A System of Ontology Based on Identity and Partial Ordering as an Adequate Logical Apparatus for Describing Taxonomical Structures of Concepts

Abstract

Concepts can be arranged in a taxonomical order by means of a partial ordering relation, and a taxonomical structure of concepts is usually supposed to have a linkage to reality. However, here is a logico-ontological jump of significant theoretical importance: a taxonomical structure of concepts does not include in itself any logical ability to relate concepts to objects. That is to say, a mere arrangement of concepts does not logically require at all of any linkage between concepts and reality. There can logically be no direct jump from concepts to reality.
The partial ordering relation required to describe a conceptual taxonomy is in close logical relation to the particle known historically as copula in syllogistic systems. By and large the copula corresponds to the relation IS-A that is today widely used in describing a conceptual taxonomy. We use the symbol ‘⊂’ for IS-A relation. Another relation that relates concepts to objects often called today ‘an-instance-of relation’ will be introduced by definition. This relation will be expressed by the symbol ‘ε’. To define this relation we shall use the functor of (non-reflexive) identity to which we assign the symbol ‘=’.
We aim in this paper to construct a logical system that is appropriate to describing the logical relations between a taxonomy of concepts and reality, which is tantamount to constructing a version of logical Ontology (first constructed by Lesniewski in 1920) on the basis of two primitive functors, i. e. ⊂ and =.