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Possibility Theory and its applications: a retrospective and prospective view. D. Dubois, H. Prade IRIT-CNRS, Université Paul Sabatier 31062 TOULOUSE FRANCE. Outline. Basic definitions Pioneers Qualitative possibility theory Quantitative possibility theory.

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Possibility Theory and its applications: a retrospective and prospective view

D. Dubois, H. Prade IRIT-CNRS, Université Paul Sabatier 31062 TOULOUSE FRANCE


  • Basic definitions

  • Pioneers

  • Qualitative possibility theory

  • Quantitative possibility theory

Possibility theory is an uncertainty theory devoted to the handling of incomplete information.

  • similar to probability theory because it is based on set-functions.

  • differs by the use of a pair of dual set functions (possibility and necessity measures) instead of only one.

  • it is not additive and makes sense on ordinal structures.

The name "Theory of Possibility" was coined by Zadeh in 1978

The concept of possibility

  • Feasibility:It is possible to do something (physical)

  • Plausibility: It is possible thatsomething occurs (epistemic)

  • Consistency : Compatible with what is known(logical)

  • Permission: It is allowed to do something (deontic)


  • S: frame of discernment (set of "states of the world")

  • x : ill-known description of the current state of affairs taking its value on S

  • L: Plausibility scale: totally ordered set of plausibility levels ([0,1], finite chain, integers,...)

  • A possibility distribution πx attached to x is a mapping from S to L : s, πx(s)  L, such that s, πx(s) = 1 (normalization)

  • Conventions:

    πx(s) = 0 iff x = s is impossible, totally excluded

    πx(s) = 1 iff x = s is normal, fully plausible, unsurprizing


  • If I do not know the age of the president, I may have statistics on presidents ages… but generally not, or they may be irrelevant.

  • partial ignorance :

    • 70 ≤ x ≤ 80 (sets,intervals)

      a uniform possibility distribution

      π(x)= 1x  [70, 80]

      = 0otherwise

  • partial ignorance with preferences : May have reasons to believe that72 > 71  73 > 70  74 > 75 > 76 > 77


  • Linguistic information described by fuzzy sets: “ he is old ” : π = µOLD

  • If I bet on president's age:I may come up with a subjective probability !

    But this result is enforced by the setting of exchangeable bets (Dutch book argument). Actual information is often poorer.

A possibility distribution is the representation of a state of knowledge: a description of how we think the state of affairs is.

  • π' more specific than π in the wide senseif and only if π' ≤ π

    In other words: any value possible for π' should be at least as possible for πthat is, π' is more informative than π

  • COMPLETE KNOWLEDGE : The most specific ones

  • π(s0) = 1 ; π(s) = 0 otherwise

  • IGNORANCE : π(s) = 1,  s  S


  • A possibility distribution on S (the normal values of x)

  • an event A

    How confident are we that x  A  S ?

  • (A) = maxuAπ(s); The degree of possibility that x  A

  • N(A) = 1 – (Ac)=min uA 1 – π(s)The degree of certainty (necessity) that x  A

Comparing the value of a quantity x to a threshold when the value of x is only known to belong to an interval [a, b].

  • In this example, the available knowledge is modeled by p(x) = 1 if  x  [a, b], 0 otherwise.

  • Proposition p = "x > " to be checked

  • i) a > : then x >  is certainly true : N(x >  ) = P(x >  ) = 1.

  • ii) b < : then x >  is certainly false ; N(x >  ) = P(x >  ) = 0.

  • iii) a ≤  ≤ b: then x >  is possibly true or false; N(x >  ) = 0; P(x >  ) = 1.

Basic properties

(A) = to what extent at least one element in A is consistent with π (= possible)

N(A) = 1 – (Ac) = to what extent no element outside A is possible = to what extent π implies A

(A  B) = max((A), (B)); N(A  B) = min(N(A), N(B)).

Mind that most of the time : (A  B) < min((A), (B)); N(A B) > max(N(A), N(B)

Corollary N(A) > 0 (A) = 1

Pioneers of possibility theory

  • In the 1950’s, G.L.S. Shackle called "degree of potential surprize" of an event its degree 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 the event to which it is assigned.

  • The degree of surprize of an event is the degree of surprize of its least surprizing realization.

  • He introduces a notion of conditional possibility

Pioneers of possibility theory

  • In his 1973 book, the philosopherDavid Lewisconsiders a relation between possible worlds he calls "comparative possibility".

  • He relates this concept of possibility to a notion of similarity between possible worlds for defining the truth conditions of counterfactual statements.

  • for events A, B, C, A B C  A  C  B.

  • The ones and only ordinal counterparts to possibility measures

Pioneers of possibility theory

  • The philosopherL. J. Cohenconsidered the problem of legal reasoning (1977).

  • "Baconian probabilities" understood as degrees of provability.

  • It is hard to prove someone guilty at the court of law by means of pure statistical arguments.

  • A hypothesis and its negation cannot both have positive "provability"

  • Such degrees of provability coincide with necessity measures.

Pioneers of possibility theory

  • Zadeh(1978) proposed an interpretation of membership functions of fuzzy sets as possibility distributions encoding flexible constraints induced by natural language statements.

  • relationship between possibility and probability: what is probable must preliminarily be possible.

  • refers to the idea of graded feasibility ("degrees of ease") rather than to the epistemic notion of plausibility.

  • the key axiom of "maxitivity" for possibility measures is highlighted (also for fuzzy events).

Qualitative vs. quantitative possibility theories

  • Qualitative:

    • comparative: A complete pre-ordering ≥πon UA well-ordered partition of U: E1 > E2 > … > En

    • absolute: πx(s)  L = finite chain, complete lattice...

  • Quantitative: πx(s)  [0, 1], integers...

    One must indicate where the numbers come from.

    All theories agree on the fundamental maxitivity axiom(A  B) = max((A), (B))

    Theories diverge on the conditioning operation

Ordinal possibilistic conditioning

  • A Bayesian-like equation: A) = min(A), )A) is the maximal solution to this equation.

    (B | A)= 1 if A, B ≠ Ø, (A) = (A  B) > 0 = (A  B) if (A) > (A  B)

    N(B | A) = 1 – (Bc| A)

    • Independence(B | A) = (B) impliesA) = min(), )

    Not the converse!!!!


  • The set of states of affairs is partitioned via π into a totally ordered set of clusters of equally plausible states

    E1 (normal worlds) > E2 >... En+1 (impossible worlds)

  • ASSUMPTION: the current situation is normal.

    By default the state of affairs is in E1

  • N(A) > 0 iff P(A) > P(Ac)

    iff A is true in all the normal situations

    Then, A is accepted as an expected truth

  • Accepted events are closed under deduction


(B) ≥(C) means « Comparing propositions on the basis of their most normal models »

  • ASSUMPTION for computing (B): the current situation is the most normal where B is true.

  • PLAUSIBLE REASONING = “ reasoning as if the current situation were normal” and jumping to accepted conclusions obtained from the normality assumption.



• If B is learned to be true, then the normal situations become the most plausible ones in B, and the accepted beliefs are revised accordingly

  • Accepting A in the context where B is true:

  • P(AB) > P(Ac B) iff N(A | B) > 0(conditioning)

    • One may have N(A) > 0 , N(Ac | B) > 0 :



Given a non-dogmatic possibility distribution π on S (π(s) > 0, s)

Propositions A, and B

  • A |=πB iff (A  B) > (A Bc)

    It means that B is true in the most plausible worlds where A is true

  • This is a form of inference first proposed by Shoham in nonmonotonic reasoning


(in A)

Example (continued)

  • Pieces of knowledge like ∆ = {b f, p  b, p  ¬f}

    can be expressed by constraints

    (b  f) > ( b ¬f)

    (p  b) > (p  ¬b)

    (p  ¬f) > (p  f)

    • the minimally specific π* ranks normal situations first:

      ¬p  b  f, ¬p ¬b

    • then abnormal situations: ¬f  b

    • Last, totally absurd situations f  p , ¬b p

Example (back to possibilistic logic)

 = material implication

  • Ranking of rules: b f has less priority that others according to p*: N*(b f ) = N*(p  b) > N*(b f)

  • Possibilistic base :

    K = {(b f ), (p  b), (p  ¬f)},with  < 

Applications of qualitative possibility theory

  • Exception-tolerant Reasoning in rule bases

  • Belief revision and inconsistency handling in deductive knowledge bases

  • Handling priority in constraint-based reasoning

  • Decision-making under uncertainty with qualitative criteria (scheduling)

  • Abductive reasoning for diagnosis under poor causal knowledge (satellite faults, car engine test-benches)


  • A set of states S;

  • A set of consequences X.

  • A decision = a mapping f from S to X

  • f(s) is the consequence of decision f when the state is known to be s.

  • Problem : rank-order the set of decisions in XS when the state is ill-known and there is a utility function on X.

  • This is SAVAGE framework.


  • Uncertainty on states is possibilistica function π: S  L

    L is a totally ordered plausibility scale

  • Preference on consequences:

    a qualitative utility function µ: X  U

    • µ(x) = 0totally rejected consequence

    • µ(y) > µ(x)y preferred to x

    • µ(x) = 1preferred consequence

Possibilistic decision criteria

  • Qualitative pessimistic utility (Whalen):

    UPES(f) = minsS max(n(π(s)), µ(f(s)))

    where n is the order-reversing map of V

    • Low utility : plausible state with bad consequences

  • Qualitative optimistic utility (Yager):

    UOPT(f) = maxsS min(π(s), µ(f(s)))

    • High utility: plausible states with good consequences

The pessimistic and optimistic utilities are well-known fuzzy pattern-matching indices

  • in fuzzy expert systems:

    • µ = membership function of rule condition

    • π = imprecision of input fact

  • in fuzzy databases

    • µ = membership function of query

    • π = distribution of stored imprecise data

  • in pattern recognition

    • µ = membership function of attribute template

    • π = distribution of an ill-known object attribute

Assumption: plausibility and preference scales L and U are commensurate

  • There exists a common scale V that contains both L and U, so that confidence and uncertainty levels can be compared.

    • (certainty equivalent of a lottery)

  • If only a subset E of plausible states is known

    • π = E

    • UPES(f) = minsE µ(f(s)) (utility of the worst consequence in E)

      criterion of Wald under ignorance

    • UOPT(f)= maxsE µ(f(s))

On a linear state space

Pessimistic qualitative utility of binary acts xAy, with µ(x) > µ(y):

  • xAy (s) = x if A occurs= y if its complement Ac occurs

    UPES(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 the consequence is y: UPES(f) = µF(y)

    Otherwise, utility reflects certainty: UPES(f) = N(A)

  • WITH UOPT(f) : replace N(A) by (A)

Representation theorem for pessimistic possibilistic criteria

  • Suppose the preference relation a on acts obeys the following properties:

    • (XS, a) is a complete preorder.

    • there are two acts such that f a g.

    •  A, f, x, y constant,x  a y  xAfyAf

    • if f >a h and g >a h imply f g >a h

    • if x is constant, h >a x and h >a g imply h >a xg

      then there exists a finite chain L, an L-valued necessity measure on S and an L-valued utility function u, such thata is representable by the pessimistic possibilistic criterion UPES(f).

Merits and limitations of qualitative decision theory

  • Provides a foundation for possibility theory

  • Possibility theory is justified by observing how a decision-maker ranks acts

  • Applies to one-shot decisions (no compensations/ accumulation effects in repeated decision steps)

  • Presupposes that consecutive qualitative value levels are distant from each other (negligibility effects)

Quantitative possibility theory

  • Membership functions of fuzzy sets

    • Natural language descriptions pertaining to numerical universes (fuzzy numbers)

    • Results of fuzzy clustering

      Semantics: metrics, proximity to prototypes

  • Upper probability bound

    • Random experiments with imprecise outcomes

    • Consonant approximations of convex probability sets

      Semantics: frequentist, subjectivist (gambles)...

Quantitative possibility theory

  • Orders of magnitude of very small probabilities

    degrees of impossibility k(A) ranging on integersk(A) = n iff P(A) = en

  • Likelihood functions (P(A| x), where x varies) behave like possibility distributions

    P(A| B) ≤ maxx B P(A| x)


  • Given a numerical possibility distribution p, defineP(p) = {Probabilities P | P(A) ≤ (A) for all A}

  • Then, generally it holds that (A) = sup {P(A) | P P(p)}N(A) = inf {P(A) | P P(p)}

  • So p is a faithful representation of a family of probability measures.

From confidence sets to possibility distributions

Consider a nested family of sets E1E2 …  En

a set of positive numbers a1 …an in [0, 1]

and the family of probability functions

P = {P | P(Ei) ≥ ai for all i}.

Pis always representable by means of a possibility measure. Its possibility distribution is precisely

πx = mini max(µEi, 1 – ai)

Random set view

  • Let mi = i – i+1 then m1 +… + mn = 1

    A basic probability assignment (SHAFER)

  • π(s)= ∑i: sAi mi (one point-coverage function)

  • Only in the consonant case can m be recalculated from π


  • A Coxian axiom(A C) = (A |C)*(C), with * = product

    Then: (A |C) = (A C)/ (C)

    N(A|C) = 1 – (Ac | C)

    Dempster rule of conditioning (preserves s-maxitivity)

    For the revision of possibility distributions: minimal change 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 measure π(s |b C) = π(s)max(1, 1/( π(s) + N(C)))

It can be shownthat:

(A |b C) = (A C)/ ((A C) + N(AcC))

N(A|bC) = N(A C) / (N(A C) + P(AcC))

= 1 – (Ac |b C)

For inference from generic knowledge based on observations

Possibility-Probability transformations

  • Why ?

    • fusion of heterogeneous data

    • decision-making : betting according to a possibility distribution leads to probability.

    • Extraction of a representative value

    • Simplified non-parametric imprecise probabilistic models

Elementary forms of probability-possibility transformations exist for a long time

  • POSS PROB: Laplace indifference principle  “ All that is equipossible is equiprobable ” = changing a uniform possibility distribution into a uniform probability distribution

  • PROB POSS: Confidence intervals Replacing a probability distribution by an interval A with a confidence level c.

    • It defines a possibility distribution

    • π(x) = 1 if x  A,

      = 1 – c if x  A

Possibility-Probability transformations : BASIC PRINCIPLES

  • Possibility probability consistency: P ≤ 

  • Preserving the ordering of events : P(A) ≥ P(B) (A) ≥ (B)or elementary events only(x) > (x') if and only if p(x) > p(x')(orderpreservation)

  • Informational criteria:

    from  to P: Preservation of symmetries

    (Shapley value rather than maximal entropy)

    from P to : optimize information content

    (Maximization or minimisation of specificity

From OBJECTIVE probability to possibility :

  • Rationale : given a probability p, try and preserve as much information as possible

  • Select a most specific element of the set PI(P) = {:  ≥ P} of possibility measures dominating P such that  (x) >  (x') iff p(x) > p(x')

  • may be weakened into : p(x) > p(x')implies (x) >  (x')

  • The result is i = j=i,…n pi

    (case of no ties)

From probability to possibility : Continuous case

  • The possibility distribution  obtained by transforming p encodes then family of confidence intervals around the mode of p.

  • The a-cut of  is the (1- a)-confidence interval of p

  • The optimal symmetric transform of the uniform probability distribution is the triangular fuzzy number

  • The symmetric triangular fuzzy number (STFN) is a covering approximation of any probability with unimodal symmetric density p with the same mode.

  • In other words the a-cut of a STFN contains the (1- a)-confidence interval of any such p.

From probability to possibility : Continuous case

  • IL = {x, p(x) ≥ } =[aL, aL+ L] is the interval of length L with maximal probability

  • The most specific possibility distribution dominating p is π such that L > 0, π(aL) = π(aL+ L) = 1 – P(IL).


Possibilistic view of probabilistic inequalities

  • Chebyshev inequality defines a possibility distribution that dominates any density with given mean and variance.

  • The symmetric triangular fuzzy number (STFN) defines a possibility distribution that optimally dominates anysymmetric density with given mode and bounded support.

From possibility to probability

  • Idea (Kaufmann, Yager, Chanas):

    • Pick a number  in [0, 1] at random

    • Pick an element at random in the -cut of π.

      a generalized Laplacean indifference principle : change alpha-cuts into uniform probability distributions.

  • Rationale : minimise arbitrariness by preserving the 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 game theory.


    • Starting point : exploit the betting approach to subjective probability

    • A critique: The agent is forced to be additive by the rules of exchangeable bets.

      • For instance, the agent provides a uniform probability distribution on a finite set whether (s)he knows nothing about the concerned phenomenon, or if (s)he knows the concerned phenomenon is purely random.

    • Idea : It is assumed that a subjective probability supplied by an agent is only a trace of the agent's belief.


    • Assumption 1: Beliefs can be modelled by belief functions

      • (masses m(A) summing to 1 assigned to subsets A).

    • Assumption 2: The agent uses a probability function induced by his or her beliefs, using the pignistic transformation (Smets, 1990) or Shapley value.

    • Method : reconstruct the underlying belief function from the probability provided by the agent by choosing among the isopignistic ones.


    • There are clearly several belief functions with a prescribed 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 belief function whose Shapley value is p is unique and consonant.


    • The least specific belief function in the sense of maximizing I(m) is characterized by

    • i = j=1,n min(pj, pi).

    • It is a probability-possibility transformation, previously suggested in 1983: This is the unique possibility distribution whose Shapley value is p.

    • It gives results that are less specific than the confidence interval approach to objective probability.

    Applications of quantitative possibility

    • Representing incomplete probabilistic data for uncertainty propagation in computations

    • (but fuzzy interval analysis based on the extension principle differs from conservative probabilistic risk analysis)

    • Systematizing some statistical methods (confidence intervals, likelihood functions, probabilistic inequalities)

    • Defuzzification based on Choquet integral (linear with fuzzy number addition)

    Applications of quantitative possibility

    • Uncertain reasoning : Possibilistic nets are a counterpart to Bayesian nets that copes with incomplete data. Similar algorithmic properties under Dempster conditioning (Kruse team)

    • Data fusion : well suited for mergingheterogeneous information on numerical data (linguistic, statistics, confidence intervals) (Bloch)

    • Risk analysis : uncertainty propagation using fuzzy arithmetics, and random interval arithmetics when statistical data is incomplete (Lodwick, Ferson)

    • Non-parametric conservative modelling of imprecision in measurements (Mauris)


    Quantitative possibility is not as well understood as probability theory.

    • Objective vs. subjective possibility (a la De Finetti)

    • How to use possibilistic conditioning in inference tasks ?

    • Bridge the gap with statistics and the confidence interval literature (Fisher, likelihood reasoning)

    • Higher-order modes of fuzzy intervals (variance, …) and links with fuzzy random variables

    • Quantitative possibilistic expectations : decision-theoretic characterisation ?


    • Possibility theory is a simple and versatile tool for modeling uncertainty

    • A unifying framework for modeling and merging linguistic knowledge and statistical data

    • Useful to account for missing information in reasoning tasks and risk analysis

    • A bridge between logic-based AI and probabilistic reasoning

    Properties of inference |=

    • A |=π A if A ≠ Ø (restricted reflexivity)

    • if A ≠ Ø, then A |=πØ never holds (consistency preservation)

    • The set {B: A |=π 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 Cthen A |=π B C

      (Right AND)

    Properties of inference |=

    • If A |=π C ; B |=π C then A  B |=π C (Left OR)

    • If A |=π B and A  B |=π C then A |=π C

      (cut, weak transitivity )

      (But if A normally implies B which normally implies C, then A may not imply C)

    • If A |=π B and if A |=π Cc is false, then A  C |=π B(rational monotony RM)

      If B is normally expected when A holds,then B is expected to hold when both A and C hold, unless it is that A normally implies not C


    • Let |= be a consequence relation on 2S x 2S

    • Define an induced partial relation on subsets as

      A > B iff A  B |= Bc for A ≠ 

    • Theorem: If |= satisfies restricted reflexivity, right weakening, rational monotony, Right AND and Left OR, then A > B is the strict part of a possibility relation on events.

      So a consequence relation satisfying the above properties is representable by possibilistic inference, and induces a complete plausibility preordering on the states.


    • A generic rule « if A then B » is modelled by P(AB) > P(Ac B).

      • This is a constraint that delimits a set of possibility

      • distributions on the set of interpretations of the language

  • Applying the minimal specificity principle:P(AB) = P(ABc ) = P(Ac Bc ) > P(Ac B).


    • ∆ = {Ai Bi, i = 1,n}

    • ∆ defines a set of constraints on possibility distributions (Ai Bi) > (Ai ¬Bi), i = 1,…n

    • • (∆) = set of feasible π's with respect to ∆

      • One may compute * : the least specific possibility distribution in (∆)

    Plausible inference from a set of default rules

    What « ∆ implies A  B » means

    • Cautious inference

      ∆ = A  B iff

      For all P  (∆), P(AB) > P(Ac B).

    • Possibilistic inference

      ∆ =* A  B iff *(AB) > *(Ac B) for the least specific possibility measure in (∆).

      Leads to a stratification of ∆ according to N*(Ac B)

    Possibilistic logic

    • A possibilistic knowledge base is an ordered set of propositional or 1st order formulas pi

    • K = {(pii), i = 1,n} where i > 0 is the level of priority or validity of pi

      i = 1 means certainty.

      i = 0 means ignorance

    • Captures the idea of uncertain knowledge in an ordinal setting

    Possibilistic logic

    • Axiomatization:

      All axioms of classical logic with weight 1

      Weighted modus ponens{(p ), (¬p  q )} |- (q min(,))

      OLD! Goes back to Aristotle school

      Idea: the validity of a chain of uncertain deductions is the validity of its weakest link

      Syntactic inference K |-(p ) is well-defined

    Possibilistic logic

    • Inconsistency becomes a graded notion inc(K) = sup{, K |- (,)}

    • Refutation and resolution methods extendK |- (p ) iffK {(p 1)} |- (,)

    • Inference with a partially inconsistent knowledge base becomes non-trivial and nonmonotonicK |-nt p iff K |- (p ) and  > inc(K)

    Semantics of possibilistic logic

    • A weighted formula has a fuzzy set of models .

    • If A = [p] is the set of models of p (subset of S),

    • |-(p a) means N(A) ≥ 

      The least specific possibility distribution induced by |-(p a) is:

      π(p a)(s) = max(µA(s), 1 – )

      = 1 if p is true in state s

      = 1 –  if p is false in state s

    Semantics of possibilistic logic

    • The fuzzy set of models of K is the intersection of the fuzzy sets of models of {(pii), i = 1,n}

    • πK(s)= mini=1,n {1 – i | s [pi]}

      determined by the highest priority formula violated by s

    • The p. d. πK is the least informed state of partial knowledge compatible with K

    Soundness and completeness

    • Monotonic semantic entailment follows Zadeh’s entailment principleK |= (p, ) stands for πK ≤ π(p a)

      Theorem: K |- (p, ) iff K |= (p )

    • For the non-trivial inference under inconsistency:{(p 1)}  K |-nt q iff (q  p) > (¬q  p)

    Possibilistic logic

    Formulas are Boolean

    Truth is 2-valued

    Weighted formulas have fuzzy sets of models

    Validity is many-valued

    degrees of validity are not compositional except for conjunctions

    Represents uncertainty

    Fuzzy logic (Pavelka)

    Formulas are non-Boolean

    Truth is many-valued

    Weighted formulas have crisp sets of models (cuts)

    Validity is Boolean

    degrees of truth are compositional

    represents real functions by means of logical formulas

    Possibilistic vs. fuzzy logics


    • K = {b  f, p  b, p  ¬f}

     = material implication

    • K  {b} |- f; K  {p} |- contradiction

  • using possibilistic logic:  < min(,)

    K = {(b  f ), (p  b ), (p  ¬f )}

    then K  {(b, 1)} |- (f ) and K  {(b, 1)} |-nt f

  • Inc(K{(p, 1), (b, 1)} = 

  • K  {(p, 1), (b, 1)} |- (¬f, min(,))

  • Hence K  {(p, 1), (b, 1)} |-nt ¬f

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