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How Different are Quantitative and Qualitative Consequence Relations for Uncertain Reasoning?. David Makinson (joint work with Jim Hawthorne). I. Uncertain Reasoning. Consequence Relations. Many ways of studying uncertain reasoning

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how different are quantitative and qualitative consequence relations for uncertain reasoning

How Different are Quantitative and QualitativeConsequence Relations for Uncertain Reasoning?

David Makinson

(joint work with Jim Hawthorne)

slide2

I

Uncertain Reasoning

consequence relations
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
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
Tricks and Traps

On quantitative side

  • Can simulate qualitative constructions

On qualitative side

  • Behaviour varies considerably according to mode of generation
policy
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
slide7

II

Qualitative Side

recall main qualitative account
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
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
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 <

example
S = {s1, s2}

s1 s2

s2 :p,q,r

s1 : p,q, r

p |~ r, but pq |~/ r

Monotony fails

Some other classical rules fail

What remains?

Example
klm family p of rules
KLM 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 |~ xy then ax |~ y

VCM: very cautious monotony

When a |~ x and b |~ x, then ab |~ x

OR: disjunction in the premises

When a |~ x and a |~ y, then a |~ xy

AND: conjunction in conclusion

all horn rules for with side conditions
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
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

slide15

III

Quantitative Side

ingredients and definition
Ingredients and Definition
  • Fix a probability function p
    • Finitely additive, Kolgomorov postulates
  • Conditionalization as usual: pa(x) = p(ax)/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
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 |~ xy then ax |~ y VCM: very cautious monotony

Fail (two-premise rules of P)

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

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

slide18

IV

Closer Comparison

two directions
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
Yes and No

Question

Probabilistically sound  Preferentially sound ?

Answer

Yes and No – depends on what kind of rule

specifics
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
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
Alternate-Conclusion Rules

Negation rationality (weaker than disjunctive rationality and rational monotony)

When a |~ x, then ab |~ xorab |~ x

Well-known:

  • Probabilistically sound
  • Not preferentially sound - fails in some stoppered preferential models
countable premise horn rules
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
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…qiqi

(1) a |/~ 

(2) a |~ ai for all i 

(3) ai |~ xi for all i 

(4) xi pairwise inconsistent

Fails in this Preferential Model
corollary
Corollary
  • 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
Other Direction

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

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

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

  • 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
Weakened Versions of OR, AND

XOR:When a |~ x,b |~ x anda |b then ab |~ x

  • Requires that the premises be exclusive
  • Well-known

WAND:When a |~ x,ay |~ , then a |~ xy

  • Requires a stronger premise
  • Hawthorne 1996
proposed axiomatization for probabilistic consequence
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
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)

slide31

V

No-Man’s Land

between O and P

more about wand when a x a y then a x y
More about WAND: When a |~ x, ay |~ , then a |~ xy

Second condition equivalent in O to each of:

  • ay |~ y
  • ay |~ z for all z
  • ab |~ y for all b (a |~ y ‘holds monotonically’)
  • (ay)b |~ y for all b
what does a y mean
What Does ay |~  mean ?
  • Quantitatively: Either t = 0 or p(ay) = 0
  • Qualitatively: Preferential model has no (minimal) ay states
  • Intuitively: a givesindefeasiblesupport to y (certain but not logically certain)
between o and p
Between O and P

Modulo rules in O:

OR CM

CT

AND

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

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

Modulo O: PAND  {CM, OR}  {CM, CT}

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

moral
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
slide36

VI

Open Questions

mathematical
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
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
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
slide40

VII

References

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

slide42

VIII

Appendices

what is stoppering
What is Stoppering?

To validate VCM: When a |~ xy then ax |~ 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

derivable from family p
Derivable from Family P

Can derive

SUP: supraclassicality:

When a |x, then a |~ x

CT: cumulative transitivity:

When a |~ x and ax |~ y, then a |~ y

Can’t derive

Plain transitivity:

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

Monotony

When a |~ x then ab |~ x

vcm versus cm
VCM versus CM

KLM (1990) use CM: cautious monotony:

When a |~ x and a |~ y, then ax |~ y

instead of VCM

When a |~ xy then ax |~ y

These are equivalent in P (using AND and RW)

But not equivalent in absence of AND

kolmogorov postulates
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(xy)= p(x) p(y) whenever x |- y

conditionalization
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(ax)/p(a)
  • pacalled the conditionalization of p on a
what is system b
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 |~ xy

  • AND WAND  VWAND
what is adams truth table test
What is Adams’ Truth-Table Test ?

There is some subset I {1,..,n} such that both by |iI(ai xi) and iI(aixi) |by

  • When n = 0 this reduces to: b |y
  • For n = 1, reduces to: either b |y or both ax |by and ax |by
  • Proof of 134ain Adams 1996 has serious gap
some alternate conclusion rules
Some Alternate-Conclusion Rules
  • Negation rationality

when a |~ x then ab |~ xorab |~ x

  • Disjunctive rationality

when ab |~ x then a |~ xorb |~ x

  • Rational monotony

when a |~ x then ab |~ xora |~ b

  • Conditional Excluded Middle

a |~ xora |~ x

Of these, NRalone holds for probabilistic consequence