how different are quantitative and qualitative consequence relations for uncertain reasoning l.
Skip this Video
Download Presentation
How Different are Quantitative and Qualitative Consequence Relations for Uncertain Reasoning?

Loading in 2 Seconds...

play fullscreen
1 / 50

How Different are Quantitative and Qualitative Consequence Relations for Uncertain Reasoning? - PowerPoint PPT Presentation

  • Uploaded on

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

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
Download Presentation

PowerPoint Slideshow about 'How Different are Quantitative and Qualitative Consequence Relations for Uncertain Reasoning?' - ura

Download Now An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
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)



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

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
  • 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
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 <

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?

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)


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


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


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



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



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


Probabilistically sound  Preferentially sound ?


Yes and No – depends on what kind of rule



  • Prob. sound  Pref. sound ?


Yes and No – depends on what kind of rule


  • 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


  • Probabilistically sound
  • Not preferentially sound - fails in some stoppered preferential models
countable premise horn rules
Countable-Premise Horn Rules

Archimedian rule (Hawthorne & Makinson 2007)


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


(1) a |/~ 

(2) a |~ ai for all i 

(3) ai |~ xi for all i 

(4) xi pairwise inconsistent

Fails in this Preferential Model
  • 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)



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:




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)

  • 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

  • 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

  • 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




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.,




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


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

  • 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

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