THE DOXASTIC ENGINEERING OF ACCEPTANCE RULES

{ Kevin T. Kelly , Hanti Lin }Carnegie Mellon University

This work was supported by a generous grant by the Templeton Foundation.

Two Models of Belief

Propositions

A CB

Two Models of Belief

(0, 1, 0)

(0, 0, 1)(1, 0, 0)

(1/3, 1/3, 1/3)

PropositionsProbabilities

A CB

How Do They Relate?

(0, 1, 0)

(0, 0, 1)(1, 0, 0)

(1/3, 1/3, 1/3)

PropositionsProbabilities

A CB

?

Acceptance

(0, 1, 0)

(0, 0, 1)(1, 0, 0)

(1/3, 1/3, 1/3)

PropositionsProbabilities

A CB

Acpt

Acceptance as Inference

You condition on whatever you accept (Kyburg, Levi, etc.)

Very serious business! Does it ever happen?

“It’s not Sunday, so let’s buy beer at the super market”.

You would never bet your life against nothing that what you say to yourself in routine planning are true.

Acceptance as Apt Representation

The geometry of probabilities is much richer than the lattice of propositions.

Aim: represent Bayesian credences as aptly as possible with propositions.

Acceptance as Apt Representation

The geometry of probabilities is much richer than the lattice of propositions.

Aim: represent Bayesian credences as aptly as possible with propositions.

A

B

C

Acpt B

Acceptance as Apt Representation

The geometry of probabilities is much richer than the lattice of propositions.

Aim: represent Bayesian credences as aptly as possible with propositions.

B CAcpt B v C

A

Familiar Puzzle for Acceptance

Suppose you accept propositions more probable than 1/2.

Consider a 3 ticket lottery. For each ticket, you accept that it loses. That entails that every ticket loses. (Kyburg)

-B -C-A1/2

Thresholds as Logic

High probability is like truth value 1.

B

CA

1

Thresholds as Geometry(0, 1, 0)

(0, 0, 1)(1, 0, 0)

(1/3, 1/3, 1/3)

Thresholds as GeometryA

B C

p(A) = 0.8

Thresholds as GeometryA

B C

A v B

p(A v B) = 0.8

Thresholds as GeometryA

B C

A v B A v C

Thresholds as GeometryA

B C

A v B A v C

AClosure under conjunction

Thresholds as Geometry

A v B A v C

A

B CB v C

Thresholds as Geometry

A v B A v C

A

B CB v C

T

The Lottery Paradox

A v B A v C

A

B CB v C

T2/3

t

The Lottery Paradox

A v B A v C

A

B v C

T

B C

2/3

t

The Lottery Paradox

A v B A v C

A

B v CB C

T 2/3

t

Two Solutions

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

(Levi 1996)

Two Solutions

A v C

B v C

T

A v B A v C

A

CB v C

T

B

A v B

LMU CMU

A

CB

(Levi 1996)

Two Solutions

A v C

B v C

T

A

CB

LMU CMU

A

CB

A v B

(Levi 1996)

Why Cars Look Different

That’s junk!I want a smoother ride!I want tighter handling!

consumer designer

Why Cars Look Different

consumer designer

Grow up.We can optimize one or the other but not both.

Why Cars Look Different

consumer designerresponsive

Grow up.We can optimize one or the other but not both.

steady

LMU

CMU

Why Cars Look Different

= Change what you accept only when it is logically refuted.

responsive

steady

LMU

CMU= Track probabilistic conditioning exactly.

End of Roundtable Segment 1

Steadiness

Responsive-ness

responsive

steady

LMU

CMU