CMU DESIGN GOALS
{ Kevin T. Kelly , Hanti Lin }Carnegie Mellon University
Responsive-ness
CMU
Qualitative Reasoning that Tracks Conditioning
Qualitative Reasoning that Tracks Conditioning
Qualitative Reasoning that Tracks Conditioning
Probabilistic
conditioning
Qualitative Reasoning that Tracks Conditioning
Probabilistic
conditioning
Acceptance
Qualitative Reasoning that Tracks Conditioning
Probabilistic
conditioning
AcceptancePropositional
belief revision
Qualitative Reasoning that Tracks Conditioning
Probabilistic
conditioning
AcceptancePropositional
belief revision
Acceptance
Qualitative Reasoning that Tracks Conditioning
Probabilistic
conditioning
AcceptancePropositional
belief revision
Acceptance=
Conditioning + acceptance = acceptance + revision
Pre-established Harmony
Acceptance
Propositional
belief revisio
n
Probabilistic
conditioning
Cheap Bayes With Harmony
Acceptance
Tie shoes?
Probabilistic
conditioning
Eat breakfast?
Get out of bed?
When You Need Bayes…
Acceptance
Probabilistic
conditioning
Help! Bayes!
Invest?
Tie shoes?
Eat breakfast?
Get out of bed?
Call Him Then
Acceptance
Condition only once
Tie shoes?
Eat breakfast?
Get out of bed?
Invest?
Call Him Then
Acceptance
Thanks.I’ll take it from here
Tie shoes?
Eat breakfast?
Get out of bed?
Invest?Condition only once
TV?
Expensive Bayes Without Harmony
Acceptance
Repeated conditioning
Tie shoes?
Eat breakfast?
Get out of bed?
Invest?
TV?
Cheap Bayes with Harmony
Acceptance
Tie shoes?
Eat breakfast?
Get out of bed?
Invest?Condition only once
TV?
LMU Design Principle: Steadiness
Steadiness = “Just conjoin the new data with
your old propositions if the two are consistent”
EB
LMU
AGM is Steady
B C
A
AGM is Steady
C
A
Non-steady Revision Rule
A
B C
Yoav Shoham
Non-steady Revision Rule
A
C
Yoav Shoham
Non-steady Revision Rule
A C
Yoav Shoham
Some Shared Design Principles
LMU
CMU
Consistency
Inconsistency is accepted nowhere.
Non-skepticism
Every atom A is accepted over some open neighborhood.
Non-OpinionationThere is an open neighborhood over which you accept a non-atom and nothing stronger.
A v B
Corner-monotonicity
C
If an atom is accepted, it continues to be accepted along the straight line to the corresponding corner.
C
CC
C
C
Corner-monotonicity
If an atom is accepted, it continues to be accepted along the straight line to the corresponding corner.
Sensible Rules
Sensible = all four properties.
C
CC
C
C
A v B
Both are Sensible!
A v C
A
CB v C
T
B
A v B A v C
A
CB v C
T
B
A v B
LMU CMU
Incompatibility Theorem
No sensible acceptance rule is both steady and tracks conditioning.
consumer designer
Sorry. You can’t have both.
A New Paradox of AcceptanceA
B C
A
A v B
p
p(.|A v B)
A New Paradox of AcceptanceA
B C
A
A v B
p
p(.|A v B)
Accept A.
Learn its consequence A v B.
If you track, you retract A!
“Cautious” Monotonicity= Hypothetico-Deductive Monotonicity
If you accept a hypothesis, don’t retract it when you learn what it entails (i.e. predicts).
A Better Idea?
A v B A v C
A
B CB v C
T
0.8
0.9
Another New Paradox of Acceptance
p
A
B
Another New Paradox of AcceptanceA
B
B
p
p(.|B)
Another New Paradox of AcceptanceA
B
A
p
p(.|B)
Another New Paradox of AcceptanceA
B
A
B
T
p
p(.|B)
p(.|B)
You will accept A v B no matter whether B or B is learned.
But if you track, you don’t accept A v B.
Case Reasoning
Accept a hypothesis, if you will accept it no matter whether E is learned or E is learned.
Theorem
The CMU rule + Shoham revision (non-steady) satisfies:
sensible tracks conditioning avoids both new paradoxes
Partial Converse
Shoham revision sensible tracks conditioningImpliesCMU rule + avoidance of the 2 new paradoxes.
Gettier Without False Lemmas
Nogot
Nobody
Somebody
Gettier case
Havit= the Truth
CMU Rule Represents it
Havit= the Truth
Nogot
Nobody
Somebody
CMU Rule is Unsteady!
HavitNogot
Nobody
Somebody
“Somebody”is retracted but not refuted.
Gettier/Unsteadiness Zones
HavitNogot
Nobody
Somebody
Shoham Revision vs. AGM Revision
Havit
Nogot Havit
Nobody
Nogot
Nobody
Shoham Revision vs. AGM Revision
“Trust what you accepted”
“Re-examine your reasons”Havit
Nogot Havit
Nobody
Nogot
Nobody
Structure Preservation
(0, 1, 0)
(0, 0, 1)(1, 0, 0)
(1/3, 1/3, 1/3)
LogicGeometry
A CB
Acpt
Some Clear CasesA
B C
InterpolationA
B C
What About Here?A
B C
Probability Lives in the Unit Cube111
100 010 001
000
011110 101
Classical Logic Lives on the Corners111
100 010 001
000
011110 101
But What if Logic Filled the Cube?111
100 010 001
000
011110 101
Classical Negation111
100 010 001
000
011110 101
Partial Negation111
100 001
000
011110
010
101
GeologicClose classical logic underPartial negation
Geological EntailmentLogical Closure =Sub-crystals
Probability is a Surface in Geologic
Classical Principle of Indifference
Principle of Indifference Completed
Probalogic = Projection of Geologic
Probalogic as Geologic in Perspective
Probalogic as Geologic in Perspective
Projection of Geological Consequences
= Probalogic
Acceptance Should Preserve Logical Structure
Acpt
Representation TheoremThe CMU rule is the only rule that preserves logical structure (entailment, disjunction and consistent conjunction).
Acpt
Feature Checklist for the CMU Rule
The CMU rule + Shoham revision satisfies
sensible tracks conditioning avoids both new paradoxes represents no-false-lemma Gettier cases unique geo-logical representation
THANK YOU!
The CMU rule + Shoham revision satisfies
sensible tracks conditioning avoids both new paradoxes represents no-false-lemma Gettier cases unique geo-logical representation