Generalized quantifiers are the mathematical analogue to natural language determiners. The natural language determiner binds with a noun to form a noun phrase, and the result binds with a verb phrase to form a sentence. For example, consider the following sentences
It has been proposed that the logical analogue for such natural language determiners is a generalized quantifier. Lindstrøm came up with a generalization of quantifier terms based on their role in a sentence. His definition had the function take the extension of the subject to a set of acceptable extensions of the predicate. The extension of the subject and predicate he took to be an element of the power set of the universe, so if the generalized quantifier took the subject (a member of the power set of the universe) to a set of acceptable predicates (which was also a member of the power set), then the domain would be the power set of the universe, and the range would be the power set of the power set of the universe.
€Historical Definition 1: A binary generalized quantifier is a function from the power set of the universe to the power set of the power set of the universe. I.e. a binary generalized quantifier is a function Qm where Qm : P (M) -> P (P (M)).
I will save the term 'generalized quantifier' for a generalization of the definition above, and when I have need, call the above function a model based quantifier. The sentence 'QAB' will be true when, for an appropriate universe M, B is in Qm(A).
Such a classification will include many functions that do not appear to be quantificational in character, however, so some natural language constraints were suggested by Barwise & Cooper, and by Keenan & Stavi. The constraints are as follows:
€ Conservativity: B is in Qm(A) iff A&B is in Qm(A).
€ Extension: If M and N are universes such that M contains N, and A and B are elements of P (N), then B is in Qm(A) iff B is in Qn(A).
€ Quantity: For A, B, C, D in P (M), if |A&B|=|C&D|, |A-B|=|C-D|, |B-A|=|D-C|,
and |(A&B)*|=|(C&D)*|, where |A| is the number of elements in A, and A* is the complement of A in M,then B is in Qm(A) iff D is in Qm(C).
€ Permutation Invariance: For any bijection, ¼, from M to N, B is in Qm(A) iff ¼(B) is in Qn( ¼(A)).
€ Finiteness: |M| < ƒ.
A model-based quantifier which has the properties listed above will be called a BC (Barwise&Cooper) quantifier. Such constraints will rule out some natural language determiners. For example, possession violates Quantity and Permutation Invariance, and the determiners 'only' and 'mostly' violate Conservativity. Finiteness is the most objectionable of the constraints, but it does provide greater simplicity. Under these constraints, the truth of a quantified statement, 'QAB', can be judged by looking at the size of the intersection set, |A&B|, and the size of the difference set, |A-B|.
This leads to an alternative characterization of the BC quantifiers, and a graphical representation due to van Benthem.
€Definition 2: The characteristic of a BC quantifier is a function from NxN -> {T,F}.
In general, the quantifier characteristic maps the ordered pair (|A&B|,|A
B|) to true or false. For example, All will map ordered pairs to true when, and only when, the second coordinate is zero, while Some will map ordered pairs to true when, and only when, the first coordinate is not zero. Additional quantifiers would include half, which maps the ordered pair (|A&B|,|A-B|) to true when, and only when, |A&B|/(|A&B|+|A-B|) = 1/2. As for the graphical representation, I will use a stylistic variant of van Benthem's tree of numbers, and we will have the following table, with points labeled.
(0,0) (1,0) (2,0) (3,0) (4,0) (5,0) ...
(0,1) (1,1) (2,1) (3,1) (4,1) (5,1) ...
(0,2) (1,2) (2,2) (3,2) (4,2) (5,2) ...
(0,3) (1,3) (2,3) (3,3) (4,3) (5,3) ...
(0,4) (1,4) (2,4) (3,4) (4,4) (5,4) ...
(0,5) (1,5) (2,5) (3,5) (4,5) (5,5) ...
: : : : : :
All Some
+ + + + + ... - + + + + ...
- - - - - ... - + + + + ...
- - - - - ... - + + + + ...
- - - - - ... - + + + + ...
: : : : : : : : : :
Half Three
- - - - - ... - - - + - ...
- + - - - ... - - - + - ...
- - + - - ... - - - + - ...
- - - + - ... - - - + - ...
: : : : : : : : : :In addition to the graphs, there will be what I call the solution set of the BC quantifier characteristic. For a given BC quantifier characteristic, Q, the solution set will be T (Q), so for example, T (All) = {(m,n) | n = 0}, T (Some) = {(m,n) | m ‚ 0}, T (Half) = {(m,n) | m/(m+n) = 0} and T (Three) = {(m,n) | m = 3}.
The BC quantifier characteristics will be closed under the normal Boolean connectives, but what I have been studying lately has been the role of relational operators on the BC quantifier characteristics. In natural language the relational operators are 'at least', 'at most', 'more than' and 'less than'. In terms of the generalized quantifiers they can be quite difficult to define. My goal of late has been to define these relational operators as a global function carrying BC quantifier characteristics to BC quantifier characteristics in a manner that accords with natural language usage. So, for 'at least', the function should give the following tables.
At least Half at least Three - + + + + ... - - - + + ... - + + + + ... - - - + + ... - - + + + ... - - - + + ... - - - + + ... - - - + + ... : : : : : : : : : :
at least all but one at least one of the three + + + + + ... - - - + - ... + + + + + ... - - + - - ... - - - - - ... - + - - - ... - - - - - ... - - - - - ... : : : : : : : : : :
With these as hints, all that remains is to find a pattern. One common suggestion is that for 'at least', from each '+' you fill both up and to the right.
€Hypothesis 1: T (at least Q) = {(m,n)|(m',n') in T (Q) and m >= m' and n >= n' .
This will work for many of the BC quantifier characteristics, though it does not obviously apply to the operator on the partitive 'at least one of the three'.
This difficulty could be solved by saying that in the case of the partitive, the operator has minimal scope, thus distinguishing between a typical determiner, and one that is never used, as indicated below.
At least (one of the three) (at least one) of the three - + + + + ... - - - + - ... - + + + + ... - - + - - ... - + + + + ... - + - - - ... - - - - - ... - - - - - ... : : : : : : : : : :
This is not the end of our difficulties. Consider the BC quantifier characteristic
Three of five - - - - - - - - - - - ... - - - - - - - - - - - ... - - - + - - - - - - - ... - - - - - - - - - - - ... - - - - - - + - - - - ... - - - - - - - - - - - ... - - - - - - - - - + - ... - - - - - - - - - - - ... : : : : : : : : : : :
Up-right closure of three of five - - - + + + + + + + + ... - - - + + + + + + + + ... - - - + + + + + + + + ... - - - - - - + + + + + ... - - - - - - + + + + + ... - - - - - - - - - + + ... - - - - - - - - - + + ... - - - - - - - - - - - ... : : : : : : : : : : :
This is clearly not the table for at least one of the three, which follows with the changes blinking.
At least three of five - + + + + + + + + ... - - + + + + + + + + ... - - - + + + + + + + + ... - - - - - + + + + + ... - - - - - - + + + + + ... - - - - - - - - + + ... - - - - - - - - - + + ... - - - - - - - - - - - ... : : : : : : : : : : :
In this case, since T (Three of five) = {(m,n) | m/(m+n) = 3/5}, the temptation is great to say that T (At least three of five) = {(m,n) | m/(m+n) >= 3/5}. I believe this move to be illegitimate, however, since there are many ways of expressing the solution set. In particular,
T (Three of five) = T ((Three of the five) or (60% of the at least 10)).
This makes them the same BC quantifier characteristic, so relational operators should work equivalently on either formulation. Given that there are different formulas for picking out the same solution set, how can we justify using the formula m/(m+n) >= 3/5?
An additional problem involves a property on relational operators that I would like to be true. It seems intuitively true that 'at least' distributes over disjunctions, that is that
T (>=(Q1 or Q2)) = T ((>=Q1) or (>=Q2))
If this is the case, however, then we would have the following table for
>=((Three of the five) or (60% of the at least 10)). - - - + + + + + + + + + ... - - - + + + + + + + + + ... - - - + + + + + + + + + ... - - - - - + + + + + + ... - - - - - - + + + + + + ... - - - - - - - - + + + + ... - - - - - - - - - + + + ... : : : : : : : : : : : :
To make the problem more vivid, consider the BC quantifier characteristic for at least ¼/4. While it may seem strange to suggest such a characteristic, involving an irrational proportion, there are uses for it. Suppose that we inscribe a square with a circle, and drop ball bearings in a random distribution on the square. We then check to see if at least ¼/4 of the ball bearings fell within the circle. This seems to make sense, and I can give the table for this quantifier.
>=(¼-4ths) - + + + + + + + + + + + ... - - - - + + + + + + + + ... - - - - - - - - + + + + ... - - - - - - - - - - - + ... - - - - - - - - - - - - ... - - - - - - - - - - - - ... - - - - - - - - - - - - ... : : : : : : : : : : : :
These were the problems I found with attempting to give a formulation for a relational operator on BC quantifiers. First, relational operators did not apply naturally to partitives. Second, the solution set could be given by different mathematical formula s, so the relations could not focus on any one of the formulas. Lastly, there are BC quantifiers that use the relational operators which have no apparent pre-image.
At the same time that I was grappling with these problems, I was also trying to find
a technique for dealing with natural language determiners for mass nouns. These occur
in such sentences as:
Generalized quantifiers which map measure spaces to measured generalized quantifiers which fit the natural constraints given above are called natural language (NL) quantifiers, and as with the BC quantifiers, there is a characteristic function for NL quantifiers.
€Definition 3: The characteristic of a NL quantifier is a function from R*xR* -> {T,F}.
Where R* stands for the non-negative reals. Instead of using a table to symbolize the solution sets, I will use a graph, with Cartesian style coordinates. Instead of putting '+'s on the graph, I will use points, lines and shaded areas to show what points are mapped to true. Some examples of graphs of NL quantifier characteristics are given below.
two and a half ¼-4ths all but 20
€definition 4: T (>=Q) = {(m,n) | (m',n') is in T (Q) s.t. m+n = m'+n' & m >= m'}.
What this amounts to is including every point that is above, and on the same diagonal as, a point which the characteristic maps to true. I will call this the closure of the graph up the count diagonal.
Let us return to the previous cases in which relational operators on BC quantifiers caused problems. First the partitive one of the three.
one of the three at least one of the three
Next, the case where there are different functions which pick out the same collection of ordered pairs of integers. In this case, the function differs on the non-integral coefficients, and this can be used to explain the difference on the integral coefficients for the relational operators.
>=(3 of 5) >=((3 of the 5)or(60% of the >=10))
Thirdly, the case where the pre-image of the relational operator, has no integral coefficient.
¼-4ths >=(¼-4ths)
The literature on generalized quantifiers currently considers only integral values. I think that the current notion of a generalized quantifier can be made richer by using a broader definition, which includes measure spaces other than those using the count measure, and which uses non-integral values. Not only can it be used for situations where mass nouns are used, it can also be used to define the relational operators on BC quantifiers. Even in the case where you are using count-nouns, relational operators work on functions with non-integral coefficients.
For the record, though I have only used the relational operator 'at least' until now, the definition of the other relational operators is as follows.
€ definition 5:
€a) T (at most Q) = {(m,n) | (m',n') is in T (Q) s.t. m+n = m'+n' & m ¾ m'}
€b) T (less than Q)= {(m,n) | (m,n) is in T (at least Q)}
€c) T (more than Q) = {(m,n) | (m,n) is in T (at most Q)}
This definition, as well as definition 4, will work on any natural language quantifier, but will match the intuitions of natural language only in the case where we look at the non-integral coefficients of natural language quantifiers.
Bibliography
€ Barwise, J. and Cooper, R., (1981), 'Generalized Quantifiers and Natural Languages', Linguistics and Philosophy 4 , 159-219
€ van Benthem, J.,(1985), 'Themes from a Workshop', in van Benthem and ter Meulen (eds), Generalized Quantifiers in Natural Language .
€ van Benthem, J.,(1987), 'Towards a Computational Semantics', in P. Gardenfors (ed) Generalized Quantifiers: Linguistic and Logical Approaches .
€ Keenan, E.L. and Stavi, J., (1986) 'A Semantic Characterization of Natural Language Determiners', Linguistics and Philosophy 9 , 253-326
€ Lindstrøm, Per,(1966), 'First Order Predicate Logic with Generalized Quantifiers', Theoria 32, 186-195
€ Westerstahl, D., (1989), 'Quantifiers in Formal and Natural Languages', in Gabbay and Guenther (eds.), Handbook of Philosophical Logic, Volume IV , D. Reidel, Dordrecht 1989, 1-131
cwkurtz@cctr.umkc.edu