Transcript austinregress.ppt

The Way Not Taken
or
How to Do Things with an
Infinite Regress
Epistemic Naturalism
The efficacy of science is a scientific question
Giere, Laudan, etc.
Impasse
Coherentism
...
...
...
Foundationalism
regress
Two Methodological Paradigms
Certainty
Entailment by evidence = “perfect support”
Two Methodological Paradigms
Problem of
induction
Certainty
Entailment by evidence = “perfect support”
Two Methodological Paradigms
Problem of
induction
Confirmation
Partial support
Certainty
Entailment by evidence = “perfect support”
Two Methodological Paradigms
Problem of
induction
Eureka!
Confirmation
data
Partial support
M
H
method
Certainty
Entailment by evidence
Halting with the right answer
Two Methodological Paradigms
Problem of
induction
Confirmation
Learning
Partial support
Convergent procedure
Eureka!
Certainty
Entailment by evidence
Halting with the right answer
Two Methodological Paradigms
Problem of
induction
Confirmation
Learning
Partial support
Convergent procedure
Certainty
Entailment by evidence
Halting with the right answer
Two Methodological Paradigms
Problem of
induction
Confirmation
Learning
Partial support
Convergent procedure
Certainty
Entailment by evidence
Halting with the right answer
Two Methodological Paradigms
Short-run perspective
“Internal”
Strategic perspective
Problem of
induction
“External”
Process paramount
No logic of discovery
Computation is special case
Computation is extraneous
Confirmation
Learning
Partial support
Convergent procedure
Certainty
Entailment by evidence
Halting with the right answer
Learning Theory Primer
epistemically relevant worlds
K
correctness
H
M
data stream
method
hypothesis
Putnam
Weinstein
Gold
Freiwald
Barzdin
Blum
Case
Smith
Sharma
Daley
Osherson
Etc.
1010?1?10???
conjecture stream
Convergence
finite
With certainty:
M
? ? 0 ? 1 0 1 0 halt!
finite
In the limit:
M
forever
? ? 0 ? 1 0 1 0 1 1 1 1…
Reliability
= Guaranteed convergence to the right answer
conjectures
data
worlds
1-sided
H
K
2-sided
Verification
Refutation
Decision
Converge to 1
Don’t
converge to 0
Converge to 1
Don’t
converge to 1
Converge to 0
Converge to 0
M
Some Reliability Concepts
Two-sided
One-sided
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
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
Underdetermination =
Degrees of Unsolvability
•Catastrophism
•The coin is unfair
•Computability
Verifiable in
the limit
Refutable in
the limit
•Uniformitarianism
•The coin is fair
•Uncomputability
Decidable in
the limit
•Observable existence
Verifiable
with certainty
Refutable
with certainty
Decidable
with certainty
•Phenomenal laws
•It will rain tomorrow
Underdetermination = Complexity
AE
EA
•The coin is fair
•Computability
•Catastrophism
Verifiable in
the limit
Refutable in
the limit
•The coin is unfair
•Uncomputability
•Uniformitarianism
Decidable in
the limit
E
A
•Existence claims
Verifiable
with certainty
Refutable
with certainty
clopen/recursive
Decidable
with certainty
•Universal laws
•It will rain tomorrow
Refinement: Retractions
You are a fool not to invest in technology
Retractions
NASDAQ
0110?11?????
Expansions
AE
EA
Verifiable in
the limit
Refutable in
the limit
Decidable in
the limit
A
A vE
...
E
•Exactly n…
2 retractions
starting with 0
v
Retractions
as
Complexity
Refinement
2 retractions
starting with 1
Boolean
combinations
of universals and
existentials
1 retraction
starting with ?
E
A
= verifiability
with certainty
1 retraction
starting with 0
1 retraction
starting with 1
0 retractions
starting with ?
= refutability
with certainty
= decidability
with certainty
A Learnability Analysis
of Methodological Regress

Presupposition of M =
presupposition
H
conjectures
data
worlds
the set of worlds in which M succeeds.
M
success
Methodological Regress
H
Same data to all
M1
P1
M2
P2
M3
P3
M4
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
Empirical Conversion

An empirical conversion is a method that
produces conjectures solely on the basis of the
conjectures of the given methods.
H
M1
M
P1
M2
P2
M3
Methodological Equivalence

Reduction: B < 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.
Simple Illustration

P1 is the presupposition under which M1 refutes H with
certainty.
 M2 refutes P1 with certainty.
H
M1
P1
M2
Worthless Regress

M1 alternates mindlessly between

acceptance and rejection.
M2 always rejects a priori.
Pretense

M pretends to refute H with certainty iff M
never retracts a rejection.
Modified Example

Same as before
 But now M1 pretends to refute H with certainty.
H
M1
P1
M2
Reduction
H
Reject when just one rejects
M
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
Starts not rejecting
2 retractions in worst case
Reliability
H
M
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
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.
Reliability
Retractions
used by M
0
1
2
H
true
false
true
M1
never rejects
rejects
rejects
M2
never rejects never rejects
P1
true
true
rejects
false
Regress Tamed
regress
H
H
M1
Pretends to
refute with
certainty
method
P1
M1
M2
2 retractions
starting with 1
Refutes
with certainty
Complexity
classification
Finite Regresses Tamed
P0
P0
Pretends :
n1 retractions
starting with c1
M
M1
P1
Pretends :
n2 retractions
starting with c2
M2
Sum all the retractions.
Start with 1 if an even
number of the regress
methods start with 0.
P2
Pn
n2 retractions
starting with c2
Mk+1
H
Infinite Popperian Regresses
P0
M1
P0
P1
M2
Each pretends
to refute
with certainty
M
P2
Pn
Mk+1
Pn + 1
Refutes with
certainty
over UiPi
Example
Regress of deciders:
“2 more = forever”
P0
M1
Halt after 2
and project
final obs.
K
Grue5
P3
Grue4
Grue3
Grue2
Grue1
Grue0
Mi
Halt after 2i
and project
final obs.
P2
P1
P0
Pi-1
Green
...
Example
Equivalent single refuting method
P0
1 if only green emeralds have been seen.
P0
Grue5
Grue4
Grue3
Grue2
0 otherwise
Grue1
Grue0
M
Green
K
H
Other Infinite Regresses
P0
M1
P0
P1
M2
Each pretends to
Verify with certainty
Use bounded retractions
Decide in the limit
Refute in the limit
M
P2
Pn
Mk+1
Pn+1
Refutes
in the limit
over UiPi
Example: Uniformitariansm
Michael Ruse, The Darwinian Revolution
Complexity
Uniformitarianism (steady-state)
Stonesfield mammals
Complexity
Catastrophism (progressive):
creation
Example: Uniformitariansm
Michael Ruse, The Darwinian Revolution
Regress of 2-retractors equivalent to a single limiting refuter:
P0
Halt with acceptance when surprised by early complexity.
M1
Keep rejecting until then.
Pi = any number of surprises except i.
Pi-1
Mi
Accept before the ith surprise is encountered.
Reject when the ith surprise is encountered.
Accept and halt when the i+1th surprise is encountered.
H
AEA
EAE
Gradual
refutability
Gradual
verifiability
Similar results
EA
AE
E
Verifiable with
certainty
Refutable
in the limit
Decidable in
the limit
A
Verifiable in
the limit
Refutable with
certainty
Decidable with
certainty
The Power
of
Nested
Infinite
Empirical
Regresses