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?

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P is a subjective probability measure.

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P is a subjective probability measure, A is a simple sentence.

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The Lockean thesis LT>r : BelP(X) iff P(X)> r ≥ 12 .

BelP is logically closed (in the sense of doxastic logic).

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P is a subjective probability measure, A→ B is a conditional.

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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

BelP(·|·) is logically closed (in the sense of AGM 1985 on belief revision).

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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 .

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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

Can we satisfy all of these desiderata simultaneously? (Yes.)

Plan of the talk:

Can we satisfy all of these desiderata simultaneously? (Yes.)

Plan of the talk:

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).

E.g., a probability measure P:!

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P conditionalized on C:!

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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 .

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 ∈[

), then X is P-stable

(cf. Skyrms 1977, 1980 on resiliency.)

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 ∈[

), then X is P-stable

(cf. Skyrms 1977, 1980 on resiliency.)

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.

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

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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)

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.

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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.

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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.

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

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.

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.

“→” 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 .

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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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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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.)

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

(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”.

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

(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”.

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).

Bel rP(Yi |Yi ∨ (X ∧¬(Y1∨Y2∨ . . .∨Yn)))

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