Church’s Thesis and
Hume’s Problem:
Justification as Performance
Kevin T. Kelly
Department of Philosophy
Carnegie Mellon University
kk3n@andrew.cmu.edu
Further Reading
(With C. Glymour) “Why You'll Never Know Whether Roger
Penrose is a Computer”, Behavioral and Brain Sciences, 13:
1990.
•The Logic of Reliable Inquiry, Oxford: Oxford University Press,
1996.
•(with O. Schulte) “Church's Thesis and Hume's Problem,” in
Logic and Scientific Methods, M. L. Dalla Chiara, et al., eds. Dordrecht:
Kluwer, 1997.
•(with O. Schulte and C. Juhl) “Learning Theory and the
Philosophy of Science”, Philosophy of Science 64: 1997.
“The Logic of Success”, British Journal for the Philosophy of Science,
51:2000.
“Uncomputability: The Problem of Induction Internalized,”
Theoretical Computer Science 317: 2004.
Computability and Learnibility, Textbook manuscript, 2005.
Fateful First Cut
Relations of Ideas
Matters of fact
Analytic
Synthetic
Certain
Uncertain
A Priori
A Posteriori
Philosophy of Math
Philosophy of Science
Proofs and algorithms
Confirmation
Truth-finding
“Rationality”
Computability
Probability
Painful Progress
Relations of Ideas
Matters of fact
Analytic
Synthetic
Certain
Uncertain
A Priori
A Posteriori
Philosophy of Math
Philosophy of Science
Proofs and algorithms
Confirmation
Truth-finding
“Rationality”
Computability
Probability
Slight Rearrangement
Philosophy of Math
Philosophy of Science
Certain
Uncertain
A Priori
A Posteriori
Proofs and algorithms
Confirmation
Truth-finding
“Rationality”
Computability
Probability
Philosophy of Science 101
Formal Questions
Scientific Questions
Unverifiable
“The sun will never
stop rising”
Verifiable
“The given number
is even”
Wrong Twice!
Formal Questions
Unverifiable
Scientific Questions
Unverifiable
Verifiable
“The sun will never
stop rising”
Verifiable
“The given number
is even”
“The next emerald
is green”
“The given computation
will never halt”
Classical Excuse
Relations of Ideas
Completed analysis
Rational
Rational ?
Reason
Matters of fact
Unbounded
experience
Animal
??
Always black?
Observation
But Maybe…
Relations of Ideas
Unbounded
experience
Unbounded
analysis
??
Matters of fact
Rational ?
Reason
??
Always
black?
Never white?
Observation
Kant
Synthetic A Priori
3
3
2
4
A Posteriori
Unbounded
experience
5
temporal intuition
3+2 = 5 ?
Intuition
??
Always
black?
Never white?
Observation
Oops!
Synthetic A Priori
spatial intuition
??
Never cross?
Intuition
A Posteriori
Unbounded
experience
??
Always
black?
Never white?
Observation
After Turing
Formal Problems
Unbounded
Input
computation
Empirical Problems
Unbounded
reality
n
??
Never halts?
Computer
??
Always
black?
Never white?
Observation
Bad First Cut
Formal Questions
Unverifiable
Scientific Questions
Unverifiable
Verifiable
“The sun will never
stop rising”
Verifiable
“The given number
is even”
“The next emerald
is green”
“The given computation
will never halt”
Good First Cut
Formal Questions
Unverifiable
Scientific Questions
Unverifiable
Verifiable
“The sun will never
stop rising”
Verifiable
“The given number
is even”
“The next emerald
is green”
“The given computation
will never halt”
Computational Epistemology
For formal and empirical questions:
Justification = truth-finding performance
Find
the truth in the best possible sense
And then as efficiently as possible.
Verification as “Support”
1.0
Deductive verification
.5
Inductive verification
Paradigms Evolve
•Objective support by evidence.
Paradigms Evolve…
•Objective support by evidence.
•Coherent personal opinion.
•“Rational” update.
Paradigms Evolve…
•Objective support by evidence.
•Coherent personal opinion.
•“Rational” update.
But Leading Metaphors Matter
•Myopic
•Bedrock = “rationality” intuitions
•Success = being “rational”
•Truth-finding performance deferred or ignored
•Problem complexity ignored
•Little resemblance to computability
Verification as Halting
Conclusion
Data
Verification as Halting
Conclusion
Data
Yes!
Verification as Halting
Conclusion
Data
Verification as Halting
Conclusion
Data
Verification as Halting
Conclusion
Data
Verification as Halting
Conclusion
Data
Can’t take it back.
Verifiability
Yes!
Conclusion true
Conclusion false
Laws are Not Verifiable
Always green?
Laws are Not Verifiable
Always green?
Laws are Not Verifiable
Always green?
Laws are Not Verifiable
Always green?
Laws are Not Verifiable
Always green?
You must halt with “yes” if it’s true!
Laws are Not Verifiable
Always green?
You must halt with “yes” if it’s true!
Laws are Not Verifiable
Yes!
Always green?
Laws are Not Verifiable
Always green?
Laws are Not Verifiable
Always green?
A Purely Formal Problem
Never shoots?
Analogy?
I ain’t a’ shootin’, yeller belly!
Never shoots?
Apparently Not
Game’s up, J.R.!
Yer program’s showin’!
Never shoots?
666
Apparently Not
Now shaddup! I’m calculatin’
a-preeorey-like what yer gonna do.
Never shoots?
666
Apparently Not
?!!!
Never shoots?
666
23455
21456
But Then Again…
I got it! Two kin play
that opry oary game!
Never shoots?
666
23455
21456
But Then Again…
I won’t
shoot ‘til I
see him shoot
in this here
opry oary
sim-yooo-laytion!
Never shoots?
666
23455
21456
666
23455
21456
But Then Again…
666
23455
21456
I ain’t a’
shootin’.
Never shoots?
666
23455
21456
But Then Again…
666
23455
21456
I still ain’t
a’ shootin’.
Never shoots?
666
23455
21456
But Then Again…
666
You check that there
opry oary all you
want…
Never shoots?
23455
21456
666
23455
21456
But Then Again…
666
23455
21456
An’ all you’ll see is I
ain’t a-shootin’
Never shoots?
666
23455
21456
But Then Again…
Aw, shucks, J.R. I reckon
You ain’t never gonna shoot.
666
23455
21456
Aw, shucks, J.R. I reckon
you ain’t never gonna shoot.
Never shoots?
666
23455
21456
But Then Again…
666
23455
Never shoots?
666
23455
But Then Again…
666
Never shoots?
666
But Then Again…
666
See ya in
the grand ol’ opry
oary, sucker!
Never shoots?
666
But Then Again…
Take that!
Never shoots?
666
But Then Again…
Never shoots?
But Then Again…
Never shoots?
Rice-Shapiro Theorem
Whatever a computable cognitive scientist
could verify about an arbitrary computer’s
input-output behavior by formally analyzing
its program…
45863
Q
D
Qoc
Rice-Shapiro Theorem
…could also have been verified by a
behaviorist computer who empirically
studies only the arbitrary computer’s inputoutput behavior!
45863
Q
D
Qoc
Uncomputable Induction
H
N
Yes!
Will see a non-halting machine
Verifiable by “ideal agent”.
Uncomputable Induction
H
N
?
Will see a non-halting machine
Q
Not verifiable by computable agent.
Similarity
Halting Problem
Universal Law
Unverifiable
Unverifiable
Demon can fool
computable agent
Demon can fool
ideal agent
Answer runs beyond
empirical experience
Answer runs beyond
formal experience
A Difference
Halting Problem
Universal Law
Input ends
Input never ends
Can handle some
examples a priori…
Problem of induction
“Internal” problem of
induction eventually
catches up!
Give and Take
Formal input
Empirical input
Jaw Breaker
Noodle
Fits in mouth but may
never melt
May never fit in mouth
May never swallow
May never swallow
Another Difference?
Formal Science
Empirical Science
Incompleteness
Problem of induction
Not Necessarily
Formal Science
Empirical Science
Incompleteness
Problem of induction
Add more powerful axiom
Conjecture a theory
Who knows if new axiom
is consistent with old?
Assume it is, keep
checking, and hope
Who knows if theory is
consistent with data?
Assume it is, keep
checking, and hope
Thesis
The problem of induction and the problem
of uncomputability are essentially similar.
So the two should be understood similarly.
From the ground upward.
Philosophy: A House Divided
•Unverifiability
•“Confirmation”
•“Support”
•“Coherence”
•“Rational” update, etc.
•Uncomputability.
•Suck it up
•Logic vs. engineering
•Try to approximate, etc.
?!
Unified Approach: Gun Control
•Find the truth in the best feasible sense.
•So drop the halting requirement if infeasible.
Q
Start with Problems, Not Methods
Partition Q over set K of inputs
A
Finite input
B
C
D
Infinite input
Convergence
Convergence
with halting
Convergence
in limit
(in limit)
?
?
?
? A
?
?
No more
revisions
? A
? ? B C
? C C
revisions
...
Success
Converge to
right answer
A
B
C
D
Don’t converge
to wrong answer
Special Cases
Verify A
A
Refute A
B
A
Decide A
A
B
B
Special Cases
Compute partial f
f
f=0
f=1
f=2
...
Dom(f)
Theory selection
M0
M1
M2
...
Solutions and Optimality
•Solution: method that succeeds on each input.
•Optimal solution: solves in best possible sense
•Solvable problem: has solution.
•Problem complexity: best sense of solvability
Halting Bounds Revisions
A
B
Verifiability 
convergence with 1 revision ending with “no”.
Refutability 
convergence with 1 revision ending with “yes”.
Decidability 
convergence with 0 revisions.
Generalization to n Revisions
A
n-Refutation =
n+1 rev ending with -A
B
n-Verification =
n+1 rev ending with A
n-Decision =
n rev
Empirical Complexity
Lim-ver
Lim-ref

Lim-dec
...
Underdetermination
...

 and 
1-ver
1-ref
 or 
0-ref

closed
1-dec
0-ver

open
0-dec
clopen
Formal Complexity
...
Lim-ver
Lim-ref

Lim-dec
...
Uncomputability

 and 
1-ver
1-ref
 or 
0-ref

Co-r.e.
1-dec

r.e.
0-ver
0-dec
decidable
Formal + Empirical Complexity
Lim-ver
Lim-ref

Lim-dec
...
Uncomputability +
Underdetermination
...

 and 
1-ver
1-ref
 or 
0-ref

Co-r.e.
1-dec

r.e.
0-ver
0-dec
decidable
Ship-shape Epistemology
...
Ideal
empirical
Effective
empirical
Effective
formal
Ship-shape Epistemology
...
Topological
invariants
Recursive
invariants
Recursive
invariants
Ship-shape Epistemology
...
Lower
Bound
Combination
Lower
Bound
Messy, Odd Distinction
...
Empirical
Borel
Mixed
effective
Borel
Formal
arithmetical
Neat, Natural Distinctions
...
Empirical
Borel
Mixed
effective
Borel
Formal
arithmetical
Example: Kantian Antinomy
Infinitely Finitely
divisible divisible


Matter
Upper Complexity Bound
Lim-ref
Infinitely Finitely
divisible divisible

Lim-ver

I say inf when
you let me cut.
...
fin  fin
...
inf  fin
Lower Complexity Bound
Lim-ref
-Lim-ver
I let you cut
when you say fin.
Infinitely Finitely
divisible divisible

Lim-ver
-Lim-ref

...
 inf  fin
...
 inf  inf
Purely Formal Analogue
Lim-ref
-Lim-ver
Infinite
domain
Finite
domain


666
Lim-ver
-Lim-ref
23455
21456
Purely Formal Analogue
Lim-ref
-Lim-ver
Infinite
domain
Finite
domain
Lim-ver
-Lim-ref
I halt on another input each
time he says fin in my opry oary
simulation of him ganderin’
at my program.
666
23455
21456
Analogies
Finite
divisibility
Finite
divisibility
Finite
domain
...
Ideal
empirical
Effective
empirical
Effective
formal
Mixed Example: Penrose
Computable
Uncomputable
???
.
Human subject
Cognitive scientist
Empirical Irony
Computable
Uncomputable
You can verify human computability
in the limit…
UCUCCCCCCCC…
Human subject
Cognitive scientist
Empirical Irony
Computable
Uncomputable
But only if you aren’t computable!
UCUCUCCUCCU…
Human subject
Cognitive scientist
Computability
Ideal


 Lim-ver
Lim-ref 
Lim-dec
Computability
We can verify our computability in the limit…
only if we are not computable!
Computable


 Lim-ver
Lim-ref 
Lim-dec
Gold/Putnam
Even assuming computability,
you can converge to the true computable
behavior only if you are not computable!
Total computable functions
f0
f 32
f 243
...
Uncomputable Predictions
(with Oliver Schulte)
T
-T
•There exists T such that
•T makes a unique prediction at each stage;
•The predictions are very uncomputable…
Uncomputable Predictions
(with Oliver Schulte)
T
-T
•There exists T such that
•T makes a unique prediction at each stage;
•The predictions are very uncomputable…
•But T is refutable by a computable method!
Moral 1: Rational Hobgoblins
Ideal
rationality
No
consistent
computable convergent
Exists inconsistent computable refuting
Exists consistent
method.
method.
computable non-convergent method.
So “a foolish consistency” precludes truth-finding!
Moral 2: Coup de Grace
•The very idea of insulating deduction from empirical
data restricts the truth-finding power of effective
science.
Empirical data
Theorem
Prover
Yes
n
Q
?
Moral 2: Coup de Grace
•The very idea of insulating deduction from empirical
data restricts the truth-finding power of effective
science.
Empirical data
Theorem
Prover
Yes
n
Q
n
Q
Conclusions
•Justification = truth-finding performance.
•Performance depends on problem complexity.
•Formal and empirical complexity are similar.
•Computational epistemology is unified
•Standard epistemology is divided.
•Insistence on division weakens truth-finding
performance of effective science.
Closing Image
Q
Standard epistemology
Computational epistemology
I’m rational
Closing Image
Q
Standard epistemology
Computational epistemology
I’m rational
Closing Image
Q
Standard epistemology
Computational epistemology
THE END
Pure Form of Naturalism
actual world
actual agent
actual problem
pure structure
How does inquiry proceed?
Have we found a correct answer?
Will we?
Are we guaranteed to?
How might we improve?
Is guaranteed success achievable?
Is guaranteed success achievable in a
completely specified, hypothetical problem?
Which possible methods achieve solutions?
Which maxims preclude solutions?
What is necessary for guaranteed success to
be achievable?
Every Problem in Its Place
more “revolutionary”.
Puzzle-solving adequacy
grad ref
Infinity (Kant’s Antinomy)
Quantized conservation
Uniformitarian geology
Parallel postulate
Probability, given existence
lim ref
Phenomenal laws
cert ref
grad ver Probability existence
Conservation laws
lim ver
lim dec
Finiteness
Computability
Global warming
Chaos (M. Harrell)
Duhem’s problem
cert ver Experimental effects
cert dec
more“normal”.
th
19
c. Geology
Advancement
Uniformitarianism (steady-state world)
Stonesfield mammals 1814
Advancement
Catastrophism (progressively cooling world):
Stonesfield mammals 1814
creation
th
19
c. Geology
Uniformitarianism lim ref
schedule
 new violation.
U
C
Early mammals
schedule
lim ver Catastrophism
 schedule
 future fossils, the
schedule stands
Early reptiles
new schedule
new schedule
6. The Reliability Regress
Reliability = success over a broad range of possibilities.
But but how can we tell we are in the range of reliability?
“method succeeds”
?
actual input stream
Answer:
Check with Another Method!
H
Same inputs to all
1
1 succeeds
2
2 succeeds
3
3 succeeds
Popperian Regress Solved
H
Methods
never take
back a
rejection.
1
1 succeeds
2
2 succeeds
3
3 succeeds
…
Later methods
are as reliable
as earlier.
H
Refutes H as
reliably as
any method
in the
regress
Verification Regress Solved
H
Methods
never take
back an
acceptance.
1
1 succeeds
2
2 succeeds
3
3 succeeds
…
Later methods
are as reliable
as earlier.
H
Refutes H in
the limit as
reliably as
any method
in the
regress.
Frequentist Probability

Background: a probability exists.

Hypothesis: its value is exactly p.
p
=p
Observed freq
too close to p
“too close”
set closer
lim ref
Observed freq
too close to p
“too close”
set closer
Duhem’s Problem

Ref
Background: there exist correct
auxiliaries with respect to which isolated
H is refutable.
H
H
Refuted with
auxiliaries
New auxiliaries
Ver
lim ver
Refuted with
auxiliaries
New auxiliaries
Antinomy of Pure Reason
…neither [the world’s] finitude nor its infinitude can be contained in
experience, because [the following are impossible]:
1.
experience … of an infinite space…,
2.
or again, of the boundary of the world by an empty space …
Prolegomena, Section 52c.
?
Experience
?
Antinomy of Pure Reason
Infinity
position
 Intuition of
more space.
Infinite
Finite
lim ref
lim ver Boundedness
 position
 intuitions, the
position is not
extended.
more
more
wait for more
wait for more
Parallel Postulate
?
lim ref
pp
pp
intersect
intersect
new pair
new pair
Paradigm Choice
Ref
Puzzle-solving adequacy: some articulation
resolves by its standards all puzzles that
arise within it.
ad
ad
“unsolvable
puzzle”
Rearticulate
Ver
lim ver
“unsolvable
puzzle”
Rearticulate
Paradigms
“Internal Puzzle-solving adequacy”:
 articulation such that
 puzzle that arises within the paradigm
 Stage by which diligent effort would
resolve the puzzle by the paradigm’s
standards without rearticulation.
A1 solves 1st
puzzle
ad
A1 solves 2nd
puzzle
A1 never
solves 3rd
puzzle
grad ref
A1
A2
A3
ad
Kuhnian Moral
lim ref
lim ver
gap
Gradual refutability of each alternative does not
suffice for convergence to an effective
paradigm!
Conservation
“Q is conserved in each reaction”:
 reaction
 stage at which the Q in = the Q out and
 Later stage, no more Q is observed in or
out
C
Grad ver
R1 unbalanced R1 unbalanced
R1
R2 unbalanced
R2
R3
C
Cognitivism
Ref
Ver
lim ver
mismatch
Computability
 program such that
 input the program
produces the right
output.
mismatch
C
C
new program
new program
“Empirical Irony of Cog Sci”
Computability
 program such that
 input
 stage by which the
right output is
produced.
Prog 1
outputs 1st
datum
C
Prog 2
outputs 2nd
datum
computer
grad ref
ideal
Prog 3 never
outputs 3rd
datum
Prog 1
Prog 2
Prog 3
C
Transcendental Deductions
•What kind of problem structure is
necessary for a given sort of success?
Certain Verifiability and Refutability
Certain verifiability:
H is a union of “cones” of serious possibilities.
Certain refutability:
H is an intersection of cones of serious possibilities.
“blue will be seen.”
cone = all
possible
extensions
...
“green forever”
Sextus Empiricus!
Non-verifiability:
Some input stream in H never enters a fan included in H.
H
H
H
...
H
Limiting Verifiability
Limiting verifiability:
H is a countable union of intersections of fans.
“The inputs will eventually stabilize to blue.”


Transcendental System
   grad ver

grad ref   
lim ver  
lim ref
lim dec
 cert ref
cert ver 
cert dec
Computable Case Similar!
 grad ver
grad ref 
 lim ref
lim ver 
lim dec
 cert ref
cert ver 
cert dec
Turing-decidable relations