In Frege and Russell’s line, the main tool to propagate ontology toward higher orders is realized by quotienting sets by equivalence relations. Indeed, this leads to produce, at the next order, separated individualities (disjoint non empty subsets covering the original set) and this in such a way that this separation is conceptually generated from a relation at the initial order.
This way to relate ontology and equivalence relations is deeply Parmenidian in inspiration : non only because it considers individualities independently of time (first point), but also because it propagates at the next order the idea of an autonomy of individualities, i.e. the idea that the individuality of individuals is independent of any relational commitment between them (second point).
"Denotational semantics of programs" (the research programme launched by Strachey and Scott around 1969) can be seen as a first step to substitute to Parmenidian ontology an Heraclitean ontology, in the sense that it faces the first point (individualities in time): in a computational universe, transformation comes first and separation phenomenas are generated by (and compatible with) these transformation.
In that talk, I will present my current work on a complementary program toward a non Parmenidian ontological point of view facing the second point (Agonal ontology) which starts by introducing a generalization of equivalence relations, collusions (total and surjective collusive relations). After having presented their main properties, notably with respect to the partitioning purpose, I will conclude about logical potential interpretations.