The Making of a System of Systems: Ontology Reveals the True Nature of Emergence Gary Langford Teresa Langford Systems Engineering Program Department of Business Administration Portland State University Portland Community College Portland, Oregon 97201, USA Portland, Oregon 97217, USA gary.langford@pdx.edu teresa.langford@pcc.edu Abstract - A consistent, cohesive, and predictive framework that harbors the ontology and taxonomy of systems and system of systems is described that exposes the true nature of emergence as resulting from interactions that change physical objects. It is shown that the part-whole relationship is represented through the language syntax and semantics of integration and interoperability for systems and system of systems to build ontologies of objects and processes with primitives for objects of objects, functions, and behaviors; and for processes with primitives of cognitions, developing procedures, and modeling. Index Terms – ontology, emergence, system of systems I. INTRODUCTION This paper grew out of the ideas in Stanislaw Leśniewski’s “Foundations of the General Theory of Sets” [1] used to develop a formal mereotopology foundation for a general theory of system and system of systems integration [2]. The profundity of Leśniewski’s works form fundamental ontology of systems and system of systems. The objective of this paper is to introduce a consistent, cohesive, and predictive framework that harbors ontology of systems and system of systems to expose the true nature of emergence. This research fills a gap – a gap that has excited and motivated thinkers who have tried to define core principles and elemental actions of emergence. The aim and utility of studying the nature of emergence is to improve the practice of making and using systems and system of systems. By applying an expository framework of objects-related and process-related concepts, ontology which represents all aspects of knowledge [3] can be applied to determine taxonomy that illuminates the normative constructs of systems and system of systems. This paper introduces an integrative framework that codifies mereology of objects and processes, each explained in terms of an ontological frame that supports a taxonomy that classifies and demonstrates relationships within the context of systems and system of systems. The problem of not having a valid ontology and taxonomy causes costly lifecycle expenses and delays in designing, making, using, sustaining, and repurposing systems and system of systems. The advantages of developing a valid formal ontology are to provide flexible dimensionalities and proper atomic form to allow for integration according to the rules of part-whole mereology so as to capture all relationships between structures, delineate all processes, and stipulate every interaction between objects in both event-based and time-based contexts. A. Mereology of Objects and Processes Beginning with a most benign portrayal, a system of systems is comprised of systems that are both integrated and made interoperable to achieve a set of metasystem functions in which all the systems may participate to varying degrees [4]. From this elemental departure, the relations between a system of systems and its constituent systems are described. Those relations are expressed in terms of part-whole constructs. The system of systems (whole) is made up of systems (parts). The part-whole relationship had its early beginnings with Socrates. Justifiably, every systems thinker-theorist, philosopher- mathematician, economist-accountant, social scientist- biologist, physicist-engineer, and practitioner-manager, recognizes the socio-economic importance of improving communications, fostering interdisciplinary work, identifying and solving system of systems problems, and predicting emergence. The premise of this paper is ontology must be construed through natural language syntax and semantics and substantiated through formal mathematics in order for the syntactic form to be expressed in general, widely applicable taxonomy. Natural language syntax for part-whole relationships is set in accordance with language of integration and interoperability. The goal is for universal ontology. B. Leśniewski Part-Whole Categorial Grammar The seminal works of Stanisław Leśniewski establish categorial grammar that lay out rules of inference, falsification, and logic based on a consistent set of definitions, axioms, theorems, and proofs [5]. Leśniewski’s fundamental axiom formed the foundation from which to construct modern-day mathematics and Boolean algebra [6]. Leśniewski’s axiom and is noncontradictory – which means not false. Leśniewski’s cardinal axiom is comprised of objects and processes. The formulation of relations between objects and processes forms essential ontology of systems and system of systems. The conjunction of parts and wholes is stipulated by the rules of noncontradictory mathematical logic. Leśniewski’s logicism forms the limits and conditions by which objects and processes interact or not. Within the logic of objects and processes [7-12], two frames are constructed – Objective and Subjective [4], Figure 1. The frame for objects has three dimensions – physical objects, interactions of objects that result in functions, and behavioral consequences due to or lack of objects or interacting objects. The frame for processes has 3 dimensions – cognitive activity, procedures (mechanisms), and