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A Semantic Model for Vague Quantifiers Combining Fuzzy Theory and Supervaluation Theory Ka Fat CHOW The 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(X 1, … X n ) 1. Choose a cut level γ (0, 1] 2. 2 crisp sets: X γ min = X 0.5 + 0.5γ ; X γ max = X > 0.5 – 0.5γ 3. Family of crisp sets: T γ (X) = {Y: X γ min Y X γ max } 4. Aggregation Formula: Q γ (X 1, … X n ) = m 0.5 ({Q(Y 1, … Y n ): Y 1 T γ (X 1 ), … Y n T γ (X n )}) m 0.5 (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(X 1, … X n ) = [0, 1] Q γ (X 1, … X n )dγ

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Glöckners Method and ST The combination of crisp sets Y 1 T γ (X 1 ), … Y n T γ (X n ) can be seen as complete specifications of the fuzzy arguments X 1, … X n 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 Glöckners Method (MGM)

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Handling Internal Penumbral Connections A new definition for Family of Crisp Sets: T γ (X) = {Y: X γ min Y X γ 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 X 1, … X n (n 2) and a number of MPs. For every γ (0, 1], every i (1 i n) and every combination of Y 1 T γ (X 1 ), … Y i–1 T γ (X i– 1 ), Y i+1 T γ (X i+1 ), … Y n T γ (X n ), there must exist at least one Y i T γ (X i ) such that Y 1, … Y i–1, Y i, Y i+1, … Y n satisfy every MP A new Aggregation Formula: Q γ (X 1, … X n ) = m 0.5 ({Q(Y 1, … Y n ): Y 1 T γ (X 1 ), … Y n T γ (X n ) such that Y 1, … Y n 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 Y 1 = {j, m} as a complete specification of TALL and Y 2 = {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 Q 1 (A 1 )({x: Q 2 (A 2 )({y: B(x, y)})}) 1. For each possible x, determine {y: B(x, y)} 2. Determine Q 2 (A 2 )({y: B(x, y)}) using MGM 3. Obtain the vague set: {x: Q 2 (A 2 )({y: B(x, y)})} = {Q 2 (A 2 )({y: B(x i, y)})/x i, …} 4. Calculate Q 1 (A 1 )({x: Q 2 (A 2 )({y: B(x, y)})}) using MGM

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A Property of MGM Suppose the membership degrees wrt the vague sets X 1, … X n 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(X 1, … X n ) 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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