Possibility Theory and its applications: a retrospective and prospective view

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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 that something 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) = 1 x  [70, 80]
  • = 0 otherwise
  • partial ignorance with preferences : May have reasons to believe that 72 > 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 sense if and only if π' ≤ π
  • In other words: any value possible for π' should be at least as possible for π that 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A π(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 (x) = 1 if x [a, b], 0 otherwise.
  • Proposition p = "x > " to be checked
  • i) a > : then x >  is certainly true : N(x >  ) = (x >  ) = 1.
  • ii) b < : then x >  is certainly false ; N(x >  ) = (x >  ) = 0.
  • iii) a ≤  ≤ b: then x >  is possibly true or false; N(x >  ) = 0; (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 philosopher David Lewis considers 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 philosopher L. J. Cohen considered 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 ≥π on U A 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 (A) > (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:
  • (AB) > (Ac B) iff N(A | B) > 0 (conditioning)
  • • One may have N(A) > 0 , N(Ac | B) > 0 :
  • non-monotony


      • Given a non-dogmatic possibility distribution π on S (π(s) > 0, s)
      • Propositions A, and B
  • A π B 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)

Exa mple (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 *: 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 possibilistic a function π: S  L
  • L is a totally ordered plausibility scale
  • Preference on consequences:
  • a qualitative utility function µ: X  U
    • µ(x) = 0 totally rejected consequence
    • µ(y) > µ(x) y preferred to x
    • µ(x) = 1 preferred 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 integers k(A) = n iff P(A) = n
  • 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 , define P() = {Probabilities P | P(A) ≤ (A) for all A}
      • Then, generally it holds that (A) = sup {P(A) | P  P()} N(A) = inf {P(A) | P  P()}
      • So  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}.
  • P is 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 -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}
  • (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 shownthat:
  • (A |b C) (A C)/ ((A C) + (Ac C))
  • N(A|b C) = (A C) / ((A C) + (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

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

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

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') (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 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 -cut of  is the (1)-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 -cut of a STFN contains the (1)-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 any symmetric 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(
      • 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 merging heterogeneous 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 |=π Ø 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 C then 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 (AB) > (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: (AB) = (ABc ) = (Ac Bc ) > (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 ∆
  • •ne 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 (∆), (AB) > (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 extend K |(p ) iff K {(p 1)} |- (,)
  • Inference with a partially inconsistent knowledge base becomes non-trivial and nonmonotonic K |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 ) means N(A) ≥ 
  • The least specific possibility distribution induced by |(p ) is:
  • π(p )(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 principle K |= (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 vs. fuzzy logics

  • 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


      • • 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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