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

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The Way Not TakenorHow to Do Things with an
Infinite Regress
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Two Methodological Paradigms
Certainty
perfect support
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Two Methodological Paradigms
Certainty
perfect support
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Two Methodological Paradigms
Confirmation
Partial support
Certainty
perfect support
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Two Methodological Paradigms
Confirmation
Partial support
Eureka!
Certainty
Halting with the right answer
perfect support
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Two Methodological Paradigms
Confirmation
Learning
Partial support
Convergent procedure
Eureka!
Certainty
Entailment by evidence
Halting with the right answer
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Two Methodological Paradigms
Confirmation
Learning
Partial support
Convergent procedure
Certainty
Entailment by evidence
Halting with the right answer
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Two Methodological Paradigms

• Relational state
• Internal
• Relation screens off process
• Computation is extraneous

• Dynamical stability
• External
• Process paramount
• Computational perspective

Confirmation
Learning
Partial support
Convergent procedure
Certainty
Entailment by evidence
Halting with the right answer
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The Ultimate Question
Methods justify conclusions. What justifies
methods?
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The Ultimate Question
What justifies methods?
Confirmation
What justifies the confirmation relation?
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Options
Coherentism Methods justify themselves
. . .
. . .
. . .
Foundationalism A priori justification
Regress Always respond with a new method
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Alternative Approach
What Justifies Science?
Confirmation
Learning
What justifies the confirmation relation?
How can we learn whether we are learning?
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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
worlds
conjectures
data
H
K
M
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Some Reliability Concepts
One-sided
Two-sided
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Example UniformitariansmMichael Ruse, The
Darwinian Revolution
Uniformitarianism (steady-state)
Complexity
Stonesfield mammals
Catastrophism (progressive)
Complexity
creation
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Example UniformitariansmMichael Ruse, The
Darwinian Revolution

• Uniformitarianism is refutable in the limit
• Side with uniformitarianism each time the current
  progressive schedule is violated.
• Eventually slide back to progressionism after the
  revised schedule stands up for a while.

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Historicism Explained

• Articulations refutable only in a paradigm.
• Articulations crisply refutable with a paradigm.
• No time at which a paradigm must be rejected.
• Method inputs neednt be cognized.
• Propositions forced by inputs relative to a
  paradigm may be paradigm-relative.

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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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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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Standard Objection

• Very nice, but every method is reliable only
  under background presuppositions.
• How do we know that the background assumptions
  are true?
• Every assumption should be subject to empirical
  review.

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Plausible Fallacy

• If you could learn whether your pressupositions
  were true, you could chain this ability together
  with your original method to learn without them.
• Therefore, if you need the presuppositions, you
  cant reliably assess them.
• Therefore, learning theory is an inadmissible
  version of foundationalism.

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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.

• Duhem no hypothesis is refutable in isolation.
• Popper to avoid coddling a false hypothesis
  forever, establish rejection conditions in
  advance even though the hypothesis is not really
  refutable.

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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
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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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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
data
0
output
. . .
1
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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
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
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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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
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Example UniformitariansmMichael Ruse, The
Darwinian Revolution
Regress of 2-retractors equivalent to a single
limiting refuter
Pi any number of surprises except i.
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)
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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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
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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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Infinitary Reduction
P0
P0
M
Increment if pointed method rejects.
0
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.

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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
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Naturalism Logicized

• Unlimited Fallibilism every presupposition is
  held up to the tribunal of experience.
• No free lunch captures objective power of
  empirical regresses.
• Progress-oriented convergence is the goal.
• Feasibility reductions are computable, so
  analysis applies to computable regresses.
• Historicism dovetails with a logical viewpoint
  on paradigms and articulations.