Preface

Ontology in the philosophical sense is concerned with the nature of being as well as with respective basic concepts. Aristotle called ontology the first philosophy. Despite its metaphysical nature, this first philosophy or ontology is a significant attempt to introduce a systematic approach to the process of thinking about the world surrounding us, about our conceptual thinking and about us ourselves.

Today's advanced science inherited the original ontology's efforts to systemize and conceptualize. In this respect the concept of ontology nowadays is a non-speculative methodology for studying reality objects and used tools, both of which are important for our orientation in the physical, technical, mental, and social worlds. As such, we deal with the process of studying, as well as with the outcomes of such study, of the objects observed or created by man and their respective concepts, relations between them, and relations between their systems in different fields of science.

Understood in this modern and precise way, this book applies the meaning of ontology in a search for semantic meanings of objects and concepts. It examines the relations between general or abstract terms and their meanings using a variety of logics such as automated indexation of databases. Therefore, the meaning of "to be" and "how to be" is exercised in complex or abstracted cases, for example, in number theory. The objective is to model, categorize, and conceptualize the knowledge available, and to design relevant schemes such as function-related diagrams or tree structures. Then, pending on the field of study, it is possible to discuss, introduce, and compare a variety of ontologies as the formal models representing our knowledge of the field or the problem. As such, it is possible to discuss ontology as it relates to experimental organization, software and systems engineering, artificial intelligence, the Semantic Web, and informatics and information sciences.

Through these kinds of ontologies we understand the models with standard structures of entities, classes, qualities, and relations. Such models are explicit, created using suitable language and formalized descriptions of the given systems of the objects or their respective concepts and relations. Then we speak about a model or a data model of the problems. The language used for these purposes can be formal (e.g., languages of physical or mathematical theories), semiformal, or informal, especially in the initial stages of studying the problems.

However, ontologies are not only the final formal and declarative representations, models, or data models of given problems but they are also the methodology and, consequently, the method and the process of creating these declarations or models. Ontology as a process creates, uses, and provides, as its output, a model or descriptive ontology. Such a model is a glossary of definitions of concepts corresponding to the objects, a thesaurus of definitions of the relations among the concepts and the respective objects. Thus, the ontology in this descriptive sense is both vocabulary and grammar that are used to keep and pass over the knowledge of the problems studied.

This book *Ontological Analyses in Science, Technology and Informatics* is the illustration of the modern and scientific application of ontology.

Chapter 1 deals with logic inference ontology used in the theory of proof based upon the language of physics; Chapter 2 examines knowledge-based pattern ontology in the Resource Description Framework (RDF) language in Semantic Web applications; Chapter 3 deals with implementation of the ontological XOL language in cross-application communication; Chapter 4 discusses ontological studies in the fields of diagnosis and expert systems in health and food; and Chapter 5 covers management ontology and taxonomy.

This book is designed for theorists as well as those persons dealing with designing, developing, managing, and decision-making in the field of ontology.

I would like to thank the publishing process managers and technical staff at IntechOpen for their commitment, friendly effort, and support throughout the production of this book. Finally, I express my thanks to all the contributing authors for their valuable chapters.

> **Ing. Bohdan Hejna, Ph.D., dr. h. c.** Department of Mathematics, University of Chemistry and Technology, Prague, Czech Republic

**Chapter 1**

Theorems

**self-referential and is consistent**.

*Bohdan Hejna*

**Abstract**

inference

**1. Introduction**

*Thermodynamics* [4].

Common Gnoseological Meaning

We will demonstrate that the *I.* and *the II. Caratheodory theorems* and their common formulation as the *II. Law of Thermodynamics* are physically analogous with the *real* sense of the *Gödel*'s wording of his *I.* and *II. incompleteness theorems*. By using physical terms of the *adiabatic changes* the Caratheodory theorems express the properties of the *Peano Arithmetic inferential process* (and even properties of any *deductive and recursively axiomatic* inference generally); as such, they set the physical and then logical limits of any real inference (of the sound, not paradoxical thinking), which can run only on a physical/thermodynamic basis having been compared with, or translated into the formulations of the Gödel's proof, they represent the first historical and *clear* statement of gnoseological limitations of the deductive and recursively axiomatic inference and sound thinking generally. We show that **semantically understood** and with the language of logic and metaarithmetics, the full meaning of the **Gödel proof expresses the universal validity**

**of the** *II.* **law of thermodynamics and that the Peano arithmetics is not**

1

**Keywords:** arithmetic formula, thermodynamic state, adiabatic change,

To show that the real/physical sense of the Gödel incompleteness theorems that the very real sense of them—is the meta-arithmetic-logical analog of the Caratheodory's claims about the *adiabatic system* (that they are the analog of the sense of the *II.* Law of Thermodynamics), we compare the states in the *state space* of an *adiabatic thermodynamic system* with *arithmetic formulas* and the *Peano inference* is compared with the *adiabatic changes* within this state space. The *whole set of the states* now *not achievable adiabatically* represents the existence of the states on an adiabatic path, but this fact is not expressible adiabatically. This property of which is the

<sup>1</sup> The reader of the paper should be familiar with the Gödel proof's way and terminology; *SMALL CAPITALS* in the whole text mean the Gödel numbers and working with them. This chapter is based, mainly, on the [1–4]. This paper is the continuation of the lecture *Gödel Proof, Information Transfer and*

of Gödel and Caratheodory

## **Chapter 1**
