Around 1900, set theory—the foundation of mathematics—was threatened by paradox. Russell discovered a set that gives rise to a contradiction: the set of all sets that are not a member of themselves. The solution turned out to be to regard sets as successively formed (or “constructed”) by gathering available objects into a single set. This “process” is iterated a great (in fact, infinite) number of times. By contrast, analogous paradoxes still pose a threat to intensional entities, such as propositions, properties, and various kinds of groups, which are needed in formal semantics and formal ontology, as well as philosophy. Here there is still no agreed-upon solution, a century after set theory was placed on a secure footing.
To remedy this, we extend the “constructional” approach to provide a secure foundation for these intensional entities as well. We use philosophy to make sense of the highly idealized form of “construction” that is invoked, which far outstrips what we humans (or even our computers) can in fact construct. Additionally, we seek inspiration from ideas in constructive mathematics to develop and apply some logical-mathematical tools appropriate for the study of intensional entities. One such idea is so-called predicativity: very roughly, we ban even some mildly circular definitions, which generalize over a domain to which the defined entity would belong. Another such idea is non-instantial generality: having a single generic explanation of why some generalization holds, as opposed to a highly conjunctive explanation that passes through each instance of the generalization. For example, we can explain why every object a has a singleton set {a}, not by considering each and every object, but in terms of a general recipe for constructing singleton sets.
Our aim is to provide consistent foundations for the mentioned intensional entities, user-friendly enough to be helpful in formal semantics, formal ontology, and mathematics. If successful, our work has the potential to do to these disciplines what the constructional approach to sets did to set theory. We aim to provide a foundation for groups (e.g. teams, committees), nominalization (properties derived from adjectives or verbs, such as wisdom or running), and structured propositions.
