General Axioms Research Articles

When I Use the Term Humanistic Psychology...

All societies need ways to account for the existential facts of life. Humanistic psychology may be viewed as a recent attempt to provide a coherent story acceptable to an increasingly secular society. The author believes that even after 25 years, humanistic psychology is not yet a discipline but is what Kuhn refers to as "preparadigmatic." It is an amorphous field of inquiry in which there are no general axioms, no consensus as to the important questions or standards of evaluation, and wherein quite different forms of utterance are made about the nature of human reality. The boundaries between religious-type utterances and propositions made as scientific fact are frequently blurred. The author discusses how this creates the possibility both for fanaticism and for the recent court ruling in Alabama declaring secular humanism a religious belief and thereby prohibited in the public schools. Also discussed is the concept of an autonomous self that is held by humanistic psychology as a psychological "fact" even though it can be shown to be culturally given.

The "Prime Number Theorem" for the Periodic Orbits of a Bernoulli Flow

(1988). The “Prime Number Theorem” for the Periodic Orbits of a Bernoulli Flow. The American Mathematical Monthly: Vol. 95, No. 5, pp. 385-398.

On active linear compartments

Active linear compartments are capable of transforming input substances into each other. Based on several general axioms of functioning of these compartments their mathematical description in the form of matrix convolution operators is derived. Results are provided, regarding relationships between two alternative modes of mathematical description of linear active compartments. Then, properties of systems of active compartments are considered, based on results from the renewal theory.

Computing protected circumscription

This paper deals with computing circumscription in the case of Horn data with additional protection (indefinite data), an intermediate investigation between Reiter's result on predicate completion and Lifschitz's efforts to make general (formula) circumscription more efficient as a computational tool. Reiter has shown a close tie between McCarthy's circumscription and Clark's predicate completion. Here we investigate a similar tie between an extended version of circumscription involving protected data, and an extended version of predicate completion. When we have a fully ground atomic protected theory, we show that an extension to the relational algebra can be used to obtain all (and only) correct answers. When general Horn axioms are added to the protected theory, we show that Horn axioms also can be used to compute sound answers; however, some correct answers will not be found.

Classical thermodynamics of homogeneous systems based upon Carnot's general axiom

Classical thermodynamics of homogeneous systems based upon Carnot's general axiom

An axiomatic scheme more general than quantum theory

An axiomatic scheme generalizing the operational approach to quantum theory is described. Only quite general axioms ensuring the existence of well-behaved probabilities are postulated. The space-time location of macroscopic apparatus interacting with the object is explicitly taken into consideration. The states and observables are defined and their time development is considered. The classification of physical processes with respect to their reversibility or irreversibility in time is given. The conditions of Lorentz and translational invariance are formulated. Linear transformations corresponding to operations on the object are introduced. In the case of reversible processes these transformations form an algebra and linear representations of the Poincaré group arise naturally. These results are, in general, invalid for irreversible processes. The position of quantum theory in the scheme described is clarified.

Lexicographic additive differences

Several authors have identified sets of axioms for a preference relation ≻ on a two-factor set A × X which imply that ≻ can be represented by specific types of numerical structures. Perhaps the two best-known of these are the additive representation, for which there are real valued functions f A on A and f X on X such that ( a, x) ≻ ( b, y) if and only if f A ( a) + f X ( x) > f A ( b) + f X ( y), and the lexicographic representation which, with A as the dominant factor, has ( a, x) ≻ ( b, y) if and only if f A ( a) > f A ( b) or { f A ( a) = f A ( b) and f X ( x) > f X ( y)}. Recently, Duncan Luce has combined the additive and lexicographic notions in a model for which A is the dominant factor if the difference between a and b is sufficiently large but which adheres to the additive representation when the difference between a and b lies within what might be referred to as a lexicographic threshold. The present paper specifies axioms for ≻ which lead to a numerical model which also has a lexicographic component but whose local tradeoff structure is governed by the additive-difference model instead of the additive model. Although the additive-difference model includes the additive model as a special case, the new lexicographic additive-difference model is not more general than Luce's model since the former has a “constant” lexicographic threshold whereas Luce's model has a “variable” lexicographic threshold. Realizations of the new model range from the completely lexicographic representation to the regular additive-difference model with no genuine lexicographic component. Axioms for the latter model are obtained from the general axioms with one slight modification.

Absolute temperatures as a consequence of Carnot's general axiom

1. Program In our book, Concepts and Logic of Classical Thermodynamics,1 Mr. BHARATHA and I developed classical thermodynamics on the basis of Part I of Carnot's General Axiom, namely, the motive power of a Carnot cycle is positive and is determined by its operating temperatures and by the amount of heat it absorbs. For a given body, then, there is a function G such that for any Carnot cycle # L(%) = G(0+,0 ,C + (^j)>0. (1) The domain of G is the set of operating temperatures and heats absorbed that may appertain to Carnot cycles for the body in question. It is part of the definition of a Carnot cycle that 6+>6~ and that C+(^)>0. This definition and 1 C. Truesdell & S. Bharatha, Concepts and Logic of Classical Thermodynamics as a Theory of Heat Engines, Rigorously Constructed upon the Foundation Laid by S. Carnot and F. Reech, N.Y. etc., Springer-Verlag, 1977.

Search strategy and selection function for an inferential relational system

An inferential relational system is one in which data in the system consists of both explicit facts and general axioms (or “views”). The general axioms are used together with the explicit facts to derive the facts that are implicit (virtual relations) within the system. A top-down algorithm, as used in artificial intelligence work, is described to develop inferences within the system. The top-down approach starts with the query, a conjunction of relations, to be answered. Either a relational fact solves a given relation in a conjunct, or the relation is replaced by a conjunct of relations which must be solved to solve the given relation. The approach requires that one and only one relation in a conjunction be replaced (or expanded) by the given facts and general axioms. The decision to expand only a single relation is termed a selection function. It is shown for relational systems that such a restriction still guarantees that a solution to the problem will be found if one exists. The algorithm provides for heuristic direction in the search process. Experimental results are presented which illustrate the techniques. A bookkeeping mechanism is described which permits one to know when subproblems are solved. It further facilitates the outputting of reasons for the deductively found answer in a coherent fashion.

A Spatial-Discounting Theory of Interaction Preferences

In a variety of situations involving human interaction, actors compare alternative interaction opportunities in terms of their perceived distance or accessibility to opportunities within some relevant space of attributes. In this paper, a theory of interaction preferences is proposed in which actors implicitly discount potential opportunity interactions in terms of these perceived distances. This form of implicit discounting behavior is shown to be characterized by three general axioms of observable preference behavior. The special case of exponential spatial discounting is also characterized explicitly. In addition, conditions are established under which discounted preferences are representable by utility functions. Finally, the dependence of discounted preferences on the underlying configuration of distances is examined in depth.

Municipal Interfaces in the Third Sector: A Negative View

Functions performed by governments vary widely from state to state and city to city. There truly is no exclusive function, except law making. And today, after the Watergate disclosures, even that assumption is under attack as being erroneous. Local governmental functions are determined by state law, policies of the governing body, or by public referenda. What is a governmentally provided service, and what is performed by the private sector, varies widely from city to city. There is no real rationale as to what factors combine to make these local determinations. But, over a quarter decade, this author has devised a general axiom which may explain most local decisions:

A modification of the axiom system of quantum mechanics

A general axiom system, including both classical and quantum mechanics as special cases, is proposed. On the basis of the axioms assumed it is shown that the logic of experimentally verifiable propositions concerning any (classical or quantum) physical system may be embedded into an atomistic complete lattice. Moreover, in the quantum case (characterized in the paper by validity of the superposition principle) a generalization of the Piron's representation theorem for the logic is stated.

MRPPS?An interactive refutation proof procedure system for question-answering

In this paper a system termed the Maryland refutation proof procedure system (MRPPS) is described. MRPPS is an interactive system which gives the user the ability to create and maintain a core-bound data base and to input queries either as well-formed formulas in the first-order predicate calculus or as clauses. MRPPS consists of three basic components: (1) a set of inference rules for making logical deductions, many of which are based on the resolution principle; (2) a search strategy for heuristically determining the sequence in which to select base clauses and to perform deductions on clauses already generated; and (3) a base clause selection strategy that uses heuristic and semantic information for determining which data axioms and general axioms are to be brought to bear on the problem. These three components are described and MRPPS is compared to related work.

Duality Theory for Infinite Horizon Convex Models

Often it is desirable to formulate certain decision problems without specifying a cut-off date and terminal conditions (which are sometimes felt to be arbitrary). This paper examines the duality theory that goes along with the kind of open-ended convex programming models frequently encountered in mathematical economics and operations research. Under a set of general axioms, duality conditions necessary and sufficient for infinite horizon optimality are derived. The proof emphasizes the close connection between duality theory for infinite horizon convex models and dynamic programming. Dual prices with the required properties are inductively constructed in each period as supports to the state evaluation function.

Cohomology of Sheaves of Holomorphic Functions Satisfying Boundary Conditions on Product Domains

This paper considers sheaves of germs of holomorphic functions which satisfy certain boundary conditions on product domains in ${{\mathbf {C}}^n}$. Very general axioms for boundary behavior are given. This includes as special cases ${L^p}$ boundary behavior, $1 \leq p \leq \infty$; continuous boundary behavior; differentiable boundary behavior of order $m,0 \leq m \leq \infty$, with an additional Hölder condition of order $\alpha ,0 \leq \alpha \leq 1$, on the $m$th derivatives. A fine resolution is constructed for those sheaves considered, and the main result of the paper is that all higher cohomology groups for these sheaves are zeŕo.

Cohomology of sheaves of holomorphic functions satisfying boundary conditions on product domains

This paper considers sheaves of germs of holomorphic functions which satisfy certain boundary conditions on product domains in ${{\mathbf {C}}^n}$. Very general axioms for boundary behavior are given. This includes as special cases ${L^p}$ boundary behavior, $1 \leq p \leq \infty$; continuous boundary behavior; differentiable boundary behavior of order $m,0 \leq m \leq \infty$, with an additional Hölder condition of order $\alpha ,0 \leq \alpha \leq 1$, on the $m$th derivatives. A fine resolution is constructed for those sheaves considered, and the main result of the paper is that all higher cohomology groups for these sheaves are zeŕo.

Computer Simulation of Multiple Strategies in Human Binary Prediction

Most models of human binary-prediction behavior (two-choice probability learning) have been of the mathematical learning theory variety and have given little consideration to S's memory for the event sequence and to the kinds of strategic behavior he may be using. This study considers reasons for dissatisfaction with that approach and proposes a new type of model which is based on a memory structure like that described by Atkinson and Shiffrin. The model is of the computer-simulation type and proposes that S uses each of several prediction strategies in approximate proportion to its relative frequency of success. The theory is developed from a number of general assumptions and specific axioms and is tested on the data from a new binary-prediction experiment. The discussion concludes with comparisons to results from studies in mathematical learning theory and suggestions for further development of the new type of model.

Relativistic Theories of Gravitation

THE results which general relativity predicts for the three well known tests (the motion of the perihelion, the deflexion of light rays and the gravitational red-shift) and for new tests which we have proposed1,2 in the one-body problem are also obtained in certain Lorentz-invariant theories of gravitation. Whitrow and Morduch3,4 have examined the Lorentz-invariant theories of gravitation developed earlier and have calculated most of the known empirical tests for these theories in order to analyse the possibility of deciding experimentally between them and Einstein's general theory of relativity. We have continued the work of Whitrow and Morduch by classifying all possible Lorentz-invariant theories of gravitation that satisfy certain general axioms and by analysing the one-body problem in all these theories.

The independence of a weak axiom of choice

1. The purpose of this paper is to show that, if the axioms of a system G of set theory are consistent, then it is impossible to prove from them the following weak form of the axiom of choice: (H1) For every denumerable set x of disjoint two-element sets, there is a set y, called a choice set for x, which contains exactly one element in common with each element of x. Among the axioms of the system G, we take, with minor modifications, Axioms A, B, C of Gödel [6]. Clearly, the independence of H1 implies that of all stronger propositions, including the general axiom of choice and the generalized continuum hypothesis.The proof depends upon ideas of Fraenkel and Mostowski, and proceeds in the following manner. Let a be a denumerable set of objects Δ0, Δ1, Δ2, …, the exact nature of which will be specified later. Let μj = {Δ2j, Δ2j+1} for each j, c = {μ0, μ1, μ2, …}, and b = [the sum set of a]. By transfinite induction, construct the class Vc which is the closure of b under the power-set operation. For each j, it is possible to define a 1–1 mapping of Vc onto itself with the following properties. The mapping preserves the ε-relation, or, more precisely, .

Quantal aspects of scientific information

This paper is an attempt to clarify some aspects of the approach to experimentation suggested by the author in a recent publication in the Philosophical Magazine, (hereafter referred to as P.M.). The concept of "amount of information" is shown to have three distinct senses in current literature. Two of these are definable as numerical features of the logical pattern of propositional relations which we make to represent a result. The third measures the relative unexpectedness of the pattern, which may or may not be connected with its numerical features. Since logical patterns can be built up from discrete quantal elements, the information enabling them to be built is quantised by our use of 'yes-or-no' logical forms as scientific statements. The number of discrete 'elementary propositions' in a given description cannot be altered by any complete reformulation. This is seen to be the basis of our ability to 'barter' certain quantities for one another - e.g. accuracy for speed of response in a galvanometer or a communication-channel. The term "selective information" is suggested to distinguish the third sense of 'information' (called amount of detail in P.M.) from the first two, whioh measure respectively the number of independent features (structural information) and the weight of evidence or precision (metrical information) in a result. 'Selective information' is the measure of information currently used by communication engineers, and a distinguishing title appears to be essential. From the standpoint here adopted the various uncertainty-relations of physics illustrate a general axiom expressing the quantal nature of the logical descriptions which we make. Some other practical and theoretical implications of the theory are examined.