One of my colleagues called the other day and asked if we still relied on the distinction between intensional and extensional sets (really, intensionally and extensionally defined sets). Yes, even more so now.

An extensional set is one whose members are enumerated. An intensional set is one where individuals gain membership through some sort of rule. The employees in your organization are extensionally defined (e.g., someone puts you in the employee master file and you’re an employee). Cheap hotels within ten miles of the Denver Airport would be an intensionally defined set. No one is maintaining this specific list, and membership will fluctuate based on rates.)

Why bother with the distinction? In traditional design, almost everything is extensional. Each entity in your design has its own table and/or its own OO class. Individuals gain membership by someone sticking the individual into the table or class (through a user interface to be sure, but it’s just as overt as if they were dealing directly with the underlying classes).

In OWL, with inference, we have the possibility of inferring membership (of defining intensional sets). Yet, as I look at existing ontologies, it’s rare to find designers taking advantage of this. I think that a few decades of designing exclusively with extensional sets have blunted our thinking.

On the flip side, you can’t make all your classes intensional. There is a regress problem here. I can say an employee is a Person with an unexpired Employment Contract. This is an intensionally defined set, but it relies on the Person and Employment Contract sets. These may or may not be intensionally defined. If they are, you need to look a bit further to find the definition behind them.

Our experiments and review of other ontologies suggest that in useful ontologies it’s hard to get beyond about half and half, but this is a worthwhile goal to shoot for. A lot of the power in semantics comes from consistently using and applying inference.

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