Formal Axiomatic System Research Articles

Mülahizələr hesabında deduksiya metodunun tətbiqinin elmi-nəzəri metodoloji problemləri

The issue of the emergence of formal axiomatic logical systems due to the emergence of logical antinomies in formal axiomatic systems, specifically the issue of developing formal logical axiomatics in the calculus of considerations was investigated in the considered research. At the same time, in order to determine the characteristics of the implementation of the logical-methodological principles and provisions of the deductive reasoning obviously, conceptual-logical foundations of the calculus of considerations was studied and the main propositions of the calculus of considerations and the analysis of the initial logical operations on them were given in the study. The basic laws of logic and the expression of one logical act with another one were also explained in the research work by giving the concepst of proportional form and tautology in the calculus of considerations. At the same time, main properties and the procedure of setting of the formal deductive theory in the calculus of considerations were studied in the article. In order to provide an adequate characterization of the conceptual and methodological bases of the calculus of considerations in the research work, the basic logical laws of the calculus of considerations developed by the German mathematician and thinker P. Hilbert were also given. In addition, the main principles and methodological provisions of the deductive method were investigated, and the rules of deriving new conclusions from the axioms were interpreted. At the end of the reviewed article, the main principles and methods of the formal axiomatic mathematical systems built on a deductive basis, specifically the calculus of considerations, were studied and a logical-methodological analysis of the calculus of considerations was carried out on this basis. In this framework, the syntactic and semantic analysis of formalized language in formal axiomatic logical systems was carried out separately, the constituent parts of those systems, including the construction scheme, were given.

Classical consequences of constructive systems.

This is a survey of formal axiomatic systems for the three main varieties of constructive analysis, in a common language and with intuitionistic logic, which are as nearly as possible compatible with classical analysis and with one another. Classically sound consequences of principles of intuitionistic mathematics are emphasized. Compatibility with classical analysis is of two kinds. On the one hand, Bishop's constructive mathematics and a very substantial part of intuitionistic analysis are classically correct, sharing with constructive recursive mathematics a neutral subsystem adequate for recursive function theory and elementary real analysis. On the otherhand, each constructive system considered here is separately consistent with the negative interpretation of each of its classically sound subsystems, establishing internal compatibility with the classical context it is intended to refine. A possibly new criterion for the transposability of unique existential quantifiers and a recent theorem of Vafeiadou on the internal compatibility of classical and intuitionistic analysis are included. This article is part of the theme issue 'Modern perspectives in Proof Theory'.

Indeterminacy after AI: Introduction to Special Section

AbstractThere is a close connection between the avant-garde—Dadaist, Surrealist, and Constructivist—use of indeterminate procedures and the early-twentieth-century scientific theories of indeterminacy, such as those of Niels Bohr, Kurt Gödel, and Alan Turing. The Dadaist disruption of ingrained logics, through collage, chance operations, and the simultaneous poem, transcended causality to liberate “an infinite set of processes and possibilities” [1]. Surrealist psycho-archaeological techniques—automatic drawing, writing, and frottage—redefined automaticity as a source of a deeper, nonconscious surreality. Constructivism, similarly, made use of incongruous micro and macro fragments, collapsing orders of magnitude to “reassemble” the world along different axes. In quantum physics, Bohr demonstrated the entanglement of the observer, phenomenon, conceptual framework, and measuring apparatus. Given that all quantum experiments presuppose a question about either position (the particle aspect of a phenomenon) or movement (its wave aspect), but not about both at the same time, they could not deterministically predict an(y) individual atomic process, only statistical regularities. Gödel similarly demonstrated the incompleteness of formal axiomatic systems by showing that there are propositions that can neither be proven nor disproven. Elaborating mathematician David Hilbert’s Entscheidungsproblem—the question about the final decidability or predictability of universal logics—Turing likewise demonstrated that some functions cannot be computed.

Formal System of Categorical Syllogistic Logic Based on the Syllogism <i>AEE-4</i>

Adopting a different method from the previous scholars, this article deduces the remaining 23 valid syllogisms just taking the syllogism AEE-4 as the basic axiom. The basic idea of this study is as follows: firstly, make full use of the trichotomy structure of categorical propositions to formalize categorical syllogisms. Then, taking advantage of the deductive rules in classical propositional logic and the basic facts in the generalized quantifier theory, we deduce the remaining 23 valid categorical syllogisms by taking just one syllogism (that is, AEE-4) as the basic axiom. This article not only reveals the reducible relations between the syllogism AEE-4 and the other 23 valid syllogisms, but also establishes a concise formal axiomatic system for categorical syllogistic logic. We hope that the results and methods will provide a good mathematical paradigm for studying other kinds of syllogistic logics, and that the project will appeal to specialists in logic, linguistic semantics, computational semantics, cognitive science and artificial intelligence.

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”.

On the centenary of Zygmunt Janiszewski (1888-1920): ideals of mathematical practice and the constitution of continuum theory

This paper shows the importance of applying a certain approach to the history and philosophy of mathematical practice to the study of Zygmunt Janiszewski's contribution to the topological foundations of Continuum theory. In the first part, a biography of Janiszewski is presented. It emphasizes his role as one of the founders of the Polish School of Mathematics, and the social, political and military facets in which his intellectual character was revealed, as well as the values that guided his academic and scientific life. Kitcher's view of mathematical practice is then adopted to examine the philosophical conceptions and epistemological style of Janiszewski in relation to the construction of the formal axiomatic system of knowledge about the continua. Finally, it is shown the convenience of differentiating in Kitcher's approach, the methods, procedures, techniques and strategies of practice, and the aesthetic values of mathematics. Keywords: Zygmunt Janiszewski; Continuum theory; Philosophy of mathematical practice; Polish school of mathematics.

Kurt Gödel’s Religious Worldview

Kurt Gödel is well-known as a first-class logician-mathematician, but less well for his proof of God. Godel's Incompleteness Theorems proved that all formal axiomatic systems have inherent limitations. He created also “Gödel numbering,” a special code for writing mathematical formulae. His proof of God was presented logically on the basis of modal axioms. Gödel was sure of God’s personal influence and believed in eternal life of the human soul. He was more than only a “Baptized Lutheran” whose belief was “theistic.” Yet Gödel’s individual assurance of God’s “personal existence“ cannot be viably presented on an interpersonal basis being a “first-person“ type of knowledge and, thus, outside interpersonal conditions for an objective construction beyond a “verbal proof.“ There are categories of reality not easily translatable without a shift in their meaning or a simplifying reduction. The metaphor of an analogy between the brain and its mind as against a computer’s hard- and software does not adequately consider the polarity between the message and its meaning. Gödel’s God was not a modally conceived formal-logical abbreviation of something unattainable for the believer, but a personal Security which does not require any proof.

Relationship between drawing and figures on students’ argumentation and proof

In this article, we wish to explore the influence of the figure and the drawing on students’ argumentation and proof during the solving problem. Our research is based on both Toulmin’s model and Vinner’s concept image and concept definition. After analysing the arguments with the Toulmin model, we analyse the personnel concept definition, concept image evoked about the figure and the effect of drawing which intervene in students’ arguments. Our data suggest that wrong student’s arguments seem to be based on their concept image evoked on the figure manipulated which is in contradiction with the formal axiomatic system. Moreover, the data of the arguments seem to come from an abusive interpretation of the drawings. Observations that emerge from students’ oral and written speeches reveal continuities and gaps between their argumentation and proof.

Axiomatics of Cosmology

The Truth admits of being learned no further than the system theory of mathematical continuum which is the only a priori existing reality. The rest can and must be derivable more deductively than inductively. The symbiosis of both ways does necessarily lead to the unified theory which is a formal axiomatic system. As such, it is subject to the Kurt Goedel’s Meta theorem implying that the final scientific knowledge ever achievable by the human civilization will not be without metaphysical kind of component. It is as if Providence vindicates.

Axiomatics of Cosmology

The Truth admits of being learned no further than the system theory of mathematical continuum which is the only a priori existing reality. The rest can and must be derivable more deductively than inductively. The symbiosis of both ways does necessarily lead to the unified theory which is a formal axiomatic system. As such, it is subject to the Kurt Goedel’s Meta theorem implying that the final scientific knowledge ever achievable by the human civilization will not be without metaphysical kind of component. It is as if Providence vindicates.

Can a machine think (anything new)? Automation beyond simulation

This article will rework the classical question ‘Can a machine think?’ into a more specific problem: ‘Can a machine think anything new?’ It will consider traditional computational tasks such as prediction and decision-making, so as to investigate whether the instrumentality of these operations can be understood in terms of the creation of novel thought. By addressing philosophical and technoscientific attempts to mechanise thought on the one hand (e.g. Leibniz’s mathesis universalis and Turing’s algorithmic method of computation), and the philosophical and cultural critique of these attempts on the other, I will argue that computation’s epistemic productions should be assessed vis-à-vis the logico-mathematical specificity of formal axiomatic systems. Such an assessment requires us to conceive automated modes of thought in such a way as to supersede the hope that machines might replicate human cognitive faculties, and to thereby acknowledge a form of onto-epistemological autonomy in automated ‘thinking’ processes. This involves moving beyond the view that machines might merely simulate humans. Machine thought should be seen as dramatically alien to human thought, and to the dimension of lived experience upon which the latter is predicated. Having stepped outside the simulative paradigm, the question ‘Can a machine think anything new?’ can then be reformulated. One should ask whether novel behaviour in computing might come not from the breaking of mechanical rules, but from following them: from doing what computers do already, and not what we might think they should be doing if we wanted them to imitate us.

The Constitution as an Axiomatic System

The Constitution is considered as an informal axiomatic system. The strategy proposed by the authors rests on the following propositions: (1) axioms are considered as contextual definitions of those concepts by means of which they are formulated; and (2) the main requirement for this type of system is internal consistency. The first proposition is necessary for considering the Constitution as an informal axiomatic system, while the second is sufficient, because the approach proposed, apart from consistency, must certainly consider the requirements for formal axiomatic systems, such as independence and completeness. The authors argue that the Constitution can be compared to axiomatic constructions in modern science in the sense that is given in the research on the logic and methodology of deductive sciences. This analogy is appropriate to the extent to which constitutional provisions are interpreted as the basic elements of the legal system, just as in the formal sciences axioms are regarded as basic principles that define the main features of the formal system. This means that the Constitution itself is seen as coherent, consistent discourse that contextually defines the meaning of the basic terms of the legal system.

On a Geometric Foundation of Mathematics (Su una Fondazione Geometrica della Matematica)

Frege with Grundlagen der Arithmetik and Hilbert with Grundlagen der Geometrie are two outstanding figures that are attributed to a fundamental role in the arithmetization of mathematics. However, the latest writings of Frege, released posthumously, testify to his reflection on the nature of mathematics. In them Frege argues that mathematics is all about geometry and begins a theory that aims to define complex numbers geometrically. For this purpose he introduced a notion of identical relationships that tends to set up a geometric aspect ratio. In addition, Grundlagen der Geometrie can be given a radically different reading from that which emphasizes Hilbert's exclusive intention to found geometry on a purely formal axiomatic system. Several authors argue that by his work, and in particular through the arithmetic of the segment s introduced in it, Hilbert wanted to emancipate the geometry from instruments outside her, such as numbers, finding them within a substantially synthetic geometry.

Bridging the gap between analytic and synthetic geometry: Hilbert’s axiomatic approach

The paper outlines an interpretation of one of the most important and original contributions of David Hilbert’s monograph Foundations of Geometry (1899), namely his internal arithmetization of geometry. It is claimed that Hilbert’s profound interest in the problem of the introduction of numbers into geometry responded to certain epistemological aims and methodological concerns that were fundamental to his early axiomatic investigations into the foundations of elementary geometry. In particular, it is shown that a central concern that motivated Hilbert’s axiomatic investigations from very early on was the aim of providing an independent basis for geometry. Accordingly, these concerns about an independent grounding for elementary geometry determined very clear methodological constraints in the process of embedding it into a formal axiomatic system. It is argued that Hilbert not only sought to show that geometry could be considered a pure mathematical theory, once it was presented as a formal axiomatic system; he also aimed at showing that in the construction of such an axiomatic system one could proceed purely geometrically, avoiding concept formations borrowed from other mathematical disciplines like arithmetic or analysis.

Elementary Process Theory: a formal axiomatic system with a potential application as a foundational framework for physics supporting gravitational repulsion of matter and antimatter

AbstractTheories of modern physics predict that antimatter having rest mass will be attracted by the earth's gravitational field, but the actual coupling of antimatter with gravitation has not been established experimentally. The purpose of the present research was to identify laws of physics that would govern the universe if antimatter having rest mass would be repúlsed by the earth's gravitational field. As a result, a formalized axiomatic system was developed together with interpretation rules for the terms of the language: the intention is that every theorem of the system yields a true statement about physical reality. Seven non‐logical axioms of this axiomatic system form the Elementary Process Theory (EPT): this is then a scheme of elementary principles describing the dynamics of individual processes taking place at supersmall scale. It is demonstrated how gravitational repulsion functions in the universe of the EPT, and some observed particles and processes have been formalized in the framework of the EPT. Incompatibility of Quantum Mechanics (QM) and General Relativity (GR) with the EPT is proven mathematically; to demonstrate applicability to real world problems to which neither QM nor GR applies, the EPT has been applied to a theory of the Planck era of the universe. The main conclusions are that a completely formalized framework for physics has been developed supporting the existence of gravitational repulsion and that the present results give rise to a potentially progressive research program.

Principia iuris de Luigi Ferrajoli

El texto desarrolla dos observaciones y evidencia, sucesivamente, algunos puntos críticos. La primera observación es que la construcción propuesta tiene fundamento en la afirmación que señala una intrínseca interdependencia entre la democracia y el Derecho y entre el Derecho y la razón: de aquí nace la elección de una teoría axiomatizada del Derecho. El sistema axiomático formal, por tanto, no es parte accesoria, eliminable de la obra; es, por el contrario, su intrínseca estructura y razón. La segunda observación es que la estructura que surge es la de un «constitucionalismo fuerte» que identifica un sistema de límites y también de obligaciones al poder, que sin ellas sería soberano, de la mayoría. A partir de estas observaciones preliminares, se analizan algunos puntos. El primero reside en el carácter intrínsecamente inacabado del sistema, característica que FERRAJOLI presenta como un valor adjunto. El segundo es la relación de la formalización con la explicabilidad del lenguaje y, por tanto, con la comunicabilidad del Derecho. El tercero es el doble rol del sistema axiomático: definir un posible universo de discurso o describir rigurosamente un universo ya definido de otro modo. El último nodo es la aplicabilidad del método propuesto a los sistemas de Derecho contemporáneo, fuertemente influenciados por el contacto directo con los sistemas del common law.

Are There Absolutely Unsolvable Problems? Godel's Dichotomy

This is a critical analysis of the first part of Godel's 1951 Gibbs lecture on certain philosophical consequences of the incompleteness theorems. Godel's discussion is framed in terms of a distinction between objective mathematics and subjective mathematics, according to which the former consists of the truths of mathematics in an absolute sense, and the latter consists of all humanly demonstrable truths. The question is whether these coincide; if they do, no formal axiomatic system (or Turing machine) can comprehend the mathematizing potentialities of human thought, and, if not, there are absolutely unsolvable mathematical problems of diophantine form. Either ... the human mind ... infinitely surpasses the powers of any finite machine, or else there exist absolutely unsolvable diophantine problems.

Skolem and pessimism about proof in mathematics

Attitudes towards formalization and proof have gone through large swings during the last 150 years. We sketch the development from Frege's first formalization, to the debates over intuitionism and other schools, through Hilbert's program and the decisive blow of the Gödel Incompleteness Theorem. A critical role is played by the Skolem-Lowenheim Theorem, which showed that no first-order axiom system can characterize a unique infinite model. Skolem himself regarded this as a body blow to the belief that mathematics can be reliably founded only on formal axiomatic systems. In a remarkably prescient paper, he even sketches the possibility of interesting new models for set theory itself, something later realized by the method of forcing. This is in contrast to Hilbert's belief that mathematics could resolve all its questions. We discuss the role of new axioms for set theory, questions in set theory itself, and their relevance for number theory. We then look in detail at what the methods of the predicate calculus, i.e. mathematical reasoning, really entail. The conclusion is that there is no reasonable basis for Hilbert's assumption. The vast majority of questions even in elementary number theory, of reasonable complexity, are beyond the reach of any such reasoning. Of course this cannot be proved and we present only plausibility arguments. The great success of mathematics comes from considering 'natural problems', those which are related to previous work and offer a good chance of being solved. The great glories of human reasoning, beginning with the Greek discovery of geometry, are in no way diminished by this pessimistic view. We end by wishing good health to present-day mathematics and the mathematics of many centuries to come.

Refined common knowledge logics or logics of common information

In terms of formal deductive systems and multi-dimensional Kripke frames we study logical operations know, informed, common knowledge and common information. Based on [6] we introduce formal axiomatic systems for common information logics and prove that these systems are sound and complete. Analyzing the common information operation we show that it can be understood as greatest open fixed points for knowledge formulas. Using obtained results we explore monotonicity, omniscience problem, and inward monotonocity, describe their connections and give dividing examples. Also we find algorithms recognizing these properties for some particular cases.

Husserl's Two Notions Of Completeness

In this paper I discuss Husserl's solution of the problem of imaginary elements in mathematics as presented in the drafts for two lectures hegave in Gottingen in 1901 and other related texts of the same period,a problem that had occupied Husserl since the beginning of 1890, whenhe was planning a never published sequel to Philosophie der Arithmetik(1891). In order to solve the problem of imaginary entities Husserl introduced,independently of Hilbert, two notions of completeness (definiteness in Husserl'sterminology) for a formal axiomatic system. I present and discuss these notionshere, establishing also parallels between Husserl's and Hilbert's notions ofcompleteness.