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Proto-Probability

A Qualitative Part of Conditional Probability

Düsseldorf, May 2011

David Makinson, LSE

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Story Line

• Concerns the logic of uncertain inference• In particular, of qualitatively formulated but

probabilistically sound inference • The qualitative approach leads us to generalize

usual concepts of conditional probability (ratio/unit, Hosiasson-Lindenbaum, Popper, van Fraassen,...)

• We call it proto-probability, outline some basic concepts, examples, comparisons

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Compare with Jim's Talk

Inter alia, Jim looked at a generalization of conditional probability, which is:

Qualitative: no numbersNon-linear: ordering of values need not be complete (elements may be incomparable)

I look at a generalization of conditional probability that is:

Qualitative: no numbersNon-linear: Order of values need not be antisymmetric, but still complete

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I Hawthorne's System QII Kinds of Conditional ProbabilityIII Enter Proto-ProbabilityIV ComparisonsV Take-Home Message

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‘Nonmonotonic’ Logics

A number of qualitative logics failing monotony have been developed for uncertain reasoning

Notably preferential consequence, also default consequence relations (nonmonotonic logics)

But they cannot be interpreted straightforwardly in terms of probability

Because they satisfy several Horn rules that are not probabilistically sound

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Probabilistic Soundness

Definition used here

For each prob. function p: L→ [0,1] and threshold t ∈ [0,1] we define a relation |~p,t

a |~p,t x iff p(x|a) ≥ t

A rule for |~ (Horn or not) is prob. sound iff it holds for every choice of relation |~p,t

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Examples

The most central failure is the rule of conjunction in the conclusion (∧+) Whenever a ~ x and a ~ y then a ~ x∧y

Others includeDisjunction in the premises (∨+) Whenever a ~ x and b ~ x then a∨b ~ x

Cumulative transitivity (cut)Whenever a ~ x and a∧x ~ y then a ~ y

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System Q

In 1996, Jim Hawthorne formulated a system Q for probabilistically sound qualitative inference

Three groups of axioms:

1-premise Horn rules (as for preferential logics)

2-premise Horn rules (weaken ∧+ and ∨+)

A non-Horn rule (∨−) that does not hold for all preferential logics (and not for default logic)

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One-premise Horn rules of Q(three structural, one for ∧)

a |~ aReflexivity

Whenever a |~ x and x├ y, then a |~ yRight weakening

Whenever a |~ x and a ╫ b, then b |~ xLeft classical equivalence

Whenever a |~ x∧y, then a∧x |~ yVery cautious monotony

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Two-Premise Horn Rules for Q

Exclusive ∨+Whenever a |~ x, b |~ x, a ├ ¬b then a∨b |~ x

Weak ∧+ Whenever a |~ x and a∧¬y |~ y then a |~ x∧y

The underlining indicates where they weaken the versions that hold for preferential (and classical) consequence

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Almost-Horn Rule for Q

Exclusive ∨− (negation rationality) When a∨b |~ x and a ├ ¬b, either a |~ x or b |~ x

EquivalentlyWhen a∨b |~ x and a ├ ¬b and a |/~ x then b |~ x

Fails for some preferential consequence relationsLike a converse (but not quite) of exclusive ∨+Not a Horn rule

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Sound? Yes Complete? Open

Fact: All the postulates of Hawthorne's system Q are probabilistically sound

Open question: Is the system complete for finite-premise Horn rules? That is: Is every prob. sound finite-prem. Horn rule derivable in the system?

Answer not known!

But negative for countable premise Horn rules.

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I Hawthorne's System QII Kinds of Conditional ProbabilityIII Enter Proto-ProbabilityIV ComparisonsV Take-Home Message

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Broad Categories

The soundness theorem for Q holds wrt all of the usual kinds, so need not really distinguish hereBut just as a reminder, two broad categories:

Defined from unconditional– Familiar ratio (ratio unit)Irreducibly conditional (two-place)–Three main classes

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Irreducibly Conditional

Three main classesHosiasson-Lindenbaum 1940 (narrowest)Popper/Rényi 1959 (broader)van Fraassen 1976/1995 (broadest by singleton)

So soundness of system Q over vF class implies same for the others

Reminder: Using ideas of Rényi, can give vF a very transparent and elegant axiomatization

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System of van Fraassen

Functions p: L2 → [0,1] such that:

(vF1) p(x,a) = p(x,b) whenever a ≈ b

(vF2) For each a, pa is either a 1-place Kolmogorov function with pa(a) = 1, or else is the unit function i.e. pa(x) = 1 for all x

(vF3) p(x∧y,a) = p(x,a)⋅p(y,a∧x) for all a,x,y

Comment: VF2 even simpler to state if we widen dfn of Kolmogorov fn to include the 1-place unit fn

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I Hawthorne's System QII Kinds of Conditional ProbabilityIII Enter Proto-ProbabilityIV ComparisonsV Take-Home Message

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How Much is Needed?

Feeling: To verify the soundness of Hawthorne's system Q wrt conditional prob fns, it seems that we don’t need the full power even of van Fraassen axioms

But if we just drop one of the three vF axioms, we allow functions for which Q is not sound

Question: How much of vF cndl prob do we really need to validate system Q of uncertain inference?

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Modest Goal

Formulate a qualitative notion of probability that is broader than even the van Fraassen functions, yet

Is restrained enough to ensure that Hawthorne's system Q is still sound wrt it, and

Broad enough to allow a representation theorem for Q

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Domain and Range of Proto-Probability Functions

Domain: L2 Range: Any non-empty set D equipped with a relation ≤ that is transitive and complete (d ≤ e or e ≤ d, for all d,e ∈ D) with a greatest element 1 and a least element 0Comment: Don’t need anti-symmetry (so not nec. linear) but do require trans and complete (ranking) Conditions on functions: A proto-probability function is any p: L2→ D satisfying following six conditions...

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Five Conditions (all but ∨)

Hawthorne's Q

a |~ a

If a |~ x then a |~ y provided x├ y

If a |~ x then b |~ x provided a ≈ b

If a |~ x∧y then a∧x |~ y

If a |~ x then a |~ x∧y provided a∧¬y |~ y

Proto-Prob p

p(a,a) = 1

p(x,a) ≤ p(y,a) provided x├ y

p(x,a) = p(x,b) provided a ≈ b

p(x∧y,a) ≤ p(y,a∧x)

p(x,a) ≤ p(x∧y,a) provided p(y,a∧¬y) > 0

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Condition for ∨

Hawthorne's Q

Exclusive ∨+ If a |~ x, b |~ x, a ├ ¬b then a∨b |~ x

Exclusive ∨− If a∨b |~ x and a ├ ¬b then a |~ x or b |~ x

Proto-Prob p

Disjunctive Interpolationp(x,a) ≤ p(x,a∨b) ≤ p(x,b) whenever p(x,a) ≤ p(x,b) and a ├ ¬b

Comment1st / 2nd ≤ validates ∨+ / ∨−

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Ancestry

Recall disjunctive interpolationp(x,a) ≤ p(x,a∨b) ≤ p(x,b) whenever p(x,a) ≤ p(x,b) and a ├ ¬bKoopmanSecond ≤ was discussed by Koopman 1940, called 'alternative presumption'GärdenforsDouble ≤ extracts qualitative content of an axiom of Gärdenfors 1988 ch 5 for revision of one-place probability functions

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Gärdenfors' Axiom

Translated to language of 2-place prob fns:when ¬b ∈ Cn(a)p(x,a∨b) = p(x,a)⋅k + p(x,b)⋅(1−k) where k = p(a,a∨b)

Immediately implies disjunctive interpolation

In his language of revision of 1-place prob fns:p∗(a∨b) = (p∗a)⋅k +(p∗b)⋅(1−k), when ¬b ∈ Cn(a), where k = (p∗(a∨b))(a)

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Broader than vF

Every van Fraassen function is a proto-probability functionConverse fails: There are proto-probability functions that are not van Fraassen functionsAs expected: No reals, no anti-symmetry,no addition, no multiplicationStriking example: Characteristic function of classical logical consequence is not a vF function but is a proto-probability fn into {0,1} !

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Soundness of Q wrt Proto

Take any proto-probability function p over a set D equipped with transitive, complete relation ≤ and choose t ∈ D

Define relation |~p,t as expected:

a |~p,t x iff p(x,a) ≥ t.

Then all postulates of Hawthorne’s system Q hold for |~p,t

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Representation for Q wrt Proto

Theorem For every relation |~ between formulae satisfying the postulates of Hawthorne's system Q, there is a proto-probability function p and threshold t in its range such that |~ = |~pt

ProofTrivial: Just take D = {0,1}, put p to be the characteristic function of |~, and put t = 1

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I Hawthorne's System QII Kinds of Conditional ProbabilityIII Enter Proto-ProbabilityIV ComparisonsV Take-Home Message

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Two Examples

Spohn's conditional ranking functions (with order converted)Dubois and Prade's conditional possibility measures

Comments In these two examples, disjunctive interpolation holds without needing the hypothesis that a ├ ¬bThey also validate conjunction in the conclusion

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Halpern's Plausibility Measures

Halpern 2001 looked for a 'most general' kind of conditional probability that would include all (or almost all) those known in the literature

Will not give full definition here

He called his class plausibility measures

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Incomparable (Strictly Speaking)

Central: Orderings differentHalpern's ≤ is a partial ordering whereas our ≤ is a ranking (reflex, trans, complete, not nec. anti-sym)Detail: Domains Halpern's 2nd domain may be proper subset of firstLimiting Case: Unit functionHalpern excludes the 2-place unit function (incompatible with his axiom p(⊥,a) = 0 for all a)

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Nevertheless we Compare

When we restrict both kinds of construction toFull language L for both domains Linear relations in value structureand Add unit function to Halpern (and adjust one of his axioms to permit this)thenHalpern's class is considerably more general

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Comments

This is in accord with different motivations Halpern: search for a notion that covers just

about all kinds of probability-like functions that have appeared in the literature Proto: Most general that validates the logic Q

However, all Halpern's examples appear to be proto-probability functions

Room for further comparison, also with Jim’s classes

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I Hawthorne's System QII Kinds of Conditional ProbabilityIII Enter Proto-ProbabilityIV ComparisonsV Take-Home Message

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Conclusions

We may extract a qualitative part of conditional probability, sufficient to validate Hawthorne's system Q of inference

The resulting notion of proto-probability functions generalizes beyond the van Fraassen conditional probability functions

Uses no numbers, addition, or multiplication, and the order need not be fully linear (complete, but not necessarily anti-symmetric)

Less general (roughly) than Halpern's plausibility measures, but includes all his examples

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Reference

“Conditional probability in the light of qualitative belief change” section 5

J. Phil. Logic 40: 2011 121-153