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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.

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”.

Historically, our theme is situated within the triangle of three of Kant's students: Hegel (1770-1831), Bolzano (1781-1848) and Peirce (1839-1914). All three wanted to change Kant's strict separation of philosophy and science by developing a new conception of logic. Bolzano inaugurated the so-called linguistic turn of philosophy which became the guiding principle of all analytical philosophy (Dummett, 2014) and he opposed Hegel’s unity of concept and object of knowledge. Charles Peirce took a middle position, a position that is expressed in his so-called Pragmatic Maxim (Peirce, CP 5.3). Taken together we might say that a universal principle of complementarity of meaning and reference, or of meaning and information (in the sense of Shannon) finds its origin in Post-Kantian philosophy. We encounter here the very same approach of principled thinking endorsed by Einstein in physics (special theory of relativity) or by the formal axiomatic approach in mathematics (Hilbert)! Key Words: Bolzano, Hegel, Peirce; Complementarity of sense and reference; Geometry from Euclid to Einstein; Hilbert. Historicamente, nosso tema se situa dentro do triângulo de três alunos de Kant: Hegel (1770-1831), Bolzano (1781-1848) e Peirce (1839-1914). Todos os três queriam mudar a separação estrita de Kant entre filosofia e ciência, desenvolvendo uma nova concepção de lógica. Bolzano inaugurou a chamada virada linguística da filosofia, que se tornou o princípio orientador de toda filosofia analítica (Dummett, 2014) e se opôs à unidade de conceito e objeto de conhecimento de Hegel. Charles Peirce assumiu uma posição intermediária, posição que se expressa em sua assim chamada Máxima Pragmática (Peirce, CP 5.3). Tomados em conjunto, podemos dizer que um princípio universal de complementaridade de significado e referência, ou de significado e informação (no sentido de Shannon) encontra sua origem na filosofia pós-kantiana. Encontramos aqui a mesma abordagem do pensamento baseado em princípios endossada por Einstein na física (teoria da relatividade especial) ou pela abordagem axiomática formal na matemática (Hilbert)! Palavras-chave: Bolzano, Hegel, Peirce; Complementaridade de sentido e referência; Geometria de Euclides a Einstein; Hilbert.

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. Este artigo mostra a importância de aplicar uma certa abordagem da história e da filosofia da prática matemática ao estudo da contribuição de Zygmunt Janiszewski para os fundamentos topológicos da Teoria do continuum. Na primeira parte, é apresentada uma biografia de Janiszewski. Ela enfatiza seu papel como um dos fundadores da Escola de Matemática da Polônia, e as facetas sociais, políticas e militares nas quais seu caráter intelectual foi revelado, assim como os valores que guiaram sua vida acadêmica e científica. A visão de Kitcher sobre a prática matemática é então adotada para examinar as concepções filosóficas e o estilo epistemológico de Janiszewski em relação à construção do sistema axiomático formal de conhecimento sobre o contínuo. Finalmente, mostra-se a conveniência de diferenciar na abordagem de Kitcher, os métodos, procedimentos, técnicas e estratégias da prática, e os valores estéticos da matemática. Palavras-chave: Zygmunt Janiszewski; Teoria do Continuum; Filosofia da prática matemática; Escola de Matemática da Polônia.

This paper looks into the writings of Jurgen Rodig (1942–1975) on legal logic with the aim of identifying those results which still appear to be essential contributions to legal theory. Of particular relevance are (i) the reasons why propositional and predicate logic in its classsical form should be considered adequate and by and large sufficient to represent formal legal reasoning. Other issues discussed in this paper are (ii) Rodig’s views concerning specifically legal variants of logical arguments, (iii) the axiomatic method in legal theory, (iv) ways of substantiating general norms as a central step in what constitutes the legal syllogism, (v) the correct method of interpreting written law, (vi) the question as to whether there are formal characteristics of general norms, and (vii) structural properties of legal definitions and of similar legislative drafting tools.

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.

Classic of Changes is a Chinese cultural classic born more than 3000 years ago. Its profound philosophical thoughts and the use of divination have brought Classic of Changes to a strong oriental mysticism. The view of the heaven and man of yin and yang and the five elements states of Classic of Changes are completely different from the Western elemental theory of ancient Greece. The latter gave birth to classical and modern scientific theories, and the yin and yang and the eight trigrams symbol has become synonymous with oriental mysticism. In fact, the cosmology of the Holism of Classic of Changes is a precious scientific heritage of mankind. The axiomatization of the symbolic formal system of Classic of Changes aims to unveil the veil of oriental mysticism and provide oriental wisdom for the development of modern science. Transforming the symbolic system of Classic of Changes into a formal axiom system is the crystallization of the fusion of wisdom between the East and the West. The axiomatization of the symbolic formal system of Classic of Changes shows that the oriental and the western scientific tradition harmony but not sameness and there is no conflict. Classic of Changes can also be interpreted by the axiomatic system like Euclid’s Elements. The main contribution of this paper is that the author skillfully uses mathematical language to formulate the system of Classic of Changes, reconstructs the ideological system of Classic of Changes with the axiomatic method and realizes the scientificalization of Chinese classical natural philosophy. The formal axiom system of Classic of Changes may give us such a revelation: the gap between the oriental and western scientific traditions is mainly in the axiomatization, but the lack of oriental scientific tradition may be the bottleneck of the development of modern science.

En este trabajo se exploran los principales planteamientos filosóficos de la teoría de la prueba de David Hilbert. Específicamente, el trabajo se centra en sus ideas respecto de la lógica, el concepto de prueba, la axiomática, el concepto de verdad, la metamatemática, el conocimiento a priori y la naturaleza general del conocimiento científico. El objetivo principal es mostrar y caracterizar su enfoque epistemológico acerca de la fundamentación del conocimiento, donde la lógica aparece como garantía de dicha fundamentación. Hilbert supone que el apriorismo proposicional, que propone para sustentar las matemáticas, sustenta —a su vez— un método general para el tratamiento del problema en otros ámbitos como lo es el de las ciencias naturales. Dicho método es la axiomática. Se trata entonces de contribuir a recuperar y actualizar el pensamiento filosófico de Hilbert acerca del papel que juega la lógica para la totalidad del conocimiento científico.

The paper aims at coping with the difficult problem of rationally uniting astonishingly huge amount of qualitatively different modal logics. For realizing this aim artificial languages of symbolic logic and the axiomatic methodology are used. Therefore, the method of constructing and studying formal logic inferences within the axiom system under investigation is exploited systematically. Inventing and elaborating a hitherto not-considered axiomatic system of epistemology uniting normal and not-normal modal logics is the new nontrivial scientific result of this work. History of philosophy and systematical philosophy, formal ethics and formal aesthetics, philosophical epistemology and analytical theology, philosophy of law and philosophy of science are among the important fields of application of the nontrivial abstract-theoretic principles demonstrated in this paper. Using the above-indicated machinery the author has arrived to the following main conclusion: the famous philosophical principles of utilitarianism, hedonism, optimism, pragmatism, fideism, falsifiability, verifiability, “Hume’s Guillotine”, “naturalistic fallacies” et al have not absolutely indefinite (unlimited) but quite definite (limited) sphere of relevant applicability; the precise formal definition of the border-line of mentioned sphere of relevance is the axiomatic one submitted and discussed in the paper. This general conclusion is instantiated in the text by several particular conclusions concerning explication and clarification of specific philosophical ideas and principles, for example, the one of kalokagathia. The author concludes that constructing and investigating the axiomatic systems of universal philosophical epistemology is indispensable for adequate representing human knowledge in artificial intellectual systems, for instance, in autonomous AI‑robots

Set theory faces two difficulties: formal definitions of sets/subsets are incapable of assessing biophysical issues; formal axiomatic systems are complete/inconsistent or incomplete/consistent. To overtake these problems reminiscent of the old-fashioned principle of individuation, we provide formal treatment/validation/operationalization of a methodological weapon termed “outer approach” (OA). The observer’s attention shifts from the system under evaluation to its surroundings, so that objects are investigated from outside. Subsets become just “holes” devoid of information inside larger sets. Sets are no longer passive containers, rather active structures enabling their content’s examination. Consequences/applications of OA include: a) operationalization of paraconsistent logics, anticipated by unexpected forerunners, in terms of advanced truth theories of natural language, anthropic principle and quantum dynamics; b) assessment of embryonic craniocaudal migration in terms of Turing’s spots; c) evaluation of hominids’ social behaviors in terms of evolutionary modifications of facial expression’s musculature; d) treatment of cortical action potentials in terms of collective movements of extracellular currents, leaving apart what happens inside the neurons; e) a critique of Shannon’s information in terms of the Arabic thinkers’ active/potential intellects. Also, OA provides an outer view of a) humanistic issues such as the enigmatic Celestino of Verona’s letter, Dante Alighieri’s “Hell” and the puzzling Voynich manuscript; b) historical issues such as Aldo Moro’s death and the Liston/Clay boxing fight. Summarizing, the safest methodology to quantify phenomena is to remove them from our observation and tackle an outer view, since mathematical/logical issues such as selective information deletion and set complement rescue incompleteness/inconsistency of biophysical systems.

This paper explores archival material concerning the reception of Leonard J. Savage’s foundational work of rational choice theory in its subjective-Bayesian form. The focus is on the criticism raised in the early 1960s by Daniel Ellsberg, William Fellner, and Cedric Smith, who were supporters of the newly developed subjective approach but could not understand Savage’s insistence on the strict version he shared with Bruno de Finetti. The episode is well known, thanks to the so-called Ellsberg Paradox and the extensive reference made to it in current decision theory. But Savage’s reaction to his critics has never been examined. Although Savage never really engaged with the issue in his published writings, the private exchange with Ellsberg and Fellner, and with de Finetti about how to deal with Smith, shows that Savage’s attention to the generalization advocated by his correspondents was substantive. In particular, Savage’s defense of the normative value of rational choice theory against counterexamples such as Ellsberg’s did not prevent him from admitting that he would give careful consideration to a more realistic axiomatic system, should the critics be able to provide one.

The positivistic views that dominated the early debate on the foundations of mathematics, at the beginning of the 20th century, survived the “negative results” that have shown the limits of the axiomatic approach since the 1930s. Rigour, abstraction and symbolism have been confused with formalism, based on finite strings of signs, pre-given axioms, and potentially mechanisable rewriting rules. This contributed to major clarifications in the mathematical praxes but obscured the limits of formalisms due to the exclusion of the historical creation of sense proper to any science. We expand on this sometimes fruitful confusion with some case studies. We then hint to the historical creation of sense as a component of an epistemology of mathematics. We continue with an analogy with genocentric approaches in biology, as similar positivistic views resurfaced there fifty years later. Finite sequences of letters in the DNA would completely determine ontogenesis and phylogenesis, according to the Central Dogma of molecular biology. Limits and “negative evidence” have been disregarded while searching for the “gene for” everything. Alternative perspectives require a reconstruction of the sense of history as locus for the constitution of any object of biological knowledge. In particular, the historicity of biological evolution will be understood in terms of changing phase spaces and of the role of rare events in all phylogenetic trajectories. The analysis of the evolutionary production of variability, adaptivity and ecosystemic diversity is a key component of the project we hint to, as part of a renewed relation to the biological environment.

Aristotle’s syllogistic is the first ever deductive system. After centuries, Aristotle’s ideas are still interesting for logicians who develop Aristotle’s work and draw inspiration from his results and even more from his methods. In the paper we discuss the essential elements of the Aristotelian system of syllogistic and Łukasiewicz’s reconstruction of it based on the tools of modern formal logic. We pay special attention to the notion of completeness of a deductive system as discussed by both authors. We describe in detail how completeness can be defined and proved with the use of an axiomatic refutation system. Finally, we apply this methodology to different axiomatizations of syllogistic presented by Łukasiewicz, Lemmon and Shepherdson.

Fil: Fernández de Castro, Max. Universidad Autónoma Metropolitana. Departamento de Filosofía; México. Fil: Torres Falcón, Yolanda. Universidad Autónoma Metropolitana. Departamento de Filosofía; México. Fernández de Castro, M. y Torres Falcón, Y. (2020). La constitución del programa de Hilbert. Metatheoria, 10(2), 31-50. En las páginas que siguen pretendemos dar una visión panorámica y esquemática de la evolución del programa formalista que resulta de los estudios recientes de notas de curso hasta hace poco inéditas. Analizaremos primeramente ciertos elementos del programa (la preferencia por el método axiomático, el estructuralismo y el logicismo). En segundo lugar observaremos cómo, una vez el programa establecido en 1920 (aunque con cierta vaguedad), diversos finitismos con una base común fueron ensayados por Hilbert y Bernays hasta 1931, en una tentativa por definir con precisión su programa y llevarlo a buen término. El resultado es el de un complejo programa de investigación en continua evolución. In the pages that follow, it is our intention to present a panoramic and schematic view of the evolution of the formalist program, which derives from recent studies of lecture n otes that were unknown until very recently. Firstly, we analyze certain elements of the program (the preference for the axiomatic method, as well as structuralism and logicism). Secondly, we observe how, once the program was established in 1920 (albeit somewhat vaguely), in the period up to 1931, different types of finitism with a common basis were tried out by Hilbert and Bernays, in an effort to define their program precisely and bring it successfully to fruition. The result is a complex research program in constant evolution.

We characterize four types of agentive knowledge using a stit semantics over branching discrete-time structures. These are ex ante knowledge, ex interim knowledge, ex post knowledge, and know-how. The first three are notions that arose from game-theoretical analyses on the stages of information disclosure across the decision making process, and the fourth has gained prominence both in logics of action and in deontic logic as a means to formalize ability. In recent years, logicians in AI have argued that any comprehensive study of responsibility attribution and blameworthiness should include proper treatment of these kinds of knowledge. This paper intends to clarify previous attempts to formalize them in stit logic and to propose alternative interpretations that in our opinion are more akin to the study of responsibility in the stit tradition. The logic we present uses an extension with knowledge operators of the Xstit language, and formulas are evaluated with respect to branching discrete-time models. We also present an axiomatic system for this logic, and address its soundness and completeness.

This chapter proposes to extend Smedslund’s axiomatic system of psycho-logic (PL) into a psycho/socio-logic theory that is based upon insights from social theory and from the so-called linguistic turn in the social sciences. It will be argued that developing a conceptual system for psychology as Smedslund did makes only sense if it is embedded in a broader context of social theory since psychological phenomena cannot be separated from the social realm. Smedslund focusses upon the agency of people, but one needs also to take into account the impact of structures on people. To this end, a reformulation of some of the axioms of PL will be presented that take as a starting point the notion that persons are social and moral beings and that the study of persons should start with the conversational context in which they operate.

The end of the nineteenth century and the beginning of the twentieth century marked a turning point in the way mathematicians approach their discipline. Even more than the attempts to refound the corpus on the basis of set theory, the successes of the axiomatic method challenged the traditional conception of the meaning of mathematical statements, which had hitherto been largely based on a spontaneous belief in an adequacy between these statements and their empirical and physical correlates.

Purpose Considering current development of Grey Systems Theory (GST), we can come up with the following thesis: practical applications are a dominant subject of research. Thus, what seems to be symptomatic for relatively young knowledge disciplines, the authors observe the presence of imbalance between the development of GST application tools and theory’s epistemological and methodological background. As for GST, epistemological and methodological problems are becoming visible especially in the issues of determining a clear criterion of demarcation of this kind of a theory from others. In other words, this problem can be reduced to the issue of a precise determination of what the category of a grey system and grey information is. This problem is of great importance for further development and popularisation of GST in the world of science. Realising its significance, the purpose of this paper is to create a general overview of Grey Systems epistemology and afterwards create axiomatic and formal frames for a category of greyness. Design/methodology/approach In order to achieve set goals, two research approaches were accepted. In the area of inference about epistemology of GST an approach characteristic of an analytical philosophy was used, whereas in the case of axiomatic and formal frames for a category of greyness the authors referred to terms of a set theory and the principles of a pragmatic logic. Findings The result of research is to formulate a concept of a grey system and a concept of grey information in the context of a process of cognition. Moreover, a function of greyness and other fundamental categories of GST will be defined in an axiomatic way. Originality/value The paper presents a new consistent frame for the issues of methodological and epistemological backgrounds of GST. An original concept is to refer in considerations to a newly proposed grey space. This space was used for a formal justification of such elementary categories as grey numbers, a weight function of whitenization or grey sequences. The value of achievements shown in the paper is underlined by the fact that proposed theoretical constructions require further development and they can potentially open up new research trends in the GST.

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.

How should one conceive of the method of mathematics, if one takes a naturalist stance? Mathematical knowledge is regarded as the paradigm of certain knowledge, since mathematics is based on the axiomatic method. Natural science is deeply mathematized, and science is crucial for any naturalist perspective. But mathematics seems to provide a counterexample both to methodological and ontological naturalism. To face this problem, some naturalists try to naturalize mathematics relying on Darwinism. But several difficulties arise when one tries to naturalize in this way the traditional view of mathematics, according to which mathematical knowledge is certain and the method of mathematics is the axiomatic method. This paper suggests that, in order to naturalize mathematics through Darwinism, it is better to take the method of mathematics not to be the axiomatic method.

For the first time in analytic philosophy the universal theory of knowledge is precisely formulated and defined as an axiomatic system ?. For formulating it four axiom schemes are used. Exploiting meta-language gave a possibility to make the axiomatic definition (of the system ?) more simple and compact (in comparison with the options submitted by the author before). The system ? under construction and definition consistently unites in one whole normal and non-normal modal logics; this gives a possibility consistently to synthesize in one conceptual scheme the rationalistic and empiricist paradigms of philosophizing about knowledge in general. Systematical constructing and investigating discrete mathematical models of philosophy of knowledge in general and further perfecting axiomatic systems of universal philosophical epistemology (in particular, studying the system ?) is indispensable for constructing adequate metaphysic sub-system of artificial intelligence of autonomous robots.

The subject matter of this research is Kant’s apriorism underlying Hilbert’s formalism in the philosophical grounding of mathematics as a self-sufficing system. The research aim is the invention of such a logically formalized axiomatic epistemology system, in which it is possible to construct formal deductive inferences of formulae—modeling the formalism ideal of Hilbert—from the assumption of Kant’s apriorism in relation to mathematical knowledge. The research method is hypothetical–deductive (axiomatic). The research results and their scientific novelty are based on a logically formalized axiomatic system of epistemology called Σ + C, constructed here for the first time. In comparison with the already published formal epistemology systems Ξ and Σ, some of the axiom schemes here are generalized in Σ + C, and a new symbol is included in the object-language alphabet of Σ + C, namely, the symbol representing the perfection modality, C: “it is consistent that…”. The meaning of this modality is defined by the system of axiom schemes of Σ + C. A deductive proof of the consistency of Σ + C is submitted. For the first time, by means of Σ + C, it is deductively demonstrated that, from the conjunction of Σ + C and either the first or second version of Gödel’s theorem of incompleteness of a formal arithmetic system, the formal arithmetic investigated by Gödel is a representation of an empirical knowledge system. Thus, Kant’s view of mathematics as a self-sufficient, pure, a priori knowledge system is falsified.

The value of the axiomatic method in conjunction with Muncaster's PropCalc Workplace to analyze and increase the precision of social science theory is illustrated through applications that include political philosophy, ethical reasoning, and theories of attitude change. All of the examples were originally expressed through verbal statements. A formal logical analysis is applied to the US Declaration of Independence which provides a set of propositions upon which governments are based, Einstein's essay on the laws of science and the laws of ethics, and basic concepts of balance and dissonance theories of attitude change. In each case, symbolic representations of the verbal language are then subjected to analysis for consistency and redundancy among the propositions and sample conclusions from each model are presented. The Muncaster's PropCalc is a valuable aid in identifying the integrity and consequences of the models. The discussion includes an examination of the relationship between the models and their correspondence with reality.

The origin of the attempt to calculate infinity in mathematics goes back to a century. God died because of the attempt of mathematics to draw the Infinite, that is, the divine into the world of worldly calculation. But as the infinite secularization process progressed, the paradox of set theory brought mathematics into crisis again, and new questions such as “How can we eliminate these paradoxes?” At this point, Gilles Deleuze questions whether this paradox is a disease that must be eliminated. He tells us that the moment we face the limit of paradox, our thinking begins. The meaning is not already given, but it depends on sequencing, and paradox can only be positive if we accept it as the beginning of the thought. From this point of view, the paradox that has been regarded as a disease to be removed from mathematics can serve as an opportunity to create healthy thoughts. Could it be a reason for health, according to the ‘way of dealing with disease’? This paper starts from these questions. First of all, I will briefly review the theorem of Gehdel, which caused a crisis in the completeness or recklessness of the truth in mathematics and what Cohen"s imperfections of the axiomatic system prove. And I would like to see the points where the paradoxes appear in mathematics as the starting point of new thought and the place where ‘externality’ appears, and to relate this to the ‘event’ of Alain Badiou and the reason of ‘being a relic’. Through this, we try to find the possibility of the reason that can be caught at the point of impossibility of paradox and abyss.

Tyler Andrew Wunder, in his article, “Alvin Plantinga on Paul Draper’s evolutionary atheology: implications of theism’s non-contingency (Wunder in International Journal for the Philosophy of Religion 74: 67–75 2013),” argues that Plantinga makes a serious error regarding probabilities in his critique of Draper. Properly modified, Wunder believes the argument “works,” but only in a trivial sense. This paper argues that Wunder’s objection, based on an assumed probability calculus, is merely asserted; whereas, there are other competing axiomatic systems consistent with Plantinga’s treatment of probability. As to the modified argument, it is demonstrated that Wunder mistakenly concludes that two key propositions are contradictory. The consequence of this is not that Plantinga’s argument “works” in a trivial sense, but rather that the argument becomes incoherent. Lastly, this paper will explore the consequences of both Wunder’s and Plantinga’s assumptions concerning conditional probability for Draper’s evidentiary argument and Plantiga’s Evolutionary Argument Against Naturalism.

Nowadays, a very important theory, the mental models theory, is demonstrating that it is able to explain most of the results in reasoning experiments reported by the cognitive science literature. However, this has a consequence. The mental models theory is mainly focused on content and meaning, and its theses can lead to reject the idea that syntax plays a role in the human mind and that reasoning is logical. But, in this paper, I try to show that it is possible to accept the basic framework of the mental models theory and, at the same time, to continue to claim that there are syntactic and formal logical processes coherent with the way our mind works. To do that, I argue that, even accepting that the mental models theory describes correctly the processes why certain combinations of possibilities are detected, it can be stated that the relationships between such combinations indicated by the theory are consistent with, for example, the modal axiomatic system K.

The present topic is that of limits—the metamathematical limits of axiomatic systems, epistemic limits of explanation, linguistic limitations of expression, conceptual limits of conceivability, and, ultimately, questions of ontological and metaphysical limits as well. The limitations of axiomatic demonstration and of mechanical computation are clear from the Turing and Godelian traditions. Section 16.2 pursues extensions and analogies to limitations intrinsic in the structure of explanation, restrictive on a Principle of Sufficient Reason (PSR). In Sect. 16.3 we consider the limitations on expression entailed by recursive linguistic structure, extending the argument from single languages to sets of possible languages and showing that even the properties of languages inevitably outstrip the properties expressible within those languages. Section 16.4 considers epistemic implications, extending the discussion beyond language to incompleteness of any body of conceivable truths in the face of a demonstrably larger realm of fact. Notwithstanding this amorphousness, something can nevertheless be shown regarding its general character that merits further consideration.

Presented article is the second of two articles, the aim of which is to introduce the reader has no special mathematical training, with the possibilities of application of mathematical methods developed in the scientific direction of “Conceptual analysis and design of systems of organizational management (CAD SOM)”, designed to solve a variety of tasks, such as technical and humanitarian spheres on the basis of the proposed methodological approach to the mathematization of the theoretical knowledge. At the heart of this methodological approach is a process of conceptualization, which is understood as a theoretical study of qualitative aspects of a selected domain using mathematical forms (axiomatic) theory, the locking connection between the concepts of logical derivability characterizing this subject area. Designed axiomatic theory – conceptual scheme – is the basis for building database structures, decision-making processes, a variety of phenomena subject area, structure and genesis of domain analysis and other tasks. One of the main advantages of the sending of methodological approach is the ability to work with complex regions based on the controlled synthesis tool terminal theory of conceptual schemes, explicated simple fragments of the subject area. Given the non-mathematical preparation of the reader, the contents of the methods illustrated by conceptualizing a conceptually simple subject areas – areas related relations, as well as the choice of one of the most simple goals conceptualization – structuring the domain and build a variety of its phenomena. The first article was given a brief description of mathematical methods, describes the main stages of the conceptualization of the subject areas, ranging from the definition of the boundaries of the domain and ending with the theory of the synthesis of the terminal and determine its compliance with the tasks of conceptualizing. In the chosen example – areas related relations – have been considered in detail first, most important, the steps taken and the impact of technology on these stages of conceptualizing solutions. They discussed the definition of family relationship and the concept of “relative”, identifies the major errors and their consequences are discussed various options for positing the core of the conceptual scheme, carried out a precise definition of the content to rely in this case the basic concepts and their properties. In the second article concludes with a discussion of issues related to the behest of basic sets – namely, issues of formation of so-called “Declaration on the base set” terminological problems and questions refer to basic sets. Described final core of the conceptual scheme, carried out an axiomatic study of selected domain. The basic attention in second article is given to expansion of the theory and difficulties with which thus the conceptualist – researcher collides. By the example of the entered terms – “blood family”, “the connected family”, “family balls”, “generations of ancestors”, “genealogical trees and rhizomes”, “binding generations”, “related clans” – are shown opportunities of directions of expansion of the axiomatic theory. Influence of definition of concept “relative” on carrying out of border between relatives and nonrelatives is shown. For demonstration of variants of erosion of kinship, concepts of species “sorodstvennikov” are entered. At the end of the article briefly reviews the steps required for the synthesis of conceptual schemes. In the presentation of possible mathematical omitted entries, and the deployment of the theory described in the attributive form – in terms of empirically determined characteristics.

We characterize four types of agentive knowledge using a stit semantics over branching discrete-time structures. These are \emph{ex ante} knowledge, \emph{ex interim} knowledge, \emph{ex post} knowledge, and know-how. The first three are notions that arose from game-theoretical analyses on the stages of information disclosure across the decision making process, and the fourth has gained prominence both in logics of action and in deontic logic as a means to formalize ability. In recent years, logicians in AI have argued that any comprehensive study of responsibility attribution and blameworthiness should include proper treatment of these kinds of knowledge. This paper intends to clarify previous attempts to formalize them in stit logic and to propose alternative interpretations that in our opinion are more akin to the study of responsibility in the stit tradition. The logic we present uses an extension with knowledge operators of the Xstit language, and formulas are evaluated with respect to branching discrete-time models. We also present an axiomatic system for this logic, and address its soundness and completeness.

This paper proposes a logical framework for representing and reasoning about imperfect information games. We first extend Game Description Language (GDL) with the standard epistemic operators and provide it with a semantics based on the epistemic state transition model. We then demonstrate how to use the language to represent the rules of an imperfect information game and formalize common game properties as well as epistemic properties. We also show how to use the framework to reason about players' own and each others' knowledge during game playing. Furthermore, we prove that the model-checking problem of the framework is in Δ 2 P , even though its lower bound is Θ 2 P . These results indicate that the framework makes a good balance between expressive power and computational efficiency. Finally we provide a sound and complete axiomatic system for this logic. With action, temporal and epistemic operators, the completeness proof requires a novel combination of techniques used for completeness of dynamic logic and epistemic temporal logics. The proof theory provides a feasible tool to analyze properties of a family of games.

Disquotational theories of truth are often criticised for being too weak to prove interesting generalisations about truth. In this paper we will propose a certain formal theory to serve as a framework for a solution of the generalisation problem. In contrast with Horwich’s original proposal, our framework will eschew psychological notions altogether, replacing them with the epistemic notion of believability. The aim will be to explain why someone who accepts a given disquotational truth theory Th, should also accept various generalisations not provable in Th. The strategy will consist of the development of an axiomatic theory of believability, one permitting us to show how to derive the believability of generalisations from basic axioms that characterise the believability predicate, together with the information that Th is a theory of truth that we accept.

In early analytic philosophy, one of the most central questions concerned the status of arithmetical objects. Frege argued against the popular conception that we arrive at natural numbers with a psychological process of abstraction. Instead, he wanted to show that arithmetical truths can be derived from the truths of logic, thus eliminating all psychological components. Meanwhile, Dedekind and Peano developed axiomatic systems of arithmetic. The differences between the logicist and axiomatic approaches turned out to be philosophical as well as mathematical. In this paper, I will argue that Dedekind’s approach can be seen as a precursor to modern structuralism and as such, it enjoys many advantages over Frege’s logicism. I also show that from a modern perspective, Frege’s criticism of abstraction and psychologism is one-sided and fails against the psychological processes that modern research suggests to be at the heart of numerical cognition. The approach here is twofold. First, through historical analysis, I will try to build a clear image of what Frege’s and Dedekind’s views on arithmetic were. Then, I will consider those views from the perspective of modern philosophy of mathematics, and in particular, the empirical study of arithmetical cognition. I aim to show that there is nothing to suggest that the axiomatic Dedekind approach could not provide a perfectly adequate basis for philosophy of arithmetic.

Do other earthly forms of life evolved to the level of intelligent life? Cancer and resistance to antibiotics obliged to ask this question. Signs of intelligence are found at its simplest levels. We try to see if logic is used at these levels. Peter of Spain's suppositio materialis is applied to the chemical signals of cells. Dynamic Logic is used to understand these chemical communications. is used, instead of “language”. The development of life appears as the development of an axiomatic system. The rights of unborn humans and God's rights over His own creation appear as the most powerful arguments for conservationism.

The question that is the subject of this article is not intended to be a sociological or statistical question about the practice of today’s mathematicians, but a philosophical question about the nature of mathematics, and specifically the method of mathematics. Since antiquity, saying that mathematics is problem solving has been an expression of the view that the method of mathematics is the analytic method, while saying that mathematics is theorem proving has been an expression of the view that the method of mathematics is the axiomatic method. In this article it is argued that these two views of the mathematical method are really opposed. In order to answer the question whether mathematics is problem solving or theorem proving, the article retraces the Greek origins of the question and Hilbert’s answer. Then it argues that, by Godel’s incompleteness results and other reasons, only the view that mathematics is problem solving is tenable.

This paper proposes a simple, flexible, axiom-based mechanism for facilitating the comparison between a scholar's citation count and the visibility of the journals wherein the scholar's articles were published. The goal is to help research review bodies easily grasp the distinction these two forms of scholarly accomplishment and to provide a transparent way to articulate expectations about them to scholars. The approach is demonstrated using a widely applied and cooperative functional form that can reflect, via different parameter values, a wide range of possible beliefs about the relative merits of citation counts and journal visibility.

The paper entitled “I.P.Podlasii’s contributions to the reconstruction of pedagogic normativity” highlights the contribution of the contemporary educator of the eastern area, I.P.Podlasii in demonstrating the scientificity of pedagogy. I.P.Podlasii is one of the few contemporary educators who noticed the need to address the issue of normativity, not by starting from the didactic principles but from the axioms of pedagogy. The axioms of pedagogy, fundamental truths which no longer need to be proven, are epistemological guidelines for the construction of the scientific knowledge system. The model of pedagogical normativity suggested by I.P.Podlasii, adaptable to the paradigm of the curriculum, includes: the axioms of pedagogy, the regularities of pedagogy, specific laws, concrete laws, principles of pedagogy, pedagogic rules which have an operational nature. I.P.Podlasii suggests the promotion and application of a three-step axiomatic method which creates the necessary context for the affirmation of a system of pedagogical axioms. It refers to systematising the basic concepts, stabilised at the level of the basic matrix of pedagogy. The second step consists in correlating the basic concepts and the third step in concentrated organisation of the theories of the field through formula, symbols. The contributions of the mentioned educator are remarkable. given that they are not reduced to the didactic principles, but create a coherent system. The scientific novelty and originality arises from its specificity, hermeneutic research of a theoretical type, which focuses on identifying the epistemic structures of pedagogy.

International audience; Philosophical systems have developed for centuries, but only in the nineteenth century did the notion of scientific philosophy emerge. This notion presented two dimensions in the early twentieth century. One dimension arose from scientists' concern with conceptual foundations for their disciplines, while another arose from philosophers' appetite for more rigorous philosophy. In the current paper, I will focus on David Hilbert's construct of "critical mathematics" and Edmund Husserl' and Jules Vuillemin's systematic philosophy. All these three thinkers integrated Kant's legacy with the axiomatic method. However, they did so in different ways, with Hilbert's goal being the opposite of that of Husserl or Vuillemin. Specifically, I will show how the scientism of Hilbert's mathematical epistemology aimed at shattering the ambition of philosophy to submit mathematical practices and problems to philosophy's own principles and methods, be they transcendental or metaphysical. On the other hand, phenomenology promoted the idea of a non-exact philosophical rigour and highlighted the need of a point of view encompassing positive sciences, ontology, and ethical values in connection with the dominant category of sense/meaning, and Jules Vuillemin built on from the inseparability of thought-scientific or philosophical-from the metaphysics of free will and choice.

Kazimierz Ajdukiewicz was an eminent representative of the Lvov-Warsaw School. His main interest was the cognitive role of language. His radical conventionalism intended to explain rapid and fundamental changes in science. He used the method of philosophical paraphrase to make traditional metaphysical questions decidable. Then he drew metaphysical conclusions from the so called “semantic epistemology” based—according to his programme—on semantics (which played an important role in his research) and formal logic. His categorial grammar aiming to formulate the general criteria of syntactic coherence was the first grammar based exclusively on the structural properties of expressions. He also undertook a number of methodological issues, both general and detailed. He was interested in the theory of definition, the theory of questions, the problem of rationality of fallible inferential methods, the foundation of sentences, classification of reasonings, of sciences and of axiomatic systems, and in the reconstruction and evaluation of scientific procedures.

In recent years there has been a surge of interest in Godel’s ontological proof of the existence of God. In spite of all this extensive concern, it is not certain whether there is any improvement in understanding the motivations of Godel’s ontological proof. Why was Godel so preoccupied with completing his own ontological proof? To the best of my knowledge, no one has dealt with this basic question seriously enough to answer it. In this chapter, I propose to examine Godel’s ideas against a somewhat larger background in order to understand his motivation for establishing the ontological proof. I shall point out that the value of Godel’s proof is to be found in the possible role of his proof of the existence of God in his philosophy as a whole as well as in its relative merit as an ontological proof. Hopefully, my guiding question as to Godel’s motivation will turn out to be extremely fruitful by enabling us to fathom his mind regarding God and mathematics.

In the thesis Zygmunt Zawirski’s concept of scientific metaphysics two questions were posed as the means of presenting an explorative issue: Can axiomatic method give rise to philosophy as a science, study of universal “order”? Does scientific metaphysics constitute solid, mathematical representation of our intuitions and instinctive strivings and development of previous attempts to demonstrate the existence of unity among multiplicity of phenomena. The article aims to reconstruct Zygmunt Zawirski’s metaphilosophy, among other things: relations between the sciences and philosophy.

Zygmunt Zawirski was one of the greatest member of the Lvov-Warsaw School. The paper presents his life and the list of his main works, the greatest achievements in philosophy of nature, methodology of science and metaphysics. He pointed at the possibility of studying the scientific metaphysics with the use of the axiomatic method. His important work was the philosophical issue of time: The evolution of the notion of time. Zawirski looked for the justification of the scientific research, listing the problem of induction and deduction, development of the scientific theories and relations of science and metaphysics.

Bernays has not drawn scholarly attention that he deserves. Only quite recently, the reevaluation of his philosophy, including the projects of editing, translating, and reissuing his writings, has just started. As a part of this renaissance of Bernays studies, this chapter tries to distinguish carefully between Hilbert’s and Bernays’ views regarding the axiomatic method. We shall highlight the fact that Hilbert was so proud of his own axiomatic method on textual evidence. Bernays’ estimation of the place of Hilbert’s achievements in the history of the axiomatic method will be scrutinized. Encouraged by the fact that there are big differences between the early middle Bernays and the later Bernays in this matter, we shall contrast them vividly. The most salient difference between Hilbert and Bernays will shown to be found in the problem of the uniformity of the axiomatic method. In the same vein, we will discuss the later Bernays’ criticism of Carnap, for Carnap’s project of philosophy of science in the late 1950s seems to be a continuation and an extension of Hilbert’s faith in the uniformity of the axiomatic method.

Seldom has a mathematical axiom engendered the kind of criticism and controversy as did Zermelo’s (1904) Axiom of Choice (henceforth, AC). In this paper, we intend to place the development of the Axiom of Choice in its proper historical context relative to the period often called “the crisis in the foundations of mathematics.” To this end, we propose that the nature of the controversy surrounding AC warrants a division of the Grundlagenkrise der Mathematik into two separate horns: (1) an ontological crisis related to the nature and status of mathematics itself (viz., the nature of its foundation and the logical paradoxes that surrounded early attempts to logically formalize mathematics); and (2) a methodological branch concerned rather with the nature of mathematical practice (viz., the nature of mathematical proofs). These two strands are inexorably intertwined and, though it is not new to suggest that the controversy surrounding AC was related either to the foundational crisis or to a polemic about the nature of mathematical demonstration, it is perhaps new to state that the question of the validity of AC not only was a central question of this period but also, furthermore, was one of its primary drivers—one which led to a profound paradigm shift in the way we construe mathematical reasoning, whether it has led us down a path of embracing realism/Platonism or intuitionism/constructivism.

This paper compares the axiomatic method of David Hilbert and his school with Rudolf Carnap's general axiomatics that was developed in the late 1920s, and that influenced his understanding of logic of science throughout the 1930s, when his logical pluralism developed. The distinct perspectives become visible most clearly in how Richard Baldus, along the lines of Hilbert, and Carnap and Friedrich Bachmann analyzed the axiom system of Hilbert's Foundations of Geometry—the paradigmatic example for the axiomatization of science. Whereas Hilbert's axiomatic method started from a local analysis of individual axiom systems in which the foundations of mathematics as a whole entered only when establishing the system's consistency, Carnap and his Vienna Circle colleague Hans Hahn instead advocated a global analysis of axiom systems in general. A primary goal was to evade, or formalize ex post, mathematicians' 'material' talk about axiom systems for such talk was held to be error-prone and susceptible to metaphysics.

In the section of the Antinomy of pure Reason Kant presents three notions of infinity. By investigating these concepts of infinity, this paper highlights important ‘building blocks’ of the structure of the mathematical antinomies, such as the ability of reason of producing ascending and descending series, as well as the notions of given and givable series. These structural features are discussed in order to clarify Ernst Zermelo’s reading of Kant’s antinomy, according to which the latter is deeply rooted in the tendency of the mind of producing “creative progress” and “inclusive closure”. The aim of this paper is to explain in which sense and why Kant’s treatment of the antinomies attracts the attention of Zermelo in the early 1900s and which aspects of his second axiomatic system have been inspired by Kant’s philosophy. Thus, by reading Kant’s antinomy ‘through Zermelo’s eyes’—with emphasis on the concept of regressive series in indefinitum and on that of regressive series ad infinitum – this paper identifies the echoes of Kant’s work in the making of the ZFC set theory.

This article concerns the split between syntactic and semantic approaches to scientific theories. It aims at showing that an axiomatic representation of a scientific theory is a precondition of comprehending if the models of contain infinite entities. This result is established on the basis of the proposition that the human mind—which is finitely bounded for all we know—is not capable of directly grasping infinite entities. In view of this cognitive limitation, an indirect and finite representation of possibly infinite components of the models of a scientific theory proves to be indispensable. Sets of axioms and sets of axiom schemes provide such a representation. These considerations will be cast into an argument for an axiomatic conception of scientific theories. The article concludes with a case study of the ideal gas model.

We discuss the interplay between the axiomatic and the semantic approach to truth. Often, semantic constructions have guided the development of axiomatic theories and certain axiomatic theories have been claimed to capture a semantic construction. We ask under which conditions an axiomatic theory captures a semantic construction. After discussing some potential criteria, we focus on the criterion of ℕ-categoricity and discuss its usefulness and limits.