from the lockean thesis to conditionals · 2011. 5. 25. · belp(j) is logically closed (in the...
TRANSCRIPT
From the Lockean Thesis to Conditionals
Hannes Leitgeb
LMU Munich
May 2011
Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 1 / 19
When is a conditional rationally acceptable to an agent?
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure.
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A is a simple sentence.
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A is a simple sentence.
The Lockean thesis LT>r : BelP(X) iff P(X)> r ≥ 12 .
BelP is logically closed (in the sense of doxastic logic).
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A→ B is a conditional.
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A→ B is a conditional.
The Lockean thesis LT>r : BelP(Y |X) iff P(Y |X)> r(X?) ≥ 12 .“→” of the Lockean thesis: If BelP(Y |X) then P(Y |X)> r ≥ 1
2 .
BelP(·|·) is logically closed (in the sense of AGM 1985 on belief revision).
Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 2 / 19
When is a conditional rationally acceptable to an agent?
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure.
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A is a simple sentence.
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A is a simple sentence.
The Lockean thesis LT>r : BelP(X) iff P(X)> r ≥ 12 .
BelP is logically closed (in the sense of doxastic logic).
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A→ B is a conditional.
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A→ B is a conditional.
The Lockean thesis LT>r : BelP(Y |X) iff P(Y |X)> r(X?) ≥ 12 .“→” of the Lockean thesis: If BelP(Y |X) then P(Y |X)> r ≥ 1
2 .
BelP(·|·) is logically closed (in the sense of AGM 1985 on belief revision).
Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 2 / 19
When is a conditional rationally acceptable to an agent?
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure.
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A is a simple sentence.
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A is a simple sentence.
The Lockean thesis LT>r : BelP(X) iff P(X)> r ≥ 12 .
BelP is logically closed (in the sense of doxastic logic).
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A→ B is a conditional.
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A→ B is a conditional.
The Lockean thesis LT>r : BelP(Y |X) iff P(Y |X)> r(X?) ≥ 12 .“→” of the Lockean thesis: If BelP(Y |X) then P(Y |X)> r ≥ 1
2 .
BelP(·|·) is logically closed (in the sense of AGM 1985 on belief revision).
Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 2 / 19
When is a conditional rationally acceptable to an agent?
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure.
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A is a simple sentence.
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A is a simple sentence.
The Lockean thesis LT>r : BelP(X) iff P(X)> r ≥ 12 .
BelP is logically closed (in the sense of doxastic logic).
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A→ B is a conditional.
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A→ B is a conditional.
The Lockean thesis LT>r : BelP(Y |X) iff P(Y |X)> r(X?) ≥ 12 .“→” of the Lockean thesis: If BelP(Y |X) then P(Y |X)> r ≥ 1
2 .
BelP(·|·) is logically closed (in the sense of AGM 1985 on belief revision).
Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 2 / 19
When is a conditional rationally acceptable to an agent?
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure.
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A is a simple sentence.
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A is a simple sentence.
The Lockean thesis LT>r : BelP(X) iff P(X)> r ≥ 12 .
BelP is logically closed (in the sense of doxastic logic).
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A→ B is a conditional.
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A→ B is a conditional.
The Lockean thesis LT>r : BelP(Y |X) iff P(Y |X)> r(X?) ≥ 12 .“→” of the Lockean thesis: If BelP(Y |X) then P(Y |X)> r ≥ 1
2 .
BelP(·|·) is logically closed (in the sense of AGM 1985 on belief revision).
Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 2 / 19
When is a conditional rationally acceptable to an agent?
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure.
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A is a simple sentence.
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A is a simple sentence.
The Lockean thesis LT>r : BelP(X) iff P(X)> r ≥ 12 .
BelP is logically closed (in the sense of doxastic logic).
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A→ B is a conditional.
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A→ B is a conditional.
The Lockean thesis LT>r : BelP(Y |X) iff P(Y |X)> r(X?) ≥ 12 .“→” of the Lockean thesis: If BelP(Y |X) then P(Y |X)> r ≥ 1
2 .
BelP(·|·) is logically closed (in the sense of AGM 1985 on belief revision).
Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 2 / 19
When is a conditional rationally acceptable to an agent?
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure.
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A is a simple sentence.
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A is a simple sentence.
The Lockean thesis LT>r : BelP(X) iff P(X)> r ≥ 12 .
BelP is logically closed (in the sense of doxastic logic).
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A→ B is a conditional.
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A→ B is a conditional.
The Lockean thesis LT>r : BelP(Y |X) iff P(Y |X)> r(X?) ≥ 12 .“→” of the Lockean thesis: If BelP(Y |X) then P(Y |X)> r ≥ 1
2 .
BelP(·|·) is logically closed (in the sense of AGM 1985 on belief revision).
Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 2 / 19
When is a conditional rationally acceptable to an agent?
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure.
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A is a simple sentence.
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A is a simple sentence.
The Lockean thesis LT>r : BelP(X) iff P(X)> r ≥ 12 .
BelP is logically closed (in the sense of doxastic logic).
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A→ B is a conditional.
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A→ B is a conditional.
The Lockean thesis LT>r : BelP(Y |X) iff P(Y |X)> r(X?) ≥ 12 .“→” of the Lockean thesis: If BelP(Y |X) then P(Y |X)> r ≥ 1
2 .
BelP(·|·) is logically closed (in the sense of AGM 1985 on belief revision).
Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 2 / 19
When is a conditional rationally acceptable to an agent?
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure.
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A is a simple sentence.
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A is a simple sentence.
The Lockean thesis LT>r : BelP(X) iff P(X)> r ≥ 12 .
BelP is logically closed (in the sense of doxastic logic).
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A→ B is a conditional.
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A→ B is a conditional.
The Lockean thesis LT>r : BelP(Y |X) iff P(Y |X)> r(X?) ≥ 12 .
“→” of the Lockean thesis: If BelP(Y |X) then P(Y |X)> r ≥ 12 .
BelP(·|·) is logically closed (in the sense of AGM 1985 on belief revision).
Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 2 / 19
When is a conditional rationally acceptable to an agent?
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure.
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A is a simple sentence.
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A is a simple sentence.
The Lockean thesis LT>r : BelP(X) iff P(X)> r ≥ 12 .
BelP is logically closed (in the sense of doxastic logic).
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A→ B is a conditional.
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A→ B is a conditional.
The Lockean thesis LT>r : BelP(Y |X) iff P(Y |X)> r(X?) ≥ 12 .
“→” of the Lockean thesis: If BelP(Y |X) then P(Y |X)> r ≥ 12 .
BelP(·|·) is logically closed (in the sense of AGM 1985 on belief revision).
Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 2 / 19
When is a conditional rationally acceptable to an agent?
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure.
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A is a simple sentence.
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A is a simple sentence.
The Lockean thesis LT>r : BelP(X) iff P(X)> r ≥ 12 .
BelP is logically closed (in the sense of doxastic logic).
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A→ B is a conditional.
!
"!
"! #!
"! #!
$%&"'#(!
"! #!!!$!
$%&"'$!)!#(!
P is a subjective probability measure, A→ B is a conditional.
The Lockean thesis LT>r : BelP(Y |X) iff P(Y |X)> r(X?) ≥ 12 .
“→” of the Lockean thesis: If BelP(Y |X) then P(Y |X)> r ≥ 12 .
BelP(·|·) is logically closed (in the sense of AGM 1985 on belief revision).
Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 2 / 19
From the Lockean Thesis to Conditionals
Can we satisfy all of these desiderata simultaneously?
(Yes.)
If so, how, and does this lead to a unified theory of
quantitative belief and qualitative belief,
absolute belief and conditional belief,
the acceptablity of simple statements and conditionals?
(Yes, or so we hope.)
Plan of the talk:
1 The Lockean Thesis and Absolute Belief
2 “→” of the Lockean Thesis and Conditional Belief
3 Epilogue: Solving A Problem Contextually
Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 3 / 19
From the Lockean Thesis to Conditionals
Can we satisfy all of these desiderata simultaneously? (Yes.)
If so, how, and does this lead to a unified theory of
quantitative belief and qualitative belief,
absolute belief and conditional belief,
the acceptablity of simple statements and conditionals?
(Yes, or so we hope.)
Plan of the talk:
1 The Lockean Thesis and Absolute Belief
2 “→” of the Lockean Thesis and Conditional Belief
3 Epilogue: Solving A Problem Contextually
Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 3 / 19
From the Lockean Thesis to Conditionals
Can we satisfy all of these desiderata simultaneously? (Yes.)
If so, how, and does this lead to a unified theory of
quantitative belief and qualitative belief,
absolute belief and conditional belief,
the acceptablity of simple statements and conditionals?
(Yes, or so we hope.)
Plan of the talk:
1 The Lockean Thesis and Absolute Belief
2 “→” of the Lockean Thesis and Conditional Belief
3 Epilogue: Solving A Problem Contextually
Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 3 / 19
The Lockean Thesis and Absolute Belief
Let W be a set of possible worlds, and let A be an algebra of subsets of W(propositions) in which an agent is interested at a time.
We assume that A is closed under countable unions (σ-algebra).
Let P be an agent’s degree-of-belief function at the time.
P1 (Probability) P : A→ [0,1] is a probability measure on A.
P(Y |X) = P(Y∩X)P(X) , when P(X)> 0.
P2 (Countable Additivity) If X1,X2, . . . ,Xn, . . . are pairwise disjoint membersof A, then
P(⋃n∈N
Xn) =∞
∑n=1
P(Xn).
Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 4 / 19
E.g., a probability measure P:!
!"! #!
$!
%&'()! %&*(! %&%*+!
%!%&%,+! %&%%%%-!
%&%%)!%&%'..(!
P conditionalized on C:!
!"! #!
$!
%! %! %!
%!%&'()! %&%%*!
%&+!%!
Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 5 / 19
Accordingly, let Bel express an agent’s beliefs.
B1 (Logical Truth) Bel(W ).
B2 (One Premise Logical Closure) For all Y ,Z ∈ A:If Bel(Y ) and Y ⊆ Z , then Bel(Z ).
B3 (Finite Conjunction) For all Y ,Z ∈ A:If Bel(Y ) and Bel(Z ), then Bel(Y ∩Z ).
B4 (General Conjunction) For Y = {Y ∈ A |Bel(Y )},⋂
Y is a member of A,and Bel(
⋂Y ).
It follows: There is a strongest proposition BW , such that Bel(Y ) iff Y ⊇ BW .
Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 6 / 19
In order to spell out under what conditions these postulates are consistent withthe Lockean thesis, we will need the following probabilistic concept:
Definition(P-Stability) For all X ∈ A:
X is P-stabler iff for all Y ∈ A with Y ∩X , ∅ and P(Y )> 0: P(X |Y )> r .
So P-stabler propositions have stably high probabilities under salientsuppositions. (Examples: All X with P(X) = 1; X = ∅; and many more!)
Remark: If X is P-stabler with r ∈[
12 ,1
), then X is P-stable
12 .
(cf. Skyrms 1977, 1980 on resiliency.)
Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 7 / 19
In order to spell out under what conditions these postulates are consistent withthe Lockean thesis, we will need the following probabilistic concept:
Definition(P-Stability) For all X ∈ A:
X is P-stabler iff for all Y ∈ A with Y ∩X , ∅ and P(Y )> 0: P(X |Y )> r .
So P-stabler propositions have stably high probabilities under salientsuppositions. (Examples: All X with P(X) = 1; X = ∅; and many more!)
Remark: If X is P-stabler with r ∈[
12 ,1
), then X is P-stable
12 .
(cf. Skyrms 1977, 1980 on resiliency.)
Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 7 / 19
Then the following representation theorem can be shown:
Theorem
Let Bel be a class of ordered pairs of members of a σ-algebra A, and letP : A→ [0,1]. Then the following two statements are equivalent:
I. P and Bel satisfy P1, B1– B4, and LT≥P(BW )> 12 .
II. P satisfies P1 and there is a (uniquely determined) X ∈ A, such that
– X is a non-empty P-stable12 proposition,
– if P(X) = 1 then X is the least member of A with probability 1, and:
For all Y ∈ A:
Bel(Y ) if and only if Y ⊇ X
(and hence, BW = X).
This does not yet presuppose P2.
Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 8 / 19
Example: Let W = {w1, . . . ,w7}, let P be as in the example before.
6. P({w7}) = 0.00006
5. P({w6}) = 0.002
4. P({w5}) = 0.018
3. P({w3}) = 0.058, P({w4}) = 0.03994
2. P({w2}) = 0.342
1. P({w1}) = 0.54
!!
"! #!
$!
%&'()! %&*(! %&%*+!
%!%&%,+! %&%%%%-!
%&%%)!%&%'..(!
This yields the following P-stable12 sets:
{w1,w2,w3,w4,w5,w6,w7} (≥ 1.0)
{w1,w2,w3,w4,w5,w6} (≥ 0.99994)
{w1,w2,w3,w4,w5} (≥ 0.99794)
{w1,w2,w3,w4} (≥ 0.97994)
{w1,w2} (≥ 0.882)
{w1} (≥ 0.54)
Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 9 / 19
With P2 one can prove: The class of P-stabler propositions X in A withP(X)< 1 is well-ordered with respect to the subset relation.
!
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!
*+,#-!
*+,$-!
*+,%-!!
*+,&-!
*+,&'$-!
!
$!
This implies: If there is a non-empty P-stabler X in A with P(X)< 1 at all, thenthere is also a least such X.
Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 10 / 19
!
"#! "$!
"%!
$!
%!
#!
%!
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#!
!!$!
#&!%!
!!%!
#&!$!
!!%!
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!!#!
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!!$!
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!!#!
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!!#!
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And almost all P over finite W have a least P-stable12 set X with P(X)< 1!
Hence, for almost all P there is an r , such that there is a Bel , where
B1–3 Logical closure (with W finite).LT>r For all X : Bel(X) iff P(X)> r .
NT There is an X , such that Bel(X) and P(X)< 1.
Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 11 / 19
!
"#! "$!
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And almost all P over finite W have a least P-stable12 set X with P(X)< 1!
Hence, for almost all P there is an r , such that there is a Bel , where
B1–3 Logical closure (with W finite).LT>r For all X : Bel(X) iff P(X)> r .
NT There is an X , such that Bel(X) and P(X)< 1.
Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 11 / 19
Moral:
Given P and a threshold r , the agent’s Bel is determined uniquely by theLockean thesis.
Bel is even closed logically iff
Bel is given by a P-stable12 set X with P(X) = r > 1
2 .
So the Lockean thesis and the logical closure of belief are jointlysatisfiable as long as the threshold r is co-determined by P.
This almost never collapses into: Bel(X) iff P(X) = 1.
Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 12 / 19
And finally, of course:
Lottery Paradox: Given a uniform measure P on a finite set W of worlds,W is the only P-stabler set with r ≥ 1
2 ; so only W is to be believed then.
This makes good sense: the agent’s degrees of belief don’t give her muchof a hint of what to believe. That is why the agent ought to be cautious.
Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 13 / 19
“→” of the Lockean Thesis and Conditional Belief
TheoremThe following two statements are equivalent:
I. P and Bel satisfy P1, the AGM axioms for belief expansion, and BPr−.
II. P satisfies P1, and there is a (uniquely determined) X ∈ A, such that X isa non-empty P-stabler proposition, and Bel(·|·) is given by X.
TheoremThe following two statements are equivalent:
I. P and Bel satisfy P1–P2, the AGM axioms for belief revision, and BPr .
II. P satisfies P1–P2, and there is a (uniquely determined) chain X ofnon-empty P-stabler propositions in A, such that Bel(·|·) is given by X .
BPr− (“→” of Lockean thesis) For all Y ∈ A, s.t. P(Y )> 0 [and Y ∩BW , ∅] :
For all Z ∈ A, if Bel(Z |Y ), then P(Z |Y )> r .
Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 14 / 19
Example: Let W = {w1, . . . ,w7}, let P be again as in the example before.
Then if Bel(·|·) satisfies AGM, and if P and Bel(·|·) jointly satisfy BP12 , then
Bel(·|·) must be given by some coarse-graining of the ranking in red below.
Choosing the maximal (most fine-grained) Bel(·|·) yields the following:
Bel(A∧B|A) (A→ A∧B)
Bel(A∧B|B) (B→ A∧B)
Bel(A∧B|A∨B) (A∨B→ A∧B)
Bel(A|C) (C→ A)
¬Bel(B|C) (C9 B)
Bel(A|C∧¬B) (C∧¬B→ A)
!!
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Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 15 / 19
For three worlds again (and r = 12 ), the maximal Bel(·|·) as being determined
by P and r are given by these rankings:
!
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Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 16 / 19
Moral:
Given P and a threshold r , the agent’s Bel(·|·) is not determined uniquelyby the “→” of the Lockean thesis.
But any such Bel(·|·) is closed logically iff it is given by a sphere system ofP-stabler sets.
Given P and a threshold r , the agent’s maximal Bel(·|·) amongst thosethat satisfy all of our postulates is determined uniquely.
(And there is always such a unique maximal choice Bel rP given a ratherweak auxiliary assumption.)
Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 17 / 19
Two remarks:
B1→ C1, . . . ,Bn→ Cn ∴ X → Y is logically valid iff for all P, r ≥ 12 holds:
If Bel rP(C1|B1), . . . ,Bel rP(Cn|Bn)
then Bel rP(Y |X).
The resulting logic is exactly Adams’ logic of conditionals again! E.g.:
X → Y , X → ZX → (Y ∧Z )
(And)X → Z , Y → Z(X ∨Y )→ Z
(Or)
(X ∧Y )→ Z , X → YX → Z
(Cautious Cut)X → Y , X → Z(X ∧Y )→ Z
(Cautious M.)
Conditionalization on Zero Sets:
P∗, with P∗(Y |X) = P(Y |BX ), determines a Popper function.cf. van Fraassen (1995), Arlo-Costa & Parikh (2004) on “belief cores”.
Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 18 / 19
Two remarks:
B1→ C1, . . . ,Bn→ Cn ∴ X → Y is logically valid iff for all P, r ≥ 12 holds:
If Bel rP(C1|B1), . . . ,Bel rP(Cn|Bn)
then Bel rP(Y |X).
The resulting logic is exactly Adams’ logic of conditionals again! E.g.:
X → Y , X → ZX → (Y ∧Z )
(And)X → Z , Y → Z(X ∨Y )→ Z
(Or)
(X ∧Y )→ Z , X → YX → Z
(Cautious Cut)X → Y , X → Z(X ∧Y )→ Z
(Cautious M.)
Conditionalization on Zero Sets:
P∗, with P∗(Y |X) = P(Y |BX ), determines a Popper function.cf. van Fraassen (1995), Arlo-Costa & Parikh (2004) on “belief cores”.
Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 18 / 19
Epilogue: Solving A Problem Contextually
A challenge to the theory:
Intuitively, Expansion/Revision can be problematic:
!
"!
#!
Bel rP(Y1∨Y2∨ . . .∨Yn |X), ¬Bel rP(¬Yi |X)
Bel rP(Yi |Yi ∨ (X ∧¬(Y1∨Y2∨ . . .∨Yn)))
Lottery’s revenge: For the same reason, if both P and Bel represent thesame large finite lottery, then P(BW ) must be very close to 1!
In both cases, the solution is to make qualitative belief relativized to partitions:
Possible: Bel rP,{Zj}(Y1∨Y2∨ . . .∨Yn |X), ¬Bel rP,{Z ′j }(Y1∨Y2∨ . . .∨Yn |X).
Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 19 / 19
Epilogue: Solving A Problem Contextually
A challenge to the theory:
Intuitively, Expansion/Revision can be problematic:
!
"!
#!
Bel rP(Y1∨Y2∨ . . .∨Yn |X), ¬Bel rP(¬Yi |X)
Bel rP(Yi |Yi ∨ (X ∧¬(Y1∨Y2∨ . . .∨Yn)))
Lottery’s revenge: For the same reason, if both P and Bel represent thesame large finite lottery, then P(BW ) must be very close to 1!
In both cases, the solution is to make qualitative belief relativized to partitions:
Possible: Bel rP,{Zj}(Y1∨Y2∨ . . .∨Yn |X), ¬Bel rP,{Z ′j }(Y1∨Y2∨ . . .∨Yn |X).
Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 19 / 19
Epilogue: Solving A Problem Contextually
A challenge to the theory:
Intuitively, Expansion/Revision can be problematic:
!
"!
#!
Bel rP(Y1∨Y2∨ . . .∨Yn |X), ¬Bel rP(¬Yi |X)
Bel rP(Yi |Yi ∨ (X ∧¬(Y1∨Y2∨ . . .∨Yn)))
Lottery’s revenge: For the same reason, if both P and Bel represent thesame large finite lottery, then P(BW ) must be very close to 1!
In both cases, the solution is to make qualitative belief relativized to partitions:
Possible: Bel rP,{Zj}(Y1∨Y2∨ . . .∨Yn |X), ¬Bel rP,{Z ′j }(Y1∨Y2∨ . . .∨Yn |X).
Hannes Leitgeb (LMU Munich) From the Lockean Thesis to Conditionals May 2011 19 / 19