John F. Randolph

Psycho-logics: 3. The structure of empathy

A new general theory of the structure of human psycho-dynamics has been proposed ( Hilgartner, 1965 , Hilgartner, 1968 ), and has been translated into the form of an axiomatic system stated in a set-theory notation ( Hilgartner and Randolph, 1969a , Hilgartner and Randolph, 1969b ). This paper constitutes the third instalment of our presentation of this theory. (3) The structure of empathy The ‘awareness’ (Awi) and ‘consciousness’ (Csi) which we examined were derived from contact with inanimate environmental objects. By applying our constructs to the topic of contact between human organismis, we show the structure of the directive correlations which serve as the foundations of human social relations. We then use this extended notation to analyze a fabricated “encounter” in which two strangers meet in an otherwise deserted hallway, and exchange a glance and a warm smile, but no words. This “encounter” is shown as being important to the two participants, and in the notation, it is shown just how and why it proves important.

Psycho-logics: An axiomatic system describing human behavior

Abstract We have listed our undefined terms, stated our premises, given what we claim to be an adequate account of the structure of biological ‘purposiveness’. and set up an algebraic set-theory notation which describes the transactional nature of ‘perception’. We have used this notation to describe the structure of any (mainly ‘perceptual’) encounter, and have used the resulting model to define our key terms. Finally, in order to test the adequacy of our model, we have deployed this logical calculus in the not-mainly-‘perceptual’ situations of ordinary physiological ‘need’ and satisfaction, of frustration, and of danger; and as judged by its handling of these situations, our model appears adequate. In the next paper of this series, we shall analyze the structure of ‘undistorted’ or ‘unimpaired’ human behavior, starting with the intrinsically interesting situation of finding a contradiction between what we expect and what we observe , and the consequent process of changing our premises . In a later paper, we shall explore the structure of ‘unimpaired’ human inter-personal transactions, leading up to a detailed analysis of the neurobiological events which occur when two strangers meet in an otherwise deserted hallway, and exchange a glance and warm smile, but no words. Subsequently, we shall explore the structure of ‘distorted’ or ‘impaired’ human behavior and experience, and the structure of the processes by which “distortions” or “impairments” of behavior and experience can be corrected.

Psycho-logics 2. The structure of ‘unimpaired’ human behavior

At this point, we have provisionally demonstrated the self-consistency of our axiomatic system. We started by stating our undefined terms, and our premises; we set up our logical machinery, we turned this logical machinery one complete revolution, and got our premises back. Further, by analyzing in detail just what our hypothetical organism learned from his encounter with the Ames trapezoidal window, we have provided a representation of ‘sane’ behavior which can be directly compared with the perceivable structure of human experience. The next step will be extend this system so as to show the structure of the integrated social relations between human organisms. Then we shall explore the structure of ‘impaired’ subsets of Cs (‘unsane behavior’). Finally, we shall explore the mechanisms whereby a human organism can learn how to ‘complete’ his hitherto ‘impaired’ subsets of Cs.

SETS AND SPACES

This chapter provides an overview of sets and spaces. It reviews equivalent sets and presents various theorems and definitions. A theorem presented in the chapter states that every superset of an infinite set is infinite and that every subset of a countable set is countable. The union of a countable collection of countable sets is countable, and the set of rational numbers is countably infinite. The chapter describes sequences of sets and metric spaces. It also discusses open sets. The intersection of two open sets is open, and the intersection of a finite number of open sets is open. Every compact set in a metric space is bounded and closed, and every compact subset of a metric space is separable. The chapter also reviews cardinals and connected sets.

SEQUENCES AND SERIES

This chapter provides an overview of sequences and series and discusses the extended real number system and inferior and superior limits. It discusses the limit of a real sequence. The chapter describes the properties of sequences of points in a metric space. A metric space in which every Cauchy sequence converges is said to be a complete metric space. A complete normed vector space is called a Banach space. A complete inner product space is called a Hilbert space. The chapter discusses sequences of complex numbers, the absolute and conditional convergence, and double sequences and series. The chapter also presents various theorems and definitions.

MEASURE AND INTEGRATION

This chapter provides an overview of measure and integration and presents various theorems and definitions. In 1904, Henri Lebesgue introduced a generalization of the notion of length, which is both intuitive and has many applications, extensions, and abstractions. The chapter reviews outer measures and measurability and rings and additivity. It also discusses the Lebesgue integration and the Lebesgues monotone convergence theorem used in many practical and theoretical situations. The chapter also reviews various problems on convergence.

Chapter 5 – MEASURE THEORY

Publisher Summary This chapter provides an overview of measure theory and discusses metric outer measure. If S is any space and μ* is any outer measure on S, then μ* has the additivity property expressed by μ*(A ∪ B ) = μ*A + μ*B provided A ∩ B = O and at least one of A or B is μ-measurable. The theorem that Lebesgue outer measure m* is a metric outer measure is proved. Various properties of lebesgue measure, and σ-algebras are discussed in the chapter. All Borel sets are Lebesgue measurable. The chapter also discusses Lebesgue outer k-measure, Fubinis theorem, outer ordinate sets, and Ergodic theory. Problems on inner measure, and outer measures from measures are also presented in the chapter.

Chapter 7 – DERIVATIVES

Publisher Summary This chapter provides an overview of derivatives. The chapter discusses Dini derivatives. The mean values such as local minimum and local maximum are reviewed. The chapter presents the theorem of Rolles theorem, generalized law of the mean, law of the mean, and l’Hospitals rule. The chapter discusses trigonometry. Trigonometry with angles—intuitive—led to the concept of trigonometric functions of a real variable, radians, and in this extension: (1) an arc of a circle has a length, (2) arcs of any length can be laid off on a circle, and (3) arc length is given by the usual formula, was assumed in the chapter. The chapter reviews the fundamental theorem of algebra. This theorem is important because it shows that the complex number field is algebraically complete in the sense that every nonconstant polynomial with complex coefficients has a complex root. The Fourier series is also reviewd in the chapter.

Chapter 6 – CONTINUITY

Publisher Summary This chapter provides an overview of continuity. The chapter discusses limits and continuity of functions. A function which is continuous at each point of its domain is said to be a continuous function. From many results presented in the chapter, it seems that a measurable function is obtained by any reasonable operations using measurable functions or sequences of measurable functions. The chapter reviews uniformity, Tietze extension theorem, and Weierstrass approximation theorem. A real-valued continuous function on an interval may be approximated uniformly by a polynomial. A polynomial is said to be a rational polynomial if all its coefficients are rational numbers. The chapter also reviews absolute continuity. A theory is presented that shows how intimately absolute continuity and Lebesgue integrability are related. It further discusses equicontinuity, and semicontinuity. Limits inferior and superior for sequences of sets and sequences of real numbers are extended to real-valued functions. All continuous functions with measurable domains are measurable functions. The chapter shows how nearly the converse comes to being true.

Chapter 8 – STIELTJES INTEGRALS

Publisher Summary This chapter provides an overview of Stieltjes integrals. The chapter reviews the Riemann-Stieltjes integrals, the Darboux-Stieltjes integrals, and the Riemann integrals. The integrators of bounded variation are discussed, and a few of the ways as to how Lebesgue integrals occur in connection with either variation or Stieltjes integrals are reviewed in the chapter. The Lebesgue-Stieltjes integrals, and the Lebesgue decomposition and Radon-Nikodym theorems are discussed. Various problems on an abortive idea are also discussed in the chapter.