comparative probability.15in - esslli 2014 - logic and...
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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
Wes Holliday and Thomas Icard: Comparative Probability 2
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.
Wes Holliday and Thomas Icard: Comparative Probability 3
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.
Wes Holliday and Thomas Icard: Comparative Probability 4
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.
Wes Holliday and Thomas Icard: Comparative Probability 5
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.
Wes Holliday and Thomas Icard: Comparative Probability 6
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: 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.
Wes Holliday and Thomas Icard: Comparative Probability 7
Basic Notions
Non-strict as basic:
A % B – “A is 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.“A and B are equally probable.”
Strict as basic:
A > B – “A is more probable than B.”
Define: A ≈ B iff A 6> B and B 6> A;
Define: A ' B either A > B or A ≈ 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. . .
Wes Holliday and Thomas Icard: Comparative Probability 8
Basic Notions
Non-strict as basic:
A % B – “A is 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.“A and B are equally probable.”
Strict as basic:
A > B – “A is more probable than B.”
Define: A ≈ B iff A 6> B and B 6> A;
Define: A ' B either A > B or A ≈ B.
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.
Wes Holliday and Thomas Icard: Comparative Probability 9
Basic Notions
Non-strict as basic:
A % B – “A is 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.“A and B are equally probable.”
Strict as basic:
A > B – “A is more probable than B.”
Define: A ≈ B iff A 6> B and B 6> A;
Define: A ' B either A > B or A ≈ B.
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.
Wes Holliday and Thomas Icard: Comparative Probability 10
Basic Notions
Outline
1. Basic Notions X
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
Wes Holliday and Thomas Icard: Comparative Probability 11
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 �.)
Wes Holliday and Thomas Icard: Comparative Probability 12
Considering Axioms: Rationality and Realism
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 .
Wes Holliday and Thomas Icard: Comparative Probability 13
Considering Axioms: Rationality and Realism
de Finetti’s Axioms
de Finetti’s proposed the following axioms for %:
I nonnegativity: A % ∅;
I nontriviality: ∅ 6%W ;
I totality: A % B of B % A;
I transitivity: if A % B and B % C , then A % C ;
I quasi-additivity: if (A∪ B) ∩ C = ∅, thenA % B ⇔ A∪ C % B ∪ C .
Wes Holliday and Thomas Icard: Comparative Probability 14
Considering Axioms: Rationality and Realism
de Finetti’s Axioms
de Finetti’s proposed that % is a total preorder satisfying:
I nonnegativity: A % ∅;
I nontriviality: ∅ 6%W ;
I quasi-additivity: if (A∪ B) ∩ C = ∅, thenA % B ⇔ A∪ C % B ∪ C .
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.
Wes Holliday and Thomas Icard: Comparative Probability 15
Considering Axioms: Rationality and Realism
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?
Wes Holliday and Thomas Icard: Comparative Probability 16
Considering Axioms: Rationality and Realism
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.
Wes Holliday and Thomas Icard: Comparative Probability 17
Considering Axioms: Rationality and Realism
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.
Wes Holliday and Thomas Icard: Comparative Probability 18
Considering Axioms: Rationality and Realism
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?
Wes Holliday and Thomas Icard: Comparative Probability 19
Considering Axioms: Rationality and Realism
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.
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?
Wes Holliday and Thomas Icard: Comparative Probability 20
Considering Axioms: Rationality and Realism
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.
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.
Wes Holliday and Thomas Icard: Comparative Probability 21
Considering Axioms: Rationality and Realism
de Finetti’s Axioms
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:
I nonnegativity: A % ∅;
I nontriviality: ∅ 6%W ;
I quasi-additivity: if (A∪ B) ∩ C = ∅, thenA % B ⇔ A∪ C % B ∪ C .
Wes Holliday and Thomas Icard: Comparative Probability 22
Considering Axioms: Rationality and Realism
Outline
1. Basic Notions X
2. Considering Axioms: Rationality and Realism X
• de Finetti’s Axioms X
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
Wes Holliday and Thomas Icard: Comparative Probability 23
Comparative Probability as Modal Logic
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. . .
Wes Holliday and Thomas Icard: Comparative Probability 24
Comparative Probability as Modal Logic
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).
Wes Holliday and Thomas Icard: Comparative Probability 25
Comparative Probability as Modal Logic
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.
Wes Holliday and Thomas Icard: Comparative Probability 26
Comparative Probability as Modal Logic
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}.
Wes Holliday and Thomas Icard: Comparative Probability 27
Comparative Probability as Modal Logic
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)
Wes Holliday and Thomas Icard: Comparative Probability 28
Comparative Probability as Modal Logic
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)
Wes Holliday and Thomas Icard: Comparative Probability 29
Comparative Probability as Modal Logic
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)
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).
Wes Holliday and Thomas Icard: Comparative Probability 30
Comparative Probability as Modal Logic
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)
Wes Holliday and Thomas Icard: Comparative Probability 31
Comparative Probability as Modal Logic
de Finetti’s Axioms
de Finetti’s proposed the following axioms for %:
I nonnegativity: A % ∅;
I nontriviality: ∅ 6%W ;
I totality: A % B of B % A;
I transitivity: if A % B and B % C , then A % C ;
I quasi-additivity: if (A∪ B) ∩ C = ∅, thenA % B ⇔ A∪ C % B ∪ C .
Wes Holliday and Thomas Icard: Comparative Probability 32
Comparative Probability as Modal Logic
de Finetti’s System FA
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 ¬(⊥ > >)
Tot (ϕ > ψ) ∨ (ψ > ϕ)
Tran (ϕ > ψ)→ ((ψ > χ)→ (ϕ > χ))
A ¬3((ϕ ∨ ψ) ∧ χ)→ (ϕ > ψ↔ ((ϕ ∨ χ) > (ψ ∨ χ)))
Wes Holliday and Thomas Icard: Comparative Probability 33
Comparative Probability as Modal Logic
Outline
1. Basic Notions X
2. Considering Axioms: Rationality and Realism X
• de Finetti’s Axioms X
3. Comparative Probability as Modal Logic X
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
Wes Holliday and Thomas Icard: Comparative Probability 34
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?
Wes Holliday and Thomas Icard: Comparative Probability 35
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.
Wes Holliday and Thomas Icard: Comparative Probability 36
Probabilistic Representability – The KPS Counterexample
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).
Wes Holliday and Thomas Icard: Comparative Probability 37
Probabilistic Representability – The KPS Counterexample
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).
Wes Holliday and Thomas Icard: Comparative Probability 38
Probabilistic Representability – The KPS Counterexample
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(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}.
Wes Holliday and Thomas Icard: Comparative Probability 39
Probabilistic Representability – The KPS Counterexample
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. 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.
Wes Holliday and Thomas Icard: Comparative Probability 40
Probabilistic Representability – The KPS Counterexample
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 some 0 < ε < 1/3, define
ν(a) = 4− ε ν(b) = 1− ε ν(c) = 3− ε ν(d) = 2 ν(e) = 6.
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. 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
Probabilistic Representability – The KPS Counterexample
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}.
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.
Wes Holliday and Thomas Icard: Comparative Probability 42
Probabilistic Representability – The KPS Counterexample
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. 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. 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!
Wes Holliday and Thomas Icard: Comparative Probability 43
Probabilistic Representability – The KPS Counterexample
We have shown:
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.
Wes Holliday and Thomas Icard: Comparative Probability 44
Probabilistic Representability – The KPS Counterexample
The KPS ordering is such that:
{a} � {b, c} {c , d} � {a, b} {b, e} � {a, c} {a, b, c} � {d , e}.
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?”
Wes Holliday and Thomas Icard: Comparative Probability 45
Probabilistic Representability – The KPS Counterexample
Outline
1. Basic Notions X
2. Considering Axioms: Rationality and Realism X
• de Finetti’s Axioms X
3. Comparative Probability as Modal Logic X
4. Probabilistic Representability
• The Kraft-Pratt-Seidenberg Counterexample X• 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
Wes Holliday and Thomas Icard: Comparative Probability 46
Probabilistic Representability – The KPS Counterexample
We have shown:
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.
QUESTION: what further conditions on % must we assume forprobabilistic representability?
Wes Holliday and Thomas Icard: Comparative Probability 47
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.
Wes Holliday and Thomas Icard: Comparative Probability 48
Probabilistic Representability – Cancellation Axioms and Scott’s Theorem
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.
Wes Holliday and Thomas Icard: Comparative Probability 49
Probabilistic Representability – Cancellation Axioms and Scott’s Theorem
That pragmatic argument does not exactly work with FC : for anybalanced sequences A1, . . . ,An and B1, . . . ,Bn of subsets of W ,
if Ai % Bi for all i < n, then Bn % An.
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?
Wes Holliday and Thomas Icard: Comparative Probability 50
Probabilistic Representability – Cancellation Axioms and Scott’s Theorem
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.
Proposition (Finite Cancellation)
If a relation % on ℘(W ) is probabilistically representable, then %satisfies FC .
Proof. On next slide. . .
Wes Holliday and Thomas Icard: Comparative Probability 51
Probabilistic Representability – Cancellation Axioms and Scott’s Theorem
Proposition (Finite Cancellation)
If a relation % on ℘(W ) is prob. representable, then for anybalanced sequences A1, . . . ,An and B1, . . . ,Bn of subsets of W ,
if Ai % Bi for all i < n, then Bn % An.
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.
Wes Holliday and Thomas Icard: Comparative Probability 52
Probabilistic Representability – Cancellation Axioms and Scott’s Theorem
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.
Wes Holliday and Thomas Icard: Comparative Probability 53
Probabilistic Representability – Cancellation Axioms and Scott’s Theorem
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. 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.
Wes Holliday and Thomas Icard: Comparative Probability 54
Probabilistic Representability – Cancellation Axioms and Scott’s Theorem
KPS and Finite Cancellation
Recall that the KPS ordering had the following inequalities:
{a} � {b, c} {c , d} � {a, b} {b, e} � {a, c} {a, b, c} � {d , e}.
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.
Wes Holliday and Thomas Icard: Comparative Probability 55
Probabilistic Representability – Cancellation Axioms and Scott’s Theorem
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.
Wes Holliday and Thomas Icard: Comparative Probability 56
Probabilistic Representability – Cancellation Axioms and Scott’s Theorem
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.
Wes Holliday and Thomas Icard: Comparative Probability 57
Probabilistic Representability – Cancellation Axioms and Scott’s Theorem
Outline
1. Basic Notions X
2. Considering Axioms: Rationality and Realism X
• de Finetti’s Axioms X
3. Comparative Probability as Modal Logic X
4. Probabilistic Representability X
• The Kraft-Pratt-Seidenberg Counterexample X• Cancellation Axioms, Scott’s Representation Theorem X
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
Wes Holliday and Thomas Icard: Comparative Probability 58
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).
Wes Holliday and Thomas Icard: Comparative Probability 59
Segerberg and Gardenfors’ Completeness Theorem
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.
Wes Holliday and Thomas Icard: Comparative Probability 60
Segerberg and Gardenfors’ Completeness Theorem
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))
Wes Holliday and Thomas Icard: Comparative Probability 61
Segerberg and Gardenfors’ Completeness Theorem
System FP
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 ¬(⊥ > >)
Tot (ϕ > ψ) ∨ (ψ > ϕ)
FC ϕ1 . . . ϕnEψ1 . . . ψn → ((∧i<n
(ϕi > ψi ))→ (ψn > ϕn))
Wes Holliday and Thomas Icard: Comparative Probability 62
Segerberg and Gardenfors’ Completeness Theorem
System FA
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 ¬(⊥ > >)
Tot (ϕ > ψ) ∨ (ψ > ϕ)
Tran (ϕ > ψ)→ ((ψ > χ)→ (ϕ > χ))
A ¬3((ϕ ∨ ψ) ∧ χ)→ (ϕ > ψ↔ ((ϕ ∨ χ) > (ψ ∨ χ)))
Wes Holliday and Thomas Icard: Comparative Probability 63
Segerberg and Gardenfors’ Completeness Theorem
System FP
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 ¬(⊥ > >)
Tot (ϕ > ψ) ∨ (ψ > ϕ)
FC ϕ1 . . . ϕnEψ1 . . . ψn → ((∧i<n
(ϕi > ψi ))→ (ψn > ϕn))
Wes Holliday and Thomas Icard: Comparative Probability 64
Segerberg and Gardenfors’ Completeness Theorem
Comparing FA and FP
Recall that earlier we showed:
FactA relation % on ℘(W ) satisfies FC≤3 iff % is transitive andquasi-additive.
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.
Wes Holliday and Thomas Icard: Comparative Probability 65
Segerberg and Gardenfors’ Completeness Theorem
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.
Wes Holliday and Thomas Icard: Comparative Probability 66
Segerberg and Gardenfors’ Completeness Theorem
Review: Probabilistic Kripke Models
Definition (Probabilistic Kripke Model)
A probabilistic Kripke model is a tupleM = 〈W ,R, {µw}w∈W ,V 〉 such that:
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. 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 ).
.
Wes Holliday and Thomas Icard: Comparative Probability 67
Segerberg and Gardenfors’ Completeness Theorem
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 ),
where JχKMw = {v ∈ R(w) | M, v � χ}.
The definition of consequence is standard.
Wes Holliday and Thomas Icard: Comparative Probability 68
Segerberg and Gardenfors’ Completeness Theorem
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.
Wes Holliday and Thomas Icard: Comparative Probability 69
Segerberg and Gardenfors’ Completeness Theorem
Outline
1. Basic Notions X
2. Considering Axioms: Rationality and Realism X
• de Finetti’s Axioms X
3. Comparative Probability as Modal Logic X
4. Probabilistic Representability X
• The Kraft-Pratt-Seidenberg Counterexample X• Cancellation Axioms, Scott’s Representation Theorem X
5. Segerberg and Gardenfors’ Completeness Theorem X
6. Bonus 1: Imprecise Representability (by a set of measures)
7. Bonus 2: Extending an Ordering on a Set to the Powerset
Wes Holliday and Thomas Icard: Comparative Probability 70
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?
Wes Holliday and Thomas Icard: Comparative Probability 71
Bonus 1: Imprecise Representability (by a set of measures)
Generalized Finite Cancellation
Recall 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.
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
if Ai % Bi for all i < n, then Bn % An.
Wes Holliday and Thomas Icard: Comparative Probability 72
Bonus 1: Imprecise Representability (by a set of measures)
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
if Ai % Bi for all i < n, then Bn % An.
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 .
Wes Holliday and Thomas Icard: Comparative Probability 73
Bonus 1: Imprecise Representability (by a set of measures)
System FG
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 ¬(⊥ > >)
GFC ϕ1 . . . ϕn−1 ϕn . . . ϕn︸ ︷︷ ︸r times
Eψ1 . . . ψn−1 ψn . . . ψn︸ ︷︷ ︸r times
→
((∧i<n
(ϕi > ψi ))→ (ψn > ϕn))
Wes Holliday and Thomas Icard: Comparative Probability 74
Bonus 1: Imprecise Representability (by a set of measures)
Soundness and Completeness
Bot ϕ > ⊥
BT ¬(⊥ > >)
GFC ϕ1 . . . ϕn−1 ϕn . . . ϕn︸ ︷︷ ︸r times
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.
Wes Holliday and Thomas Icard: Comparative Probability 75
Bonus 1: Imprecise Representability (by a set of measures)
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:
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. 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 ).
.
Wes Holliday and Thomas Icard: Comparative Probability 76
Bonus 1: Imprecise Representability (by a set of measures)
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 ),
where JχKMw = {v ∈ R(w) | M, v � χ}.
The definition of consequence is standard.
Wes Holliday and Thomas Icard: Comparative Probability 77
Bonus 1: Imprecise Representability (by a set of measures)
Soundness and Completeness
Bot ϕ > ⊥
BT ¬(⊥ > >)
GFC ϕ1 . . . ϕn−1 ϕn . . . ϕn︸ ︷︷ ︸r times
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.
Wes Holliday and Thomas Icard: Comparative Probability 78
Bonus 1: Imprecise Representability (by a set of measures)
Outline
1. Basic Notions X
2. Considering Axioms: Rationality and Realism X
• de Finetti’s Axioms X
3. Comparative Probability as Modal Logic X
4. Probabilistic Representability X
• The Kraft-Pratt-Seidenberg Counterexample X• Cancellation Axioms, Scott’s Representation Theorem X
5. Segerberg and Gardenfors’ Completeness Theorem X
6. Bonus 1: Imprecise Representability X
7. Bonus 2: Extending an Ordering on a Set to the Powerset
Wes Holliday and Thomas Icard: Comparative Probability 79
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?
Wes Holliday and Thomas Icard: Comparative Probability 80
Bonus 2: Extending an Ordering on a Set to the Powerset
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.”
Wes Holliday and Thomas Icard: Comparative Probability 81
Bonus 2: Extending an Ordering on a Set to the Powerset
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).
Wes Holliday and Thomas Icard: Comparative Probability 82
Bonus 2: Extending an Ordering on a Set to the Powerset
Following Lewis, we can lift �w to a relation �lw on ℘(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.
Wes Holliday and Thomas Icard: Comparative Probability 83
Bonus 2: Extending an Ordering on a Set to the Powerset
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.
Wes Holliday and Thomas Icard: Comparative Probability 84
Bonus 2: Extending an Ordering on a Set to the Powerset
Kratzer’s Semantics
Definition (World-Ordering Model)
A (total) world-ordering model M = 〈W ,R, {�w | w ∈ W },V 〉has for each w ∈ W a (total) preorder �w on R(w).
Following Lewis, we can lift �w to a relation �lw on ℘(W ):
A �lw B iff ∀b ∈ Bw ∃a ∈ Aw : a �w b.
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.
Wes Holliday and Thomas Icard: Comparative Probability 85
Bonus 2: Extending an Ordering on a Set to the Powerset
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)
Wes Holliday and Thomas Icard: Comparative Probability 86
Bonus 2: Extending an Ordering on a Set to the Powerset
Kratzer’s Semantics
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
Wes Holliday and Thomas Icard: Comparative Probability 87
Bonus 2: Extending an Ordering on a Set to the Powerset
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))
Wes Holliday and Thomas Icard: Comparative Probability 88
Bonus 2: Extending an Ordering on a Set to the Powerset
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.
Wes Holliday and Thomas Icard: Comparative Probability 89
Bonus 2: Extending an Ordering on a Set to the Powerset
Kratzer’s Semantics
Definition (World-Ordering Model)
A (total) world-ordering model M = 〈W ,R, {�w | w ∈ W },V 〉has for each w ∈ W a (total) preorder �w on R(w).
Following Lewis, we can lift �w to a relation �lw on ℘(W ):
A �lw B iff ∀b ∈ Bw ∃a ∈ Aw : a �w b.
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.
Wes Holliday and Thomas Icard: Comparative Probability 90
Bonus 2: Extending an Ordering on a Set to the Powerset
Kratzer’s Semantics
Definition (World-Ordering Model)
A (total) world-ordering model M = 〈W ,R, {�w | w ∈ W },V 〉has for each w ∈ W a (total) preorder �w on R(w).
Following Lewis, we can lift �w to a relation �lw on ℘(W ):
A �lw B iff ∃ function f : Bw → Aw s.th. ∀x ∈ Bw : f (x) �w x .
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.
Wes Holliday and Thomas Icard: Comparative Probability 91
Bonus 2: Extending an Ordering on a Set to the Powerset
An Alternative Lifting
Definition (World-Ordering Model)
A (total) world-ordering model M = 〈W ,R, {�w | w ∈ W },V 〉has for each w ∈ W a (total) preorder �w on R(w).
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.
Wes Holliday and Thomas Icard: Comparative Probability 92
Bonus 2: Extending an Ordering on a Set to the Powerset
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 �↑
Wes Holliday and Thomas Icard: Comparative Probability 93
Bonus 2: Extending an Ordering on a Set to the Powerset
System FG
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 ¬(⊥ > >)
GFC ϕ1 . . . ϕn−1 ϕn . . . ϕn︸ ︷︷ ︸r times
Eψ1 . . . ψn−1 ψn . . . ψn︸ ︷︷ ︸r times
→
((∧i<n
(ϕi > ψi ))→ (ψn > ϕn))
Wes Holliday and Thomas Icard: Comparative Probability 94
Bonus 2: Extending an Ordering on a Set to the Powerset
Soundness and Completeness
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 .
Wes Holliday and Thomas Icard: Comparative Probability 95
Bonus 2: Extending an Ordering on a Set to the Powerset
Soundness and Completeness
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 .
Wes Holliday and Thomas Icard: Comparative Probability 96
Bonus 2: Extending an Ordering on a Set to the Powerset
Outline
1. Basic Notions X
2. Considering Axioms: Rationality and Realism X
• de Finetti’s Axioms X
3. Comparative Probability as Modal Logic X
4. Probabilistic Representability X
• The Kraft-Pratt-Seidenberg Counterexample X• Cancellation Axioms, Scott’s Representation Theorem X
5. Segerberg and Gardenfors’ Completeness Theorem X
6. Bonus 1: Imprecise Representability X
7. Bonus 2: Extending an Ordering on a Set to the Powerset X
Wes Holliday and Thomas Icard: Comparative Probability 97
Bonus 2: Extending an Ordering on a Set to the Powerset
Thank you!
Wes Holliday and Thomas Icard: Comparative Probability 98