Possibility Theory and its applications a retrospective and可能性理論及其應(yīng)用

Possibility

Theory

and

itsapplications:

a

retrospectiveand

prospective

viewD.

Dubois,

H.

PradeIRIT-CNRS,

Université

PaulSabatier31062

TOULOUSEFRANCEOutlineBasic

definitionsPioneersQualitative

possibility

theoryQuantitative

possibility

theoryPossibility

theory

is

an

uncertaintytheory

devoted

to

the

handling

ofincomplete

information.structures.similar

to

probability

theory

because

it

is

basedon

set-functions.differs

by

the

use

of

a

pair

of

dual

setfunctions

(possibility

and

necessity

measures)instead

of

only

one.it

is

not

additive

and

makes

sense

on

ordinalThe

name

"Theory

of

Possibility"

was

coined

by

Zadehin

1978The

concept

of

possibilityFeasibility:somethingItispossible

to

do(physical)Plausibility:occursItispossible

that

something(epistemic)Consistency

:

Compatible

with

what

isknown(logical)Permission:

It

is

allowed

to

do

something(deontic)POSSIBILITY

DISTRIBUTIONS(uncertainty)S:

frame

of

discernment

(set

of

"states

of

theworld")x

:

ill-known

description

of

the

current

state

ofaffairs

taking

its

value

on

SL:

Plausibility

scale:

totally

ordered

set

ofplausibility

levels

([0,1],

finite

chain,

integersA

possibility

distribution

πx

attached

to

x

is

amapping

from

S

to

L

:s,

πx(s)L,

such

thats,

πx(s)

=

1

(normalization)Conventions:πx(s)

=

0

iff

x

=

s

is

impossible,

totally

excludedπx(s)

=

1

iff

x

=

s

is

normal,

fully

plausible,unsurprizingEXAMPLE

:

x

=

AGE

OF

PRESIDENTIf

I

do

not

know

the

age

of

the

president,I

may

have

statistics

on

presidents

ages…but

generally

not,

or

they

may

beirrelevant.partial

ignorance

:70

≤

x

≤

80

(sets,

intervals)a

uniform

possibility

distributionπ(x)

=

1=

0x

[70,

80]otherwisepartial

ignorance

with

preferences

:

Mayhave

reasons

to

believe

that

72>

71

73

>

70

74

>

75

>

76

>

77EXAMPLE

:

x

=

AGE

OF

PRESIDENTLinguistic

information

described

by“

he

is

old

”

:

π

=

μfuzzy

sets:OLDIf

I

bet

on

president"s

age:

I

maycome

up

with

a

subjective

probability

!But

this

result

is

enforced

by

thesetting

of

exchangeable

bets

(Dutchbook

argument).

Actual

information

isA

possibility

distribution

is

the

representation

ofstate

of

knowledge:a

description

of

how

we

think

the

state

of

affairs

isπ"

more

specific

than

π

in

the

widesense

if

and

only

ifπIn

other

words:

any

value

possible

for

π"

should

beat

least

as

possible

for

πthainformative

than

πCOMPLETE

KNOWLEDGE

:

The

most

specificonesπ(s0)

=

1

;

π(s)

=

0

otherwiseIGNORANCE

:

π(s)

=

1,s

SPOSSIBILITY

AND

NECESSITYOF

AN

EVENTA

possibility

distribution

on

Sof

x)(thenormal

valuesan

event

AHow

confident

are

we

that

xAS

?(A)

=

maxuAπ(s);The

degree

of

posN(A)

=

1

–(s)(Ac)=minu

A1

–

πThe

degree

of

certainty

(neceComparing

the

value

of

a

quantity

x

to

a

thresholdwhen

the

value

of

x

is

only

known

to

belong

to

an

interv[a,

b].In

this

example,

the

available

knowledge

ismodeled

by

p(x)

=

1

if

x

[a,

b],

0

otherwisProposition

p

=

"x

>i)

a

>

:

then

x

>"

to

be

checkedis

certainly

true

:N(x

>

)

=

P(x

>b

<

:

then

x

>

is

certainly

false

;N(x

>

)

=

P(x

>a

≤

≤

b:

then

x

>

is

possibly

true

or

f N(x

>

)

=

0;

P(x

>

)

=

1.Basic

properties(A)

=

to

what

extent

at

least

one

element

in

Ais

consistent

with

π

(=

possible)N(A)

=

1

–

(Ac)

=

to

what

extent

no

element=

to

what

extent

πoutside

A

is

possibleimplies

A(A

B)

=

max(

(A),

(B));

N(AN(B)).B)

=

min(N(A),Mind

that

most

of

the

time

:B)

<

min(

(A),

(B));B)

>

max(N(A),(AN(AN(B)Corollary

N(A)

>

01(A)

=Pioneers

of

possibility

theoryIn

the

1950’s,

G.L.S.

Shackle

called"degree

of

potential

surprize"

of

an

event

itsdegree

of

impossibility.Potential

surprize

is

valued

on

a

disbelief

scale,namely

a

positive

interval

of

the

form

[0,

y*],where

y*

denotes

the

absolute

rejection

of

theevent

to

which

it

is

assigned.The

degree

of

surprize

of

an

event

is

the

degreeof

surprize

of

its

least

surprizing

realization.He

introduces

a

notion

of

conditional

possibilityPioneers

of

possibility

theoryIn

his

1973

book,

the

philosopher

DavidLewisconsiders

a

relation

betweenforeventsA,B,C,ABCACB.possible

worlds

he

calls

"comparativepossibility".He

relates

this

concept

of

possibility

to

anotion

of

similarity

between

possible

worldsfor

defining

the

truth

conditions

ofcounterfactual

statements.The

ones

and

only

ordinal

counterparts

topossibility

measuresPioneers

of

possibility

theoryThe

philosopher

L.

J.

Cohen

considered

theproblem

of

legal

reasoning

(1977)."Baconian

probabilities"

understood

as

degrees

ofprovability.It

is

hard

to

prove

someone

guilty

at

the

court

oflaw

by

means

of

pure

statistical

arguments.A

hypothesis

and

its

negation

cannot

both

havepositive

"provability"Such

degrees

of

provability

coincide

withnecessity

measures.Pioneers

of

possibility

theoryZadeh(1978)

proposed

an

interpretation

ofmembership

functions

of

fuzzy

sets

as

possibilitydistributions

encoding

flexible

constraints

induceby

natural

language

statements.relationship

between

possibility

and

probability:what

is

probable

must

preliminarily

be

possible.refers

to

the

idea

of

graded

feasibility

("degreesof

ease")

rather

than

to

the

epistemic

notion

ofplausibility.the

key

axiom

of

"maxitivity"

for

possibilitymeasures

is

highlighted

(also

for

fuzzy

events).Qualitative

vs.

quantitative

possibilitytheoriesQualitative:comparative:

A

complete

pre-ordering

≥π

onU

A

well-ordered

partition

of

U:

E1

>

E2

>

…

>

Enabsolute:

πx(s)lattice...Quantitative:

πx(s)L

=

finite

chain,

complete[0,

1],

integers...One

must

indicate

where

the

numbers

come

from.All

theories

agree

on

the

fundamental

maxitivityaxiom(A

B)

=

max(

(A),

(B))Theories

diverge

on

the

conditioningoperationOrdinal

possibilisticconditioningA

Bayesian-like

equation:A)

=

min(A)

is

the

maximal

solution

to

this

equation.A(B

|

A) =

1

if

A,

B

≠

?,0N(B

|

A)

=

1

–

(Bc|

A)(A)

=

(A

B)

>=

(A B)

if(A)

>?Independence)Not

the

converse!!!!QUALITATIVE

POSSIBILISTIC

REASONINGThe

set

of

states

of

affairs

is

partitioned

via

πinto

a

totally

ordered

set

of

clusters

of

equallyplausible

statesE1

(normal

worlds)

>

E2

>...

En+1

(impossibleworlds)ASSUMPTION:

the

current

situation

is

normal.By

default

the

state

of

affairs

is

in

E1N(A)

>

0

iff

P(A)

>

P(Ac)iff

A

is

true

in

all

the

normal

situationsThen,

A

is

accepted

as

an

expected

truthAccepted

events

are

closed

under

deductionA

CALCULUS

OF

PLAUSIBLE

INFERENCE(B)

≥(C)

means

?

Comparing

propositions

onthe

basis

of

their

most

normal

models

?ASSUMPTION

for

computing

(B):

the

currentsituation

is

the

most

normal

where

B

is

true.PLAUSIBLE

REASONING

=

“

reasoning

as

ifthe

current

situation

were

normal”

and

jumpingto

accepted

conclusions

obtained

from

thenormality

assumption.DIFFERENT

FROM

PROBABILISTICREASONING

BASED

ON

AVERAGINGACCEPTANCE

IS

DEFEASIBLE

If

B

is

learned

to

be

true,

then

thenormal

situations

become the

mostplausible

ones

in

B,

and

the

acceptedbeliefs

are

revised

accordinglyAccepting

A

in

the

context

where

B

is

true:B)

iff

N(A

|

B)

>

0N(A)

> 0

,

N(Ac

|

B)

>P(A

B)

>

P(Ac(conditioning)One

may

have 0

:non-monotonyWITH

APLAUSIBLE

INFERENCEPOSSIBILITY

DISTRIBUTIONGiven

a

non-dogmatic

possibility

distribution

πon

S

(π(s)

>

0,

s)Propositions

A,

and

BA

|=π

B

iff

(AIt

means

thatB)

>

(A

Bc)B

is

truA

is

trueThis

is

a

form

of

inference

firstproposed

by

Shoham

in

nonmonotonicreasoningWITH

APLAUSIBLE

INFERENCEPOSSIBILITY

DISTRIBUTION(in

A)Exa

mple

(continued)Pieces

of

knowledge

like

?

=

{b

f,

p?f}can

be

expressed

by

constraintsb,

p(bf)

>(

b?f)(pb)

>(p?b)(p?f)

>(pf)the

minimally

specific

π*

ranks

normalsituations

first:?pbf,

?p?bthen

abnormalLast,

totallysituations:

?fabsurd

situationsbfp

,

?bpExample

(back

to

possibilistic

logic)=

material

implicationRanking

of

rules:

bf

has

lesspriority

that

others

according

to

p*:N*(b

f

)

=

N*(p

b)

>N*(b

f)Possibilistic

base

:K

=

{(b)},f

),

(pwith

<b),

(p?fApplications

of

qualitativepossibility

theoryException-tolerant

Reasoning

in

rule

basesBelief

revision

and

inconsistency

handlingin

deductive

knowledge

basesHandling

priority

in

constraint-basedreasoningDecision-making

under

uncertainty

withqualitative

criteria

(scheduling)Abductive

reasoning

for

diagnosis

underpoor

causal

knowledge

(satellite

faults,

carengine

test-benches)ABSOLUTE

APPROACHTO

QUALITATIVE

DECISIONA

set

of

states

S;A

set

of

consequences

X.A

decision

=

a

mapping

f

from

S

to

Xf(s)

is

the

consequence

of

decision

fwhen

the

state

is

known

to

be

s.Problem

:

rank-order

the

set

of

decisionsin

XS

when

the

state

is

ill-known

and

there

is

a

utility

function

on

X.This

is

SAVAGE

framework.ABSOLUTE

APPROACHTO

QUALITATIVE

DECISIONUncertainty

on

states

ispossibilistica

function

π:

SL

is

a

totally

ordered

plausibility

scalePreference

on

consequences:a

qualitative

utility

function

μ:

XUtotally

rejectedμ(x)

=

0consequenceμ(y)

>

μ(x)μ(x)

=

1y

preferred

to

xpreferred

consequencePossibilistic

decision

criteriQualitative

pessimistic

utility

(Whalen):UPES(f)

=

minsSmax(n(π(s)),

μ(f(s)))where

n

is

the

order-reversing

map

of

VLow

utility

:

plausible

state

with

badconsequencesQualitative

optimistic

utility

(Yager):UOPT(f)

=

maxsHigh

utility:consequencesSmin(π(s),

μ(f(s)))plausible

states

with

goodThe

pessimistic

and

optimistic

utilities

are

well-known

fuzzy

pattern-matching

indicesin

fuzzy

expert

systems:μ

=

membership

function

of

rule

conditionπ

=

imprecision

of

input

factin

fuzzy

databasesμ

=

membership

function

of

queryπ

=

distribution

of

stored

imprecise

datain

pattern

recognitionμ

=

membership

function

of

attribute

templateπ

=

distribution

of

an

ill-known

object

attribuAssumption:plausibility

and

preferenceand

U

are

commensuratescales

LThere

exists

a

common

scale

V

that

containsboth

L

and

U,

so

that

confidence

and

uncertaintylevels

can

be

compared.(certainty

equivalent

of

a

lottery)If

only

a

subset

E

of

plausible

states

isknownπ

=EEUPES(f)

=

mins

μ(f(s))

(utility

of

the

worstconsequence

in

E)criterion

of

Wald

under

ignoranceEUOPT(f)=

maxs

μ(f(s))On

a

linear

state

spacePessimistic

qualitativeutility

of

binaryactsxAy,

with

μ(x)xAy

(s)

=

x

if

Aoccurs>

μ(y):=

y

if

its

coUPES(xAy)

=

median

{μ(x),

N(A),

μ(y)}Interpretation:

If

the

agent

is

sure

enough

of

A,it

is

as

if

the

consequence

is

x:

UPES(f)

=

μF(x)If

he

is

not

sure

about

A

it

is

as

if

theconsequence

is

y:

UPES(f)

=

μF(y)Otherwise,

utility

reflects

certainty:

UPES(f)

=

N(AWITH

UOPT(f)

:

replace

N(A)

by

(A)Representation

theoremfor

pessimistic

possibilistic

criteriaaSuppose

the

preference

relation

on

acts

obeysthe

following

properties:a(XS,

a)

is

a

complete

preorder.there

are

two

acts

such

that

fA,

f,

x,

y

constant,

xag.y

xAfyAfif

f

>a

h

and

g

>a

h

imply

f g

>a

hif

x

is

constant,

h

>a

x

and

h

>a

g

imply

h

>a

x

gthen

there

exists

a

finite

chain

L,

an

L-valuednecessity

measure

on S

and

an

L-valued

utilityafunction

u,

such

that

is

representable

by

thepessimistic

possibilistic

criterion

UPES(f).Merits

and

limitationsof

qualitative

decision

theoryProvides

a

foundation

for

possibility

theoryPossibility

theory

is

justified

by

observinghow

a

decision-maker

ranks

actsApplies

to

one-shot

decisions

(nocompensations/

accumulation

effects

inrepeated

decision

steps)Presupposes

that

consecutive

qualitativevalue

levels

are

distant

from

each

other(negligibility

effects)Quantitative

possibilitytheoryMembership

functions

of

fuzzy

setsNatural

language

descriptions

pertaining

tonumerical

universes

(fuzzy

numbers)Results

of

fuzzy

clusteringSemantics:

metrics,

proximity

toprototypesUpper

probability

boundRandom

experiments

with

imprecise

outcomesConsonant

approximations

of

convex

probabilitysetsSemantics:

frequentist,

subjectivist(gambles)...Quantitative

possibilitytheoryOrders

of

magnitude

of

very

smallprobabilitiesdegrees

of

impossibility

k(A)

ranging

onintegers

k(A)

=

n

iff

P(A)

=

enLikelihood

functions

(P(A|

x),

wherex

varies)

behave

like

possibilitydistributionsBP(A|

B)

≤

maxx

P(A|

x)POSSIBILITY

ASUPPER

PROBABILITYGiven

a

numerical

possibility

distribution

pdefineP(p)

=

{Probabilities

P

|

P(A)

≤

(A)A}Then,

generally

it

holdsthat(A)

=

sup

{P(A)

|

PP(p)}

N(A)

=

inf

{P(A)

|

P

P(p)So

p

is

a

faithful

representation

of

a

familyof

probability

measures.From

confidence

sets

to

possibilitydistributionsConsider

a

nested

family

of

sets

E1

E2 …

Ena

set

of

positive

numbers

a1

…an

in

[0,

1]and

the

family

of

probability

functionsP

=

{P

|

P(Ei)

≥

ai

for

all

i}.P

is

always

representable

by

means

of

a

possibilitymeasure.

Its

possibility

distribution

is

preciselyπx

=

mini

max(μEi,

1

–

ai)Random

set

viewLet

mi

=

i

–

i+1then

m1

+…

+

mn

=

1A

basic

probabilityassignment

(SHAFER)π(s)=

∑i:

sAimi

(one

point-coverage

function)Only

in

the

consonant

case

can

m

berecalculated

from

πCONDITIONAL

POSSIBILITYMEASURESA

Coxian

axiom(C),(Awith

*C)

==

product(A

|C)*Then:

(A

|C)

=(AC)/(C)N(A|

C)

=

1

–

(Ac

|

C)Dempster

rule

of

conditioning

(preserves

s-maxitivity)For

the

revision

of

possibility

distributions:

minimachange

of

when

N(C)

=

1.It

improves

the

state

of

information(reduction

of

focal

elements)Bayesian

possibilistic

conditioning(A

|b

C)

=

sup{P(A|C),

P

≤

,

P(C)

>

0}N(A

|b

C)

=

inf{P(A|C),

P

≤

,

P(C)

>

0}It

is

still

a

possibility

measureIt

can

be

shown

that:π(s|b

C)

=

π(s)max(1,(A

|b

C)

=

(AC)/

(

(AC)

+

N(AcC))N(A|b

C)

=

N(A

C)

/

(N(A=

1

–

(Ac

|b

C)C)

+

P(AcC))For

inference

from

generic

knowledge

based

onobservationsPossibility-Probability

transformationsWhy

?fusion

of

heterogeneous

datadecision-making

:

betting

according

to

apossibility

distribution

leads

to

probabiliExtraction

of

a

representative

valueSimplified

non-parametric

impreciseprobabilistic

modelsPOSS

PROB:

Laplace

indifference

principle“

All

that

is

equipossible

is

equiprobable

”=

changing

a

uniform

possibility

distribution

intouniform

probability

distributionPROB

POSS:

Confidence

intervalsReplacing

a

probability

distribution

by

an

intervalA

with

a

confidence

level

c.It

defines

a

possibility

distributionπ(x)

=

1

if

xA,=

1

–

c

if

x

AElementary

forms

of

probability-possibility

transformations

exist

for

along

timePossibility-Probability

transformations

:BASIC

PRINCIPLESPossibility

probability

consistency:

P

≤Preserving

the

ordering

of

events

:P(A)

≥

P(B)(A)

≥or

elementary

even(B)onlyp(x")(x)

>

(x")

if(order

preservation)Informational

criteria:from

to

P:

Preservation

of

symmetries(Shapley

value

rather

than

maximal

entropy)from

P

to:

optimize

information

content(Maximization

or

minimisation

ofspecificityFrom

OBJECTIVE

probability

to

possibility

:Rationale

:

given

a

probability

p,

try

andpreserve

as

much

information

as

possibleSelect

a

most

specific

element

of

the

setPI(P)

= { :

≥

P}

of

possibilitymeasures

dominating

P

such

that(x)

>(x")

iff

p(x)

>

p(x")may

be

weakened

into

:p(x)

>

p(x")

implies

(x)

>The

result

is

=i

j=i,…npi(case

of

no

ties)From

probability

to

possibility

:Continuous

caseThe

possibility

distribution

obtained

bytransforming

p

encodes

then

family

of

confidenceintervals

around

the

mode

of

p.The

a-cut

of

is

the

(1-

a)-confidence

intervalof

pThe

optimal

symmetric

transform

of

the

uniformprobability

distribution

is

the

triangular

fuzzynumberThe

symmetric

triangular

fuzzy

number

(STFN)

isa

covering

approximation

of

any

probability

withunimodal

symmetric

density

p

with

the

samemode.In

other

words

the

a-cut

of

a

STFN

contains

the(1-

a)-confidence

interval

of

any

such

p.IL

=

{x,

p(x)

≥

}=

[aL,

aL+

L]is

the

interval

oflength

L

with

maximalprobabilityThe

most

specificpossibility

distributiondominating

p

is

πsuch

that

L

>

0,

π(aL)

=

π(aL+

L)

=

1

–P(IL).From

probability

to

possibility

:Continuous

casebPossibilistic

view

of

probabilisticinequalitiesChebyshev

inequality

defines

apossibility

distribution

that

dominateany

density

with

given

mean

andvariance.The

symmetric

triangular

fuzzy

number(STFN)

defines

a

possibility

distributithat

optimally

dominates

any

symmetricdensity

with

given

mode

andbounded

support.From

possibility

to

probabilitIdea

(Kaufmann,

Yager,

Chanas):Pick

a

number

in

[0,

1]

at

randomPick

an

element

at

random

in

the-cut

of

π.a

generalized

Laplacean

indifference

principle

:change

alpha-cuts

into

uniform

probabilitydistributions.Rationale

:

minimise

arbitrariness

by

preservingthe

symmetry

properties

of

the

representation.The

resulting

probability

distribution

is:The

centre

of

gravity

of

the

polyhedron

P(p)The

pignistic

transformation

of

belief

functions

(Smets)The

Shapley

value

of

the

unanimity

game

N

in

gametheory.SUBJECTIVE

POSSIBILITYDISTRIBUTIONSStarting

point

:

exploit

the

bettingapproach

to

subjective

probabilityA

critique:

The

agent

is

forced

to

beadditive

by

the

rules

of

exchangeable

bets.For

instance,

the

agent

provides

a

uniformprobability

distribution

on

a

finite

set

whether(s)he

knows

nothing

about

the

concernedphenomenon,

or

if

(s)he

knows

the

concernedphenomenon

is

purely

random.Idea

:

It

is

assumed

that

a

subjectiveprobability

supplied

by

an

agent

is

only

atrace

of

the

agent"s

belief.SUBJECTIVE

POSSIBILITYDISTRIBUTIONSAssumption

1:

Beliefs

can

be

modelledby

belief

functions(masses

m(A)

summing

to

1

assigned

tosubsets

A).Assumption

2:

The

agent

uses

aprobability

function

induced

by

his

or

herbeliefs,

using

the

pignistic

transformation(Smets,

1990)

or

Shapley

value.Method

:

reconstruct

the

underlying

belieffunction

from

the

probability

provided

bythe

agent

by

choosing

among

theisopignistic

ones.SUBJECTIVE

POSSIBILITYDISTRIBUTIONSThere

are

clearly

several

belief

functions

withprescribed

Shapley

value.Consider

the

least

informative

of

those,in

the

sense

of

a

non-specificity

index(expected

cardinality

of

the

random

set)I(m)

=

∑

m(A)

card(A).RESULT

:

The

least

informative

belieffunction

whose

Shapley

value

is

p

isunique

and

consonant.SUBJECTIVE

POSSIBILITYDISTRIBUTIONSThe

least

specific

belief

function

in

thesense

of

maximizing

I(m)

is

characterizedbyi

j=1,n=

min(pj,

pi).It

is

a

probability-possibility

transformatipreviously

suggested

in

1983:

This

is

theunique

possibility

distribution

whoseShapley

value

is

p.It

gives

results

that

are

less

specific

thanthe

confidence

interval

approach

toobjective

probability.Applications

of

quantitativepossibilityRepresenting

incomplete

probabilistic

data

foruncertainty

propagation

in

computations(but

fuzzy

interval

analysis

based

on

theextension

principle

differs

from

conservativeprobabilistic

risk

analysis)Systematizing

some

statistical

methods(confidence

intervals,

likelihood

functions,probabilistic

inequalities)Defuzzification

based

on

Choquet

integral

(linearwith

fuzzy

number

addition)Applications

of

quantitativepossibilityUncertain

reasoning

:

Possibilistic

nets

are

acounterpart

to

Bayesian

nets

that

copes

withincomplete

data.

Similar

algorithmic

propertiesunder

Dempster

conditioning

(Kruse

team)Data

fusion

:

well

suited

for

mergingheterogeneous

information

on

numerical

data(linguistic,

statistics,

confidence

intervals)

(BloRisk

analysis

:

uncertainty

propagation

usingfuzzy

arithmetics,

and

random

interval

arithmeticswhen

statistical

data

is

incomplete

(Lodwick,Ferson)Non-parametric

conservative

modelling

ofimprecision

in

measurements

(Mauris)PerspectivesQuantitative

possibility

is

not

as

wellunderstood

as

probability

theory.Objective

vs.

subjective

possibility

(a

la

DeFinetti)How

to

use

possibilistic

conditioning

in

inferencetasks

?Bridge

the

gap

with

statistics

and

the

confidenceinterval

literature

(Fisher,

likelihood

reasoning)Higher-order

modes

of

fuzzy

intervals(variance,

…)

and

links

with

fuzzy

randomvariablesQuantitative

possibilistic

expectations

:

decisiontheoretic

characterisation

?ConclusionPossibility

theory

is

a

simple

and

versatiletool

for

modeling

uncertaintyA

unifying

framework

for

modeling

andmerging

linguistic

knowledge

andstatistical

dataUseful

to

account

for

missing

informationin

reasoning

tasks

and

risk

analysisA

bridge

between

logic-based

AI

andprobabilistic

reasoningProperties

of

inference

|=A

|=π

A

if

A

≠

?

(restricted

reflexivity)if

A

≠

?,

then

A

|=π

?

never

holds

(consistencypreservation)πThe

set

{B:

A

|=

B}

is

deductively

closed-If

A

B

and

C

|=π

A

then

C

|=π

B(right

weakening

rule

RW)-If

A

|=π

B

and

A

|=π

C

then

A

|=π

B

C(Right

AND)Properties

of

inference

|=IfA

|=πC

;

B

|=π

C

then

AB

|=π

C(LeftOR)IfA

|=πB

and

AB

|=π

Cthen

A

|=π(cut,Cweak

transitivity)(But

if

A

normally

implies

B

which

normally

implies

C,then

A

may

not

imply

C)CcIf

A

|=π

B

and

if

A

|=π

is

false,

then

A

C

|=πB

(rational

monotony

RM)If

B

is

normally

expected

when

A

holds,then

B

is

expected

to