The distinction between abstract and concrete entities, especially its precise characterization, remains underexplored in foundational ontology research. This paper aims to constitute the initial steps towards a formal ontology of abstracta and concreta. We begin by presenting three existing criteria (epistemic, spatiotemporal and causal) for the abstract/concrete distinction. We illustrate them with some well-known upper ontologies. After examining the shared assumption by the three criteria that any entity is either abstract or concrete but not both, we develop an alternative and more general framework for formally representing abstract and concrete entities. In particular, we propose a relational account of them by introducing the relation of “concretization”. The pivotal idea is that being abstract (or concrete) amounts to being concretized by (or concretizing) some other entity. We also briefly discuss a concretization-based reinterpretation of the spatiotemporal criterion and universals.

Abstract The semantics in ordered abelian groups Scott proposed for Łukasiewicz’s many-valued logic fails to be sound for one direction of one of the rules Scott gave for implication. We show this by a counterexample Urquhart has used to justify that in his own semantics, every formula has to have a least point at which it is valid. While this condition would make Scott’s semantics sound, it would cause a problem with its completeness. The question arises whether one can still amend Scott’s semantics so as to make it both sound and complete or better stick to Urquhart’s semantics anyway.