1 John D Norton Department of History and Philosophy of Science University of Pittsburgh June 29 2022 Mangoletsi Potts Lectures 2022 Material Theory of Induction 2 3 This Talk 4 There is no ID: 935842
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Slide1
There is No Complete
Calculus of Inductive Inference
1
John D. Norton
Department of History and Philosophy of Science
University of PittsburghJune 29, 2022
Mangoletsi
-Potts Lectures 2022
Slide2Material Theory of Induction
2
Slide33
Slide4This Talk
4
There is no
non-trivial,
complete calculus of inductive inference.
What is
completeness and why would we want it?
Why can
’t
we have it?
Really?
Isn
’t there some way out?
Slide5The Path to Completeness
5
Slide6What IS inductive inference?
6
?
?
?
An appealing possibility
Good
inductive inference
=
Woes of
inductive inference
Mechanical calculation
strength of support
other calculus
or some other calculus
Inference that conforms with the probability calculus
Particular facts of inductive support
Computation of a particular conditional probability
General properties of inductive inference
Demonstration of a theorem in the probability calculus
Slide7Particular Facts
7
Is 7,919 the thousand
’
s prime number?
Compute mechanically by the sieve or Eratosthenes
2
3
4
5
6
7
8
9
10
11
12
13
…
1st
Is evolution well supported by the fossil record?
Compute mechanically
P( evolution | fossils ) = 0.99865
2nd
3rd
4th
5th
6th
or
[ evolution | fossils ] =
big
Slide8General Properties
8
Are there infinitely many prime numbers?
Yes. Euclid proves constructively that for any prime number, there is always a greater one.
If a simple hypothesis
simp
and complicated one
comp
both entail the
evid
ence, should we prefer the simpler one?
Prove as a theorem that
P(
simp
|
evid
) > P(
comp
|
evid
)
or
[
simp
|
evid
] > [
comp
|
evid
]
Slide9External Inductive Content
(illustrated)
9
Slide10Simple versus Complicated
10
P(
simp
|
evid )
P(
comp
|
evid
)
=
P(
simp
)
P(
comp
)
priors
P(
evid
|
simp
)
P(
evid
|
comp
)
x
likelihoods
1
1
Both
simp
and
comp
entail
evid
.
posteriors
Slide11Simple versus Complicated
11
P(
simp
|
evid )
P(
comp
|
evid
)
=
P(
simp
)
P(
comp
)
priors
posteriors
There are
few
simple hypotheses.
{
simp
} = {
simp
1
,
simp
2
,
…}
There are
many
complicated hypotheses.
{
comp
} = {
comp
1
,
comp
2
,
comp
3
, …}
Stipulate
P( {
simp
} ) = P( {
comp
} )
P(
simp
|
evid
) >> P(
comp
|
evid
)
P(
simp
)
P(
comp
)
=
many
few
>>1
Slide12Simple versus Complicated
12
P(
simp
|
evid )
P(
comp
|
evid
)
=
P(
simp
)
P(
comp
)
priors
posteriors
There are
few
simple hypotheses.
{
simp
} = {
simp
1
,
simp
2
,
…}
There are
many
complicated hypotheses.
{
comp
} = {
comp
1
,
comp
2
,
comp
3
, …}
Stipulate
P(
simp
1
) = P(
simp
2
) =…
… = P(
comp
1
) = P(
comp
2
) = …
P(
simp
|
evid
)
=
P(
comp
| evid )
P(
simp) = P(comp)
Slide13External Inductive Content
13
Stipulate
P( {
simp
} ) = P( {
comp
} )
Stipulate
P(
simp
1
) = P(
simp
2
) =…
… = P(
comp
1
) = P(
comp
2
) = …
The determination of inductive strengths of support by considerations external to calculations within the calculus.
=
Eliminate the influence of external inductive content
by
expanding the scope of the analysis to include the external factors.
The optimists
’
hope.
Slide14Completeness
14
Slide15Complete list of what supports what with what strength.
No inductive content introduced from outside.
Completeness
in a domain
15
inductive content
All
is fixed by the calculus from
propositions within the domain.
IN
OUT
Proper warrant
for all hypotheses of science
All background facts of science
universal calculus
Slide1616
THERE IS
NO
NON-TRIVIAL
COMPLETE CALCULUS
OF INDUCTIVE
INFERENCE.
THERE IS
NO
NON-TRIVIAL
COMPLETE CALCULUS
OF INDUCTIVE
INFERENCE.
Slide17Demonstrating the Incompleteness
17
Slide18A standard way of defining a logic of induction that expresses completeness.
Well-behaved under redescription of the propositions.
Overall…
18
IF
(a) The inductive logic is deductively definable;
(b) asymptotically stable; and
(c) continuous,
THEN
all inductive strengths of support are the same.
(the trivial inductive logic).
Proof method: principle of indifference thinking runs amok.
Slide19Deductive Definability
19
Slide2020
Deductive Structure:
Finite Boolean Algebras
Goal: Lay bare the deductive structure of ordinary sentential logic, free of the distracting duplications: A = (AvA) = (A&A) = (Av (A&A)) = ….
All finite sets of sentences belong to one of a
one, two, three, four, … atom Boolean algebra:
universally true
contradiction
Three-atom
algebra.
Atoms a
1
, a
2
, a
3
are the logically strongest propositions that are not
Finitely
many
propositions closed under
v
(or),
&
(and),
~
(negation).
Slide21Calculus
= strengths can by computed by explicit or implicit
rules
Inductive strengths of support
21
[A|B]
= Strength of inductive support that proposition
B
affords to
A
Deductive definability
Deductive Structure
fixes
Inductive strengths of support.
Slide22Deductively
Definable
Logic of Induction
22
Explicit
definition of [Ai| Ak
]for i, k = 1, ..., m [Ai|Ak] = formula that mentions only
• the deductive relations among{A1, A2, ..., A
m
}
• their deductive relations to the atoms of the algebra Ω.
E.g. classical probability
[A|B] = P(A|B) = #A&B/#B
Implicit
definition of [A
i
| A
k
]
for
i
, k = 1, ..., m
Sentences that mention only
• the strengths [
A
i
|A
k
]
• the
deductive relations among
A
1
, A
2, ..., Am• their deductive relations to the atoms of the algebra
Ω.The sentences uniquely fix the strengths.
E.g. Kolmogorov axioms plus equiprobability of atoms.
#formula = number of atoms in formula
Slide23There are many, interesting, deductively definable logics of induction
23
Slide24Deductive relations among A
1
, …, A
m
and the atoms.
Completeness
24
… must be fixed by resources fully within the algebra.
In a very big Boolean algebra…
…for some very large set of propositions
{A
1
, A
2
, …, A
m
} …
… strengths [A
1
|A
2
],
[A
1
|A
3
], …, [A
m-1
|A
m
]…
External inductive content
External inductive content
Slide25Symmetry Theorem
25
Slide2626
Symmetries of a Boolean algebra
Same
deductive
structure if we…
… relabel the atoms arbitrarily.
… permute the atomic labels arbitrarily.
Slide2727
Richness
of Symmetries
of a Boolean algebra
Cyclic permutation of atoms a
1
, a
2
, a
3
Exchange of atoms a
1
and a
3
Slide28In a deductively definable logic of induction…
28
FIX
• the deductive relations among
{A
1
, A
2
, ..., A
m
}
• their deductive relations to the atoms of the algebra Ω.
strengths
[A
i
|A
k
]
FIX
Symmetry here
Symmetry here
a
1
entails a
1
v a
2
a
2
entails a
2
v a
1
same deductive relations as
[a
1
| a
1
v a
2
]
= [a
2
| a
1
v a
2
]
Slide29…
#A
1
&A
2&A3
#A
1
&A
2
&~A
3
#~A
1
&~A
2
&~A
3
#A
1
&~A
2
&~A
3
Deductive structure
29
fixed
by number
of atoms
in the logically strongest propositions:
Slide30Symmetry Theorem
30
For propositions A
1
and A
2 in the set {A1, A2, ..., Am}
[A
1
|A
2
]
=
f
(
#A
1
&A
2
&…&A
m
,
#~A
1
&A
2
&…&A
m
,
#A
1
&~A2&…&Am, …#~A1&~A2
&…&~Am)
list of 2
m
logically strongest conjunctions
Very many
f
’
s
Very many inductive logics
Slide31Asymptotic Stability
31
Slide3232
How
to make
a Boolean Algebra Bigger
a
1, a2, a3
a
1
There is a solar eclipse on June 1.
disjunctive refinement
a
1
expanded to
b
1
v b
2
b
1
,
b
2
, a
2
, a
3
b
1
There is a solar eclipse in the morning of June 1.
b
2
There is a solar eclipse in the afternoon of June 1.
Slide33Refine for more expressive power
33
two atoms
P(inside|
Ω
) = 1/2
a
a
ten atoms
P(inside|
Ω
) = 8/10
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
100 atoms
P(inside|
Ω
) = 79/100
P(inside|
Ω
) = π/4 = 0.785398...
Asymptotic stability
[A|B]
stabilizes towards a unique limiting value
under arbitrary disjunctive refinement of A and B and the other propositions.
Slide34Asymptotic Stability
34
As disjunctive refinement continues:
[A
1
| A
3
], [A
2
| A
3
]
stabilize.
Eventually just inductive hair-splitting.
Else
no definite values for
[A
1
| A
3
], [A
2
| A
3
]
unless
refinement halted by inductive content from outside.
Slide35Triviality Deduced
35
Slide3636
The Rough Idea
refine
a
1
to b
1
v…vb
99
A deductively definable logic has no way to tell when further disjunctive refinement becomes inductively inert.
[a
1
| a
1
v a
2
] = [a
2
| a
1
v a
2
]
From earlier
“
[1
atom
|2
atoms
]
”
“
[1
atom
|2
atoms
]
”
refine
a
2
to c
1
v…vc
99
[b
1
v…vb
99
| b
1
v…vb
99
v a
2
]
vs
[a
2
| b
1
v…vb
99
v a
2
]
“
[99
atoms
|100
atoms
] > [1
atom
|100
atoms]”
[a
1
| a
1v c1v…vc99] vs [c1v…vc99 | a1
v c1v…vc99]
“
[1
atom
|100
atoms
] < [99
atoms
|100
atoms
]
”
Confuse by malicious disjunctive refinement.
Slide3737
The Rough Idea
A deductively definable logic has no way to tell when further disjunctive refinement becomes inductively inert.
Instability…
Unless all
strengths converge to the same limit under repeated disjunctive refinement.
[A|B] [C|D] [E|F] [G|H] …
same limit
!
!
!
!
!
!
Slide3838
Slide3939
Slide40In sum…
40
For
a set of propositions {A
1
, …, A
m} defined on finite Boolean algebrasand for
an inductive logic that is:(a) deductively definable,(b) asymptotically stable and(c) continuous,
all
the well-defined inductive strengths [A
i
| A
k
] converge under disjunctive refinement to a single strength.
This includes the maximum strength [Ω|Ω] and the minimum strength [Ø
|
Ω
].
Avoid triviality
Incompleteness
Introduce
external inductive
content
e.g. stipulate
priors, preferred partitions, …
Slide41Escape?
41
Slide4242
Enhanced
deductive
logic?
• infinite Boolean algebra
• predicate logic …
Enhanced Logics?
Enhanced
inductive
logic?
[A|B,C] = support for A from B with background C
[A,B|C,D] = relative support for A vs B from C with background D
…
No protection from malicious refinements.
Deductive structure remains highly symmetric.
Slight adjustment to proof yields analogous no go result.
Slide43there is a ratio of priors that delivers it
external
inductive content
…incomplete
for fixed evidence
E
for any nominated favoring of
H
1
over
H
2
“
washing out of the priors
”
??
Bayes
’
theorem
Subjective Bayesianism
43
Prior probabilities
introduced as
arbitrary opinion.
Symmetry
of symmetry theorem broken.
Malicious
disjunctive refinement blocked.
BUT
P(hypothesis
| evidence)
= arbitrary opinion
+ evidential warrant
Escape
:
Tempered Bayesianism
Use only
sensible
priors
Slide44Subjective Bayesian
Confirmation Measures
44
e. g. Distance measure
d(H,E,B) = P(H|E&B) – P(H|B)
Either
the confirmation measure
is dependent on an arbitrary prior
or
is NOT dependent on an arbitrary prior.
Fails
to
provide an objective measure of inductive support.
Is subject
to
the triviality result
or its use is incomplete.
Slide45Conclusion
45
Slide4646
It is merely inference that conforms with such-and-such a calculus.
What is
inductive
inference?
There will always be
a bit missing
in formal accounts of inductive inference.
Priors must
be set
“
sensibly.
”
Preferred partitions chosen.
…
External inductive content is always
needed to escape triviality
Slide4747
Slide4848
The End
The End
Slide49Appendices
49
Slide50The condition
50
Asymptotic Stability Under
Disjunctive Refinement
For some fixed set of propositions {A
1
, ..., Am
} of the explicit or implicit definition of a deductively definable logic of induction,for each strength [Ai| Ak], i, k = 1, ..., m,
there exists a limiting value, possibly unique to that strength, [A
i
| A
k
]
lim
to which the strength converges
under all possible disjunctive refinements of the algebra.
Slide51Continuity illustrated
51
A
B
C
D
0
atoms
10
atoms
10
atoms
15
atoms
10
atoms
15
atoms
Absolute differences in atom counts do not matter in the limit.
[A|B]
[C|D]
100
atoms
150
atoms
100
atoms
150
atoms
10000
atoms
15000
atoms
10000
atoms
15000
atoms
converge
same limit