

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Refinement Retractions
You are a fool not to invest in technology
Retractions
NASDAQ
0 1 1 0 ? 1 1 ? ? ? ? ?
Expansions
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Short-run constraints

• Goodmans new riddle

Grue3
Grue4
Grue5
Grue1
Grue2
Grue0
Green
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Short-run constraints

• Have to project Green unless Green is refuted.

Green projector at most one retraction
Grue3
Grue4
Grue5
Grue1
Grue2
Grue0
?
Green
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Short-run constraints

• Have to project Green unless Green is refuted.

Grue2 projector can be forced to make two
retractions
Grue3
Grue4
Grue5
Grue1
Grue2
Grue0
?
Green
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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

• Exactly n

Boolean combinations of universals and
existentials
1 retraction starting with ?
A
E
1 retraction starting with 0
1 retraction starting with 1
refutability with certainty
verifiability with certainty
0 retractions starting with ?
decidability with certainty
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Historicist Logic

• Feyerabends thesis
• Universal rules impede progress.
• Learning theory
• Different learners for different problems.
• Complexity depends on material facts.
• Plausible universal rules can restrict
  (computable) learning power.

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Historicism

• Kuhn
• Scientific change is not cumulative.
• Scientific changes are not rationally compelled.
• Paradigm viability, not truth
• Every problem that will arise has some normal
  scientific solution.
• Learning theory
• Convergence is not monotone.
• Paradigm viability is only refutable in the
  limit.
• Certainty case data forces efficient learners
  to reject.
• Limiting case efficient learners may reject at
  any time.

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Two Paths from Certainty
The problem of induction
Intuitive Universal End-in-itself Immediate Relati
on
Naturalistic Problem-specific Goal-directed Diachr
onic Process
Partial support
Convergent reliability
Perfect support
Reliability with halting
Certainty
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Pressing Logical Questions for Laudans Program

• What can be achieved with a chain of methods
  checking methods?
• Can the achievement be related to progress?
• Is Laudans foundationalism necessary?

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Pressing Philosophical Question for Learning
Theory

• Mi succeeds under material assumption K.
• What if K is challenged?

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How Not to Proceed
. . .
. . .
. . .
This means justifies.
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A Learnability Analysisof Methodological Regress

• Presupposition of M
• the set of worlds in which M succeeds.

worlds
conjectures
data
H
success
M
presupposition
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Methodological Regress
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.

Regress achievement
Single-method achievement
Scale of underdetermination
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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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Methodological Equivalence

• Reduction B lt A iff
• There is an empirical conversion of an arbitrary
  group of methods collectively achieving A into a
  group of methods collectively 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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Worthless Regress

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

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Pretense

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

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Modified Example

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

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Reduction
H
Reject when just one rejects
Accept otherwise
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 case
Starts not rejecting
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Reliability
H
Reject when just one rejects
Accept otherwise
H -H
P1 M1 never rejects M2 never rejects M1 rejects M2 never rejects
-P1 M1 rejects M2 rejects M1 never rejects M2 rejects
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Converse Reduction

• M decides H with at most 2 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
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
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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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Finite Regresses Tamed
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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Finitary Reduction
P0
M
data
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Finitary Reduction
P0
M
Set m count the rejections
0
1 if m is even
0 otherwise
data
0
output
. . .
1
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Ho-Hum

• The assumed reliability of the final method
  anchors the chain (backwards induction).
• Presuppositions can always be checked.
• The REAL problem is the infinite case.

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Infinite Popperian Regresses
UI
P0
UI
Ever weaker presuppositions
P2
UI
Refutes with certainty over UiPi
Pn
. . .
Each pretends to refute with certainty
UI
Pn 1
. . .
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Example
Regress of deciders 2 more forever
Halt after 2 and project final obs.
Halt after 2i and project final obs.
K
P2
P3
P1
. . .
Grue3
Grue4
Grue5
Grue1
Grue2
Grue0
P0
Green
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Example
Equivalent single refuting method
P0
1 if only green emeralds have been seen.
0 otherwise
K
Grue3
Grue4
Grue5
Grue1
Grue2
Grue0
P0
H
Green
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Infinitary Reduction
P0
M
data
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Infinitary Reduction
P0
M
0
0 if some Mi rejects
1 otherwise
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 rejects
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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Observations

• Methodological localism new methods added
  later in response to challenges.
• Collapse a single method could have succeeded
  if the regress exists.
• Regress trades convergence time for weaker
  presuppositions (than initial methods).
• Poppers dream endless refutations of
  refutations lead to the truth.

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Ho Hum

• You assume that the hypothesis entails the
  presupposition.
• For any finite number of non-inclusions, apply
  the finite sequence analysis. The nested tail
  amounts to a terminal limiting refuter.
• What about challenges to the presupposition UiPi
  of the whole infinite regress?
• The construction works for any countable ordinal
  regress. It applies to regresses of regresses of
  regresses.

68
Other Infinite Regresses
UI
P0
UI
Ever weaker presuppositions
P2
Each pretends to Verify with certainty Use
bounded retractions Decide in the limit Refute in
the limit
UI
Refutes in the limit over UiPi
Pn
. . .
UI
Pn1
. . .
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Example UniformitariansmMichael Ruse, The
Darwinian Revolution
Uniformitarianism (steady-state)
Complexity
Stonesfield mammals
Catastrophism (progressive)
Complexity
creation
70
Example UniformitariansmMichael Ruse, The
Darwinian Revolution
Regress of 2-retractors equivalent to a single
limiting refuter
Halt with acceptance when surprised by early complexity.
Keep rejecting until then.
Pi any number of surprises except i.
Accept before the ith surprise is encountered.
Reject when the ith surprise is encountered.
Accept and halt when the i1th surprise is encountered.
H
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Infinitary Reduction
P0
data
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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)
1 if pointer bumps
0 otherwise
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
1 if M said 1 by now.
0 otherwise.
P0
. . .
M
Pi1 P0 v M converges to 0 no later than i.
1 if M said 0 at i - 1 or 1 no sooner than i.
0 otherwise.
. . .
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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.
76
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
77
The Powerof NestedInfiniteEmpiricalRegresses
AEA
EAE
Gradual refutability
Gradual verifiability
Similar results
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
78
Ho-Hum

• You assume that the regress is increasingly
  reliable.
• The INTERESTING case is when the regress isnt
  even directed.
• Also, is some kind of regress methodologically
  equivalent to limiting decision?

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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
80
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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Infinitary Reduction
P0
P0
M
Increment if pointed method rejects.
0
M1(e) if an even number of methods up to the pointer reject.
Flip M1(e) o.w.
e
0
. . .
k
1
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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.

84
Naturalism Logicized

• Unlimited Fallibilism regresses of regresses of
  empirical second-guessing.
• No free lunch captures objective power of
  empirical regresses.
• Progress-oriented convergence is the goal.
• Axiology reductions are computable, so analysis
  applies to computable regresses.

85
Limiting Verification Regress
UI
P0
UI
Ever weaker presuppositions
P2
Each pretends to Decide in the limit Verify in
the limit
UI
Gradually verifies over UiPi
Pn
. . .
UI
Pn1
. . .
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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