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How Different are Quantitative and Qualitative

Consequence Relations for Uncertain Reasoning?

David Makinson

(joint work with Jim Hawthorne)

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Uncertain Reasoning

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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

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Behaviour

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

• But exactly how much do they differ, and in what respects?

• Are there any respects in which the quantititive ones are more regular?

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Tricks and Traps

On quantitative side

Can simulate qualitative constructions

On qualitative side

Behaviour varies considerably according to mode of generation

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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

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II

Qualitative Side

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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

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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 )

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Preferential Consequence - Definition

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

a |~S x iff x is satisfied by every state s that is

minimal among those satisfying a

state : in S

satisfied : under |

minimal : wrt <

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Example

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?

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KLM Family P of Rules

a |~ a reflexivity

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

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

When a |~ xy then ax |~ yVCM: very cautious monotony

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

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

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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)

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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

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III

Quantitative Side

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

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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

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IV

Closer Comparison

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Two Directions

Preferentially sound / Probabilistically sound– OR, AND– Look more closely later

Probabilistically sound Preferentially sound ?– Nobody seems to have examined– Presumed positive

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Yes and No

Question

Probabilistically sound Preferentially sound ?

Answer

Yes and No – depends on what kind of rule

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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

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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)

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Alternate-Conclusion Rules

Negation rationality (weaker than disjunctive rationality and rational monotony)

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

Well-known:

– Probabilistically sound

– Not preferentially sound - fails in some stoppered preferential models

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Countable-Premise Horn Rules

Archimedian rule (Hawthorne & Makinson 2007)

Whenevera |~ ai (premises: i )

ai |~ xi (premises: i ) xi pairwise inconsistent (side conditions)

then a |~ – Probabilistically sound

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

– But not preferentially sound

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Fails in this Preferential Model

: r, qi (i )

 

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

2 : r, q1, q2,q3, ….

1 : r, q1,q2, …

Put a r

ai q1…qi

xi q1…qiqi

(1) a |/~

(2) a |~ ai for all i

(3) ai |~ xi for all i

(4) xi pairwise inconsistent

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Corollary

• No representation theorem for probabilistic consequence relations in terms of finite-premise Horn rules

• Contrast with KLM representation theorem for preferential consequence relations

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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

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Weakened Versions of OR, AND

XOR: When a |~ x, b |~ x and a | 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

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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

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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 2001Adams 1996 (claimed)

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V

No-Man’s Land

between O and P

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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

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What Does ay |~ mean ?

• Quantitatively: Either t = 0 or p(ay) = 0

• Qualitatively: Preferential model has no (minimal) ay states

• Intuitively: a gives indefeasible support to y (certain but not logically certain)

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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: P AND {CM, OR} {CM, CT}

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

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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

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VI

Open Questions

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Mathematical

• Is Hawthorne’s family O complete for prob. consequence over finite-premise Horn rules ?

Conjecture: positive

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

Conjecture: negative

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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

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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 of each 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

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VII

References

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References

James Hawthorne & David Makinson The quantitative/qualitative watershed for rules of uncertain inference Studia Logica Sept 2007

David MakinsonCompleteness 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

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VIII

Appendices

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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 s s 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

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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

MonotonyWhen a |~ x then ab |~ x

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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

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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

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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)

• pa called the conditionalization of p on a

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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

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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 134a in Adams 1996 has serious gap

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Some Alternate-Conclusion Rules

• Negation rationalitywhen a |~ x then ab |~ x or ab |~ x

• Disjunctive rationalitywhen ab |~ x then a |~ x or b |~ x

• Rational monotonywhen a |~ x then ab |~ x or a |~ b

• Conditional Excluded Middlea |~ x or a |~ x

Of these, NR alone holds for probabilistic consequence