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How Different are Quantitative and Qualitative Consequence Relations for Uncertain Reasoning?

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### How Different are Quantitative and QualitativeConsequence Relations for Uncertain Reasoning?

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David Makinson

(joint work with Jim Hawthorne)

Uncertain Reasoning

Consequence Relations

- Many ways of studying uncertain reasoning
- One way: consequence relations (operations) and their properties
- Two approaches to their definition:
- Quantitative (using probability)
- Qualitative (various methods)
- Tend to be studied by different communities

Behaviour

Widely felt: quantitatively defined consequence relations rather less well-behaved than qualitative counterparts

- Butexactlyhow much do they differ, and in what respects?
- Are there any respects in which the quantititive ones are more regular?

Tricks and Traps

On quantitative side

- Can simulate qualitative constructions

On qualitative side

- Behaviour varies considerably according to mode of generation

Policy

- Don’t try to twist one kind of approach to imitate the other
- Take most straightforward version of each
- Compare their behaviour as they are

Qualitative Side

Recall Main Qualitative Account

- Name: preferential consequence relations
- Due to: Kraus, Lehmann, Magidor
- Status: Industry standard
- Our presentation: With single formulae (rather than sets of them) on the left

Preferential models

Structure S = (S, , |) where:

- S is an arbitrary set (elements called states)
- is a transitive, irreflexive relation over S (called a preference relation)
- | is a satisfaction relation between states and classical formulae (well-behaved on classical connectives )

Preferential Consequence - Definition

Given a preferential modelS = (S, , |), define consequence relation |~S by rule:

a |~Sx iff x is satisfied by every state s that is minimal among those satisfying a

state : in S

satisfied : under |

minimal : wrt <

S = {s1, s2}

s1 s2

s2 :p,q,r

s1 : p,q, r

p |~ r, but pq |~/ r

Monotony fails

Some other classical rules fail

What remains?

ExampleKLM Family P of Rules

a |~ a reflexivity

When a |~ x and x |y then a |~ y

RW: right weakening

When a |~ x and a||b then b |~ x

LCE: left classical equivalence

When a |~ xy then ax |~ y

VCM: very cautious monotony

When a |~ x and b |~ x, then ab |~ x

OR: disjunction in the premises

When a |~ x and a |~ y, then a |~ xy

AND: conjunction in conclusion

All Horn rules for |~(with side-conditions)

Whenever

a1 |~ x1, …., an |~ xn (premises with |~)

and

b1 |- y1, …., bm |- ym (side conditions with |-)

then

c |~ z (conclusion)

(No negative premises, no alternate conclusions; finitely many premises unless signalled)

KLM Representation Theorem

A consequence relation |~ between classical propositional formulae is a preferential consequence relation (i.e. is generated by some stoppered preferential model) iff it satisfies the Horn rules listed in system P

Quantitative Side

Ingredients and Definition

- Fix a probability function p
- Finitely additive, Kolgomorov postulates
- Conditionalization as usual: pa(x) = p(ax)/p(a)
- Fix a threshold t in interval [0,1]
- Define a consequence relation |~p,t , briefly |~, by the rule:

a |~p,tx iff either pa(x) t or p(a) 0

Successes and Failures

Succeed (zero and one premise rules of P)

a |~ a Reflexivity

When a |~ x and x |y then a |~ y RW: right weakening

When a |~ x and a||b then b |~ x LCE: left classical equivalence

When a |~ xy then ax |~ y VCM: very cautious monotony

Fail (two-premise rules of P)

When a |~ x and b |~ x, then ab |~ x OR: disjunction in premises

When a |~ x and a |~ y, then a |~ xy AND: conjunction in conclusion

Closer Comparison

Two Directions

Preferentially sound / Probabilistically sound

- OR, AND
- Look more closely later

Probabilistically sound Preferentially sound ?

- Nobody seems to have examined
- Presumed positive

Yes and No

Question

Probabilistically sound Preferentially sound ?

Answer

Yes and No – depends on what kind of rule

Specifics

Question

- Prob. sound Pref. sound ?

Answer

Yes and No – depends on what kind of rule

Specifics

- Finite-premise Horn rules: Yes
- Alternative-conclusion rules: No
- Countable-premise Horn rules: No

Finite-Premise Horn rules

Should have been shown c.1990…Hawthorne & Makinson 2007

If the rule is probabilistically sound

(i.e. holds for every consequence relation generated by a prob.function, threshold)

then it is preferentially sound

(i.e. holds for every consequence relation generated by a stoppered pref. model)

Alternate-Conclusion Rules

Negation rationality (weaker than disjunctive rationality and rational monotony)

When a |~ x, then ab |~ xorab |~ x

Well-known:

- Probabilistically sound
- Not preferentially sound - fails in some stoppered preferential models

Countable-Premise Horn Rules

Archimedian rule (Hawthorne & Makinson 2007)

Whenever

a |~ ai(premises: i )

ai |~ xi(premises: i )

xi pairwise inconsistent(side conditions)

thena |~

- Probabilistically sound

Archimedean property of reals: t 0 n: n.t 1

- But not preferentially sound

:r, qi (i )

n :r, q1,.., qn,qn+1

2 : r, q1, q2,q3, ….

1 : r, q1,q2, …

Put ar

ai q1…qi

xiq1…qiqi

(1) a |/~

(2) a |~ ai for all i

(3) ai |~ xi for all i

(4) xi pairwise inconsistent

Fails in this Preferential ModelCorollary

- No representation theorem for probabilistic consequence relations in terms of finite-premise Horn rules
- Contrast with KLM representation theorem for preferential consequence relations

Other Direction

Pref. sound but not prob. sound: two-premise Horn rules:

OR: When a |~ x and b |~ x, then ab |~ x

AND: When a |~ x and a |~ y, then a |~ xy

- Are there weakened versions that are prob. sound?
- Can we get completeness over finite-premise Horn rules?
- Representation no!, completeness maybe
- Wedge between representation and completeness
- Completeness relative to class of expressions

Weakened Versions of OR, AND

XOR:When a |~ x,b |~ x anda |b then ab |~ x

- Requires that the premises be exclusive
- Well-known

WAND:When a |~ x,ay |~ , then a |~ xy

- Requires a stronger premise
- Hawthorne 1996

Proposed Axiomatization for Probabilistic Consequence

Hawthorne’s family O(1996):

- The zero and one-premise rules of P
- Plus XOR, WAND

Open question: Is this complete for finite-premise Horn rules (possibly with side-conditions) ?

Conjecture: Yes

Partial Completeness Results

The following are equivalent for finite-premise Horn rules with pairwise inconsistent premise-antecedents

(1) Prob. sound

(2a) Pref. sound (all stoppered pref.models)

(2b) Sound in all linear pref. models at most 2 states

(3) Satisfies ‘truth-table test’ of Adams

(4a) Derivable from B{XOR} (when n 1, from B)

(4b) Derivable from family O

(4c) Derivable from family P

for n 1: van Benthem 1984, Bochman 2001

Adams 1996 (claimed)

More about WAND: When a |~ x, ay |~ , then a |~ xy

Second condition equivalent in O to each of:

- ay |~ y
- ay |~ z for all z
- ab |~ y for all b (a |~ y ‘holds monotonically’)
- (ay)b |~ y for all b

What Does ay |~ mean ?

- Quantitatively: Either t = 0 or p(ay) = 0
- Qualitatively: Preferential model has no (minimal) ay states
- Intuitively: a givesindefeasiblesupport to y (certain but not logically certain)

Between O and P

Modulo rules in O:

OR CM

CT

AND

CT: when a |~ x and ax |~ y then a |~ y

CM: when a |~ x and a |~ y then ax |~ y

Modulo O: PAND {CM, OR} {CM, CT}

(Positive parts Adams 1998, Bochman 2001; CM / AND tricky)

Moral

- AND serves as a watershed condition between family O (sound for probabilistic consequence) and family P(characteristic for qualitative consequence)
- No other single well-known rule does the same

Open Questions

Mathematical

- Is Hawthorne’s family O completefor prob. consequenceover finite-premise Horn rules ?

Conjecture: positive

- Can we give a representation theorem for prob.consequence in terms of O + NR + Archimedes + …?

Conjecture: negative

Philosophical

- Pref. consequence, as a formal modelling of qualitative uncertain consequence, validates AND
- So do most others, e.g. Reiter default consequence
- But do we really want that?
- Perhaps it should fail even for qualitative consequence relations
- Example: paradox of the preface

Paradox of the preface(Makinson 1965)

An author of a book making a large number n of assertions may check and recheck them individually, and be confident ofeach that it is correct. But experience teaches that inevitably there will be errors somewhere among the n assertions, and the preface may acknowledge this. Yet these n+1 assertions are together inconsistent.

- Inconsistent belief set, whether or not we accept AND
- Inconsistent belief, if we accept AND

References

References

James Hawthorne & David Makinson

The quantitative/qualitative watershed for rules of uncertain inference

Studia Logica Sept 2007

David Makinson

Completeness Theorems, Representation Theorems: What’s the Difference?

Hommage à Wlodek: Philosophical Papers decicated to Wlodek Rabinowicz, ed. Rønnow-Rasmussen et al., www.fil.lu.se/hommageawlodek

Appendices

What is Stoppering?

To validate VCM: When a |~ xy then ax |~ y, we need to impose stoppering (alias smoothness) condition:

Whenever state s satisfies formula a, either:

- s is minimal under among the states satisfying a
- or there is a state ss that is minimal under among the states satisfying a

Automatically true in finite preferential models. Also true in infinite models when no infinite descending chains

Derivable from Family P

Can derive

SUP: supraclassicality:

When a |x, then a |~ x

CT: cumulative transitivity:

When a |~ x and ax |~ y, then a |~ y

Can’t derive

Plain transitivity:

When a |~ x and x |~ y, then a |~ y

Monotony

When a |~ x then ab |~ x

VCM versus CM

KLM (1990) use CM: cautious monotony:

When a |~ x and a |~ y, then ax |~ y

instead of VCM

When a |~ xy then ax |~ y

These are equivalent in P (using AND and RW)

But not equivalent in absence of AND

Kolmogorov Postulates

Any function defined on the formulae of a language closed under the Boolean connectives, into the real numbers, such that:

(K1) 0 p(x) 1

(K2) p(x) = 1 for some formula x

(K3) p(x)p(y) whenever x |- y

(K4) p(xy)= p(x) p(y) whenever x |- y

Conditionalization

- Let p be a finitely additive probability function on classical formulae in standard sense (Kolmogorov postulates)
- Let a be a formula with p(a) 0
- Write pa alias p(•|a)for the probability function defined by the standard equation pa(x) = p(ax)/p(a)
- pacalled the conditionalization of p on a

What is System B?

- Burgess 1981
- May be defined as the 1-premise rules in O and P plus 1-premise version of AND:

VWAND: When a |~ x and a |y then a |~ xy

- AND WAND VWAND

What is Adams’ Truth-Table Test ?

There is some subset I {1,..,n} such that both by |iI(ai xi) and iI(aixi) |by

- When n = 0 this reduces to: b |y
- For n = 1, reduces to: either b |y or both ax |by and ax |by
- Proof of 134ain Adams 1996 has serious gap

Some Alternate-Conclusion Rules

- Negation rationality

when a |~ x then ab |~ xorab |~ x

- Disjunctive rationality

when ab |~ x then a |~ xorb |~ x

- Rational monotony

when a |~ x then ab |~ xora |~ b

- Conditional Excluded Middle

a |~ xora |~ x

Of these, NRalone holds for probabilistic consequence

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