SBVR — quantifications based on cardinality

Doc: dtc/06-03-02
Date: March 2006
Version: Interim Convenience Document
Chapter: 9.1.1.7
Pages:
Nature: Editorial
Severity: minor

Description:

General problems in clause 9.1.1.7:

a. Logical quantifications are intensive; they do not require the materialization of the set of satisfying instances. Cardinality quantifications are extensive; they require materialization of the set. It is not clear that the extensions of all SBVR concepts are sets, and it is clear that many of them cannot be materialized. In general, cardinality-based quantifications can only apply to projections.

Note: The intensive version of the "at most 1" quantification for a predicate P is a "uniqueness rule" (a Necessity): IF P AND P THEN x = y. So it is possible to specify "at most 1" and "exactly 1" quantification without using cardinalities.

b. Cardinality requires a notion of "equality" defined on the instances from which the set is drawn. Otherwise cardinality cannot be measured. SBVR appears to introduce this concept via Reference Schemes, but it does not do so explicitly in discussing cardinality.

c. The current definitions of the cardinality-based 'quantifications' (at least n, at most n, etc.) refer to undefined operations.

Other problems:

In what fact-type is Cardinality a role?
Cardinality is a property of a set, that is, a fact-type about a set, viz. 'set' has cardinality 'non-negative integer'.

The definitions of maximum and minimum cardinality are meaningless. These terms are roles in some unspecified fact types that make use of 'integer' is less than 'integer'.

Recommendation:

Before Cardinality, insert new section heading.

Change the definition of 'cardinality' to read:
non-negative integer designating the number of distinguished things in a collection

Specify the fact-types in which maximum and minimum cardinality are 'roles'. Or don't bother and specify cardinality quantifications as fact-types. E.g. at-most-n quantification:
X has at most n Ys
is interpreted:
for any given X, the cardinality of the projection = 'set of all Y such that X has Y' is-less-or-equal n
Note that such a construct IS a logical formulation, but it is NOT a "logical quantification".

Add the fact type 'set has cardinality'. Explain the relevance of distinguishability for quantifications other than universal and existential quantification.