Quantification and aggregation over concepts of the ontology

The first phase of developing an intelligent system is the selection of an ontology of symbols representing relevant concepts of the application domain. These symbols are then used to represent the knowledge of the domain. This representation should be \emph{elaboration tolerant}, in the sense that it should be convenient to modify it to take into account new knowledge or requirements. Unfortunately, current formalisms require a significant rewrite of that representation when the new knowledge is about the \emph{concepts} themselves: the developer needs to "\emph{reify}" them. This happens, for example, when the new knowledge is about the number of concepts that satisfy some conditions. The value of expressing knowledge about concepts, or "intensions", has been well-established in \emph{modal logic}. However, the formalism of modal logic cannot represent the quantifications and aggregates over concepts that some applications need. To address this problem, we developed an extension of first order logic that allows referring to the \emph{intension} of a symbol, i.e., to the concept it represents. We implemented this extension in IDP-Z3, a reasoning engine for FO($\cdot$) (aka FO-dot), a logic-based knowledge representation language. This extension makes the formalism more elaboration tolerant, but also introduces the possibility of syntactically incorrect formula. Hence, we developed a guarding mechanism to make formula syntactically correct, and a method to verify correctness. The complexity of this method is linear with the length of the formula. This paper describes these extensions, how their relate to intensions in modal logic and other formalisms, and how they allowed representing the knowledge of four different problem domains in an elaboration tolerant way.