The concept of atomicity as species of matter grounds the natural sciences of chemistry, biology, and medicine (and others too), grounding the current logical and mathematical notations with material identities.

My purpose here is to present formal arguments that connect the multiple potential semiotic ascriptions of natural language to the scientific languages of description of natural systems. Also, I seek to respond to the recent request for a systematic approach to “Integral Biomathics” (Ehresmann et al., 2012).

My arguments are designed to span the scientific and logical gaps I identified between the philosophical categorical structures introduced by Robert Rosen and by Ehresmann and Vanbremeersch (Chandler, 2009). In addition, the proposed modal logic corresponds with the abstract, referential notations in such a way that calculations follow from the syntax of the logic developed from chemical methodologies.

I constructed this molecular discourse from three novel representations of informed material causes, following both the logical roots developed by C.S. Peirce (1837-1913) and the practical modern methodologies for chemical “proof of structure":

1. The atomic number, as a semiotic object, is a mathematical structure, an electrically labeled bipartite graph, selectively attracting and repelling other relatives (Chandler, 2009).
2. The table of elements represents an ordered multi-set of unique material properties, each with a unique agency and selective combinatorial potential (Chandler, 2017).
3. A transitive copulative grammar that combines and composes propositional sentences representing material species by transforming diagrams through transition states (Chandler, 2025).

The composition of two atomic sentences into a molecular sentence is an electrical process, a co-mutative occasion that both associates and re-distributes logical particles (Chandler, 2017). This unique quantitative grammar for perplex sentences transforms inter-subjective and inter-predicative representations of matter between a priori and a posteriori representations of numbers and other mathematical objects, such as groups, matrices, graphs, and diagrams. The diagrammatic terminology for this situational generative grammar was created to represent sub-atomic events including catalysis, as observed by chemical methodologies.

The existing scientific notations for the chemical sciences are related to one another by formal group structures (semantic groups, permutation groups, identity groups, space groups, and reproductive groups), roughly related to CS Peirce’s abductive logic.

The logical consistency of chemical methodologies was necessary for the development of molecular biology and also supported the historic development of both thermodynamics and quantum electrodynamics. Therefore, the perplex number system (Chandler, 2009) appears to be formally sufficient to relate the symbolic (grammatical and mathematical) inter-dependency between Kantian a posteriori representation of atomic numbers with or without the a priori representations of space and time.