This book introduces the properties of conservative extensions of First Order Logic (FOL) to new Intensional First Order Logic (IFOL). This extension allows for intensional semantics to be used for concepts, thus affording new and more intelligent IT systems. Insofar as it is conservative, it preserves software applications and constitutes a fundamental advance relative to the current RDB databases, Big Data with NewSQL, Constraint databases, P2P systems, and Semantic Web applications. Moreover, the many-valued version of IFOL can support the AI applications based on many-valued logics.

This book analyzes the generation of the arrow-categories of a given category, which is a foundational and distinguishable Category Theory phenomena, in analogy to the foundational role of sets in the traditional set-based Mathematics, for defi nition of natural numbers as well. This inductive transformation of a category into the infinite hierarchy of the arrowcategories is extended to the functors and natural transformations. The author considers invariant categorial properties (the symmetries) under such inductive transformations. The book focuses in particular on Global symmetry (invariance of adjunctions) and Internal symmetries between arrows and objects in a category (in analogy to Field Theories like Quantum Mechanics and General Relativity). The second part of the book is dedicated to more advanced applications of Internal symmetry to Computer Science: for Intuitionistic Logic, Untyped Lambda Calculus with Fixpoint Operators, Labeled Transition Systems in Process Algebras and Modal logics as well as Data Integration Theory.

Quantum mechanics, based on the SchrAdinger equation (and its relativistic Dirac's extension) is a statistical theory, here denominated as Statistical Quantum Mechanics (SQM), to differentiate it from the new part of the quantum theory, provided in PART I and II, denominated Individual-particles Quantum Mechanics (IQM). Both of them are necessary components of the quantum theory, as are the Classical Mechanics for Individual objects (ICM), based on the Newton equations, Hamiltonian-Jacobi equations or the Euler-Lagrange equation of motion of individual objects, and the Statistical Classical Mechanics (SCM) based on the Liouville equations. The SQM tells us the various possible outcomes of experiments and the corresponding probabilities if we would do a large number of identical experiments on individual quantum systems. The SQM systems are not all identical but this is the same type of fluctuation that occurs in classical statistical descriptions in SCM. At first sight the situation may not appear very different therefore from the description provided by classical statistical mechanics. In that case however, we have an underlying description (ICM) that provides a complete (i.e. non-statistical) description of the world, which in general is far too complex, however, to be of use. The last PART III of this trilogy is dedicated to the completion of the whole theoretical mechanics, both classical and quantum inside a 9-D time-space manifold of the Universe. Only in this final third volume, this IQM theory, dedicated in the first two volumes only to the elementary particles, is extended also to the non-elementary particles (like hadrons, nucleus, atoms, molecules, and all every-day objects in our common life, up to the biggest non-elementary particles, like the planets, stars, etc.) in our unique Universe. So, each object in our Universe, from the smallest (elementary) to the biggest, can be mathematically expressed by the same mathematical 9-D complex field expression, in a unifying way at which the physical determinism holds for the individual objects at all micro-macro scales in our Universe.

This unique manuscript presents a novel approach to QM by modelling an elementary particle via 3D matter/energy density, which propagates in the open time-space continuum as a rest mass energy density wave packet. This simple idea is based on the fact that any macroscopic object of mass M that occupies a finite 3D volume V can be represented by an energy-density contained in V, so that the integration of this energy-density over V provides the total energy E = Mc^2 . This new theory is fully integrated with the theory of relativity, and completes the quantum theory of Einstein by overcoming the Copenhagen interpretation. The newly introduced partial differential equations describe the relativistic phenomena and, generally, the dependence of a particle's geometrical form (its internal matter distribution) on its velocity and acceleration. A number of well-known physical principles are obtained as derived results of this theory, and are consolidated by a number of detailed examples. Part I, which is dedicated to the completion of QM, is composed of five Chapters. In the first two chapters, the nucleus is analysed in terms of the material discussed. Chapter Three is dedicated to the development of the Lagrangian density for the complex wave packets of the rest mass energy density of an elementary particle, and to the new quantum field theory. The authors obtained the set of new non-Hamiltonian TSPF quantum operators, parameterised by the vector velocity field of energy density with corresponding Hilbert spaces for accelerated particles, and these were valid in any (infinitesimal) local Minkowski time-space. The main results are the new differential equations obtained as conservation laws for Noether currents and Euler-Lagrange equations, which express the exact form of the complex terms used in the differential equations in Chapters One and Two, and introduce the most useful concept of the velocity for any infinitesimal amount of the energy density flux of a particle (the hidden variables). In Chapter Four, a gauge theory and a new explanation of the mass gap conjecture in Yang-Mils theory and of the Higgs mechanism without necessity of the new Higgs field and its bosons, along with a new explanation of double-slit experiments are presented. Thus, the authors obtained a conservative extension of current probabilistic/statistic QM valid for an ensemble of particles, each individual particle, and which is deterministic and compatible by classical mechanics.

The proposed completion of QM theory, with the new non-probabilistic equations and a new mathematical basis for the deterministic quantum mechanics is presented here as a conservative extension of the Standard QM by 3-dimensional (of rest mass energy density) elementary particles. This theory can reshape our view of the quantum world, allowing us to include also classical gravity and to answer some of the deep unresolved questions at the heart of quantum mechanics. This new theory of particles is a constructive approach, alternative to the string theory. Thus, it avoids the infinitary problems of the inverse square for gravitational and electric forces, and may be used as a formal basis for Einsteins unification theory. This second volume is the continuation of Part I and is dedicated to its unification with General Relativity (GR) and with higher dimensions used for the particles properties, such as electrical charge, spin, colours, etc. In Chapter One, after a brief presentation of the main results obtained in Part I, the concepts of GR theory, tensors, and differential pseudo-Rimannian 4-dimensional time-space manifolds are gradually introduced sufficiently for a self-contained presentation and derivation of these new covariant equations. Then, the unification of this newly completed deterministic QM with the GR theory is provided . Chapter Two introduces and examines the extremely small, compacted higher dimensions in order to integrate the charge and spin properties of the elementary particles in a complete quantum physics theory for elementary particles. Chapter Three considers the higher dimensional wave packets of the particles and their reduction to the 4-dimensional time-space, and particles vector kets in the higher-dimensional manifold framework. These are composed of Minkowski time-space and new small, compacted dimensions: a spacelike extra dimension for the quantised spin vectors of the particles and time-like extra dimensions for the quantised charges. After that important properties of this QM theory are considered, the new quantum superposition for the states of a number of individual particles and ERP paradox (entanglement) are discussed.