A Semantic Model for
Vague Quantifiers
Combining Fuzzy Theory
and Supervaluation
Theory
Ka Fat CHOW
The Hong Kong Polytechnic University
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 nonvague predicates are replaced by vague
predicates
 Do not consider the issue of higher order
vagueness

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║
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
Some Problems of FT (2)
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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
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.
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
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

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
Glöckner’s Method for
Vague Quantifiers (VQs)
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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
Quantifier Fuzzification Mechanism (QFM)
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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 γ  (0, 1]
2. 2 crisp sets: Xγmin = X 0.5 + 0.5γ; Xγmax= X> 0.5 – 0.5γ
3. Family of crisp sets: Tγ(X) = {Y: Xγmin  Y  Xγmax}
4. Aggregation Formula: ║Qγ(X1, … Xn)║ = m0.5({║Q(Y1, … Yn)║: Y1 
Tγ(X1), … Yn  Tγ(Xn)})
m0.5(Z) = inf(Z), if |Z|  2  inf(Z) > 0.5
= sup(Z), if |Z|  2  sup(Z) < 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, 1]║Qγ(X1, … Xn)║dγ
Glöckner’s Method and ST
The combination of crisp sets Y1  Tγ(X1), … Yn 
Tγ(Xn) can be seen as complete specifications of
the fuzzy arguments X1, … Xn at the cut level γ
 No need to use fuzzy formulae for Boolean
operators
 Can handle tautologies / contradictions correctly
 To handle internal / external penumbral
connections correctly, we need Modified
Glöckner’s Method (MGM)

Handling Internal
Penumbral Connections
A new definition for Family of Crisp Sets:
Tγ(X) = {Y: Xγmin  Y  Xγmax 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

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

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 γ  (0, 1], every i (1  i  n) and
every combination of Y1  Tγ(X1), … Yi–1  Tγ(Xi–
1), Yi+1  Tγ(Xi+1), … Yn  Tγ(Xn), there must exist
at least one Yi  Tγ(Xi) such that Y1, … Yi–1, Yi,
Yi+1, … Yn satisfy every MP
 A new Aggregation Formula: ║Qγ(X1, … Xn)║ =
m0.5({║Q(Y1, … Yn)║: Y1  Tγ(X1), … Yn  Tγ(Xn)
such that Y1, … Yn satisfy the MP(s)})

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)})})
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
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.
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