1. A Semantic Model for Vague Quantifiers Combining Fuzzy Theory and Supervaluation Theory Ka Fat CHOW The Hong Kong Polytechnic University
The title of my presentation today is "A Semantic Model for Vague Quantifiers Combining Fuzzy Theory and Supervaluation Theory". Note that this talk contains some ideas not covered in the paper associated with this talk.

2. 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
First of all, I will spell out some basic assumptions and the scope of my study.
First, I assume that vagueness is manifested as degree of truth, which can be represented by a number in [0, 1].
Next, I assume that 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.
Moreover, in this paper I do not consider the issue of higher order vagueness, which will be left for further studies.

3. 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║
Now I briefly introduce some basic concepts of Fuzzy Theory (FT). FT is an important theory for vagueness. It uses fuzzy sets to model vague concepts. A fuzzy set can be characterized by membership degrees, i.e. the degree to which each individual of the universe belongs to a fuzzy set. Here I use ║p║ to denote the truth value of the statement p. Thus, ║x  S║ can be used to denote the membership degree of an individual x wrt a fuzzy set S. In many cases, the membership degrees may be specified as a function, called the membership degree function (MDF). Some examples of MDFs will be given later.
Another characteristic of FT is that it makes use of fuzzy formuale for Boolean operators. Here are the most commonly used fuzzy formulae for conjunction, disjunction and negation, where "min" and "max" represent the minimum and maximum functions, respectively.

4. 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
While the fuzzy formulae for Boolean operators are quite intuitive, they may result in incorrect truth values for certain vague propositions. In what follows, I use a model to illustrate the problems of FT. In this model the universe is composed of two persons John, represented by j, and Mary, represented by m. Two fuzzy sets, TALL and SHORT, are defined by specifying the membership degrees of j and m wrt these two sets.
First, FT fails to handle certain tautologies / contradictions correctly. For example, consider the statement "John is tall or John is not tall". Since this is a tautology, it should have truth value 1. But using the aforesaid fuzzy formulae, this statement will have truth value 0.5 wrt the model.

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
Second, FT fails to handle certain internal penumbral connections correctly. Internal penumbral connections refer to statements concerning the borderline cases of one vague set. For example, consider the statement "Mary is tall and John is not tall" concerning the vague set TALL. Since according to the membership degree specification of TALL, John is taller than Mary, this statement should have truth value 0. But using the fuzzy formulae, this statement will have truth value 0.3 wrt the model.
Third, FT fails to handle certain external penumbral connections correctly. External penumbral connections refer to statements concerning the border lines between 2 or more vague sets. For example, consider the statement "Mary is tall and Mary is short" concerning the border line between two vague sets: TALL and SHORT. Since according to intuition, a person cannot be tall and short at the same time, this statement should have truth value 0. But using the fuzzy formulae, this statement will have truth value 0.3 wrt the model.

6. 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.
Supervaluation Theory (ST) adopts a different approach towards vagueness. ST views vague concepts as truth value gaps and evaluates truth values of sentences with vague concepts by means of admissible complete specifications (ACSs). A complete specification is an assignment of the truth value 1 or 0 to every individual wrt the relevant 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.

7. 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
Here I illustrate how truth values are evaluated under ST. We consider the same model as given above. Here is an example of ACS, in which John is assigned to the set TALL and Mary is assigned to the set SHORT. Note that among all possible complete specifications arising from this model, only 6 are admissible.
Next I show how ST can rectify the flaws of FT. First, we note that under a complete specification, any vague statement will become non-vague statement whose truth value can then be evaluated using the ordinary method of Predicate Logic. Now a vague statement in the form of a tautology (contradiction) will have truth value 1 (0) under all complete specifications, because complete specifications must obey classical logical laws. Thus, the statement "John is tall or John is not tall", which has the form of a tautology, will have truth value 1 under all complete specifications, and so is true under the supervaluation method.

8. 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
Second, ST will rule out all inadmissible complete specifications related to the borderline cases of one vague set. Here is an example of an inadmissible complete specification related to TALL. This complete specification is inadmissible because Mary is assigned to the set TALL while John, being taller than Mary, is not assigned to TALL. By ruling out all such inadmissible complete specifications, we can then evaluate the truth value of the statement "Mary is tall and John is not tall" correctly.
Third, ST will rule out all inadmissible complete specifications related to the border lines between 2 or more vague sets. Here is an example of an inadmissible complete specification related to the border line between TALL and SHORT. This complete specification is inadmissible because Mary is assigned to the set TALL and SHORT at the same time. By ruling out all such inadmissible complete specifications, we can then evaluate the truth value of the statement "Mary is tall and Mary is short" correctly.

9. 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
However, there is a weakness of ST, i.e. it cannot distinguish different degrees of vagueness because it treats all borderline cases alike as truth value gaps.
Therefore, what we need is a theory that combines some features of FT and some features of ST. To be more precise, we need a theory that can distinguish different degrees of vagueness and yet avoid the flaws of FT. In what follows, I will propose such a combined theory.

10. 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
This combined theory is based on Glöckner's method for evaluating the truth values of statements with vague quantifiers (VQs). This approach distinguishes two types of VQs: semi-fuzzy quantifiers are those VQs that only take crisp (i.e. non-fuzzy) sets as arguments; while fuzzy quantifiers are those VQs that take both crisp and fuzzy sets as arguments.
For example, here is a possible MDF of a semi-fuzzy quantifier "(about 10)". Note that the truth value of a statement with this VQ is determined by the value of |A  B| / |A| – 10, where A and B are crisp sets.
Ordinary quantifiers such as "every" can also be treated as semi-fuzzy quantifiers. Their characteristic functions can be treated as MDFs which output only two truth values, i.e. 0 and 1. For example, here is the MDF of "every".

11. 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 γ; 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
Under Glöckner's approach, all linguistic quantifiers are modeled as semi-fuzzy quantifiers initially. These semi-fuzzy quantifiers are then transformed to fuzzy quantifiers by means of a quantifier fuzzification mechanism (QFM). Here I will summarize how to evaluate the truth value of a vague quantified statement by using a particular QFM.
Suppose we are to evaluate the truth value of the quantified statement Q(X1, … Xn) with n arguments X1, … Xn. First we choose a cut level γ  (0, 1]. Then for each argument X of Q, we define two crisp sets: Xγmin = X γ; Xγmax = X> 0.5 – 0.5γ. Based on these two crisp sets, we can then define a family of crisp sets: Tγ(X) = {Y: Xγmin  Y  Xγmax}. For each possible combination Y1, … Yn of these crisp sets, we can evaluate the truth value of Q(Y1, … Yn) using the MDF associated with Q. We then aggregate all these truth values by using this aggregation formula which makes use of the definition of "m0.5(Z)". This aggregation formula gives us the truth value of Q(X1, … Xn) at the cut level γ. Finally, we combine the truth values at different cut levels by means of the definite integral. The value of the integral will be taken to be the truth value of Q(X1, … Xn).

12. 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)
Although Glöckner's method is based on FT, it shares some similarities with ST. First, 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.
Second, under Glöckner's approach, there is no need to use the fuzzy formulae for Boolean operators because all Boolean operations are done on all possible combinations of crisp operands and then aggregated. Thus, Glöckner's method can handle tautologies / contradictions correctly.
However, to handle internal / external penumbral connections correctly, we need to make modifications to Glöckner's method. Glöckner's original method with the modifications will henceforth be called the Modified Glöckner's Method (MGM).

13. 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
To handle internal penumbral connections correctly, we adopt a new definition for the 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}. In this new definition, the condition after "such that" is to ensure that if an individual x is assigned to a vague set X in a complete specification, then any individual y with equal or higher membership degree wrt X should also be assigned to X.
For illustration, consider this model again. Consider the complete specification ║j  TALL║ = 0, ║m  TALL║ = 1, ║j  SHORT║ = 0, ║m  SHORT║ = 0 that corresponds to Y = {m} as a complete specification of TALL. This complete specification does not satisfy the new definition above because although m  Y and ║m  TALL║  ║j  TALL║, we do not have j  Y. Thus, this complete specification is inadmissible and will be ruled out. In fact, the new definition will rule out all inadmissible complete specifications related to the borderline cases of one vague set. You can check that under the new definition, the truth value of the statement "Mary is tall and John is not tall" will be 0, in accord with our previous discussion.

14. 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
To handle external penumbral connections, we can make use of meaning postulates (MPs) associated with a model. These are statements concerning the relationship between sets in the model. One example of MP is TALL  SHORT = , which states that no individual can be tall and short at the same time.
But the use of MPs leads to some complications. One problem is how to specify the relationship between the MPs and the set specifications of the model? In face of this problem, there are a number of positions that we can take.
First, we may allow "complete freedom" by imposing no constraint on the MPs and set specifications. But this may lead to the undesirable consequence that no ACSs of the sets can satisfy the MPs.
Second, we may allow "0 degree of freedom (0DF)" by imposing the strictest constraint that every possible ACS of the sets should satisfy every MP. But this constraint is so strict that many models in practical applications will be ruled out.

15. 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)})
In between these two extreme positions, I propose that we allow "1 Degree of Freedom (1DF)" by imposing the following constraint (henceforth "1DF constraint"): 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.
Under the assumption that our model and MP(s) satisfy the 1DF constraint, we may now adopt 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)}). In this new formula, the condition after "such that" is to ensure that the complete specifications Y1, … Yn of the fuzzy arguments satisfy every MP in the model.

16. 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
For illustration, consider this model again. This model is associated with the MP TALL  SHORT = . You can check that this model and this MP satisfy the 1DF constraint.
Consider the complete specification ║j  TALL║ = 1, ║m  TALL║ = 1, ║j  SHORT║ = 0, ║m  SHORT║ = 1 that corresponds to Y1 = {j, m} as a complete specification of TALL and Y2 = {m} as a complete specification of SHORT. You can see that Y1 and Y2 do not satisfy the MP because Y1  Y2  . Thus, this complete specification is inadmissible and will be ruled out. In fact, the new definition will rule out all inadmissible complete specifications related to the border lines of 2 or more vague sets. You can check that under the new definition, the truth value of the statement "Mary is tall and Mary is short" will be 0, in accord with our previous discussion.

17. Eg. Every boy loves every girl.
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)})})
According to modern Generalized Quantifier Theory (GQT), quantified statements with both subject and object(s) can be modeled by iterated quantifiers. For example, the sentence "Every boy loves every girl" can be represented by the expression every(BOY)({x: every(GIRL)({y: LOVE(x, y)})}), which, in daily language, means "Every boy x is such that for every girl y, x loves y". Note that this expression can be derived by the operation called "iteration". For details, please refer to references about GQT.

18. 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
MGM can be adapted to quantified statements with iterated VQs. For simplicity, here I only consider quantified statements with two iterated VQs. These statements have the form Q1(A1)({x: Q2(A2)({y: B(x, y)})}), where Q1 and Q2 are VQs.
We can evaluate the truth value of such quantified statements using the following steps. First, for each possible x, determine {y: B(x, y)}, which may be a vague set. Then determine ||Q2(A2)({y: B(x, y)})|| using MGM. By doing so, we will obtain the following vague set: {x: Q2(A2)({y: B(x, y)})} = {||Q2(A2)({y: B(xi, y)})||/xi, …}, where xi ranges over all possible xs. Finally, we can evaluate ||Q1(A1)({x: Q2(A2)({y: B(x, y)})})|| by using MGM.
An example of evaluating the truth value of a quantified statement with iterated VQs can be found in the paper associated with this talk. Note that the above method can be easily generalized to quantified statements with more than 2 VQs.

19. 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.
Finally, it can be shown that MGM possesses the following property: 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.
This property shows that MGM is a generalization of the supervaluation method. Thus, originated from Glöckner's FT framework, MGM is indeed a combination of FT and ST.