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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) 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) 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) 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 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.