Formal Axiomatic Theory Research Articles

Interactive generation of new knowledge based on automatic means of logical inference

The work relates to the field of knowledge representation and algorithmization of their processing. To describe the subject of consideration by a set of statements and their processing, logical means of the first order type are used. Unlike the well-known achievements in the field of automatic proof (AP) in formal axiomatic theories, the automatic text synthesis of statements is complicated by the poor formalizability of the value of knowledge concept contained in the text. Based on the development of the calculus of positively-formed standardized formulas (pfs-formulas) the possibilities of analysis automation of the description of the subject matter and interactive generation of meaningful statements to improve this description are investigated. The proposed method of interactive knowledge generation integrates the capabilities of AP with algorithmized human-machine synthesis of new knowledge texts. AP in the calculus of pfs-formulas helps a person not only in the analysis of properties, but also in the choice of the required properties of the description of the subject matter. When the absence of a property desirable for a person in the analyzed knowledge is revealed by means of AP, the hypotheses is formed that logically guarantees the presence of the required property. Variants of hypotheses are automatically synthesized in a preliminary form, which are then refined interactively to improve the content of the generated text of new knowledge. The rules of synthesis are justified. Their application continues the inference in situations of inapplicability of the basic rule of inference in the calculus of pfs-formulas. The extension of calculus by the rules of synthesis transforms it as a first-order calculus of the deductive type into a means of deductive-abductive inference, which differs from the known works using abduction by the absence of a number of restrictions. Illustrative examples demonstrate the representation, analysis and generation of knowledge. These examples also show in interactive knowledge processing the importance of visualization of the processed text, provided by the large-block structure of pfs-formulas constructed from typical quantifiers. In conclusion, the results of the paper and possible directions for the development of the results are formulated.

Formally Deriving the Third Newton’s Law from a Pair of Nontrivial Assumptions in a Formal Axiomatic Theory “Sigma-V”

Formally Deriving the Third Newton’s Law from a Pair of Nontrivial Assumptions in a Formal Axiomatic Theory “Sigma-V”

Formal Inferring the Law of Conservation of Energy from Assuming A-Priori-ness of Knowledge in a Formal Axiomatic Epistemology System Sigma

The research purpose is invention (construction) of a formal logical inference of the Law of Conservation of Energy within a logically formalized axiomatic epistemology-and-axiology theory Sigma from a precisely defined assumption of a-priori-ness of knowledge. For realizing this aim, the following work has been done: 1) a two-valued algebraic system of formal axiology has been defined precisely and applied to proper-philosophy of physics, namely, to an almost unknown (not-recognized) formal-axiological aspect of the physical law of conservation of energy; 2) the formal axiomatic epistemology-and-axiology theory Sigma has been defined precisely and applied to proper-physics for realizing the above-indicated purpose. Thus, a discrete mathematical model of relationship between philosophy of physics and universal epistemology united with formal axiology has been constructed. Results: 1) By accurate computing relevant compositions of evaluation-functions within the discrete mathematical model, it is demonstrated that a formal-axiological analog of the great conservation law of proper physics is a formal-axiological law of two-valued algebra of metaphysics. (A precise algorithmic definition of the unhabitual (not-well-known) notion “formal-axiological law of algebra of metaphysics” is given.) 2) The hitherto never published significantly new nontrivial scientific result of investigation presented in this article is a formal logical inference of the law of conservation of energy within the formal axiomatic theory Sigma from conjunction of the formal-axiological analog of the law of conservation of energy and the assumption of a-priori-ness of knowledge.

A Formal Deductive Inference of the Law of Inertia in a Logically Formalized Axiomatic Epistemology System Sigma from the Assumption of Knowledge A-Priori-Ness

The general purpose of the research—systematical clarifying and explicating the too vague proper philosophical concepts of space, void, matter, motion, inertia, for making a logical harmony between them and the corresponding notions of proper physics. The special purpose of the research—invention (construction) of a formal inference of the well-known Newton’s first law of mechanics within a logically formalized axiomatic epistemology system from a set of precisely defined presumptions. For realizing this aim the following work has been done: a two-valued algebraic system of metaphysics as formal axiology has been applied to philosophical epistemology and philosophy of nature; a formal axiomatic theory called Sigma has been applied to physics for realizing the above-indicated special purpose of the research. Thus, constructing a discrete mathematical model of relationship between universal epistemology and philosophy of physics has been done. Research results: The main hitherto not published significantly new nontrivial scientific result of applied investigations presented in this article is a formal inference of the well-known Newton’s first law of mechanics within the formal axiomatic epistemology system Sigma from conjunction of the formal-axiological analog of the proper-law-of-mechanics (which analog is the formal-axiological law of two-valued algebra of metaphysics) and the assumption of a-priori-ness of knowledge. For obtaining this main research result, a set of accessory nontrivial novelties has been used, for instance; a precise algorithmic definition is given for the notion “law of metaphysics” in the algebraic system of metaphysics as formal axiology; a formal-axiological equivalence in the algebraic system is defined precisely. Precise tabular definitions are given for relevant evaluation-functions determined by evaluation-arguments, for example; “movement of (what, whom) x”; “speed of x”; “vector of x”; “velocity of x”; “magnitude of x”; “finiteness (definiteness) of x”; “dynamical closed-ness (isolated-ness) of x”; “constant-ness, immutability, conservation of x”.

The Theory of Cognitive Consciousness, and Λ (Lambda)

We provide an overview of the theory of cognitive consciousness (TCC), and of [Formula: see text]; the latter provides a means of measuring the amount of cognitive consciousness present in a given cognizer, whether natural or artificial, at a given time, along a number of different dimensions. TCC and [Formula: see text] stand in stark contrast to Tononi’s Integrated information Theory (IIT) and [Formula: see text]. We believe, for reasons we present, that the former pair is superior to the latter. TCC includes a formal axiomatic theory, [Formula: see text], the 12 axioms of which we present and briefly comment upon herein; no such formal theory accompanies IIT/[Formula: see text]. TCC/[Formula: see text] and IIT/[Formula: see text] each offer radically different verdicts as to whether and to what degree AIs of yesterday, today, and tomorrow were/are/will be conscious. Another noteworthy difference between TCC/[Formula: see text] and IIT/[Formula: see text] is that the former enables the measurement of cognitive consciousness in those who have passed on, and in fictional characters; no such enablement is remotely possible for IIT/[Formula: see text]. For instance, we apply [Formula: see text] to measure the cognitive consciousness of: Descartes; and the first fictional detective to be described on Earth (by Edgar Allen Poe), Auguste Dupin. We also apply [Formula: see text] to compute the cognitive consciousness of an artificial agent able to make ethical decisions using the Doctrine of Double Effect.

Приложение логики к философской теологии: формальный дедуктивный вывод утверждения о бытии Бога из допущения априорности знания в формальной аксиоматической теории Сигма

Приложение логики к философской теологии: формальный дедуктивный вывод утверждения о бытии Бога из допущения априорности знания в формальной аксиоматической теории Сигма

Between mathematical formalism, normative choice rules, and the behavioural sciences: The emergence of rational choice theories in the late 1940s and early 1950s

This paper discusses why mathematical economists of the early Cold War period favored formal-axiomatic over behavioral choice theories. One reason was that formal-axiomatic theories allowed mathematical economists to improve the conceptual and theoretical foundations of economics and thereby to increase its scientific status. Furthermore, the separation between mathematical economics and other behavioral sciences was not as clear-cut as often argued. While economists did not modify their behavioral assumptions, some acknowledged the empirical shortcomings of their models. The paper reveals the multifaceted nature of rational choice theories reflected in the changing interpretations and roles of the theories in those early years.

Formal Axiomatic Theories Based on a Three-Valued Logic

Formal axiomatic theories based on the three-valued logic of Lukasiewicz are considered. Main notions related to these theories, in particular, those of Luk-model, Luk-consistent theory, and Luk-complete theory are introduced. Logical calculuses that describe such theories are defined; counterparts of the classical compactness and completeness theorems are proved. Theories of arithmetic based on Lukasiewicz’s logic and on its constructive (intuitionistic) variant are investigated; the theorem on effective Luk-incompleteness is proved for a large class of arithmetic systems. This theorem is a three-valued counterpart of the famous Godel theorem on incompleteness of formal theories. Three-valued counterparts of Presburger’s arithmetic system are defined and proved to be Luk-complete but incomplete in the classical sense. Bibliography: 29 titles.

Cognitive development in advanced mathematics using technology

This paper considers cognitive development in mathematics and its relationship with computer technology, with special emphasis on the use of visual imagery and symbols and the later shift to formal axiomatic theories. At each stage, empirical evidence is presented to show how these forms of thinking are enhanced, changed, or impeded by the use of technology.