From One Object to a Structured World: Individuals, Properties, and Prototime

This paper starts from an extremely sparse ontology of one individual and properties, and asks how a structured world can emerge from the growth of instantiated structure. I work with a basic language of individuals and properties, a comprehension-like scheme for properties, and a minimal existence principle for individuals. Within this framework I prove a simple “from one to two” theorem: for any individual a there is a corresponding mirror property Ma that holds of exactly those individuals that are indistinguishable from a in purely qualitative terms. I then construct configuration spaces built from a single individual a and admissible sets of properties, and show how a “prototime” can be read off the monotonic growth of instantiated structure: more instantiated properties correspond to “later” stages. On this basis I introduce ranked prototemporal frames and prove representation theorems: under a natural Richness Assumption on the property domain, every finite (and every countable) ranked frame can be represented as a configuration space of one individual plus properties. This provides a structural bridge between a minimal individuals-and-properties ontology and abstract orders of “growing structure”. The paper is the third part of a four-paper project on the genealogical structure of logic, laws, and the universe. It connects the protological work of For a Genealogy of Proto-Logic with the genealogical theory of laws and logical time in an Evolving Block Universe developed in The Genealogy of the Universe: Basic Logic, Logical Time, and Historical Laws.