For the rest of this chapter, we will restrict our attention to standard quanti- fiers; that is, monadic binary quantifiers (of type (1, 1)). In addition, we will make one further restrictions on the quantifiers to be considered in this sec- tion: we will assume that they obey EXT. This was defined in chapter 3; here we repeat definition 3.2.5 simplified to the (1, 1) type:
Definition 5.1.1 (EXT) A quantifier Q of type (1, 1) follows EXT if for all M,M', A 1,A 2 ⊆ M ⊆ M', Q M (A 1,A 2) iff Q M ' (A 1,A 2).
What will allow us to define effective ways to deal with these quantifiers is the following: when dealing with monadic quantifiers, all arguments are sets. Sets do not carry structural information, like relations do. Thus, isomorphisms between sets are simply bijections. In other words, all that matters to check if sets are isomorphic is the cardinality of the sets. When dealing with finite models, as we do, all sets are finite. Thus, we restrict ourselves to finite quan- tifiers, those where all arguments are finite sets. Hence, matters of cardinality can be seen as simply arithmetic predicates on natural numbers. To define quantifiers, then, we can give properties on natural numbers.
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Badia, A. (2009). Implementation and Optimization of Standard GQs. In: Quantifiers in Action. Advances in Database Systems, vol 37. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-09564-6_5
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