Basic Assumptions and Scope of Study
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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 STST 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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Glöckner’s 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 γ (0, 1] 2. 2 crisp sets: Xγ
min = X 0.5 + 0.5γ; Xγmax= X> 0.5 – 0.5γ
3. Family of crisp sets: Tγ(X) = {Y: Xγmin Y Xγ
max}
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
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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)
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Handling Internal Penumbral Connections
A new definition for Family of Crisp Sets:
Tγ(X) = {Y: Xγmin Y Xγ
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 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)})
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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.