A Semantic Model for Vague Quantifiers Combining Fuzzy Theory and Supervaluation Theory Ka Fat CHOW The Hong Kong Polytechnic University

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A Semantic Model for Vague Quantifiers Combining
Fuzzy Theory and Supervaluation Theory Ka Fat
CHOWThe Hong Kong Polytechnic University
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Basic Assumptions and Scope of Study

• Vagueness is manifested as degree of truth, which
  can be represented by a number in 0, 1.
• Classical tautologies / contradictions by virtue
  of classical logic / lexical meaning remain their
  status as tautologies / contradictions when the
  non-vague predicates are replaced by vague
  predicates
• Do not consider the issue of higher order
  vagueness

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Fuzzy Theory (FT)

• Uses fuzzy sets to model vague concepts
• p - truth value of p
• x ? S - membership degree of an individual x
  wrt a fuzzy set S
• Membership Degree Function (MDF)
• Uses fuzzy formulae for Boolean operators
• p ? q min(p, q)
• p ? q max(p, q)
• p 1 p

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Some Problems of FT (1)

• Model U PERSON j, m TALL 0.5/j,
  0.3/m SHORT 0.5/j, 0.7/m
• FT fails to handle tautologies / contradictions
  correctly
• E.g. John is tall or John is not tall
  max(j ? TALL, 1 j ? TALL) 0.5

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Some Problems of FT (2)

• Model U PERSON j, m TALL 0.5/j,
  0.3/m SHORT 0.5/j, 0.7/m
• FT fails to handle internal penumbral connections
  correctly
• Internal Penumbral Connections concerning the
  borderline cases of one vague set
• E.g. Mary is tall and John is not tall
  min(m ? TALL, 1 j ? TALL) 0.3
• FT fails to handle external penumbral connections
  correctly
• External Penumbral Connections concerning the
  border lines between 2 or more vague sets
• E.g. Mary is tall and Mary is short
  min(m ? TALL, m ? SHORT) 0.3

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Supervaluation Theory (ST)

• Views vague concepts as truth value gaps
• Evaluates truth values of sentences with vague
  concepts by means of admissible complete
  specifications (ACSs)
• Complete specification assignment of the truth
  value 1 or 0 to every individual wrt the vague
  sets in a statement
• If the statement is true (false) on all ACSs,
  then it is true (false). Otherwise, it has no
  truth value.

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Rectifying the Flaws of FT (1)

• Model U PERSON j, m TALL 0.5/j,
  0.3/m SHORT 0.5/j, 0.7/m
• An example of ACS j ? TALL 1, m ? TALL
  0, j ? SHORT 0, m ? SHORT 1
• A vague statement in the form of a tautology
  (contradiction) will have truth value 1 (0) under
  all complete specifications
• John is tall or John is not tall 1

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Rectifying the Flaws of FT (2)

• Model U PERSON j, m TALL 0.5/j,
  0.3/m SHORT 0.5/j, 0.7/m
• Rules out all inadmissible complete
  specifications related to the borderline cases of
  one vague set j ? TALL 0, m ? TALL 1, j
  ? SHORT 0, m ? SHORT 0
• Mary is tall and John is not tall 0
• Rules out all inadmissible complete
  specifications related to the border lines
  between 2 or more vague sets j ? TALL 1, m
  ? TALL 1, j ? SHORT 0, m ? SHORT 1
• Mary is tall and Mary is short 0

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Weakness of ST

• ST cannot distinguish different degrees of
  vagueness because it treats all borderline cases
  alike as truth value gaps
• We need a theory that combines FT and ST a
  theory that can distinguish different degrees of
  vagueness and yet avoid the flaws of FT

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Glöckners Method for Vague Quantifiers (VQs)

• Semi-Fuzzy Quantifiers only take crisp (i.e.
  non-fuzzy) sets as arguments
• Fuzzy Quantifiers take crisp or fuzzy sets as
  arguments
• MDFs of some Semi-Fuzzy Quantifiers
• (about 10)(A)(B) T-4,-1,1,4(A ? B / A
  10)
• every(A)(B) 1, if A ? B
• 0, if A ? B

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Quantifier Fuzzification Mechanism (QFM)

• All linguistic quantifiers are modeled as
  semi-fuzzy quantifiers initially
• QFM transform semi-fuzzy quantifiers to fuzzy
  quantifiers
• Q(X1, Xn)
• 1. Choose a cut level ? ? (0, 1
• 2. 2 crisp sets X?min X? 0.5 0.5? X?max Xgt
  0.5 0.5?
• 3. Family of crisp sets T?(X) Y X?min ? Y ?
  X?max
• 4. Aggregation Formula Q?(X1, Xn)
  m0.5(Q(Y1, Yn) Y1 ? T?(X1), Yn ? T?(Xn))
• m0.5(Z) inf(Z), if Z ? 2 ?
  inf(Z) gt 0.5
• sup(Z), if Z ? 2 ?
  sup(Z) lt 0.5
• 0.5, if (Z ? 2 ?
  inf(Z) ? 0.5 ? sup(Z) ? 0.5) ? (Z ?)
• r, if Z r
• 5. Definite Integral Q(X1, Xn) ?0,
  1Q?(X1, Xn)d?

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Glöckners Method and ST

• The combination of crisp sets Y1 ? T?(X1), Yn ?
  T?(Xn) can be seen as complete specifications of
  the fuzzy arguments X1, Xn at the cut level ?
• No need to use fuzzy formulae for Boolean
  operators
• Can handle tautologies / contradictions correctly
• To handle internal / external penumbral
  connections correctly, we need Modified
  Glöckners Method (MGM)

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Handling Internal Penumbral Connections

• A new definition for Family of Crisp Sets
• T?(X) Y X?min ? Y ? X?max such that for
  any x, y ? U, if x ? Y and x ? X ? y ? X,
  then y ? Y
• Model U PERSON j, m TALL 0.5/j,
  0.3/m SHORT 0.5/j, 0.7/m
• The inadmissible complete specification j ?
  TALL 0, m ? TALL 1, j ? SHORT 0, m ?
  SHORT 0 corresponds to Y m as a complete
  specification of TALL
• Mary is tall and John is not tall 0

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Handling External Penumbral Connections (1)

• Meaning Postulates (MPs)
• E.g. TALL ? SHORT ?
• How to specify the relationship between the MPs
  and the set specifications of the model?
• Complete Freedom no constraint on the MPs and
  set specifications may lead to the consequence
  that no ACSs of the sets can satisfy the MPs
• 0 Degree of Freedom (0DF) every possible ACS of
  the sets should satisfy every MP many models in
  practical applications are ruled out

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Handling External Penumbral Connections (2)

• 1 Degree of Freedom (1DF) Consider a model with
  the vague sets X1, Xn (n ? 2) and a number of
  MPs. For every ? ? (0, 1, every i (1 ? i ? n)
  and every combination of Y1 ? T?(X1), Yi1 ?
  T?(Xi1), Yi1 ? T?(Xi1), Yn ? T?(Xn), there
  must exist at least one Yi ? T?(Xi) such that Y1,
  Yi1, Yi, Yi1, Yn satisfy every MP
• A new Aggregation Formula Q?(X1, Xn)
  m0.5(Q(Y1, Yn) Y1 ? T?(X1), Yn ? T?(Xn)
  such that Y1, Yn satisfy the MP(s))

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Handling External Penumbral Connections (3)

• Model U PERSON j, m TALL 0.5/j,
  0.3/m SHORT 0.5/j, 0.7/m
• MP TALL ? SHORT ?
• This model and this MP satisfy the 1DF constraint
• The inadmissible complete specification j ?
  TALL 1, m ? TALL 1, j ? SHORT 0, m ?
  SHORT 1 corresponds to Y1 j, m as a
  complete specification of TALL and Y2 m as a
  complete specification of SHORT
• Mary is tall and Mary is short 0

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Iterated Quantifiers

• Quantified statements with both subject and
  object can be modeled by iterated quantifiers
• Eg. Every boy loves every girl.
• every(BOY)(x every(GIRL)(y LOVE(x, y)))

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Iterated VQs

• Q1(A1)(x Q2(A2)(y B(x, y)))
• 1. For each possible x, determine y B(x, y)
• 2. Determine Q2(A2)(y B(x, y)) using MGM
• 3. Obtain the vague set x Q2(A2)(y B(x,
  y)) Q2(A2)(y B(xi, y))/xi,
• 4. Calculate Q1(A1)(x Q2(A2)(y B(x, y)))
  using MGM

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A Property of MGM

• Suppose the membership degrees wrt the vague sets
  X1, Xn are restricted to 0, 1, 0.5 and the
  truth values output by a semi-fuzzy quantifier Q
  with n arguments are also restricted to 0, 1,
  0.5, then Q(X1, Xn) as calculated by MGM is
  the same as that obtained by the supervaluation
  method if we use 0.5 to represent the truth value
  gap.
• MGM is a generalization of the supervaluation
  method.