Mathematical Logic Research Articles

Duality Symmetry, Two Entropy Functions, and an Eigenvalue Problem in Generalized Gibbs’ Theory

Abstract We generalize the convex duality symmetry in Gibbs’ statistical ensemble formulation, between the Gibbs entropy φV,N(E) as a function of mean internal energy E and Massieu’s free entropy ΨV,N(β) as a function of inverse temperature β. The duality in terms of Legendre-Fenchel transform tells us that Gibbs’ thermodynamic entropy is to the law of large numbers (LLN) for arithmetic sample mean values what Shannon’s information entropy is to the LLN for empirical counting frequencies in independent and identically distributed data. Proceeding with the same mathematical logic, we identify the energy of the state {ui } as the conjugate variable to the counting of statistical occurrence {mi } and find a Hamilton-Jacobi equation for the Shannon entropy analogous to an equation of state in thermodynamics. An eigenvalue problem that is reminiscent of certain features in quantum mechanics arises in the entropy theory of statistical counting frequencies of Markov correlated data.

Remarks on skew Hilbert algebras and weak BCK*-algebras

Abstract Skew Hilbert algebras were recently introduced as a tool for a unified treatment of several ordered algebraic structures with implication important for mathematical logic. They were presented axiomatically as posets with implication, one of the axioms being stated in terms of lower and upper cones. In the paper, the important subclass of the so-called strong skew Hilbert algebras is shown to coincide with a class of weak BCK*-algebras that is axiomatized by simple quasi-equations. Several applications of skew Hilbert algebras are reconsidered in the context of weak BCK*-algebras, and the relevant classes of positive implicative and orthoimplicative weak BCK*-algebras are briefly reviewed.

The winner of the competition on the best technologies in the industry: Mathematical support and conditions for holding the competition

Introduction. The problem and the goal. The article is devoted to the search for mathematical support and conditions for conducting a competition in order to identify the winner and the best technology in the industry (economics, education, medicine, other fields and experimental sciences), the research results of which fit the framework of independent and independent samples. It is a continuation of the study published in the author's earlier works. Materials and Methods. The methodology for solving the research problem includes integration of mathematical education and mathematical logic. Results. New concepts of ‘industry technology’ and ‘competitive matrix’ have been defined, where technologies are determined by samples compiled from responses to industry technology requirements (categories). These requirements (or categories) incorporate the results of experimental work, scientific novelty, practical significance, etc., with the focus on the specifics of the industry. The register of technology requirements and their assessment scales are determined by two commissions: the organization-based one and the commission composed of industry experts. This approach, a combination of experimental and expert assessments, is more flexible, and contributes to increasing the objectivity and reliability of findings, on the one hand; on the other hand, the results are obtained in terms of technology itself, and not indirectly – by the influence of technology on subjects (in natural sciences) and on students (in education), as it was before until now. The new concepts are clarified; they are formulated in a mathematically complete form, that is, in terms of necessity and sufficiency, conditions and an algorithm for finding these types of technologies; their varieties are described. Conclusions. Mathematical support, the quality of life criterion, the algorithm, the necessary and sufficient conditions for finding the winner of the competition and the best technology in the industry contribute to: increasing the adequacy of technologies to their goals; excluding false technologies from possible implementation in any industries and purifying science from substandard results.

Efforts To Improve The Mathematical Logic Intelligence Of Children Aged 5-6 Years Through Loose Part Media At Paud Harapan Ummat, Sari Rejo Village, Medan Polonia District

This study aims to find out how to improve the mathematical logic intelligence of children aged 5-6 years through loose part media at PAUD Harapan Ummat, Sari Rejo Village, Medan Polonia District. To support the ability of mathematical logic intelligence in early childhood 5-6 years old, it can be done by utilizing loose part media, which includes natural materials, such as leaves, seeds, grains and shells. This study uses the Classroom Action Research method.The data collection techniques carried out are observation and documentation.with two cycles. Each cycle consists of 2 meetings. Then the researcher will observe the improvement of children's mathematical logic intelligence in each cycle to see the percentage increase. Observations were carried out based on research instruments in the form of BB, MB, BSH, BSB. Therefore, the conclusion of the study proves the effectiveness of loose part media in developing early childhood logic intelligence skills. Researchers recommend teachers to use loose part media to stimulate the mathematical logic intelligence of 5-6-year-old children.

Continuum Hypothesis Higher Infinity and Human Consciousness

When Centor proved that all infinities not equal completely New Era in logic and in mathematics opened. It is proved that there are more real numbers than natural numbers and there cannot be a maximum set in mathematics. If we take the power set of a given set then it always contains more members than the set itself. In this way there is no and infinities. There cannot be a collection which can be called the highest or maximum collection. This is also a limit of human reasoning because it appears in the form that if it is believed that the continuum hypothesis is true then it also works. And if we believe that the continuum hypothesis is false then it also works. Actually, Godel and Cohen proved that both continuum hypothesis and its negation can be proved as independent from the rest of the axioms of set theory. Since 1963 this is an unsolved problem in mathematical logic. Very recently the famous mathematical logician hugh woodin proved a result in which there is a particular type of model in which the continuum hypothesis is false. In our Upnishada Anant is called the Swarooplakshan of Brahma. If Anant is to be taken seriously when merely on the ground of intellect infinity cannot be grasped. We have repeatedly observed in Upnishada that true infinity can be obtained only in the state of liberation and mathematical infinity is not true infinity which can be assigned to ultimate reality or Brahma. There is a future scope of research in this direction where particularly from Taitiriya Upnishad where Satya, Gyan and Anant are stated as Swarooplakshan of Brahma. This new science of consciousness can be developed. Only Brahma is infinite in the absolute sense of the term and all other infinities which occur in logic and mathematics are relative infinities and so they cannot be the characteristic of reality.

Analysis of the Relationship Between Understanding Mathematical Logic and Managerial Decision-Making Effectiveness

Analysis of the Relationship Between Understanding Mathematical Logic and Managerial Decision-Making Effectiveness

Mediatization@Logic.com

Mediatization scholars promised to connect mediatization research with social theory, describe its interplay with other meta-processes, and create a “common theoretically based roof”. Hepp and Couldry state that the concept of logic is unable to do that, yet do not provide for the alternative. The paper argues there is a wide misunderstanding what the concept of logic actually is and what it is bringing to the table. Hepp considers logic in terms of “narrow and reductionist thinking”. Media logic is questioned in terms of its universal validity compared to mathematical or philosophical logic. There are at least two significant problems with this attitude towards logic. First, there is sociological logic. Even the theorists that provide the conceptual background of cultural or constructivist approach to mediatization use the concept of logic: Bourdieu and Elias. While they use the concept of logic to describe the organizing principle of habitus or figuration formations, it is cultural mediatization theorists that reject that. Second, even if media logic is too static or rigid, in times of digitalization and datafication the concept of logic is not institutionalist at all – it is mathematical and philosophical, embodied by digital technology itself, artifical intelligence (logical reasoning), algorithms (logic in software), social bots and automation. The paper argues media logic can help mediatization to become a roof term for media studies.

On Nilpotent Minimum logics defined by lattice filters and their paraconsistent non-falsity preserving companions

Abstract Nilpotent Minimum logic (NML) is a substructural algebraizable logic that is a distinguished member of the family of systems of Mathematical Fuzzy logic, and at the same time it is the axiomatic extension with the prelinearity axiom of Nelson and Markov’s Constructive logic with strong negation. In this paper our main aim is to characterize and axiomatize paraconsistent variants of NML and its extensions defined by (sets of) logical matrices over linearly ordered NM-algebra with lattice filters as designated values, with special emphasis on those that only exclude the falsum truth-value, called non-falsity preserving logics. We also consider turning these non-falsity preserving logics into Logics of Formal Inconsistency by expanding them with a consistency operator, and we axiomatize them as well. Finally, we provide a full description of the logics defined by finite products of matrices over finite NM-chains.

An appraisal of the competence of mathematical fuzzy logic approach via adaptive neuro-fuzzy inference system (ANFIS) in biodiesel production from algae oil

Abstract The fuel crisis and environmental problems can be resolved using biodiesel from various basic materials. This paper uses the transesterification process and segregation to produce biodiesel from algal oil. According to the four input factors (power, methanol-to-oil percentage, catalyst utilized, and process time), 29 experiments have been conducted to manufacture biodiesel. This work focuses on modeling and estimating the processes involved in producing biodiesel, specifically from algal oil, using the (ANFIS) adaptive neuro-fuzzy inference system methodology. The determination coefficient (R-square) and the mean root-mean-square error (RMSE) were employed to reconcile the ANFIS findings with the true results of the research. During training, the RMSE statistical variables were 1.209 and the R-squared was 0.9742. This instance, involves the ANFIS Framework additionally the Gaussian membership function was used and examined. This modeling approach shows promise for use in the biodiesel manufacturing process, potentially increasing the efficacy and efficiency of the generation of biodiesel from algal oil, given the high estimation accuracy shown by the ANFIS.

Coastal Sceneries of Albania, An Emerging 3S Destination: Analysis of Physical Characteristics and Human Activity Impacts

The increase in tourism economic benefits is the most common purpose along the Mediterranean coastal regions but, very often, conflicts of interest arise between short-term benefits and long-term conservation goals. This is particularly the case of Albania, a very popular emerging “Sun, Sea and Sand” (3S) destination characterized by massive fluxes of national/international visitors during the summer period. Among beach users’ preferences, global studies show that five parameters of greater importance stand out from the rest, i.e., safety, facilities, water quality, no litter, and scenery, and the latter is the main concern of this study. Albania is well known for its outstanding natural coastal beauty which was assessed at 40 sites by using the Coastal Scenic Evaluation System (CSES) method. Based on the evaluation of 26 physical/human parameters and using weighting matrix parameters and fuzzy logic mathematics, the technique enables one to obtain an Evaluation Index (D) that allows one to classify each investigated site into five scenic classes, from Class I (extremely attractive natural sites; D ≥ 0.85) to Class V (very unattractive developed urban/industrial sites; D < 0.00). Pragmatically, the higher the “D” value is, the better the site scenery is. After a long process of field testing along the whole Albanian coastline (ca. 523 km in length), selected sites were chosen in rural/remote environments (22), villages (6), and urban (4) and resort areas (8) to reflect the Albanian coastal typicity and characterize the scenic impact of human activities. Most sites belonged to Class III (14), Class IV (13), Class II (8), and Class I (1). Several sites could be upgraded to Class I or Class II with slight management efforts, e.g., by carrying out cleaning operations or by reducing intrusive beach facilities.

Addition-meet fuzzy relational inequality systems, product-join relational inequality systems and their generalizations

Real unit interval valued Addition-min FRI systems are generalized to MV-algebra valued Addition-meet and Product-join fuzzy relational inequality systems, and the latter further to residual lattice valued Product-join relational inequality system. MV-algebra valued Addition-meet FRI systems are further generalized in two ways; on the one hand they are defined on involutive residual lattices and on the other hand as Addition-MV-product systems. It is proved that each such FRI system has a minimal solution or, respectively a maximal solution. The number of such solutions is discussed. Several illustrative examples are given. Through residual lattices and MV-algebras, these systems are linked in a profound way to mathematical fuzzy logic.

The application of quantum computing in music composition

Quantum computing and artificial intelligence, two prominent topics in science and technology, are rapidly advancing and extending their influence into numerous fields, including music. Quantum computer music, which merges the strengths of quantum computing and deep learning, heralds a new era in the integration of music creation with cutting-edge technology. The interactive quantum music composition “Spinnings—Q1 Synth Trio”, created by Brazilian composer Miranda during the QuTune project at the Interdisciplinary Centre for Computer Music Research (ICCMR), in collaboration with the University of Oxford, stands as a notable example of quantum computer music. This study adopts a case study approach to thoroughly investigate the technical creative process behind this work, covering elements such as quantum computing, quantum properties, qubits, quantum gates, and quantum circuits, gradually unveiling the mathematical logic behind quantum algorithmic composition. The results of this study indicate that quantum algorithmic composition, as an emerging approach to music creation, not only generates unique music through the characteristics of quantum computing but also offers new possibilities for the integration of music, art, and technology. By applying quantum bits, quantum gates, and quantum circuits, this research demonstrates how quantum computing can provide new theoretical foundations and practical methods for music composition. Furthermore, the study discusses how to optimize the interactive creative experience in quantum music works and how to enhance the understanding and appreciation of quantum music among a broader audience of musicians and listeners. With the continuous advancement of quantum computing technology, quantum music is poised to contribute a distinctive dimension to the global flourishing of musical culture. This research offers fresh perspectives and ideas for the development of this field.

ORGANIZING ACTIVITIES FOR PRIMARY STUDENTS TO EFFECTIVELY LEARN MATHEMATICAL VOCABULARY

Using mathematical language is of great significance to students in the process of learning mathematics. Improving students' mathematical language skills contributes to improving their mathematical learning outcomes. This process must be done continuously throughout the learning process. Mathematical vocabulary is an important part of mathematical language, the mastery of which fosters students to enhance their ability to use mathematical language accurately and effectively, thereby forming their mathematical logic senses. This article presents, based on basic mathematical language activities, a number of teaching organizing measures in which teachers support students to comprehend and use vocabulary effectively in learning and in practice.

Metaphysics of Reasons and the Defense of Society

The chilef problem of metaphysics of reasons is the theoretical substantiation of the steady existence of the best government and society. Reason (Erӧrterung) contains that which presupposes the formation of a concept given a priori. Montesquieu’s theory of separation of powers inspired the Constitution of the United States. But the American founding fathers decidedly rejected Montesquieu’s experience concerning the separation of powers in England. The article deals with the problem of defending Kant’s opinion that logic had made no important step either forward or backward since Aristotle. The work raises the problem of the excrescences of the scientific empiricism of the Vienna Circle. The research attempts to solve the problem connected with the debate concerning the insights of rationalists as mere tautologies. Concerning the problem of relation between the mathematical and dialectical logic, the present work shows that the first type of logic refers mainly to natural, the second mainly to social science. The article discusses the eternal problem of middle classes clearly raised by Aristotle. This problem is successfully solved in China with the help of the differentiated theory of convergence: capitalism within the framework of socialism for the East, socialism within the framework of capitalism for the West.

Mathematical Logic Integration Scheme in Ushul Fiqh

This research is aimed to arrange an integration scheme between mathematical logic and ushul fiqh which focuses on the discussion of mafhum mukholafah. This study is designed by using a qualitative design, which conducted by using library research/literature study. It was carried out by collecting materials related to research from books, scientific journals, literature and other publications that were worthy of being used as sources for research studies by describing the data through several expert opinions. The results of this study indicate that there is an integration between mathematical logic and the science of mantiq, especially in the discussion of the inverse of a statement with mafhum mukholafah. However, there is also a gap between them, there are still objects of study in ushul fiqh that cannot be accommodated in mathematical logic.

ESG in Promoting Sustainable Development of Manufacturing Companies in China’s Selected a-share Listed Companies: A Proposed Financial Mathematical Model

In this study, the researcher proposes seven SOPs for the research objects, and establishes four basic assumptions on this basis, always focusing on the main line of how much ESG affects the sustainable development of manufacturing enterprises in China's A-share market. The four basic hypotheses correspond to the four dimensions of enterprise value, operation efficiency, business risk and investor reaction, and carry out empirical analysis, and draw their own research conclusions. It is verified that ESG promotes the construction of sustainable development ability of manufacturing enterprises in China's A-share market, and empirical evidence is also provided. In addition, these test results are through models’ regression demonstrated financial model, the basis of the four basic models provide the foundation data analysis of mathematical logic. The final conclusion supports the thesis that ESG promotes the sustainable development of manufacturing enterprises. The research conclusions of this paper are as follows: a. ESG performance has a significant positive relationship with the value of manufacturing enterprises; b. ESG performance has a significant positive relationship with the operational efficiency of manufacturing enterprises; c. There is a significant negative relationship between ESG performance and business risk of manufacturing enterprises; d. ESG performance has a significant positive relationship with investor reaction (investor preference) of manufacturing enterprises.

A Guidebook for Assessing Mathematical Logic Intelligence in Kindergarten

Many children still need help understanding numbers; children also cannot differentiate between round, triangular, and square shapes. Apart from that, teachers only carry out assessments at the end of the learning semester and do not use assessment guides, especially on aspects of children's mathematical logic intelligence in their daily activities at school. Based on this, this research aims to develop a guidebook for assessing mathematical logic intelligence in kindergarten. This type of research is development research. This research uses the ADDIE model. The research subjects are learning media experts and learning material experts. The test subjects were fifteen teachers. The methods used to collect data are interviews, documentation, and questionnaires. The data collection instrument uses a questionnaire sheet. The data analysis technique used is quantitative descriptive statistical analysis. The research results are the results of material media validation tests with a test percentage value of 96.3% and the eligibility criteria being "very valid." The assessment from media experts was 98.7%, with the eligibility criteria being "very valid." The assessment score from the effectiveness instrument expert has a test percentage value of 97%, with the eligibility criteria being "very valid." The effectiveness test of individual trials were 95.69% "very effective" and small group 91.67% "very effective". It was concluded that the mathematical logic intelligence assessment guidebook is very suitable and adequate for teachers in kindergarten. The research implication is that the developed mathematical logic intelligence assessment guidebook can be used in learning activities.

From Natural to Artificial: The Transformation of the Concept of Logical Consequence in Bolzano, Carnap, and Tarski

Our standard model-theoretic definition of logical consequence is originally based on Alfred Tarski’s (1936) semantic definition, which, in turn, is based on Rudolf Carnap’s (1934) similar definition. In recent literature, Tarski’s definition is described as a conceptual analysis of the intuitive ‘everyday’ concept of consequence or as an explication of it, but the use of these terms is loose and largely unaccounted for. I argue that the definition is not an analysis but an explication, in the Carnapian sense: the replacement of the inexact everyday concept with an exact one. Some everyday intuitions were thus brought into a precise form, others were ignored and forgotten. How exactly did the concept of logical consequence change in this process? I suggest that we could find some of the forgotten intuitions in Bernard Bolzano’s (1837) definition of ‘deducibility’, which is traditionally viewed as the main precursor of Tarski’s definition from a time before formalized languages. It turns out that Bolzano’s definition is subject to just the kind of natural features—paradoxicality of everyday language, Platonism about propositions, and dependence on the external world—that Tarski sought to tame by constructing an artificial concept for the special needs of mathematical logic.

Useless Math – The Complex Untold Story

If you don't understand mathematics, ask yourself if I'm right, because others don't understand mathematics either. By useless mathematics we mean incomplete mathematical spaces of a classical 3D+t variety that are inadequate for generating well-defined definitions and hypotheses as well as time-dependent partial differential equations. The current classical discrete 3D+t space PDE, in which time is an external controller and not integrated into the 3D geometric space, cannot be integrated digitally. This space is logically incomplete and misleading in the production of definitions and hypotheses as well as in the resolution itself of time- dependent PDEs. No wonder these definitions/assumptions are ugly and result in weak or intractable mathematics, leading to all kinds of misunderstandings, from horrible notations to undisciplined length of theorems containing a considerable amount of black magic and ending with a gray nature of the mathematical result obtained. In this article we present some of the most catastrophic inaccurate assumptions existing in current classical mathematics, resulting from the use of 3D+t manifold space to specify initial conditions, boundary conditions and the source/sink term. Fortunately, these inaccurate assumptions that start with an ugly space for boundary conditions, initial conditions and source/sink term can be spotted and analyzed via 4D unitary numerical statistical theory called Cairo techniques in the format of transition chains of matrix B to complete what is missing. In other words, we present how to spot some of the ugliest mathematical conclusions of classical 3D geometry plus t as an external control numerical space, and then show how to correct them via the 4D unit space of statistical transition matrix chains. By complex and untold history, we mean that useless and misleading mathematics dominated scientific research and education throughout the 20th century, so much so that the accumulated legacy of misconceptions became a huge, complex mountain, almost impossible to eliminate. Fortunately, the numerical theory of Cairo techniques and the Laplacian theorem constitute an advanced and exhaustive form of the energy continuity equation and thus can create new logical mathematics. This is also the case of the famous Schrödinger time- dependent PDE.  The Laplacian theorem is one of the most important products created by the numerical statistical theory called Cairo techniques. In previous articles we introduced and briefly explained the so-called Laplacian theorem in the 4D x-t unit space, while in this article we highlight its importance and how it can generate new mathematics in more detail. The Laplacian partial differential equation that interests us is the one having a well-defined exclusive form and living in an isolated sample spatial control volume surrounded by a closed surface (A) and subject to Dirichlet boundary conditions. This very particular case of Laplacian PDE is always treated mathematically in a classical D^4 variety which is lazy and misleading. Finally, this article collects, studies, identifies and analyzes the dozen most common current useless mathematical events and presents an effective and adequate alternative.

MODELING PROBLEMS OF LOGICAL THINKING IN PRIMARY CLASS MATHEMATICS LESSONS

Models are created for various purposes (analysis of the studied processes, verification or demonstration of a new idea, method, system; to study and change reality, as a means of forecasting) and in a clear and concise form the studied system, process, characteristics of the idea; allows you to get important things about the connections and relations of the object. V.V. According to Davydov, the model will contain an image of the studied object. According to the nature of the description of the modeled system, there are modeling of the structure (structural models) and modeling of the operation of the processes occurring in it (flow or process models). To solve the second problem, mathematical modeling is the most effective. Modeling involves the use of abstraction and idealization. Reflecting the essential features of the original, the model acts as a kind of implementation of abstraction. The effectiveness of modeling as an achievement of consistency between model and reality is complicated by uncertainty, an important feature of models in the humanities. E. N. Gusinsky, studying the characteristics of social and humanitarian systems, emphasizes that they always have conscious and unconscious motives, are determined not only by past experience, but also are influenced by many factors of the current environment and the external environment. A.N. Dakhin emphasizes the complexity of modeling humanitarian systems, because they do not have clear components that make up the system - each of the components can "over time" become a pedagogical bifurcation point and dominate the design of goals and teaching technology. This article talks about the modeling of logical issues, the need for its application, the diversity of educational content, and the increase in requirements for the quality of education.