Comparative Probability.15in - ESSLLI 2014 - Logic and ...philosophy.berkeley.edu/file/920/lecture5.pdf · IC: The next 1000 ips will give at least 460 heads. Are the following judgments

Comparative Probability

ESSLLI 2014 - Logic and Probability

Wes Holliday and Thomas IcardBerkeley and Stanford

August 14-15, 2014

Wes Holliday and Thomas Icard: Comparative Probability 1

Outline

1. Basic Notions

2. Considering Axioms: Rationality and Realism

• de Finetti’s Axioms

3. Comparative Probability as Modal Logic

4. Probabilistic Representability

• The Kraft-Pratt-Seidenberg Counterexample

• Cancellation Axioms and Scott’s Representation Theorem

5. Segerberg and Gardenfors’ Completeness Theorem

6. Bonus 1: Imprecise Representability (by a set of measures)

7. Bonus 2: Extending an Ordering on a Set to the Powerset


References for comparative probability:

I David Krantz, R. Dunan Luce, Patrick Suppes, Amos Tversky,Foundations of Measurement, Vol. 1, 1971.

I Terrence Fine, Theories of Probability, 1973.

I Peter Fishburn, “The Axioms of Subjective Probability,” StatisticalScience, 1986.

I Louis Narens, Theories of Probability, 2007.

References for comparative probability as modal logic:

I Krister Segerberg, “Qualitative Probability in a Modal Setting,” 1971.

I Peter Gardenfors, “Qualitative Probability as an Intensional Logic,” 1975.

I Wiebe van der Hoek, “Qualitative Modalities,” 1996. Wiebe van derHoek and John-Jules Meyer, “Modalities for Reasoning about Knowledgeand Uncertainties,” 1996.

I Wesley Holliday and Thomas Icard, “Measure Semantics and QualitativeSemantics for Epistemic Modals,” 2013.


Basic Notions

Starting Point: A Binary Relation on Events

Given a set W , we will consider a binary relation % on ℘(W ).

For “events” or “propositions” A,B ∈ ℘(W ), our interpretation is:

A % B – “A is (regarded by the agent as) at least as probable as B.”

To be more general, we could take % to be a binary relation onsome algebra A of subsets of W , not necessarily ℘(W ); but sincewe’ll mostly focus on the case of finite W , this won’t be necessary.


Basic Notions

Non-strict Relation as Basic

A % B – “A is (regarded by the agent as) at least as probable as B”;

Define: A � B iff A % B and B 6% A;

Define: A ∼ B iff A % B and B % A. This indicates that “A and Bare (regarded by the agent as) equally probable.”

Define: A ↑ B iff A 6% B and B 6% A. This indicates that “A and Bare incomparable in probablity (for the agent).”

NB: A � B doesn’t necessarily indicate “A is (regarded by theagent as) more probable than B.” One may judge that A is atleast as likely as B, but withhold judgment on whether B is atleast as likely as A. That’s not to judge that A is more probable.


Basic Notions

Strict Relation as Basic

A > B – “A is (regarded by the agent as) more probable than B.”

Define: A ≈ B iff A 6> B and B 6> A;

Define: A ' B either A > B or A ≈ B.

NB: A ≈ B doesn’t necessarily indicate that “A and B are equallyprobable,” i.e., A ∼ B. It may be that the events are incomparablefor the agent. That’s not the same as being equally probable.


Basic Notions

Strict Relation as Basic

A > B – “A is (regarded by the agent as) more probable than B.”



NB: even if we assume non-strict completeness, i.e., for all A,B,“A is at least as probable as B or B is at least as probable as A,”A ≈ B still doesn’t mean that “A and B are equally probable.”

It may be that the agent judges that A is at least as probable as B,but withholds judgment on whether A is more probable than B andwithholds judgment on whether B is more probable than A. ThenA ≈ B, but the agent does not judge them to be equally probable.

It follows that A ' B doesn’t necessarily indicate that A % B.


Basic Notions

Non-strict as basic:

A % B – “A is at least as probable as B”;


Define: A ∼ B iff A % B and B % A.“A and B are equally probable.”

Strict as basic:

A > B – “A is more probable than B.”



The moral of the foregoing seems to be that if we want to captureboth “at least as probable as” and “more probable than,” then weshould take both % and > as primitive, related by the axioms:

I if A > B, then A % B;

I if A % B, then B 6> A.

However, typically just one of % or > is taken as primitive. . .


Basic Notions





Strict as basic:




ExerciseIf % is basic and � and ∼ are defined as above, then the followingequivalences (from the def.’s in terms of >) hold iff % is complete:

1. A ∼ B iff A 6� B and B 6� A;

2. A % B iff A � B or A ∼ B.


Basic Notions





Strict as basic:




ExerciseIf > is basic and ≈ and ' defined as above, then the following(from the def.’s in terms of %) hold iff > is asymmetric:

1. A > B iff A ' B and B 6' A;

2. A ≈ B iff A ' B and B ' A.


Basic Notions

Outline

1. Basic Notions X

2. Considering Axioms: Rationality and Realism

• de Finetti’s Axioms









Considering Axioms: Rationality and Realism

What Axioms are Evident?

Since � means “at least as probable,” we should at least have:

I reflexivity: A % A.

Fishburn: the following might count as “so obvious anduncontroversial as to occasion no serious criticism.”

I nonnegativity: A % ∅;

I nontriviality: ∅ 6� W (or W > ∅);

I monotonicity: if A ⊇ B, then A > B;

I asymmetry: A > B implies B 6> A; (The asymmetry of �follows from the fact that A � B := [A � B and B 6� A].)

I inclusion monotonicity: if [A ⊇ B and B > C ] or[A > B and B ⊇ C ], then A > C . (If % is transitive andmonotonic, then inclusion monotonicity holds for �.)



What Axioms are Evident?

Fishburn: one might also suggest the following axioms as obvious,but they have been challenged.

I transitivity: if A > B and B > C , then A > C ; similarly for%. (Exercise: what axioms give you transitivity for �?);

I complementarity: if A � B, then W − B 6� W − A. (We willalso consider self-duality: A % B iff W − B %W − A.)

I quasi-additivity: if (A∪ B) ∩ C = ∅, thenA % B ⇔ A∪ C % B ∪ C .



de Finetti’s Axioms

de Finetti’s proposed the following axioms for %:


I nontriviality: ∅ 6%W ;

I totality: A % B of B % A;

I transitivity: if A % B and B % C , then A % C ;





de Finetti’s proposed that % is a total preorder satisfying:




An equivalent formulation replaces the last axiom with:

I A % B iff A− B % B − A.

ExerciseShow that if % satisfies de Finetti’s axioms, then it also satisfiesmonotonicity and self-duality.



Doubts about Totality

One of de Finetti’s axioms is:

I totality: A % B or B % A.

As Fine (1973, 18) notes, “the requirement that all events becomparable is not insignificant and has been denied by manycareful students of probability including Keynes and Koopman.”

For example:

A a leaf will fall on my shoe on the walk down to town.

B one of my kitchen appliances will break in the next 24 hours.

Must it be that either I regard A to be at least as likely as B or Iregard B to be at least as likely as A?



An Example from Fishburn

Consider the following for a slightly bent coin:

I A: The next 101 flips will give at least 40 heads.

I B: The next 100 flips will give at least 40 heads.

I C : The next 1000 flips will give at least 460 heads.

Are the following judgments unreasonable?

I A > B, A ∼ C , B ∼ C .

Fishburn says that the judgments are not unreasonable.

According to these judgments, B % C and C % A, but B 6% A, so% is not transitive.



Another Example from FishburnSue is expecting to meet Smith. She doesn’t know his age orappearance. Her judgments about the likelihood of his attributes:

I Height: 6′-0′′ > 6′-1′′ > 6′-2′′;

I Age: 40 > 50 > 60;

I Hair: bown > red > blonde.

Three possible combinations are:

A 6′-0′′ 60-year old redhead;

B 6′-1′′ 40-year old blonde;

C 6′-2′′ 50-year old brunette.

Sue judges a combination X more likely than combination Y if Xis more probable than Y on two of the attributes.

Thus, A > B, B > C , and C > A, so we have intransitivity.



Ellsberg 1961

A bag contains 90 marbles. It is known that 30 of the marbles arered. It is also known that each of the remaining 60 marbles iseither green or purple, but the proportion is not known.

You’ll draw a marble at random. It seems reasonable to assignprobability 3/9 to drawing a red marble (R). It also seemsreasonable to assign probability 6/9 to drawing either a green orpurple one (G ∪ P). But what about the probabilities of G and P?



Ellsberg 1961


For each event E consider the following bet:

I BE : pays 1 Euro if E ; 0 otherwise.

Which do you prefer: BR or BG? BR or BP? BG∪P or BR∪P?



Ellsberg 1961


For each event E consider the following bet:

I BE : pays 1 Euro if E ; 0 otherwise.

Empirically: people tend to strictly prefer BR to BG (and to BP),and they tend to prefer BG∪P to BR∪P .

Moral: if the preferences reflect comparative probability judgments(e.g., if the people are expected utility maximizers), then thejudgments violate quasi-additivity:

G ∪ P % R ∪ P, but G 6% R.




Let’s assume that the foregoing examples can be explained away.

So let’s assume that de Finetti was correct that % should be atotal preorder satisfying:






Outline

1. Basic Notions X

2. Considering Axioms: Rationality and Realism X

• de Finetti’s Axioms X









Comparative Probability as Modal Logic


As Segerberg and later Gardenfors observed, comparativeprobability can be seen as a kind of modal logic.

In this spirit, we’ll introduce a formal modal language. . .



Formal Language

DefinitionGiven a countable set At = {p, q, r , . . . } of atomic sentences, thelanguage L(2,>) is defined by the grammar:

ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | 2ϕ | (ϕ>ψ).

We define ∨, →, and ↔ as usual, and

I > := (p ∨ ¬p) for some p ∈ At; ⊥ := ¬>;

I 3ϕ := ¬2¬ϕ;

I (ϕ > ψ) := (ϕ > ψ) ∧ ¬(ψ > ϕ);

I 4ϕ := (ϕ > ¬ϕ).



Models

DefinitionA QP Kripke model for L(2,>) is a tupleM = 〈W ,R, {%w}w∈W ,V 〉 where:

1. W is a nonempty set;

2. R is a serial binary relation on W , i.e., such that for allw ∈ W , R(w) = {v ∈ W | wRv} 6= ∅;

3. %w is a binary relation on ℘(R(w));

4. V : At→ ℘(W ).

NB: Segerberg uses a more general definition where %w is a binaryrelation on some algebra of subsets of R(w), not necessarily℘(W ). Since we’ll focus on finite models, this won’t matter.



Truth

DefinitionGiven a QP Kripke model M and a ϕ ∈ L(2,>), we defineM,w � ϕ (“ϕ is true at state w in model M”) as follows:

1. M,w � p iff w ∈ V (p);

2. M,w � ¬ϕ iff M,w 2 ϕ;

3. M,w � (ϕ ∧ ψ) iff M,w � ϕ and M,w � ψ;

4. M,w � 2ϕ iff ∀v ∈ R(w): M, v � ϕ;

5. M,w � ϕ > ψ iff JϕKMw %w JψKMw ,

where JχKMw = {v ∈ R(w) | M, v � χ}.



Yalcin’s List of Intuitively Valid and Invalid Patterns

V1 4ϕ→ ¬4¬ϕ

V2 4(ϕ ∧ ψ)→ (4ϕ ∧4ψ) V3 4ϕ→ 4(ϕ ∨ ψ)

V4 ϕ > ⊥ V5 > > ϕ

V6 2ϕ→ 4ϕ V7 4ϕ→ 3ϕ

V11 (ψ > ϕ)→ (4ϕ→ 4ψ)

V12 (ψ > ϕ)→ ((ϕ > ¬ϕ)→ (ψ > ¬ψ))

I1 ((ϕ > ψ) ∧ (ϕ > χ))→ (ϕ > (ψ ∨ χ))

I2 (ϕ > ¬ϕ)→ (ϕ > ψ)

I3 4ϕ→ (ϕ > ψ)

E1 (4ϕ ∧4ψ)→ 4(ϕ ∧ ψ)



System W

Taut all tautologies MPϕ→ ψ ϕ

ψ

Necϕ2ϕ

K 2(ϕ→ ψ)→ (2ϕ→ 2ψ)

Ex (2(ϕ↔ ϕ′) ∧2(ψ↔ ψ′))→ ((ϕ > ψ)↔ (ϕ′ > ψ′))

Bot ϕ > ⊥

BT ¬(⊥ > >)

Tran (ϕ > ψ)→ ((ψ > χ)→ (ϕ > χ))

Mon 2(ϕ→ ψ)→ (ϕ > ψ)




ψ

Necϕ2ϕ

K 2(ϕ→ ψ)→ (2ϕ→ 2ψ)

Ex (2(ϕ↔ ϕ′) ∧2(ψ↔ ψ′))→ ((ϕ > ψ)↔ (ϕ′ > ψ′))

Bot ϕ > ⊥

BT ¬(⊥ > >)

Tran (ϕ > ψ)→ ((ψ > χ)→ (ϕ > χ))

Mon 2(ϕ→ ψ)→ (ϕ > ψ)

Fact (Holliday and Icard 2013)

All of Yalcin’s V1-V12, and none of I1-I3 or E1, are derivable in thesystem WS that adds to W the following axiom:

S (ϕ > ψ)→ (¬ψ > ¬ϕ).




V1 4ϕ→ ¬4¬ϕ

V2 4(ϕ ∧ ψ)→ (4ϕ ∧4ψ) V3 4ϕ→ 4(ϕ ∨ ψ)

V4 ϕ > ⊥ V5 > > ϕ

V6 2ϕ→ 4ϕ V7 4ϕ→ 3ϕ

V11 (ψ > ϕ)→ (4ϕ→ 4ψ)

V12 (ψ > ϕ)→ ((ϕ > ¬ϕ)→ (ψ > ¬ψ))

I1 ((ϕ > ψ) ∧ (ϕ > χ))→ (ϕ > (ψ ∨ χ))

I2 (ϕ > ¬ϕ)→ (ϕ > ψ)

I3 4ϕ→ (ϕ > ψ)

E1 (4ϕ ∧4ψ)→ 4(ϕ ∧ ψ)




de Finetti’s proposed the following axioms for %:



I totality: A % B of B % A;

I transitivity: if A % B and B % C , then A % C ;




de Finetti’s System FA


ψ

Necϕ2ϕ

K 2(ϕ→ ψ)→ (2ϕ→ 2ψ)

Ex (2(ϕ↔ ϕ′) ∧2(ψ↔ ψ′))→ ((ϕ > ψ)↔ (ϕ′ > ψ′))

Bot ϕ > ⊥

BT ¬(⊥ > >)

Tot (ϕ > ψ) ∨ (ψ > ϕ)

Tran (ϕ > ψ)→ ((ψ > χ)→ (ϕ > χ))

A ¬3((ϕ ∨ ψ) ∧ χ)→ (ϕ > ψ↔ ((ϕ ∨ χ) > (ψ ∨ χ)))



Outline

1. Basic Notions X



3. Comparative Probability as Modal Logic X








Probabilistic Representability

Let A be an algebra on a nonempty set W , e.g., A is ℘(W ).

Definition (Agreement)

Given a binary relation % on A and a probability measureµ : A → [0, 1], µ agrees with % iff for all X ,Y ∈ A:

X % Y ⇔ µ(X ) ≥ µ(Y ).

We also say that µ represents %.

Definition (Representability)

A binary relation % on ℘(W ) is probabilistically representable iffthere is a probability measure µ on ℘(W ) that agrees with %.

QUESTION: is any relation % that satisfies de Finetti’s threeaxioms probabilistically representable?


Probabilistic Representability – The KPS Counterexample

Answer: No!

Fact (Kraft, Pratt, and Seidenberg)

There is a total preorder % on ℘({a, b, c , d , e}) such that:

1. % is nontrivial: ∅ 6%W ;

2. % is nonnegative: A % ∅;

3. % is quasi-additive: if (A∪ B) ∩ C = ∅, thenA % B ⇔ A∪ C % B ∪ C ;

4. % is not probabilistically representable.

Let’s do the proof, expanding the presentation of Krantz et al.



Proof. Suppose there is a % satisfying de Finetti’s axioms and:

{a} � {b, c} {c , d} � {a, b} {b, e} � {a, c} {a, b, c} � {d , e}.

Then % isn’t representable. For if there were an agreeing µ, then

µ({a}) > µ({b, c}) µ({c, d}) > µ({a, b}) µ({b, e}) > µ({a, c}),

so by finite additivity of µ,

µ(a) > µ(b) + µ(c)µ(c) + µ(d) > µ(a) + µ(b)µ(b) + µ(e) > µ(a) + µ(c).

Adding the left and right sides of these inequalities,

µ(a) + µ(b) + µ(c) + µ(d) + µ(e) > 2µ(a) + 2µ(b) + 2µ(c).










µ(a) + µ(b) + µ(c) + µ(d) + µ(e) > 2µ(a) + 2µ(b) + 2µ(c).










µ(d) + µ(e) > µ(a) + µ(b) + µ(c).

Then by the assumption that µ agrees with %, we would have{d , e} � {a, b, c}, contradicting {a, b, c} � {d , e}.





Then % isn’t representable. X To show that there is such arelation %, for some 0 < ε < 1/3, define

ν(a) = 4− ε ν(b) = 1− ε ν(c) = 3− ε ν(d) = 2 ν(e) = 6,

ν(A) = ∑a∈A

ν(a), which can be normalized as µ(A) = ν(A)16−3ε .

Define %ν s.th. A %ν B iff ν(A) ≥ ν(B) (iff µ(A) ≥ µ(B)).

Thus, %ν satisfies de Finetti’s axioms, and the blue �’s hold:

ν({a}) = 4− ε ν({b, c}) = 4− 2εν({c , d}) = 5− ε ν({a, b}) = 5− 2εν({b, e}) = 7− ε ν({a, c}) = 7− 2ε.





Then % isn’t representable. For some 0 < ε < 1/3, define

ν(a) = 4− ε ν(b) = 1− ε ν(c) = 3− ε ν(d) = 2 ν(e) = 6.


Thus, %ν satisfies de Finetti’s axioms, and the blue �’s hold. X

Let %′ν =%ν −{〈{d , e}, {a, b, c}〉} ∪ {〈{a, b, c}, {d , e}〉}, soblue �’s and red � hold for %′ν. Claim: deFi’s axioms hold of %′ν.

ν({d , e}) = 8, ν({a, b, c}) = 8− 3ε, but there is no distinct A with

8 ≥ ν(A) ≥ 8− 3ε; for it requires ν(A) = 8− kε for k ∈ {0, 1, 2, 3};but 8 can be expressed as a sum of distinct members of {1, 2, 3, 4, 6} in

only two ways, 4 + 1 + 3 and 2 + 6, which pick out {a, b, c} and {d , e}.

Thus, there is no distinct A with {d , e} %ν A %ν {a, b, c}.Wes Holliday and Thomas Icard: Comparative Probability 41




Let %′ν =%ν −{〈{d , e}, {a, b, c}〉} ∪ {〈{a, b, c}, {d , e}〉}, soblue �’s and red � hold for %′ν. Claim: deFi’s axioms still hold.

We’ve shown there is no distinct A with {d , e} %ν A %ν {a, b, c}.

Check: (A∪ B) ∩ C = ∅⇒ [A∪ C %′ν B ∪ C ⇔ A %′ν B ].

· If A = B or A∪ C = B ∪ C , then the principle holds by reflexivity.

· If A = {d , e} and B = {a, b, c} or vice versa, then C = ∅, so it holds;similarly if A∪ C = {d , e} and B ∪ C = {a, b, c} or vice versa.

So we can assume A 6∈ S = {{d , e}, {a, b, c}} or B 6∈ S . Then thepurple fact above implies that A %ν B iff A %′ν B (picture it).

We can also assume A∪ C 6∈ S or B ∪ C 6∈ S . Then the purple factabove implies that A∪ C %ν B ∪ C iff A∪ C %′ν B ∪ C .

Thus, the quasi-additivity of %ν implies the quasi-additivity of %′ν.





Then % isn’t representable. X To show that there is such arelation %, for some 0 < ε < 1/3, define

ν(a) = 4− ε ν(b) = 1− ε ν(c) = 3− ε ν(d) = 2 ν(e) = 6,

ν(A) = ∑a∈A

ν(a), which can be normalized as µ(A) = ν(A)16−3ε .


Thus, %ν satisfies de Finetti’s axioms, and the blue �’s hold. X

Let %′ν =%ν −{〈{d , e}, {a, b, c}〉} ∪ {〈{a, b, c}, {d , e}〉}, soblue �’s, red � hold for %′ν. Also deFi’s axioms hold of %′ν. X

Thus, we’ve finished the proof: we have a relation %′ν that satisfiesde Finetti’s axioms but is not probabilistically representable!



We have shown:





3. % is quasi-additive: if (A∪ B) ∩ C = ∅, thenA % B ⇔ A∪ C % B ∪ C .




The KPS ordering is such that:


For example:

1. Our guest is more likely to arrive on American than on Britishor Continental;

2. She is more likely to arrive on Continental or Delta than onAmerican or British;

3. She is more likely to arrive on British or Emirates than onAmerican or Continental;

4. She is more likely to arrive on American, British, orContinental than on Delta or Emirates.

Fine, Theories of Probability, p. 24: “what in our understanding ofCP compels us to eliminate the ordering . . . as a CP relation?”



Outline

1. Basic Notions X





• The Kraft-Pratt-Seidenberg Counterexample X• Cancellation Axioms and Scott’s Representation Theorem






We have shown:





3. % is quasi-additive: if (A∪ B) ∩ C = ∅, thenA % B ⇔ A∪ C % B ∪ C ;


QUESTION: what further conditions on % must we assume forprobabilistic representability?


Probabilistic Representability – Cancellation Axioms and Scott’s Theorem

Finite Cancellation

Two sequences A1, . . . ,An and B1, . . . ,Bn of subsets of W arebalanced iff for all w ∈ W ,

|{i | w ∈ Ai}| = |{j | w ∈ Bj}|.

Consider the Finite Cancellation (FC ) axiom: for any balancedsequences A1, . . . ,An and B1, . . . ,Bn of subsets of W ,

if Ai % Bi for all i < n, then Bn % An.



Let’s momentarily take both relations % and > as primitive.

Consider the axiom FC>: for any balanced sequences A1, . . . ,An

and B1, . . . ,Bn of subsets of W ,

if Ai % Bi for all i < n, then An 6> Bn.

Fishburn’s “crude but instructive” pragmatic motivation forFC>: an agent for whom Ai % Bi for all i < n and An > Bn wouldpresumably be willing to pay a positive amount to play this game:

I for each Ai that obtains, he wins 1 Euro;

I for each Bi that obtains, he loses 1 Euro.

Since A1, . . . ,An and B1, . . . ,Bn are balanced, no matter what isthe true state of the world, his take from the game will be 0 Euros;but he paid a positive amount to play, so he’ll have a sure loss.



That pragmatic argument does not exactly work with FC : for anybalanced sequences A1, . . . ,An and B1, . . . ,Bn of subsets of W ,


An agent who violates this has Ai % Bi for all i < n, and Bn 6% An.

If we assume that Bn 6% An implies An > Bn, then we can run thesame argument. But if not, then can we claim that such an agentwould be willing to pay some positive amount to play the game?



Two sequences A1, . . . ,An and B1, . . . ,Bn of subsets of W arebalanced iff for all w ∈ W ,

|{i | w ∈ Ai}| = |{j | w ∈ Bj}|.

Consider the Finite Cancellation (FC ) axiom: for any balancedsequences A1, . . . ,An and B1, . . . ,Bn of subsets of W ,


Proposition (Finite Cancellation)

If a relation % on ℘(W ) is probabilistically representable, then %satisfies FC .

Proof. On next slide. . .



Proposition (Finite Cancellation)

If a relation % on ℘(W ) is prob. representable, then for anybalanced sequences A1, . . . ,An and B1, . . . ,Bn of subsets of W ,


Proof. We prove this for finite W , leaving the infinite case as an exercise.Let µ be the probability measure on ℘(W ) that agrees with %. By theassumption that A1, . . . ,An and B1, . . . ,Bn are balanced, i.e., everyw ∈ W is in the same number of Ai ’s and Bi ’s, we have

∑i≤n

∑w∈Ai

µ({w}) = ∑i≤n

∑w∈Bi

µ({w}). (1)

Then by the additivity of µ, (1) implies

∑i≤n

µ(Ai ) = ∑i≤n

µ(Bi ). (2)

Now if for all i < n, Ai % Bi , so µ(Ai ) ≥ µ(Bi ), then (2) implies

µ(Bn) ≥ µ(An), so Bn � An. Thus, � satisfies the cancellation axioms.



Comparing Quasi-Additivity and Finite Cancellation

FC≤m: for any n ≤ m and balanced sequences A1, . . . ,An andB1, . . . ,Bn of subsets of W , if Ai % Bi for all i < n, then Bn % An.

FactA relation % on ℘(W ) satisfies FC≤3 iff % is transitive andquasi-additive.

Proof. Let’s show FC≤3 implies transitivity . Since X ,Y ,Z andY ,Z ,X are balanced, by FC≤3, [X % Y and Y % Z ]⇒ X % Z .

To show that FC≤3 implies quasi-additivity , note that if(A∪ B) ∩ C = ∅, then A, (B ∪ C ) and B, (A∪ C ) are balanced.Thus, if A % B, then A∪ C % B ∪ C . Moreover, (A∪ C ),B and(B ∪ C ),A are balanced, if A∪ C % B ∪ C , then A % B.



Comparing Quasi-Additivity and Finite Cancellation

FC≤m: for any n ≤ m and balanced sequences A1, . . . ,An andB1, . . . ,Bn of subsets of W , if Ai % Bi for all i < n, then Bn % An.


Proof. Now let’s show that quasi-additivity implies FC≤2. (Thecase of FC≤3 is more difficult and left as an exercise.)

If A1,A2 and B1,B2 are balanced, then we have

A1 − B1 = B2 − A2

A2 − B2 = B1 − A1.

By quasi-additivity, if A1 % B1, then A1 − B1 % B1 − A1. So bythe equations above, B2 − A2 % A2 − B2, whence B2 % A2.



KPS and Finite Cancellation

Recall that the KPS ordering had the following inequalities:


This violates FC≤4. The following sequences are balanced:

I 〈A1,A2,A3,A4〉 be 〈{a}, {c, d}, {b, e}, {a, b, c}〉I 〈B1,B2,B3,B4〉 be 〈{b, c}, {a, b}, {a, c}, {d , e}〉.

Then in the KPS ordering, Ai % Bi for i < 4, but B4 6% A4.



Representation Theorem

Theorem (Scott 1964)

For any finite set W , a binary relation % on ℘(W ) isprobabilistically representable iff it satisfies the nonnegativity,nontriviality, and finite cancellation axioms.

Proof. Where |W | = n, each A ∈ ℘(W ) can be associated with avector A ∈ {0, 1}n, the “characteristic function” of A.

Let Γ be the set of strict inequalities A � B, and Σ the set ofequivalences A ∼ B. For γ = A � B, and σ = A ∼ B, let

γ = A− B and σ = A− B.



Proof. For γ = A � B, and σ = A ∼ B, let

γ = A− B and σ = A− B.

LemmaThere exists c ∈ Rn such that c · γ > 0 for all γ ∈ Γ, andc · σ = 0 for all σ ∈ Σ.Given this lemma, we set:

µ(A) =c · Ac ·W

.

Then:

I If A � B, then by the lemma, µ(A) > µ(B);

I If A ∼ B, then again by the lemma, µ(A) = µ(B).

I Showing µ is a probability measure is straightforward.



Outline

1. Basic Notions X




4. Probabilistic Representability X

• The Kraft-Pratt-Seidenberg Counterexample X• Cancellation Axioms, Scott’s Representation Theorem X





Segerberg and Gardenfors’ Completeness Theorem

Expressing FC in Our Formal Language

FCm: for any n ≤ m and balanced sequences A1, . . . ,An andB1, . . . ,Bn of subsets of W , if Ai % Bi for all i < n, then Bn % An.

Let’s now try to express the finite cancellation axioms in L(2,>).

DefinitionGiven a countable set At = {p, q, r , . . . } of atomic sentences, thelanguage L(2,>) is defined by the grammar:

ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | 2ϕ | (ϕ>ψ).



Given a sequence σ = 〈σ1, . . . , σm〉 with each σk ∈ L(>), definefor each i ≤ n a formula Ni (σ) stating that |{k | σk is true}| = i(note that this is not the same as |{σk | σk is true}| = i):

N0(σ) = ¬σ1 ∧ · · · ∧ ¬σm;

N1(σ) = (σ1 ∧ ¬σ2 ∧ · · · ∧ ¬σn) ∨ (¬σ1 ∧ σ2 ∧ ¬σ3 ∧ · · · ∧ ¬σn) . . .· · · ∨ (¬σ1 ∧ · · · ∧ ¬σm−1 ∧ σm).

etc.

Next, given sequences σ and σ′, both of length m, define

σEσ′ :=∨

0≤i≤m(Ni (σ) ∧Ni (σ

′)).

Finally, define σEσ′ := 2(σEσ′).

Thus, σEσ′ says that every accessible world satisfies the samenumber of components from σ as from σ′, as desired.



Expressing FC in Our Formal Language

FC : for any balanced sequences A1, . . . ,An and B1, . . . ,Bn ofsubsets of W , if Ai % Bi for all i < n, then Bn % An.

Now we can state finite cancellation as an infinite axiom schema:

FC ϕ1 . . . ϕnEψ1 . . . ψn → ((∧i<n

(ϕi > ψi ))→ (ψn > ϕn))



System FP


ψ

Necϕ2ϕ

K 2(ϕ→ ψ)→ (2ϕ→ 2ψ)

Ex (2(ϕ↔ ϕ′) ∧2(ψ↔ ψ′))→ ((ϕ > ψ)↔ (ϕ′ > ψ′))

Bot ϕ > ⊥

BT ¬(⊥ > >)

Tot (ϕ > ψ) ∨ (ψ > ϕ)

FC ϕ1 . . . ϕnEψ1 . . . ψn → ((∧i<n

(ϕi > ψi ))→ (ψn > ϕn))



System FA


ψ

Necϕ2ϕ

K 2(ϕ→ ψ)→ (2ϕ→ 2ψ)

Ex (2(ϕ↔ ϕ′) ∧2(ψ↔ ψ′))→ ((ϕ > ψ)↔ (ϕ′ > ψ′))

Bot ϕ > ⊥

BT ¬(⊥ > >)

Tot (ϕ > ψ) ∨ (ψ > ϕ)

Tran (ϕ > ψ)→ ((ψ > χ)→ (ϕ > χ))

A ¬3((ϕ ∨ ψ) ∧ χ)→ (ϕ > ψ↔ ((ϕ ∨ χ) > (ψ ∨ χ)))



System FP


ψ

Necϕ2ϕ

K 2(ϕ→ ψ)→ (2ϕ→ 2ψ)

Ex (2(ϕ↔ ϕ′) ∧2(ψ↔ ψ′))→ ((ϕ > ψ)↔ (ϕ′ > ψ′))

Bot ϕ > ⊥

BT ¬(⊥ > >)

Tot (ϕ > ψ) ∨ (ψ > ϕ)

FC ϕ1 . . . ϕnEψ1 . . . ψn → ((∧i<n

(ϕi > ψi ))→ (ψn > ϕn))



Comparing FA and FP

Recall that earlier we showed:


Similarly, where FP3 system obtained by deleting from FP allinstances of FC for n > 3, we have:

Fact (van der Hoek)

FA and FP3 have the same theorems.



System FP

Ex (2(ϕ↔ ϕ′) ∧2(ψ↔ ψ′))→ ((ϕ > ψ)↔ (ϕ′ > ψ′))

Bot ϕ > ⊥

BT ¬(⊥ > >)

Tot (ϕ > ψ) ∨ (ψ > ϕ)

FC ϕ1 . . . ϕnEψ1 . . . ψn → ((∧i<n

(ϕi > ψi ))→ (ψn > ϕn))

Theorem (Segerberg 1971, Gardenfors 1975)

FP is sound and complete with respect to the class of modelsM = 〈W ,R,%,V 〉 for L(2,6) such that for every w ∈ W ,%w is probabilistically representable.

FP also has the finite model property with respect to that class.



Review: Probabilistic Kripke Models

Definition (Probabilistic Kripke Model)

A probabilistic Kripke model is a tupleM = 〈W ,R, {µw}w∈W ,V 〉 such that:



3. µw : ℘(R(w))→ [0, 1] such that µw (R(w)) = 1 and ifA∩ B = ∅, then µw (A∪ B) = µw (A) + µw (B);

4. V : At→ ℘(W ).

.



Semantics

Definition (Truth and Consequence)

Where M is a probabilistic Kripke model and ϕ ∈ L(2,>), wedefine M,w � ϕ (“ϕ is true at w in M”) by:

I M,w � p iff w ∈ V (p);

I M,w � ¬ϕ iff M,w 2 ϕ;

I M,w � (ϕ ∧ ψ) iff M,w � ϕ and M,w � ψ;

I M,w � 2ϕ iff ∀v ∈ R(w): M, v � ϕ;

I M,w � ϕ > ψ iff µw (JϕKMw ) ≥ µw (JψKMw ),


The definition of consequence is standard.



System FP

Ex (2(ϕ↔ ϕ′) ∧2(ψ↔ ψ′))→ ((ϕ > ψ)↔ (ϕ′ > ψ′))

Bot ϕ > ⊥

BT ¬(⊥ > >)

Tot (ϕ > ψ) ∨ (ψ > ϕ)

FC ϕ1 . . . ϕnEψ1 . . . ψn → ((∧i<n

(ϕi > ψi ))→ (ψn > ϕn))

Theorem (Segerberg 1971, Gardenfors 1975)

FP is sound and complete with respect to the class of probabilisticKripke models for L(2,6).

FP also has the finite model property with respect to that class.



Outline

1. Basic Notions X






5. Segerberg and Gardenfors’ Completeness Theorem X




Bonus 1: Imprecise Representability (by a set of measures)

Let A be an algebra on a nonempty set W , e.g., A is ℘(W ).

Definition (Agreement)

Given a binary relation % on A and a set P of probabilitymeasures µ : A → [0, 1], P agrees with % iff for all X ,Y ∈ A:

X % Y ⇔ ∀µ ∈ P : µ(X ) ≥ µ(Y ).

We also say that P imprecisely represents %.

Definition (Representability)

A binary relation % on ℘(W ) is imprecisely representable iff it isimprecisely represented by a set of probability measures on ℘(W ).

QUESTION: what axioms on % are necessary and sufficient forimprecise representability?



Generalized Finite Cancellation

Recall the Finite Cancellation (FC ) axiom: for any balancedsequences A1, . . . ,An and B1, . . . ,Bn of subsets of W ,


Compare that to the Generalized Finite Cancellation (GFC ) axiom:

if A1, . . . ,An−1,An, . . . ,An︸︷︷︸r times

and B1, . . . ,Bn−1,Bn, . . . ,Bn︸︷︷︸r times

(r ≥ 1)

are balanced sequences of subsets of W , then




Representation Theorem

The Generalized Finite Cancellation (GFC ) axiom:

if A1, . . . ,An−1,An, . . . ,An︸︷︷︸r times

and B1, . . . ,Bn−1,Bn, . . . ,Bn︸︷︷︸r times

(r ≥ 1)

are balanced sequences of subsets of W , then


Theorem (Rios Insua 1992, Alon and Lehrer 2014)

For any finite set W , a binary relation % on ℘(W ) is impreciselyrepresentable iff it satisfies reflexivity, nontriviality, nonnegativity,and GFC .



System FG


ψ

Necϕ2ϕ

K 2(ϕ→ ψ)→ (2ϕ→ 2ψ)

Ex (2(ϕ↔ ϕ′) ∧2(ψ↔ ψ′))→ ((ϕ > ψ)↔ (ϕ′ > ψ′))

Bot ϕ > ⊥

BT ¬(⊥ > >)

GFC ϕ1 . . . ϕn−1 ϕn . . . ϕn︸︷︷︸r times

Eψ1 . . . ψn−1 ψn . . . ψn︸︷︷︸r times

→

((∧i<n

(ϕi > ψi ))→ (ψn > ϕn))



Soundness and Completeness

Bot ϕ > ⊥

BT ¬(⊥ > >)


Eψ1 . . . ψn−1 ψn . . . ψn︸︷︷︸r times

→

((∧i<n

(ϕi > ψi ))→ (ψn > ϕn))

Theorem (Alon and Heifetz 2014)

The system FG is sound and complete with respect to the class ofQP Kripke models in which every %w is imprecisely representable.



Sets-of-Measures Kripke Models

Definition (Sets-of-Measures Kripke Model)

A sets-of-measures Kripke model is a tupleM = 〈W ,R, {Pw}w∈W ,V 〉 such that:



3. Pw is a set of functions µw : ℘(R(w))→ [0, 1] such thatµw (R(w)) = 1 and if A∩ B = ∅, thenµw (A∪ B) = µw (A) + µw (B);

4. V : At→ ℘(W ).

.



Semantics

Definition (Truth and Consequence)

Where M is a set-of-measures Kripke model and ϕ ∈ L(2,>), wedefine M,w � ϕ by (with other clauses as before):

I M,w � ϕ > ψ iff for all µw ∈ Pw : µw (JϕKMw ) ≥ µw (JϕKMw ),


The definition of consequence is standard.




Bot ϕ > ⊥

BT ¬(⊥ > >)


Eψ1 . . . ψn−1 ψn . . . ψn︸︷︷︸r times

→

((∧i<n

(ϕi > ψi ))→ (ψn > ϕn))

Theorem (Alon and Heifetz 2014)

The system FG is sound and complete with respect to the class ofsets-of-measures Kripke models.



Outline

1. Basic Notions X







6. Bonus 1: Imprecise Representability X



Bonus 2: Extending an Ordering on a Set to the Powerset

Toward Natural Language

Consider the English locution ‘at least as likely as’, as in

(1) It is at least as likely that our visitor is coming in on AmericanAirlines as it is that he is coming on Continental Airlines.

What does this mean? Specifically, what is its logic?

Some entailments are clear. For instance, (1) follows from (2):

(2) American is at least as likely as Continental or Delta.

What else? How might we interpret such talk model-theoretically?



What is the relation between ordinary talk using ‘probably’ and ‘atleast as likely as’ and the mathematical theory of probability?

Is Kolmogorovian probability implicated in their semantics?

Hamblin (1959, 234): “Metrical probability theory iswell-established, scientifically important and, in essentials, beyondlogical reproof. But when, for example, we say ‘It’s probably goingto rain’, or ‘I shall probably be in the library this afternoon’, arewe, even vaguely, using the metrical probability concept?”

Kratzer (2012, 25): “Our semantic knowledge alone does not giveus the precise quantitative notions of probability and desirabilitythat mathematicians and scientists work with.”



Kratzer’s Semantics

Definition (World-Ordering Model)

A (total) world-ordering model is a tupleM = 〈W ,R, {�w | w ∈ W },V 〉:I W is a non-empty set;

I R is a (serial) binary relation on W ; R(w) = {v ∈ W | wRv};I For each w ∈ W , �w is a (total) preorder on R(w);

I V : At→ ℘(W ) is a valuation function.

Following Lewis, we can lift �w to a relation �lw on ℘(W ):

A �lw B iff ∀b ∈ Bw ∃a ∈ Aw : a �w b,

where Xw = X ∩ R(w).




A �lw B iff ∀b ∈ Bw ∃a ∈ Aw : a �w b.

Definition (Truth)

Given a world-ordering pointed model M,w and ϕ ∈ L(2,>), wedefine M,w � ϕ and JϕKM = {v ∈ W |M, v � ϕ} as follows:

M,w � p iff w ∈ V (p);

M,w � ¬ϕ iff M,w 2 ϕ;

M,w � ϕ ∧ ψ iff M,w � ϕ and M,w � ψ;

M,w � 2ϕ iff ∀v ∈ R(w) : M, v � ϕ;

M,w � ϕ > ψ iff JϕKM �lw JψKM.



As pointed out by Yalcin (2010) and Lassiter (2010), Kratzer’sapproach validates some rather dubious patterns. For instance, itpredicts that (3) should follow from (1) and (2):

(1) American is at least as likely as Continental.

(2) American is at least as likely as Delta.

(3) American is at least as likely as Continental or Delta.

It also fails to validate some intuitively obvious patterns.





A (total) world-ordering model M = 〈W ,R, {�w | w ∈ W },V 〉has for each w ∈ W a (total) preorder �w on R(w).



Kratzer gives the truth clause for > using the lifted relation �lw .

Definition (Truth)

Given a pointed world-ordering model M,w and formula ϕ, wedefine M,w �l ϕ as follows (with the other clauses as before):

M,w �l ϕ > ψ iff JϕKM �lw JψKM.




V1 4ϕ→ ¬4¬ϕ

V2 4(ϕ ∧ ψ)→ (4ϕ ∧4ψ) V3 4ϕ→ 4(ϕ ∨ ψ)

V4 ϕ > ⊥ V5 > > ϕ

V6 2ϕ→ 4ϕ V7 4ϕ→ 3ϕ

V11 (ψ > ϕ)→ (4ϕ→ 4ψ)

V12 (ψ > ϕ)→ ((ϕ > ¬ϕ)→ (ψ > ¬ψ))

I1 ((ϕ > ψ) ∧ (ϕ > χ))→ (ϕ > (ψ ∨ χ))

I2 (ϕ > ¬ϕ)→ (ϕ > ψ)

I3 4ϕ→ (ϕ > ψ)

E1 (4ϕ ∧4ψ)→ 4(ϕ ∧ ψ)




FactV1-V10 and V12 are all valid over world-ordering models accordingto Kratzer’s semantics; V11 is not valid; I1-13 are all valid. X



Systems W, F, WJ, and FJ

System W is the minimal modal logic K plus:

Ex (2(ϕ↔ ϕ′) ∧2(ψ↔ ψ′))→ ((ϕ > ψ)↔ (ϕ′ > ψ′))

Bot ϕ > ⊥ BT ¬(⊥ > >)

Tran (ϕ > ψ)→ ((ψ > χ)→ (ϕ > χ))

Mon 2(ϕ→ ψ)→ (ϕ > ψ)

System F is W plus:

Tot (ϕ > ψ) ∨ (ψ > ϕ)

System WJ (resp. FJ) is W (resp. F) plus:

J ((ϕ > ψ) ∧ (ϕ > χ))→ (ϕ > (ψ ∨ χ))



Kratzer’s Semntics

Theorem (Axiomatization of Kratzer’s Semantics)

1. WJ is sound and complete with respect to the class ofworld-ordering models with Lewis’s lifting.

2. FJ is sound and complete with respect to the class of totalworld-ordering models with Lewis’s lifting.









Definition (Truth)









A �lw B iff ∃ function f : Bw → Aw s.th. ∀x ∈ Bw : f (x) �w x .


Definition (Truth)





An Alternative Lifting



Here is another way to lift �w to a relation �↑w on ℘(W ):

A �↑w B iff ∃ injection f : Bw → Aw s.th. ∀x ∈ Bw : f (x) �w x .

Definition (Truth)

Given a pointed world-ordering model M,w and formula ϕ, wedefine M,w �↑ ϕ as follows (with the other clauses as before):

M,w �↑ ϕ > ψ iff JϕKM �↑w JψKM.



An Alternatively Lifting

Given a � b � c � d , consider the liftings:

abcd 'l abc 'l abd 'l acd 'l ab 'l ac 'l ad 'l a �l

bcd 'l bc 'l bd 'l b �l cd 'l c �l d �l ∅

abcd �↑ abc �↑ abd �↑ acd �↑ bcd�↑

�↑

�↑

ab �↑ ac �↑ bc�↑

�↑

ad �↑ bd �↑ cd�↑

�↑

�↑

a �↑ b �↑ c �↑ d �↑ ∅

Figure: Comparison of Lewis’s lifting �l and the new lifting �↑



System FG


ψ

Necϕ2ϕ

K 2(ϕ→ ψ)→ (2ϕ→ 2ψ)

Ex (2(ϕ↔ ϕ′) ∧2(ψ↔ ψ′))→ ((ϕ > ψ)↔ (ϕ′ > ψ′))

Bot ϕ > ⊥

BT ¬(⊥ > >)


Eψ1 . . . ψn−1 ψn . . . ψn︸︷︷︸r times

→

((∧i<n

(ϕi > ψi ))→ (ψn > ϕn))




Here is a better way to lift �w to a relation �↑w on ℘(W ):

A �↑ B iff ∃ injective f : B → A s.th. ∀x ∈ B : f (x) � x

Proposition (Harrison-Trainor, Holliday, and Icard)FG is sound and complete with respect to the class of path-finite1

world-ordering models with the ↑ lifting.

Moral: simply changing Kratzer’s semantics by requiring that thefunction be injective yields the same logic of ‘at least as likely as’as a semantics based on sets of probability measures.

1I.e., there is no infinite path x1 �w x2 �w x3 . . . with xi 6= xj for i 6= j .




Here is a better way to lift �w to a relation �↑w on ℘(W ):

A �↑ B iff ∃ injective f : B → A s.th. ∀x ∈ B : f (x) � x

Proposition (Harrison-Trainor, Holliday, and Icard)FG is sound and complete with respect to the class of path-finite2

world-ordering models with the ↑ lifting.

Moral: all of Yalcin’s intuitively validities are validated; none of hisintuitive invalidities are validated; thus, the semantics avoids theentailment problems raised for Kratzer’s semantics.

2I.e., there is no infinite path x1 �w x2 �w x3 . . . with xi 6= xj for i 6= j .



Outline

1. Basic Notions X







6. Bonus 1: Imprecise Representability X

7. Bonus 2: Extending an Ordering on a Set to the Powerset X



Thank you!


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