Journal articles: 'Inference. Logic, Symbolic and mathematical. Mathematics'

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NAMPALLY, ARUN, TIMOTHY ZHANG, and C. R. RAMAKRISHNAN. "Constraint-Based Inference in Probabilistic Logic Programs." Theory and Practice of Logic Programming 18, no. 3-4 (July 2018): 638–55. http://dx.doi.org/10.1017/s1471068418000273.

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AbstractProbabilistic Logic Programs (PLPs) generalize traditional logic programs and allow the encoding of models combining logical structure and uncertainty. In PLP, inference is performed by summarizing the possible worlds which entail the query in a suitable data structure, and using this data structure to compute the answer probability. Systems such as ProbLog, PITA, etc., use propositional data structures like explanation graphs, BDDs, SDDs, etc., to represent the possible worlds. While this approach saves inference time due to substructure sharing, there are a number of problems where a more compact data structure is possible. We propose a data structure called Ordered Symbolic Derivation Diagram (OSDD) which captures the possible worlds by means of constraint formulas. We describe a program transformation technique to construct OSDDs via query evaluation, and give procedures to perform exact and approximate inference over OSDDs. Our approach has two key properties. Firstly, the exact inference procedure is a generalization of traditional inference, and results in speedup over the latter in certain settings. Secondly, the approximate technique is a generalization of likelihood weighting in Bayesian Networks, and allows us to perform sampling-based inference with lower rejection rate and variance. We evaluate the effectiveness of the proposed techniques through experiments on several problems.

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Mewada, Shivlal. "Perspectives of Fuzzy Logic and Their Applications." International Journal of Data Analytics 2, no. 1 (January 2021): 99–145. http://dx.doi.org/10.4018/ijda.2021010105.

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Fuzzy logic is a highly suitable and applicable basis for developing knowledge-based systems in engineering and applied sciences. The concepts of a fuzzy number plays a fundamental role in formulating quantitative fuzzy variable. These are variable whose states are fuzzy numbers. When in addition, the fuzzy numbers represent linguistic concepts, such as very small, small, medium, and so on, as interpreted in a particular contest, the resulting constructs are usually called linguistic variables. Each linguistic variable the states of which are expressed by linguistic terms interpreted as specific fuzzy numbers is defined in terms of a base variable, the value of which are real numbers within a specific range. A base variable is variable in the classical sense, exemplified by the physical variable (e.g., temperature, pressure, speed, voltage, humidity, etc.) as well as any other numerical variable (e.g., age, interest rate, performance, salary, blood count, probability, reliability, etc.). Logic is the science of reasoning. Symbolic or mathematical logic is a powerful computational paradigm. Just as crisp sets survive on a 2-state membership (0/1) and fuzzy sets on a multistage membership [0 - 1], crisp logic is built on a 2-state truth-value (true or false) and fuzzy logic on a multistage truth-value (true, false, very true, partly false and so on). The author now briefly discusses the crisp logic and fuzzy logic. The aim of this paper is to explain the concept of classical logic, fuzzy logic, fuzzy connectives, fuzzy inference, fuzzy predicate, modifier inference from conditional fuzzy propositions, generalized modus ponens, generalization of hypothetical syllogism, conditional, and qualified propositions. Suitable examples are given to understand the topics in brief.

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Riesco, Adrián, Beatriz Santos-Buitrago, Javier De Las Rivas, Merrill Knapp, Gustavo Santos-García, and Carolyn Talcott. "Epidermal Growth Factor Signaling towards Proliferation: Modeling and Logic Inference Using Forward and Backward Search." BioMed Research International 2017 (2017): 1–11. http://dx.doi.org/10.1155/2017/1809513.

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In biological systems, pathways define complex interaction networks where multiple molecular elements are involved in a series of controlled reactions producing responses to specific biomolecular signals. These biosystems are dynamic and there is a need for mathematical and computational methods able to analyze the symbolic elements and the interactions between them and produce adequate readouts of such systems. In this work, we use rewriting logic to analyze the cellular signaling of epidermal growth factor (EGF) and its cell surface receptor (EGFR) in order to induce cellular proliferation. Signaling is initiated by binding the ligand protein EGF to the membrane-bound receptor EGFR so as to trigger a reactions path which have several linked elements through the cell from the membrane till the nucleus. We present two different types of search for analyzing the EGF/proliferation system with the help of Pathway Logic tool, which provides a knowledge-based development environment to carry out the modeling of the signaling. The first one is a standard (forward) search. The second one is a novel approach based onnarrowing, which allows us to trace backwards the causes of a given final state. The analysis allows the identification of critical elements that have to be activated to provoke proliferation.

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Chernoskutov, Yu Yu. "On the Syllogistic of G. Boole." Discourse 7, no. 2 (April 29, 2021): 5–15. http://dx.doi.org/10.32603/2412-8562-2021-7-2-5-15.

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Introduction. This article focuses on the investigation of Boole’s theory of categorical syllogism, exposed in his book “The Mathematical analysis of Logic”. That part of Boolean legacy has been neglected in the prevailed investigations on the history of logic; the latter provides the novelty of the work presented.Methodology and sources. The formal reconstruction of the methods of algebraic presentation of categorical syllogism, as it is exposed in the original work of Boole, is conducted. The character of Boolean methods is investigated in the interconnections with the principles of symbolic algebra on the one hand, and with the principles of signification, taken from R. Whately, on the other hand. The approaches to signification, grounding the syllogistic theories of Boole and Brentano, are analyzed in comparison, wherefrom we explain the reasons why the results of those theories are different so much.Results and discussion. It is demonstrated here that Boole has borrowed the principles of signification from the Whately’s book “The Elements of Logic”. The interpreting the content of the terms as classes, being combined with methods of symbolic algebra, has determined the core features of Boolean syllogism theory and its unexpected results. In contrast to Whately, Boole conduct the approach to ultimate ends, overcoming the restrictions imposed by Aristotelean doctrine. In particular, he neglects the distinction of subject and predicate among the terms of proposition, the order of premises, and provide the possibility to draw conclusions with negative terms. At the same time Boole missed that the forms of inference, parallel to Bramantip and Fresison, are legitimate forms in his system. In spite of the apparent affinities between the Boolean and Brentanian theories of judgment, the syllogistics of Boole appeared to be more flexible. The drawing of particular conclusion from universal premises is allowable in Boolean theory, but not in Brentanian one; besides, in his theory is allowable the drawing of conclusion from two negative premises, which is prohibited in Aristotelian syllogistic.Conclusion. Boole consistently interpreted signification of terms as classes; being combine with methods symbolic algebra it led to very flexible syllogism theory with rich results.

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MCCARTY, CHARLES. "BROUWER’S WEAK COUNTEREXAMPLES AND TESTABILITY: FURTHER REMARKS." Review of Symbolic Logic 6, no. 3 (March 13, 2013): 513–23. http://dx.doi.org/10.1017/s1755020313000051.

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AbstractStraightforwardly and strictly intuitionistic inferences show that the Brouwer– Heyting–Kolmogorov (BHK) interpretation, in the presence of a formulation of the recognition principle, entails the validity of the Law of Testability: that the form ¬ f V ¬¬ f is valid. Therefore, the BHK and recognition, as described here, are inconsistent with the axioms both of intuitionistic mathematics and of Markovian constructivism. This finding also implies that, if the BHK and recognition are suitably formulated, then Brouwer’s original weak counterexample reasoning was fallacious. The results of the present article extend and refine those of McCarty, C. (2012). Antirealism and Constructivism: Brouwer’s Weak Counterexamples. The Review of Symbolic Logic. First View. Cambridge University Press.

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Peckhaus, Volker. "19th Century Logic Between Philosophy and Mathematics." Bulletin of Symbolic Logic 5, no. 4 (December 1999): 433–50. http://dx.doi.org/10.2307/421117.

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AbstractThe history of modern logic is usually written as the history of mathematical or, more general, symbolic logic. As such it was created by mathematicians. Not regarding its anticipations in Scholastic logic and in the rationalistic era, its continuous development began with George Boole's The Mathematical Analysis of Logic of 1847, and it became a mathematical subdiscipline in the early 20th century. This style of presentation cuts off one eminent line of development, the philosophical development of logic, although logic is evidently one of the basic disciplines of philosophy. One needs only to recall some of the standard 19th century definitions of logic as, e.g., the art and science of reasoning (Whateley) or as giving the normative rules of correct reasoning (Herbart).In the paper the relationship between the philosophical and the mathematical development of logic will be discussed. Answers to the following questions will be provided:1. What were the reasons for the philosophers' lack of interest in formal logic?2. What were the reasons for the mathematicians' interest in logic?3. What did “logic reform” mean in the 19th century? Were the systems of mathematical logic initially regarded as contributions to a reform of logic?4. Was mathematical logic regarded as art, as science or as both?

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HAMAMI, YACIN. "MATHEMATICAL INFERENCE AND LOGICAL INFERENCE." Review of Symbolic Logic 11, no. 4 (January 8, 2018): 665–704. http://dx.doi.org/10.1017/s1755020317000326.

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AbstractThe deviation of mathematical proof—proof in mathematical practice—from the ideal of formal proof—proof in formal logic—has led many philosophers of mathematics to reconsider the commonly accepted view according to which the notion of formal proof provides an accurate descriptive account of mathematical proof. This, in turn, has motivated a search for alternative accounts of mathematical proof purporting to be more faithful to the reality of mathematical practice. Yet, in order to develop and evaluate such alternative accounts, it appears as a necessary prerequisite to first possess a clear picture of what the deviation of mathematical proof from formal proof consists in. The present work aims to contribute building such a picture by investigating the relation between the elementary steps of deduction constituting the two types of proofs—mathematical inference and logical inference. Many claims have been made in the literature regarding the relation between mathematical inference and logical inference, most of them stating that the former is lacking properties that are constitutive of the latter. Such differentiating claims are, however, usually put forward without a clear conception of the properties occurring in them, and are generally considered to be immediately justified by our direct acquaintance, or phenomenological experience, with the two types of inferences. The present study purports to advance our understanding of the relation between mathematical inference and logical inference by developing a detailed philosophical analysis of the differentiating claims, that is, an analysis of the meaning of the differentiating claims—through the properties that occur in them—as well as the reasons that support them. To this end, we provide at the outset a representative list of the different properties of logical inference that have occurred in the differentiating claims, and we notice that they all boil down to the three properties of formality, generality, and mechanicality. For each one of these properties, our analysis proceeds in two steps: we first provide precise conceptual characterizations of the different ways logical inference has been said to be formal, general, and mechanical, in the philosophical and logical literature on formal proof; we then examine why mathematical inference does not appear to be formal, general, and mechanical, for the different variations of these notions identified. Our study results in a precise conceptual apparatus for expressing and discussing the properties differentiating mathematical inference from logical inference, and provides a first inventory of the various reasons supporting the observations of those differences. The differentiating claims constitute thus a set of data that any philosophical account of mathematical inference and proof purporting to be more faithful to mathematical practice ought to be able to accommodate and explain.

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Kisielewicz, Andrzej. "Odpowiedź na polemiki i komentarze do mojej książki „Logika i argumentacja”." Studia Philosophica Wratislaviensia 13, no. 3 (December 27, 2018): 137–70. http://dx.doi.org/10.19195/1895-8001.13.3.12.

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In reply to the polemics and comments on my bookLogic and ArgumentationThis article contains the reply to the polemics and comments on my book Logic and argumentation. Since the most controversial issue among the commentators was my “radical attack on formal logic”, the first part of the text generally addresses this criticism. Firstly, I thoroughly explain what my doubts regarding the formal logic are. I am not questioning the great, even spectacular, successes of formal logic in the field of mathematical logic, nor its extraordinary contribution to the foundation of computer technology. I only find that for the traditional goal of logic — that of studying the principles of reasoning, inference and definition — formal methods have failed in confrontation with practice. I am developing this thesis in detail and I am trying to justify it, first of all by referring to the practice of reasoning in the field of mathematics. In the second part, I respond to specific criticisms and comments from individual authors. Many of the comments are correct, some are based on misunderstandings, while some, in my opinion, are wrong.

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Castillo, Oscar, Patricia Melin, Fevrier Valdez, Jose Soria, Emanuel Ontiveros-Robles, Cinthia Peraza, and Patricia Ochoa. "Shadowed Type-2 Fuzzy Systems for Dynamic Parameter Adaptation in Harmony Search and Differential Evolution Algorithms." Algorithms 12, no. 1 (January 9, 2019): 17. http://dx.doi.org/10.3390/a12010017.

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Nowadays, dynamic parameter adaptation has been shown to provide a significant improvement in several metaheuristic optimization methods, and one of the main ways to realize this dynamic adaptation is the implementation of Fuzzy Inference Systems. The main reason for this is because Fuzzy Inference Systems can be designed based on human knowledge, and this can provide an intelligent dynamic adaptation of parameters in metaheuristics. In addition, with the coming forth of Type-2 Fuzzy Logic, the capability of uncertainty handling offers an attractive improvement for dynamic parameter adaptation in metaheuristic methods, and, in fact, the use of Interval Type-2 Fuzzy Inference Systems (IT2 FIS) has been shown to provide better results with respect to Type-1 Fuzzy Inference Systems (T1 FIS) in recent works. Based on the performance improvement exhibited by IT2 FIS, the present paper aims to implement the Shadowed Type-2 Fuzzy Inference System (ST2 FIS) for further improvements in dynamic parameter adaptation in Harmony Search and Differential Evolution optimization methods. The ST2 FIS is an approximation of General Type-2 Fuzzy Inference Systems (GT2 FIS), and is based on the principles of Shadowed Fuzzy Sets. The main reason for using ST2 FIS and not GT2 FIS is because the computational cost of GT2 FIS represents a time limitation in this application. The paper presents a comparison of the conventional methods with static parameters and the dynamic parameter adaptation based on ST2 FIS, and the approaches are compared in solving mathematical functions and in controller optimization.

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Alghannam, Lu, Ma, Cheng, Gonzalez, Zang, and Li. "A Novel Method of Using Vision System and Fuzzy Logic for Quality Estimation of Resistance Spot Welding." Symmetry 11, no. 8 (August 2, 2019): 990. http://dx.doi.org/10.3390/sym11080990.

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Finding a reliable quality inspection system of resistance spot welding (RSW) has become a very important issue in the automobile industry. In this study, improvement in the quality estimation of the weld nugget’s surface on the car underbody is introduced using image processing methods and training a fuzzy inference system. Image segmentation, mathematical morphology (dilation and erosion), flood fill operation, least-squares fitting curve and some other new techniques such as location and value based selection of pixels are used to extract new geometrical characteristics from the weld nugget’s surface such as size and location, shape, and the numbers and areas of all side expulsions, peaks and troughs inside and outside the fusion zone. Topography of the weld nugget’s surface is created and shown as a 3D model based on the extracted geometrical characteristics from each spot. Extracted data is used to define input fuzzy functions for training a fuzzy logic inference system. Fuzzy logic rules are adopted based on knowledge database. The experiments are conducted on a 6 degree of freedom (DOF) robotic arm with a charge-coupled device (CCD) camera to collect pictures of various RSW locations on car underbodies. The results conclude that the estimation of the 3D model of the weld’s surface and weld’s quality can reach higher accuracy based on our proposed methods.

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Ovsyak, V. K., O. V. Ovsyak, and J. V. Petruszka. "ORDER AND ORDERING IN DISCRETE MATHEMATICS AND INFORMATICS." Ukrainian Journal of Information Technology 3, no. 1 (2021): 37–43. http://dx.doi.org/10.23939/ujit2021.03.037.

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The available means of ordering and sorting in some important sections of discrete mathematics and computer science are studied, namely: in the set theory, classical mathematical logic, proof theory, graph theory, POST method, system of algorithmic algebras, algorithmic languages of object-oriented and assembly programming. The Cartesian product of sets, ordered pairs and ordered n-s, the description by means of set theory of an ordered pair, which are performed by Wiener, Hausdorff and Kuratowski, are presented. The requirements as for the relations that order sets are described. The importance of ordering in classical mathematical logic and proof theory is illustrated by the examples of calculations of the truth values of logical formulas and formal derivation of a formula on the basis of inference rules and substitution rules. Ordering in graph theory is shown by the example of a block diagram of the Euclidean algorithm, designed to find the greatest common divisor of two natural numbers. The ordering and sorting of both the instructions formed by two, three and four ordered fields and the existing ordering of instructions in the program of Post method are described. It is shown that the program is formed by the numbered instructions with unique instruction numbers and the presence of the single instruction with number 1. The means of the system of algorithmic algebras, which are used to perform the ordering and sorting in the algorithm theory, are illustrated. The operations of the system of algorithmic algebras are presented, which include Boolean algebra operations generalized to the three-digit alphabet and operator operations of operator algebra. The properties of the composition operation are described, which is intended to describe the orderings of the operators of the operator algebra in the system of algorithmic algebras. The orderings executed by means of algorithmic programming languages are demonstrated by the hypothetical application of the modern object-oriented programming language C#. The program must contain only one method Main () from which the program execution begins. The ARM microprocessor assembly program must have only one ENTRY directive from which the program execution begins.

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Qudrat-I Elahi, Khandakar. "A difficulty in Arrow’s impossibility theorem." International Journal of Social Economics 44, no. 12 (December 4, 2017): 1609–21. http://dx.doi.org/10.1108/ijse-02-2016-0065.

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Purpose The purpose of this paper is twofold. First, it evaluates the impossibility proposition, called the “Arrow impossibility theorem” (AIT), which is widely attributed to Arrow’s social choice theory. This theorem denies the possibility of arriving at any collective majority resolution in any group voting system if the social choice function must satisfy “certain natural conditions”. Second, it intends to show the reasons behind the proliferation of this impossibility impression. Design/methodology/approach Theoretical and philosophical. Findings Arrow’s mathematical model does not seem to suggest or support his impossibility thesis. He has considered only one voting outcome, well known by the phrase “the Condorcet paradox”. However, other voting results are equally likely from his model, which might suggest unambiguous majority choice. This logical dilemma has been created by Arrow’s excessive dependence on the language of mathematics and symbolic logic. Research limitations/implications The languages of mathematics and symbolic logic – numbers, letters and signs – have definite advantages in scientific argumentation and reasoning. These numbers and letters being invented however do not have any behavioural characteristics, which suggests that conclusions drawn from the model merely reflect the author’s opinions. The AIT is a good example of this logical dilemma. Social implications The modern social choice theory, which is founded on the AIT, seems to be an academic assault to the system of democratic governance that is dominating current global village. By highlighting weaknesses in the AIT, this paper attempts to discredit this intellectual omission. Originality/value The paper offers a counter example to show that the impossibility of social choice is not necessarily implied by the Arrow’s model. Second, it uses Locke’s theory of human understanding to explain why the concerned social scientists are missing this point. This approach is probably entirely novel in this area of research.

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Simpson, Stephen G. "Partial realizations of Hilbert's program." Journal of Symbolic Logic 53, no. 2 (June 1988): 349–63. http://dx.doi.org/10.1017/s0022481200028309.

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§0. Introduction. What follows is a write-up of my contribution to the symposium “Hilbert's Program Sixty Years Later” which was sponsored jointly by the American Philosophical Association and the Association for Symbolic Logic. The symposium was held on December 29,1985 in Washington, D. C. The panelists were Solomon Feferman, Dag Prawitz and myself. The moderator was Wilfried Sieg. The research which I discuss here was partially supported by NSF Grant DMS-8317874.I am grateful to the organizers of this timely symposium on an important topic. As a mathematician I particularly value the opportunity to address an audience consisting largely of philosophers. It is true that I was asked to concentrate on the mathematical aspects of Hilbert's program. But since Hilbert's program is concerned solely with the foundations of mathematics, the restriction to mathematical aspects is really no restriction at all.Hilbert assigned a special role to a certain restricted kind of mathematical reasoning known as finitistic. The essence of Hilbert's program was to justify all of set-theoretical mathematics by means of a reduction to finitism. It is now well known that this task cannot be carried out. Any such possibility is refuted by Gödel's theorem. Nevertheless, recent research has revealed the feasibility of a significant partial realization of Hilbert's program. Despite Gödel's theorem, one can give a finitistic reduction for a substantial portion of infinitistic mathematics including many of the best-known nonconstructive theorems. My purpose here is to call attention to these modern developments.

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Kochetkov, Mihail, Dmitry Korolkov, Vladimir Petrov, Oleg Petrov, Alexey Terentev, and Sergey Simonov. "Application of Cluster Analysis with Fuzzy Logic Elements for Ground Environment Assessment of Robotic Group." SPIIRAS Proceedings 19, no. 4 (September 7, 2020): 746–73. http://dx.doi.org/10.15622/sp.2020.19.4.2.

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Emergency situations, that cause risks for human life and health, dictate elevated requirements to completeness and accuracy of the presentation of information about current ground environment. Modern robotic systems include sensors, that operate on different physical principles. This causes incrementation of information entering control system. Computing resources and technical capabilities of robotic systems are limited in range and detection probabilities of appearing objects. In case of insufficient performance of the on-board computer system and high uncertainties of ground environment, robotic systems are not able to perform without combining information from robotic group and producing a single view of ground environment. Complex information from a group of robotic systems occurs in real time and a non-deterministic environment. To solve the problem of identifying attribute vectors related to a single object, as well as to evaluate the effectiveness of obtained solutions, is possible using known formulas of the theory of statistical hypothesis testing and probability theory only under the normal distribution law with the known mathematical expectation of an attribute vector and a correlation matrix. However, these conditions are usually not met in practice. Problems also arise when methods of nonparametric statistics are used with an unknown law of probability distribution. The new method of identifying attribute vectors is proposed, that does not rely on a statistical approach and, therefore, does not require knowledge of the type of distribution law and the values of its parameters. Proposed method is based on the idea of combining cluster analysis and fuzzy logic, and is relatively simple to the basic methods of multidimensional nonparametric statistics. The results of modeling information processes are presented. The advantages of proposed method are shown. The comparative values for the number of false recognitions are given. The recommendations are given for constructing fuzzy inference rules when creating an expert system knowledge base.

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Ferreira, Fernando. "Thomas Strahm. Polynomial time operations in explicit mathematics. The journal of symbolic logic, vol. 62 (1997), pp. 575–594. - Andrea Cantini. Feasible operations and applicative theories based on λη. Mathematical logic quarterly, vol. 46 (2000), pp. 291–312." Bulletin of Symbolic Logic 8, no. 4 (December 2002): 534–35. http://dx.doi.org/10.2178/bsl/1182353929.

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Cao, Feng, Yang Xu, Jun Liu, Shuwei Chen, and Xinran Ning. "CSE_E 1.0: An Integrated Automated Theorem Prover for First-Order Logic." Symmetry 11, no. 9 (September 8, 2019): 1142. http://dx.doi.org/10.3390/sym11091142.

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First-order logic is an important part of mathematical logic, and automated theorem proving is an interdisciplinary field of mathematics and computer science. The paper presents an automated theorem prover for first-order logic, called C S E _ E 1.0, which is a combination of two provers contradiction separation extension (CSE) and E, where CSE is based on the recently-introduced multi-clause standard contradiction separation (S-CS) calculus for first-order logic and E is the well-known equational theorem prover for first-order logic based on superposition and rewriting. The motivation of the combined prover C S E _ E 1.0 is to (1) evaluate the capability, applicability and generality of C S E _ E , and (2) take advantage of novel multi-clause S-CS dynamic deduction of CSE and mature equality handling of E to solve more and harder problems. In contrast to other improvements of E, C S E _ E 1.0 optimizes E mainly from the inference mechanism aspect. The focus of the present work is given to the description of C S E _ E including its S-CS rule, heuristic strategies, and the S-CS dynamic deduction algorithm for implementation. In terms of combination, in order not to lose the capability of E and use C S E _ E to solve some hard problems which are unsolved by E, C S E _ E 1.0 schedules the running of the two provers in time. It runs plain E first, and if E does not find a proof, it runs plain CSE, then if it does not find a proof, some clauses inferred in the CSE run as lemmas are added to the original clause set and the combined clause set handed back to E for further proof search. C S E _ E 1.0 is evaluated through benchmarks, e.g., CASC-26 (2017) and CASC-J9 (2018) competition problems (FOFdivision). Experimental results show that C S E _ E 1.0 indeed enhances the performance of E to a certain extent.

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Wang, Xiao Gang. "Significance of Mathematization of Philosophical Problems from the Angle of Broadspectrum Philosophy." Advanced Materials Research 433-440 (January 2012): 6315–18. http://dx.doi.org/10.4028/www.scientific.net/amr.433-440.6315.

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Whether philosophy can realize mathematization has long been controversial. As the mathematics develops a nonquantative branch- structural mathematics, however, mathematization of philosophy has a turnaround. Broadspectrum philosophy which makes use of structural mathematics has established a generally applicable as well as precise mathematical model for many philosophical problems, giving a positive answer to whether the philosophy can be mathematized. Mathematizaiton of philosophy allows more accurate and clear distinction of people’s expression in meaning, gives ideas the visible characteristics, makes philosophy an analyzable discipline, and realizes routinization of philosophical methods. Hegel was well versed in mathematics but opposed “Extreme Mathematic Attitude”, since he thought recognizing all the objects from the mathematic standpoint of “Quantity or Quantitative Relationship” would ignore the qualitative difference among the objects.［1］P239 Hegel’s opinion was based on the traditional mathematic which takes the Quantitative Relationship as the foundation. Holding the same evidence as Hegel's, most philosophers nowadays still suspect that the philosophy can be mathematized. When the modern mathematics has developed a new nonquantative branch, the Structural Mathematics, the philosophy mathematization, however, meets a turning point. Opposed to Quantitative Mathematics, the Structural Mathematics focuses on research of mathematic relationship and structure on the basis of abstract set theory. Since the structural mathematics doesn't rely on quantity and quantitative relationship, it can be combined in research of philosophy which usually doesn’t possess quantitative characteristics. Establishment of Broadspectrum Philosophy is a successful attempt. With full application of set theory, symbolic logic, modern algebra, transformation group theory and graph theory, Broadspectrum Philosophy constructs a generally applicable as well as precise mathematical mode for many philosophical problems, bringing a fundamental change to the philosophy. This paper attempts to make some preliminary analysis on the significance of establishment of Broadspectrum Philosophy.

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Baumgartner, James E. "Thomas Jech and Karel Prikry. On ideals of sets and the power set operation. Bulletin of the American Mathematical Society, vol. 82 (1976), pp. 593–595. - F. Galvin, T. Jech, and M. Magidor. An ideal game. The journal of symbolic logic, vol. 43 (1978), pp. 284–292. - T. Jech, M. Magidor, W. Mitchell, and K. Prikry. Precipitous ideals. The journal of symbolic logic, vol. 45 (1980), pp. 1–8. - Yuzuru Kakuda. On a condition for Cohen extensions which preserve precipitous ideals. The journal of symbolic logic, vol. 46(1981), pp. 296–300. - Thomas Jech and Karel Prikry. Ideals over uncountable sets: application of almost disjoint functions and generic ultrapowers. Memoirs of the American Mathematical Society, no. 214. American Mathematical Society, Providence 1979, iii + 71 pp. - Menachem Magidor. Precipitous ideals and sets. Israel journal of mathematics, vol. 35 (1980), pp. 109–134." Journal of Symbolic Logic 50, no. 1 (March 1985): 239–40. http://dx.doi.org/10.2307/2273805.

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Gkountakou, Fani, and Basil Papadopoulos. "The Use of Fuzzy Linear Regression and ANFIS Methods to Predict the Compressive Strength of Cement." Symmetry 12, no. 8 (August 4, 2020): 1295. http://dx.doi.org/10.3390/sym12081295.

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In this paper, the prediction of compressive cement strength using the fuzzy linear regression (FLR) and adaptive neuro-fuzzy inference system (ANFIS) methods was studied. Specifically, an accurate prediction method is needed as the modeling of cement strength is a difficult task, which is based on its composite nature. However, many approaches are widely implemented in strength-predicting problems, such as the artificial neural network (ANN), Mamdani fuzzy rules in MATLAB, FLR and ANFIS models. Applying these methods and comparing the results with the corresponding observed ones, we concluded that the ANFIS method successfully decreased the level of uncertainty in predicting cement strength, as the average percentage error level was extremely low. Although the FLR method had the highest average percentage error level compared with the other methods, it provides a standard equation to estimate the output values by using symmetric triangular fuzzy numbers and determines the most important factor in increasing compressive strength, in contrast to ANFIS and ANN, which are black box models, and to the fuzzy method, which uses rules without providing the specific way by which the results come out. Thus, ANFIS and FLR are appropriate methods for dealing with engineering mathematical models by using fuzzy logic.

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Cantini, Andrea. "Thomas Strahm. First steps into metapredicativity in explicit mathematics. Sets and proofs, Invited papers from Logic Colloquium '97—European meeting of the Association for Symbolic Logic, Leeds, July 1997, edited by S. Barry Cooper and John K. Truss, London Mathematical Society lecture note series, no. 258, Cambridge University Press, Cambridge, New York, and Oakleigh, Victoria, 1999, pp. 383–402." Bulletin of Symbolic Logic 8, no. 4 (December 2002): 535–36. http://dx.doi.org/10.2178/bsl/1182353930.

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Kierstead, Henry A. "G. Metakides and A. Nerode. Recursion theory and algebra. Algebra and logic, Papers from the 1974 Summer Research Institute of the Australian Mathematical Society, Monash University, Australia, edited by J. N. Crossley, Lecture notes in mathematics, vol. 450, Springer-Verlag, Berlin, Heidelberg, and New York, 1975, pp. 209–219. - Iraj Kalantari and Allen Retzlaff. Maximal vector spaces under automorphisms of the lattice of recursively enumerable vector spaces. The journal of symbolic logic, vol. 42 no. 4 (for 1977, pub. 1978), pp. 481–491. - Iraj Kalantari. Major subspaces of recursively enumerable vector spaces. The journal of symbolic logic, vol. 43 (1978), pp. 293–303. - J. Remmel. A r-maximal vector space not contained in any maximal vector space. The journal of symbolic logic, vol. 43 (1978), pp. 430–441. - Allen Retzlaff. Simple and hyperhypersimple vector spaces. The journal of symbolic logic, vol. 43 (1978), pp. 260–269. - J. B. Remmel. Maximal and cohesive vector spaces. The journal of symbolic logic, vol. 42 no. 3 (for 1977, pub. 1978), pp. 400–418. - J. Remmel. On r.e. and co-r.e. vector spaces with nonextendible bases. The journal of symbolic logic, vol. 45 (1980), pp. 20–34. - M. Lerman and J. B. Remmel. The universal splitting property: I. Logic Colloquim '80, Papers intended for the European summer meeting of the Association for Symbolic Logic, edited by D. van Dalen, D. Lascar, and T. J. Smiley, Studies in logic and the foundations of mathematics, vol. 108, North-Holland Publishing Company, Amsterdam, New York, and Oxford, 1982, pp. 181–207. - J. B. Remmel. Recursively enumerable Boolean algebras. Annals of mathematical logic, vol. 15 (1978), pp. 75–107. - J. B. Remmel. r-Maximal Boolean algebras. The journal of symbolic logic, vol. 44 (1979), pp. 533–548. - J. B. Remmel. Recursion theory on algebraic structures with independent sets. Annals of mathematical logic, vol. 18 (1980), pp. 153–191. - G. Metakides and J. B. Remmel. Recursion theory on orderings. I. A model theoretic setting. The journal of symbolic logic, vol. 44 (1979), pp. 383–402. - J. B. Remmel. Recursion theory on orderings. II. The journal of symbolic logic, vol. 45 (1980), pp. 317–333." Journal of Symbolic Logic 51, no. 1 (March 1986): 229–32. http://dx.doi.org/10.2307/2273960.

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Lerman, M. "Carl G. JockuschJr., and David B. Posner. Double jumps of minimal degrees. The journal of symbolic logic, vol. 43 no. 4 (for 1978, pub. 1979), pp. 715–724. - Carl G. JockuschJr., and David B. Posner. Automorphism bases for degrees of unsotvability. Israel journal of mathematics, vol. 40 (1981), pp. 150–164. - Richard L. Epstein. Initial segments of degrees below 0′. Memoirs of the American Mathematical Society, no. 241. American Mathematical Society, Providence1981, vi + 102 pp. - Richard A. Shore. The theory of the degrees below 0′. The journal of the London Mathematical Society, ser. 2 vol. 24 (1981), pp. 1–14." Journal of Symbolic Logic 50, no. 2 (June 1985): 550–52. http://dx.doi.org/10.2307/2274245.

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MÜLLER, SANDRA. "J. Steel, PFA implies ADL(ℝ). The Journal of Symbolic Logic, vol. 70 (2005), no. 4, pp. 1255–1296. - G. Sargsyan, Nontame mouse from the failure of square at a singular strong limit cardinal. Journal of Mathematical Logic, vol. 14 (2014), 1450003 (47 pages). - G. Sargsyan, Covering with universally Baire operators. Advances in Mathematics, vol. 268 (2015), pp. 603–665. - N. Trang, PFA and guessing models. Israel Journal of Mathematics, vol. 215 (2016), pp. 607–667." Bulletin of Symbolic Logic 26, no. 1 (March 2020): 89–92. http://dx.doi.org/10.1017/bsl.2020.6.

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Voytetsky, Igor, Taisiya Voytetskaya, Leonid Vyshnevskyi, Igor Kozyryev, Oksana Maksymova, Maksym Maksymov, and Viktoriia Kryvda. "Improving the ship's power plant automatic control system by using a model-oriented decision support system in order to reduce accident rate under the transitional and dynamic modes of operation." Eastern-European Journal of Enterprise Technologies 3, no. 2 (111) (June 30, 2021): 57–66. http://dx.doi.org/10.15587/1729-4061.2021.234447.

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This paper proposes a method to improve the performance of a ship's power plant by reducing accidents within it under transitional operating modes. The method is based on decreasing the number of service personnel errors by using a model-oriented decision support system. In order to implement the proposed method, the structure of the system of automatic control of the ship's power plant has been improved. Such an improvement of the control system implied the integration of a modeling unit and a decision support unit into its structure. The modeling unit makes it possible to predict values of the controlled parameters under a transition mode of operation before they actually appear in the system as a result of the operator's actions. A mathematical model of the automatic control system under transitional operating modes has been built for this unit. In order to implement the decision support unit, a method has been devised to formalize the task of managing the power plant under transitional operating modes. The method essentially involves modeling a transitional operating regime, followed by an evaluation of the results based on regulatory requirements and an empirical criterion for assessing the quality of enabling the diesel generators to work in parallel. In addition, a method has been developed for the decision support unit to reduce the accident rate and improve performance with the help of a mathematical apparatus of fuzzy inference, fuzzy logic, and fuzzy sets. Transitional operating regimes resulting from actual erroneous operator actions during ship flights were investigated. As a result of using the proposed system, the power plant performance increases

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BRAHA, DAN. "Special Section: Topological representation and reasoning in design and manufacturing." Artificial Intelligence for Engineering Design, Analysis and Manufacturing 14, no. 5 (November 2000): 355–58. http://dx.doi.org/10.1017/s0890060400145019.

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The word “topology” is derived from the Greek word “τοπος,” which means “position” or “location.” A simplified and thus partial definition has often been used (Croom, 1989, page 2): “topology deals with geometric properties which are dependent only upon the relative positions of the components of figures and not upon such concepts as length, size, and magnitude.” Topology deals with those properties of an object that remain invariant under continuous transformations (specifically bending, stretching, and squeezing, but not breaking or tearing). Topological notions and methods have illuminated and clarified basic structural concepts in diverse branches of modern mathematics. However, the influence of topology extends to almost every other discipline that formerly was not considered amenable to mathematical handling. For example, topology supports design and representation of mechanical devices, communication and transportation networks, topographic maps, and planning and controlling of complex activities. In addition, aspects of topology are closely related to symbolic logic, which forms the foundation of artificial intelligence. In the same way that the Euclidean plane satisfies certain axioms or postulates, it can be shown that certain abstract spaces—where the relations of points to sets and continuity of functions are important—have definite properties that can be analyzed without examining these spaces individually. By approaching engineering design from this abstract point of view, it is possible to use topological methods to study collections of geometric objects or collections of entities that are of concern in design analysis or synthesis. These collections of objects and or entities can be treated as spaces, and the elements in them as points.

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Cummings, James W. "Arthur W. Apter. On the least strongly compact cardinal. Israel journal of mathematics, vol. 35 (1980), pp. 225–233. - Arthur W. Apter. Measurability and degrees of strong compactness. The journal of symbolic logic, vol. 46 (1981), pp. 249–254. - Arthur W. Apter. A note on strong compactness and supercompactness. Bulletin of the London Mathematical Society, vol. 23 (1991), pp. 113–115. - Arthur W. Apter. On the first n strongly compact cardinals. Proceedings of the American Mathematical Society, vol. 123 (1995), pp. 2229–2235. - Arthur W. Apter and Saharon Shelah. On the strong equality between supercompactness and strong compactness.. Transactions of the American Mathematical Society, vol. 349 (1997), pp. 103–128. - Arthur W. Apter and Saharon Shelah. Menas' result is best possible. Ibid., pp. 2007–2034. - Arthur W. Apter. More on the least strongly compact cardinal. Mathematical logic quarterly, vol. 43 (1997), pp. 427–430. - Arthur W. Apter. Laver indestructibility and the class of compact cardinals. The journal of symbolic logic, vol. 63 (1998), pp. 149–157. - Arthur W. Apter. Patterns of compact cardinals. Annals of pure and applied logic, vol. 89 (1997), pp. 101–115. - Arthur W. Apter and Moti Gitik. The least measurable can be strongly compact and indestructible. The journal of symbolic logic, vol. 63 (1998), pp. 1404–1412." Bulletin of Symbolic Logic 6, no. 1 (March 2000): 86–89. http://dx.doi.org/10.2307/421078.

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Baldwin, Stewart. "Donder Hans-Dieter. Regularity of ultrafilters and the core model. Israel journal of mathematics, vol. 63 (1988), pp. 289–322. - Donder Hans-Dieter, Koepke Peter, and Levinski Jean-Pierre. Some stationary subsets of P(λ). Proceedings of the American Mathematical Society, vol. 102 (1988), pp. 1000–1004. - Walker D. J.. On the transversal hypothesis and the weak Kurepa hypothesis. The journal of symbolic logic, vol. 53 (1988), pp. 854–877." Journal of Symbolic Logic 55, no. 3 (September 1990): 1313–15. http://dx.doi.org/10.2307/2274497.

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Eklof, Paul C. "Fred Appenzeller. An independence result in quadratic form theory: infinitary combinatorics applied to ε-Hermitian spaces. The journal of symbolic logic, vol. 54 (1989), pp. 689–699. - Otmar Spinas. Linear topologies on sesquilinear spaces of uncountable dimension. Fundamenta mathematicae, vol. 139 (1991), pp. 119–132. - James E. Baumgartner, Matthew Foreman, and Otmar Spinas. The spectrum of the Γ-invariant of a bilinear space. Journal of algebra, vol. 189 (1997), pp. 406–418. - James E. Baumgartner and Otmar Spinas. Independence and consistency proofs in quadratic form theory. The journal of symbolic logic, vol. 56 (1991), pp. 1195–1211. - Otmar Spinas. Iterated forcing in quadratic form theory. Israel journal of mathematics, vol. 79 (1992), pp. 297–315. - Otmar Spinas. Cardinal invariants and quadratic forms. Set theory of the reals, edited by Haim Judah, Israel mathematical conference proceedings, vol. 6, Gelbart Research Institute for Mathematical Sciences, Bar-Ilan University, Ramat-Gan 1993, distributed by the American Mathematical Society, Providence, pp. 563–581. - Saharon Shelah and Otmar Spinas. Gross spaces. Transactions of the American Mathematical Society, vol. 348 (1996), pp. 4257–4277." Bulletin of Symbolic Logic 7, no. 2 (June 2001): 285–86. http://dx.doi.org/10.2307/2687785.

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Kossak, Roman. "James H. Schmerl. Peano models with many generic classes. Pacific Journal of Mathematics, vol. 43 (1973), pp. 523–536. - James H. Schmerl. Correction to: “Peano models with many generic classes”. Pacific Journal of Mathematics, vol. 92 (1981), no. 1, pp. 195–198. - James H. Schmerl. Recursively saturated, rather classless models of Peano arithmetic. Logic Year 1979–80. Recursively saturated, rather classless models of Peano arithmetic. Logic Year 1979–80 (Proceedings, Seminars, and Conferences in Mathematical Logic, University of Connecticut, Storrs, Connecticut, 1979/80). edited by M. Lerman, J. H. Schmerl, and R. I. Soare, Lecture Notes in Mathematics, vol. 859. Springer, Berlin, pp. 268–282. - James H. Schmerl. Recursively saturatedmodels generated by indiscernibles. Notre Dane Journal of Formal Logic, vol. 26 (1985), no. 1, pp. 99–105. - James H. Schmerl. Large resplendent models generated by indiscernibles. The Journal of Symbolic Logic, vol. 54 (1989), no. 4, pp. 1382–1388. - James H. Schmerl. Automorphism groups of models of Peano arithmetic. The Journal of Symbolic Logic, vol. 67 (2002), no. 4, pp. 1249–1264. - James H. Schmerl. Diversity in substructures. Nonstandard models of arithmetic and set theory. edited by A. Enayat and R. Kossak, Contemporary Mathematics, vol. 361, American Mathematical Societey (2004), pp. 45–161. - James H. Schmerl. Generic automorphisms and graph coloring. Discrete Mathematics, vol. 291 (2005), no. 1–3, pp. 235–242. - James H. Schmerl. Nondiversity in substructures. The Journal of Symbolic Logic, vol. 73 (2008), no. 1, pp. 193–211." Bulletin of Symbolic Logic 15, no. 2 (June 2009): 222–27. http://dx.doi.org/10.1017/s1079898600001359.

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Сафиулина, Yu Safiulina, Шмурнов, and V. Shmurnov. "Graphical Proof of the Main Theorem of Non-Euclidean Geometry." Geometry & Graphics 3, no. 3 (November 30, 2015): 18–23. http://dx.doi.org/10.12737/14416.

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The tragedy for the pioneers of non-Euclidean geometry (N. Lobachevsky and J. Boyai) was their quarrel with the scientific tradition. Figuratively speaking, in the judgment of the scientific world they could not provide proof of their views, and substantive law of science was not on their side despite the efforts of such an influential advocate as Karl Friedrich Gauss. They lost the civil process to the scientific layman, who sincerely believed that the earth is flat. Traditionally mathematical logic considers a new idea proven, if it is derived by inference from already proven ones, or recognized as obvious, or recognized without proof (postulates). Yet the founders of non-Euclidean geometry could not imagine such traditional evidence at all desire, because it had not yet been developed, and most importantly respective starting points (axioms, postulates, and theorems) had not been recognized by mathematicians. The paper outlines the original concept of non-Euclidean geometries. Hyperbolic geometry of Lobachevsky is considered based on viewing the sphere as a surface of zero curvature. In this case, the plane will have a real curvature properties of hyperboloid or a pseudosphere depending on the absolute and space anisotropy index, which replaces the concept of curvature of space; i.e. the notion of the curvature of the surface is converted to purely analytical attributes. Parabolic geometry of Euclid with degenerate absolute becomes a special case of geometries with non-degenerate absolute. The geometry of Riemann having the absolute of imaginary surface with negative Gaussian curvature at all points is declared not real but imaginary, since, according to the authors, it is impossible for plotting. References to textbooks of mechanics and mathematics departments of universities.

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Becker, Howard S. "R. Dougherty and A. S. Kechris. Hausdorff measures and sets of uniqueness for trigonometric series. Proceedings of the American Mathematical Society, vol. 105 (1989), pp. 894–897. - Alexander S. Kechris and Alain Louveau. Covering theorems for uniqueness and extended uniqueness sets. Colloquium mathematicum, vol. 59 (1990), pp. 63–79. - Alexander S. Kechris. Hereditary properties of the class of closed sets of uniqueness for trigonometric series. Israel journal of mathematics, vol. 73 (1991), pp. 189–198. - A. S. Kechris and A. Louveau. Descriptive set theory and harmonic analysis. The journal of symbolic logic, vol. 57 (1992), pp. 413–441." Bulletin of Symbolic Logic 8, no. 1 (March 2002): 94–95. http://dx.doi.org/10.2178/bsl/1182353856.

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Jockusch, Carl. "Richard A. Shore. Determining automorphisms of the recursively enumerable sets. Proceedings of the American Mathematical Society, vol. 65 (1977), pp. 318– 325. - Richard A. Shore. The homogeneity conjecture. Proceedings of the National Academy of Sciences of the United States of America, vol. 76 (1979), pp. 4218– 4219. - Richard A. Shore. On homogeneity and definability in the first-order theory of the Turing degrees. The journal of symbolic logic, vol. 47 (1982), pp. 8– 16. - Richard A. Shore. The arithmetic and Turing degrees are not elementarily equivalent. Archiv für mathematische Logik und Grundlagenforschung, vol. 24 (1984), pp. 137– 139. - Richard A. Shore. The structure of the degrees of unsolvabitity. Recursion theory, edited by Anil Nerode and Richard A. Shore, Proceedings of symposia in pure mathematics, vol. 42, American Mathematical Society, Providence1985, pp. 33– 51. - Theodore A. Slaman and W. Hugh Woodin. Definability in the Turing degrees. Illinois journal of mathematics, vol. 30 (1986), pp. 320– 334." Journal of Symbolic Logic 55, no. 1 (March 1990): 358–60. http://dx.doi.org/10.2307/2274995.

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Ressayre, J. P. "Jon Barwise and John Schlipf. On recursively saturated models of arithmetic. Model theory and algebra, A memorial tribute to Abraham Robinson, edited by D. H. Saracino and V. B. Weispfenning, Lecture notes in mathematics, vol. 498, Springer-Verlag, Berlin, Heidelberg, and New York, 1975, pp. 42–55. - Patrick Cegielski, Kenneth McAloon, and George Wilmers. Modèles récursivement saturés de l'addition et de la multiplication des entiers naturels. Logic Colloquium '80, Papers intended for the European summer meeting of the Association for Symbolic Logic, edited by D. van Dalen, D. Lascar, and T. J. Smiley, Studies in logic and the foundations of mathematics, vol. 108, North-Holland Publishing Company, Amsterdam, New York, and London, 1982, pp. 57–68. - Julia F. Knight. Theories whose resplendent models are homogeneous. Israel journal of mathematics, vol. 42 (1982), pp. 151–161. - Julia Knight and Mark Nadel. Expansions of models and Turing degrees. The journal of symbolic logic, vol. 47 (1982), pp. 587–604. - Julia Knight and Mark Nadel. Models of arithmetic and closed ideals. The journal of symbolic logic, vol. 47 no. 4 (for 1982, pub. 1983), pp. 833–840. - Henryk Kotlarski. On elementary cuts in models of arithmetic. Fundamenta mathematicae, vol. 115 (1983), pp. 27–31. - H. Kotlarski, S. Krajewski, and A. H. Lachlan. Construction of satisfaction classes for nonstandard models. Canadian mathematical bulletin—Bulletin canadien de mathématiques, vol. 24 (1981), pp. 283–293. - A. H. Lachlan. Full satisfaction classes and recursive saturation. Canadian mathematical bulletin—Bulletin canadien de mathématiques, pp. 295–297. - Leonard Lipshitz and Mark Nadel. The additive structure of models of arithmetic. Proceedings of the American Mathematical Society, vol. 68 (1978), pp. 331–336. - Mark Nadel. On a problem of MacDowell and Specker. The journal of symbolic logic, vol. 45 (1980), pp. 612–622. - C. Smoryński. Back-and-forth inside a recursively saturated model of arithmetic. Logic Colloquium '80, Papers intended for the European summer meeting of the Association for Symbolic Logic, edited by D. van Dalen, D. Lascar, and T. J. Smiley, Studies in logic and the foundations of mathematics, vol. 108, North-Holland Publishing Company, Amsterdam, New York, and London, 1982, pp. 273–278. - C. Smoryński and J. Stavi. Cofinal extension preserves recursive saturation. Model theory of algebra and arithmetic, Proceedings of the Conference on Applications of Logic to Algebra and Arithmetic held at Karpacz, Poland, September 1–7,1979, edited by L. Pacholski, J. Wierzejewski, and A. J. Wilkie, Lecture notes in mathematics, vol. 834, Springer-Verlag, Berlin, Heidelberg, and New York, 1980, pp. 338–345. - George Wilmers. Minimally saturated models. Model theory of algebra and arithmetic, Proceedings of the Conference on Applications of Logic to Algebra and Arithmetic held at Karpacz, Poland, September 1–7, 1979, edited by L. Pacholski, J. Wierzejewski, and A. J. Wilkie, Lecture notes in mathematics, vol. 834, Springer-Verlag, Berlin, Heidelberg, and New York, 1980, pp. 370–380." Journal of Symbolic Logic 52, no. 1 (March 1987): 279–84. http://dx.doi.org/10.2307/2273884.

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Lubarsky, Robert. "Patrick Farrington. Hinges and automorphisms of the degrees of non-constructibility. The journal of the London Mathematical Society, ser. 2 vol. 28 (1983), pp. 193–202. - Petr Hájek. Some results on degrees of constructibility. Higher set theory, Proceedings, Oberwolfach, Germany, April 13–23, 1977, edited by G. H. Müller and D. S. Scott, Lecture notes in mathematics, vol. 669, Springer-Verlag, Berlin, Heidelberg, and New York, 1978, pp. 55–71. - Zofia Adamowicz. On finite lattices of degrees of constructibility of reals. The journal of symbolic logic, vol. 41 (1976), pp. 313–322. - Zofia Adamowicz. Constructive semi-lattices of degrees of constructibility. Set theory and hierarchy theory V, Bierutowice, Poland 1976, edited by A. Lachlan, M. Srebrny, and A. Zarach, Lecture notes in mathematics, vol. 619, Springer-Verlag, Berlin, Heidelberg, and New York, 1977, pp. 1–43." Journal of Symbolic Logic 54, no. 3 (September 1989): 1109–11. http://dx.doi.org/10.2307/2274781.

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Cherlin, Gregory L. "Angus Macintyre, Kenneth McKenna, and Lou van den Dries. Elimination of quantifiers in algebraic structures. Advances in mathematics, vol. 47 (1983), pp. 74–87. - L. P. D. van den Dries. A linearly ordered ring whose theory admits elimination of quantifiers is a real closed field. Proceedings of the American Mathematical Society, vol. 79 (1980), pp. 97–100. - Bruce I. Rose. Rings which admit elimination of quantifiers. The journal of symbolic logic, vol. 43 (1978), pp. 92–112; Corrigendum, vol. 44 (1979), pp. 109–110. - Chantal Berline. Rings which admit elimination of quantifiers. The journal of symbolic logic, vol. 43 (1978), vol. 46 (1981), pp. 56–58. - M. Boffa, A. Macintyre, and F. Point. The quantifier elimination problem for rings without nilpotent elements and for semi-simple rings. Model theory of algebra and arithmetic, Proceedings of the Conference on Applications of Logic to Algebra and Arithmetic held at Karpacz, Poland, September 1–7, 1979, edited by L. Pacholski, J. Wierzejewski, and A. J. Wilkie, Lecture notes in mathematics, vol. 834, Springer-Verlag, Berlin, Heidelberg, and New York, 1980, pp. 20–30. - Chantal Berline. Elimination of quantifiers for non semi-simple rings of characteristic p. Model theory of algebra and arithmetic, Proceedings of the Conference on Applications of Logic to Algebra and Arithmetic held at Karpacz, Poland, September 1–7, 1979, edited by L. Pacholski, J. Wierzejewski, and A. J. Wilkie, Lecture notes in mathematics, vol. 834, Springer-Verlag, Berlin, Heidelberg, and New York, 1980, pp. 10–19." Journal of Symbolic Logic 50, no. 4 (December 1985): 1079–80. http://dx.doi.org/10.2307/2273998.

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Nyikos, Peter J. "Andreas Blass and Saharon Shelah. Ultrafilters with small generating sets. Israel journal of mathematics, vol. 65 (1989), pp. 259–271. - Andreas Blass and Saharon Shelah. There may be simple - and -points and the Rudin–Keisler ordering may be downward directed. Annals of pure and applied logic, vol. 33 (1987), pp. 213–243. - Andreas Blass. Near coherence of filters. II: Applications to operator ideals, the Stone–Čech remainder of a half-line, order ideals of sequences, and the slenderness of groups. Transactions of the American Mathematical Society, vol. 300 (1987), pp. 557–581. - Andreas Blass and Saharon Shelah. Near coherence of filters III: a simplified consistency proof. Notre Dame journal of formal logic, vol. 30 (1989), pp. 530–538. - Andreas Blass and Claude Laflamme. Consistency results about filters and the number of inequivalent growth types. The journal of symbolic logic, vol. 54 (1989), pp. 50–56. - Andreas Blass. Applications of superperfect forcing and its relatives. Set theory and its applications. Proceedings of a conference held at York University, Ontario, Canada, Aug. 10–21, 1987, edited by J. Steprāns and S. Watson, Lecture notes in mathematics, vol. 1401, Springer-Verlag, Berlin etc. 1989, pp. 18–40. - Andreas Blass and Saharon Shelah. Ultrafilters with small generating sets. Israel journal of mathematics, vol. 65 (1989), pp. 259–271." Journal of Symbolic Logic 57, no. 2 (June 1992): 763–66. http://dx.doi.org/10.2307/2275316.

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Drápal, Aleš. "Richard Laver. The left distributive law and the freeness of an algebra of elementary embeddings. Advances in mathematics, vol. 91 (1992), pp. 209–231. - Richard Laver. A division algorithm for the free left distributive algebra. Logic Colloquium '90, ASL summer meeting in Helsinki, edited by J. Oikkonen and J. Väänänen, Lecture notes in logic, no. 2, Springer-Verlag, Berlin, Heidelberg, New York, etc., 1993, pp. 155–162. - Richard Laver. On the algebra of elementary embeddings of a rank into itself. Advances in mathematics, vol. 110 (1995), pp. 334–346. - Richard Laver. Braid group actions on left distributive structures, and well orderings in the braid groups. Journal of pure and applied algebra, vol. 108 (1996), pp. 81–98. - Patrick Dehornoy. An alternative proof of Laver's results on the algebra generated by an elementary embedding. Set theory of the continuum, edited by H. Judah, W. Just, and H. Woodin, Mathematics Sciences Research Institute publications, vol. 26, Springer-Verlag, New York, Berlin, Heidelberg, etc., 1992, pp. 27–33. - Patrick Dehornoy. Braid groups and left distributive operations. Transactions of the American Mathematical Society, vol. 345 (1994), pp. 115–150. - Patrick Dehornoy. A normal form for the free left distributive law. International journal of algebra and computation, vol. 4 (1994), pp. 499–528. - Patrick Dehornoy. From large cardinals to braids via distributive algebra. Journal of knot theory and its ramifications, vol. 4 (1995), pp. 33–79. - J. R. Steel. The well-foundedness of the Mitchell order. The journal of symbolic logic, vol. 58 (1993), pp. 931–940. - Randall Dougherty. Critical points in an algebra of elementary embeddings. Annals of pure and applied logic, vol. 65 (1993), pp. 211–241. - Randall Dougherty. Critical points in an algebra of elementary embeddings, II. Logic: from foundations to applications, European logic colloquium, edited by Wilfrid Hodges, Martin Hyland, Charles Steinhorn, and John Truss, Clarendon Press, Oxford University Press, Oxford, New York, etc., 1996, pp. 103–136. - Randall Dougherty and Thomas Jech. Finite left-distributive algebras and embedding algebras. Advances in mathematics, vol. 130 (1997), pp. 201–241." Bulletin of Symbolic Logic 8, no. 4 (December 2002): 555–60. http://dx.doi.org/10.2178/bsl/1182353941.

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Zeman, Martin. "Ernest Schimmerling. Covering properties of core models. Sets and proofs. (Leeds, 1997), London Mathematical Society Lecture Note Series 258. Cambridge University Press, Cambridge, 1999, pp. 281–299. - Peter Koepke. An introduction to extenders and core models for extender sequences. Logic Colloquium '87 (Granada, 1987), Studies in Logic and the Foundations of Mathematics 129. North-Holland, Amsterdam, 1989, pp. 137–182. - William J. Mitchell. The core model up to a Woodin cardinal. Logic, methodology and philosophy of science, IX (Uppsala, 1991), Studies in Logic and the Foundations of Mathematics 134, North-Holland, Amsterdam, 1994, pp. 157–175. - Benedikt Löwe and John R. Steel. An introduction to core model theory. Sets and proofs (Leeds, 1997), London Mathematical Society Lecture Note Series 258, Cambridge University Press, Cambridge, 1999, pp. 103–157. - John R. Steel. Inner models with many Woodin cardinals. Annals of Pure and Applied Logic, vol. 65 no. 2 (1993), pp. 185–209. - Ernest Schimmerling. Combinatorial principles in the core model for one Woodin cardinal. Annals of Pure and Applied Logic, vol. 74 no. 2 (1995), pp. 153–201. - Philip D. Welch. Some remarks on the maximality of inner models. Logic Colloquium '98 (Prague, 1998), Lecture Notes in Logic 13, Association of Symbolic Logic, Urbana, Illinois, 2000, pp. 516–540." Bulletin of Symbolic Logic 10, no. 4 (December 2004): 583–88. http://dx.doi.org/10.1017/s1079898600003681.

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Hirschfeldt, Denis R. "Five papers on reverse mathematics and Ramsey-theoretic principles - C. T. Chong, Theodore A. Slaman, and Yue Yang, The metamathematics of Stable Ramsey’s Theorem for Pairs. Journal of the American Mathematical Society, vol. 27 (2014), no. 3, pp. 863–892. - Manuel Lerman, Reed Solomon, and Henry Towsner, Separating principles below Ramsey’s Theorem for Pairs. Journal of Mathematical Logic, vol. 13 (2013), no. 2, 1350007, 44 pp. - Jiayi Liu, $RT_2^^2$ does not imply WKL0. Journal of Symbolic Logic, vol. 77 (2012), no. 2, pp. 609–620. - Lu Liu, Cone avoiding closed sets. Transactions of the American Mathematical Society, vol. 367 (2015), no. 3, pp. 1609–1630. - Wei Wang, Some logically weak Ramseyan theorems. Advances in Mathematics, vol. 261 (2014), pp. 1–25." Bulletin of Symbolic Logic 22, no. 4 (December 2016): 526–30. http://dx.doi.org/10.1017/bsl.2016.32.

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Louveau, Alain. "Jack H. Silver. Counting the number of equivalence classes of Borel and coanalytic equivalence relations. Annals of mathematical logic, vol. 18 (1980), pp. 1–28. - John P. Burgess. Equivalences generated by families of Borel sets. Proceedings of the American Mathematical Society. vol. 69 (1978), pp. 323–326. - John P. Burgess. A reflection phenomenon in descriptive set theory. Fundamenta mathematicae. vol. 104 (1979), pp. 127–139. - L. Harrington and R. Sami. Equivalence relations, projective and beyond. Logic Colloquium '78, Proceedings of the Colloquium held in Mons, August 1978, edited by Maurice Boffa, Dirk van Dalen, and Kenneth McAloon, Studies in logic and the foundations of mathematics, vol. 97, North-Holland Publishing Company, Amsterdam, New York, and Oxford, 1979, pp. 247–264. - Leo Harrington and Saharon Shelah. Counting equivalence classes for co-κ-Souslin equivalence relations. Logic Colloquium '80, Papers intended for the European summer meeting of the Association for Symbolic Logic, edited by D. van Dalen, D. Lascar, and T. J. Smiley, Studies in logic and the foundations of mathematics, vol. 108, North-Holland Publishing Company, Amsterdam, New York, and Oxford, 1982, pp. 147–152. - Jacques Stern. On Lusin's restricted continuum problem. Annals of mathematics, ser. 2 vol. 120 (1984), pp. 7–37." Journal of Symbolic Logic 52, no. 3 (September 1987): 869–70. http://dx.doi.org/10.1017/s0022481200029856.

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Bencivenga, Ermanno. "Hugues Leblanc. Preface. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. ix–x. - Hugues Leblanc. Introduction. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 3–16. - Hugues Leblanc and T. Hailperin. Non-designating singular terms. A revised reprint of XXV 87. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 17–21. - Hugues Leblanc and R. H. Thomason. Completeness theorems for some presupposition-free logics. A revised reprint of XXXVII 424. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 22–57. - Hugues Leblanc and R. K. Meyer. On prefacing (∀x) ⊃ A(Y/X) with (∀Y): a free quantification theory without identity. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 58–75. (Reprinted with revisions from Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 16 (1970), pp. 447–462. - Hugues Leblanc. Truth-value semantics for a logic of existence. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 76–90. (Reprinted with revisions from Notre Dame journal of formal logic, vol. 12 (1971), pp. 153–168.) - Hugues Leblanc and R. K. Meyer. Open formulas and the empty domain. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 91–98. (Reprinted from Archiv für mathematische Logik und Grundlagenforschung, vol. 12 (1969), pp. 78–84.) - K. Lambert, Hugues Leblanc, and R. K. Meyer. A liberated version of S5. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 99–102. (Reprinted with revisions from Archiv für mathematische Logik und Grundlagenforschung, vol. 12 (1969), pp. 151–154.) - Hugues Leblanc. On dispensing with things and worlds. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 103–119. (Reprinted with revisions from Logic and ontology, edited by Milton K. Munitz, New York University Press, New York 1973, pp. 241–259.) - Hugues Leblanc. Introduction. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 123–138. - Hugues Leblanc. A simplified account of validity and implication for quantificational logic. A revised reprint of XXXV 466. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 139–143. - Hugues Leblanc. A simplified strong completeness proof for QC=. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 144–155. (Reprinted with minor revisions from Akten des XIV. Internationalen Kongresses für Philosophie Wien, 2.-9. September 1968, vol. 3, Logik Erkenntnis- und Wissenschaftstheorie Sprachphilosophie Ontologie und Metaphysik, Universität Wien, Herder, Vienna 1969, pp. 83–96.) - Hugues Leblanc. Truth-value assignments and their cardinality. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 156–165. (Reprinted with revisions from Philosophia, vol. 7 (1978), pp. 305–316.) - Hugues Leblanc. Three generalizations of a theorem of Beth's. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 166–176. (Reprinted with revisions from Logique et analyse, n.s. vol. 12 (1969), pp. 205–220.) - Hugues Leblanc and R. K. Meyer. Truth-value semantics for the theory of types. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 177–197. (Reprinted with revisions from Philosophical problems in logic, Some recent developments, edited by Karel Lambert, Synthese library, D. Reidel Publishing Company, Dordrecht 1970, pp. 77–101.) - Hugues Leblanc. Wittgenstein and the truth-functionality thesis. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 198–204. (Reprinted with revisions from American philosophical quarterly, vol. 9 (1972), pp. 271–274.) - Hugues Leblanc. Matters of relevance. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 205–219. (Reprinted from Journal of philosophical logic, vol. 1 (1972), pp. 269–286. Also reprinted in Exact philosophy, Problems, tools, and goals, edited by Mario Bunge, Synthese library, D. Reidel Publishing Company, Dordrecht and Boston 1973, pp. 3–20.) - Hugues Leblanc and G. Weaver. Truth-functionality and the ramified theory of types. A revised reprint of XLII 313. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 220–235. - Hugues Leblanc. That Principia mathematica, first edition, has a predicative interpretation after all. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 236–239. (Reprinted with revisions from Journal of philosophical logic, vol. 4 (1975), pp. 67–70.) - H. Goldberg, Hugues Leblanc, and G. Weaver. A strong completeness theorem for three-valued logic: part I. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 240–246. (Reprinted with revisions from Notre Dame journal of formal logic, vol. 15 (1974), pp. 325–330.) - Hugues Leblanc. A strong completeness theorem for three-valued logic: part II. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 247–257. (Reprinted with revisions from Notre Dame journal of formal logic, vol. 18 (1977), pp. 107–116.) - Hugues Leblanc and R. P. McArthur. A completeness result for quantificational tense logic. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 258–266. (Reprinted with revisions from Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 22 (1976), pp. 89–96.) - Hugues Leblanc. Semantic deviations. A revised reprint of XLII 313. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 267–280. - Hugues Leblanc. Introduction. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 283–292. - Hugues Leblanc. Marginalia on Gentzen's Sequenzen-Kalkulë. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 293–300. (Reprinted with revisions from Contributions to logic and methodology in honor of J. M. Bocheński, edited by Anna-Teresa Tymieniecka in collaboration with Charles Parsons, North-Holland Publishing Company, Amsterdam 1965, pp. 73–83.) - Hugues Leblanc. Structural rules of inference. A revised reprint of XXVIII 256. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 301–305. - Hugues Leblanc. Proof routines for the propositional calculus. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 306–327. (Reprinted with revisions from Notre Dame journal of formal logic, vol. 4 (1963), pp. 81–104.) - Hugues Leblanc. Two separation theorems for natural deduction. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 328–349. (Reprinted with revisions from Notre Dame journal of formal logic, vol. 7 (1966), pp. 159–180.) - Hugues Leblanc. Two shortcomings of natural deduction. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 350–357. (Reprinted with revisions from The journal of philosophy, vol. 63 (1966), pp. 29–37.) - Hugues Leblanc. Subformula theorems for N-sequents. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 358–381. (Reprinted with minor revisions from The journal of symbolic logic, vol. 33 (1968), pp. 161–179.) - E. W. Beth and Hugues Leblanc. A note on the intuitionist and the classical propositional calculus. A revised reprint of XXV 351. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 382–384. - Hugues Leblanc and N. D. Belnap Jr. Intuitionism reconsidered. A revised reprint of XXVIII 256. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 385–389. - N. D. BelnapJr., Hugues Leblanc, and R. H. Thomason. On not strengthening intuitionistic logic. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 390–396. (Reprinted with revisions from Notre Dame journal of formal logic, vol. 4 no. 4 (for 1963, pub. 1964), pp. 313–320.) - Hugues Leblanc and R. H. Thomason. The demarcation line between intuitionist logic and classical logic. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 397–403. (Reprinted with revisions from Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 12 (1966), pp. 257–262.) - Hugues Leblanc. Boolean algebra and the propositional calculus. A revised reprint of XXXVII 755. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 404–407. - Hugues Leblanc. The algebra of logic and the theory of deduction. A revised reprint of XXXVII 755. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 408–413. - Hugues Leblanc and R. H. Thomason. All or none: a novel choice of primitives for elementary logic. A revised reprint of XXXIV 124. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 414–421. - Hugues Leblanc and R. K. Meyer. Matters of separation. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 422–430. (Reprinted with revisions from Notre Dame journal of formal logic, vol. 13 (1972), pp. 229–236.) - Hugues Leblanc. Generalization in first-order logic. Existence, truth, and provability, by Hugues Leblanc, State University of New York Press, Albany1982, pp. 431–452. (Reprinted with revisions from Notre Dame journal of formal logic, vol. 20 (1979), pp. 835–857.)." Journal of Symbolic Logic 50, no. 1 (March 1985): 227–31. http://dx.doi.org/10.2307/2273801.

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Franklin, Johanna N. Y. "André Nies. Lowness properties and randomness. Advances in Mathematics, vol. 197 (2005), no. 1, pp. 274–305. - Bjørn Kjos-Hanssen, André Nies, and Frank Stephan. Lowness for the class of Schnorr random reals. SIAM Journal on Computing, vol. 35 (2005), no. 3, pp. 647–657. - Noam Greenberg and Joseph S. Miller. Lowness for Kurtz randomness. The Journal of Symbolic Logic, vol. 74 (2009), no. 2, pp. 665–678. - Laurent Bienvenu and Joseph S. Miller. Randomness and lowness notions via open covers. Annals of Pure and Applied Logic, vol. 163 (2012), no. 5, pp. 506–518. - Johanna N. Y. Franklin, Frank Stephan, and Liang. Yu Relativizations of randomness and genericity notions. The Bulletin of the London Mathematical Society, vol. 43 (2011), no. 4, pp. 721–733. - George Barmpalias, Joseph S. Miller, and André Nies. Randomness notions and partial relativization. Israel Journal of Mathematics, vol. 191 (2012), no. 2, pp. 791–816." Bulletin of Symbolic Logic 19, no. 1 (March 2013): 115–18. http://dx.doi.org/10.1017/s1079898600009124.

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Yaacov, Itaï Ben. "Ilijas Farah, Bradd Hart, and David Sherman. Model theory of operator algebras I: stability. Bulletin of the London Mathematical Society, vol. 45 (2013), no. 4, pp. 825–838, doi:10.1112/blms/bdt014. - Ilijas Farah, Bradd Hart, and David Sherman. Model theory of operator algebras II: model theory. Israel Journal of Mathematics, vol. 201 (2014), no. 1, pp. 477–505, doi:10.1007/s11856-014-1046-7. - Ilijas Farah, Bradd Hart, and David Sherman. Model theory of operator algebras III: elementary equivalence and II1factors. Bulletin of the London Mathematical Society, vol. 46 (2014), no. 3, pp. 609–628, doi:10.1112/blms/bdu012. - Isaac Goldbring, Bradd Hart, and Thomas Sinclair. The theory of tracial von Neumann algebras does not have a model companion. Journal of Symbolic Logic, vol. 78 (2013), no. 3, pp. 1000–1004." Bulletin of Symbolic Logic 21, no. 4 (December 2015): 425–27. http://dx.doi.org/10.1017/bsl.2015.32.

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Folina, Janet. "After Non-Euclidean Geometry: Intuition, Truth and the Autonomy of Mathematics." Journal for the History of Analytical Philosophy 6, no. 3 (February 5, 2018). http://dx.doi.org/10.15173/jhap.v6i3.3438.

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The mathematical developments of the 19th century seemed to undermine Kant’s philosophy. Non-Euclidean geometries challenged Kant’s view that there is a spatial intuition rich enough to yield the truth of Euclidean geometry. Similarly, advancements in algebra challenged the view that temporal intuition provides a foundation for both it and arithmetic. Mathematics seemed increasingly detached from experience as well as its form; moreover, with advances in symbolic logic, mathematical inference also seemed independent of intuition. This paper considers various philosophical responses to these changes, focusing on the idea of modifying Kant’s conception of intuition in order to accommodate the increasing abstractness of mathematics. It is argued that far from clinging to an outdated paradigm, programs based on new conceptions of intuition should be seen as motivated by important philosophical desiderata, such as the truth, apriority, distinctiveness and autonomy of mathematics.

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Santosa, Raden Gunawan, and Junius Karel Tampubolon. "Pelatihan Cara Berpikir Simbolik-Matematik di SMA BOPKRI 2 Yogyakarta." International Journal of Community Service Learning 4, no. 1 (March 20, 2020). http://dx.doi.org/10.23887/ijcsl.v4i1.23099.

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Symbolic-Mathematical Thinking is a technique of thinking to solve mathematical problems. This technique is the result of a combination of Symbolic Logic and Mathematics. In general there are three symbolic logic activities that are applied to Mathematics, namely reading symbols, logical equivalents and logical implications. This Pengabdian kepada Masyarakat (PkM) has activities to train Symbolic-Mathematical thinking using 15 strategies for BOPKRI 2 high school students. The results of this training are three important things. The first result is the type of problem that is most difficult for students to face is the type of problem that requires drawing conclusions that refer to standard mathematical definitions. The second thing is for the type of use of model settlement strategies in a system and the type of seeing patterns, students tend to be able to solve problems after being given instructions on how the model fits the problem and the pattern the problem has. Whereas the third is from two groups of students the class turns out that the continuous class attending the training gets more symbolic-mathematical thinking skills improvement than the other classes.

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SIEG, WILFRIED, and PATRICK WALSH. "NATURAL FORMALIZATION: DERIVING THE CANTOR-BERNSTEIN THEOREM IN ZF." Review of Symbolic Logic, November 18, 2019, 1–35. http://dx.doi.org/10.1017/s175502031900056x.

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Abstract Natural Formalization proposes a concrete way of expanding proof theory from the meta-mathematical investigation of formal theories to an examination of “the concept of the specifically mathematical proof.” Formal proofs play a role for this examination in as much as they reflect the essential structure and systematic construction of mathematical proofs. We emphasize three crucial features of our formal inference mechanism: (1) the underlying logical calculus is built for reasoning with gaps and for providing strategic directions, (2) the mathematical frame is a definitional extension of Zermelo–Fraenkel set theory and has a hierarchically organized structure of concepts and operations, and (3) the construction of formal proofs is deeply connected to the frame through rules for definitions and lemmas. To bring these general ideas to life, we examine, as a case study, proofs of the Cantor–Bernstein Theorem that do not appeal to the principle of choice. A thorough analysis of the multitude of “different” informal proofs seems to reduce them to exactly one. The natural formalization confirms that there is one proof, but that it comes in two variants due to Dedekind and Zermelo, respectively. In this way it enhances the conceptual understanding of the represented informal proofs. The formal, computational work is carried out with the proof search system AProS that serves as a proof assistant and implements the above inference mechanism; it can be fully inspected at http://www.phil.cmu.edu/legacy/Proof_Site/. We must—that is my conviction—take the concept of the specifically mathematical proof as an object of investigation. Hilbert 1918

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"IVAN SLESHYNSKY AS A POPULARIZER OF THE IDEAS OF MATHEMATICAL LOGIC IN UKRAINE." Journal of V. N. Karazin Kharkiv National University, Series "Philosophy. Philosophical Peripeteias", no. 62 (2020): 99–107. http://dx.doi.org/10.26565/2226-0994-2020-62-11.

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The first half of the twentieth century was marked by the simultaneous development of logic and mathematics. Logic offered the necessary means to justify the foundations of mathematics and to solve the crisis that arose in mathematics in the early twentieth century. In European science in the late nineteenth century, the ideas of symbolic logic, based on the works of J. Bull, S. Jevons and continued by C. Pierce in the United States and E. Schroeder in Germany were getting popular. The works by G. Frege and B. Russell should be considered more progressive towards the development of mathematical logic. The perspective of mathematical logic in solving the crisis of mathematics in Ukraine was noticed by Professor of Mathematics of Novorossiysk (Odesa) University Ivan Vladislavovich Sleshynsky. Sleshynsky (1854 –1931) is a Doctor of Mathematical Sciences (1893), Professor (1898) of Novorossiysk (Odesa) University. After studying at the University for two years he was a Fellow at the Department of Mathematics of Novorossiysk University, defended his master’s thesis and was sent to a scientific internship in Berlin (1881–1882), where he listened to the lectures by K. Weierstrass, L. Kronecker, E. Kummer, G. Bruns. Under the direction of K. Weierstrass he prepared a doctoral dissertation for defense. He returned to his native university in 1882, and at the same time he was a teacher of mathematics in the seminary (1882–1886), Odesa high schools (1882–1892), and taught mathematics at the Odesa Higher Women’s Courses. Having considerable achievements in the field of mathematics, in particular, Pringsheim’s Theorem (1889) proved by Sleshinsky on the conditions of convergence of continuous fractions, I. Sleshynsky drew attention to a new direction of logical science. The most significant work for the development of national mathematical logic is the translation by I. Sleshynsky from the French language “Algebra of Logic” by L. Couturat (1909). Among the most famous students of I. Sleshynsky, who studied and worked at Novorossiysk University and influenced the development of mathematical logic, one should mention E. Bunitsky and S. Shatunovsky. The second period of scientific work of I. Sleshynsky is connected with Poland. In 1911 he was invited to teach mathematical disciplines at Jagiellonian University and focused on mathematical logic. I. Sleshynsky’s report “On Traditional Logic”, delivered at the meeting of the Philosophical Society in Krakow. He developed the common belief among mathematicians that logic was not necessary for mathematics. His own experience of teaching one of the most difficult topics in higher mathematics – differential calculus, pushed him to explore logic, since the requirement of perfect mathematical proof required this. In one of his further works of this period, he noted the promising development of mathematical logic and its importance for mathematics. He claimed that for the mathematics of future he needed a new logic, which he saw in the “Principles of Mathematics” by A. Whitehead and B. Russell. Works on mathematical logic by I. Sleszynski prompted many of his students in Poland to undertake in-depth studies in this field, including T. Kotarbiński, S. Jaśkowski, V. Boreyko, and S. Zaremba. Thanks to S. Zaremba, I. Sleshynsky managed to complete the long-planned concept, a two-volume work “Theory of Proof” (1925–1929), the basis of which were lectures of Professor. The crisis period in mathematics of the early twentieth century, marked by the search for greater clarity in the very foundations of mathematical reasoning, led to the transition from the study of mathematical objects to the study of structures. The most successful means of doing this were proposed by mathematical logic. Thanks to Professor I. Sleshynsky, who succeeded in making Novorossiysk (Odesa) University a center of popularization of mathematical logic in the beginning of the twentieth century the ideas of mathematical logic in scientific environment became more popular. However, historical events prevented the ideas of mathematical logic in the domestic scientific space from the further development.

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Marschall, Benjamin. "Carnap and Beth on the Limits of Tolerance." Canadian Journal of Philosophy, August 13, 2021, 1–19. http://dx.doi.org/10.1017/can.2021.16.

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Abstract Rudolf Carnap’s principle of tolerance states that there is no need to justify the adoption of a logic by philosophical means. Carnap uses the freedom provided by this principle in his philosophy of mathematics: he wants to capture the idea that mathematical truth is a matter of linguistic rules by relying on a strong metalanguage with infinitary inference rules. In this paper, I give a new interpretation of an argument by E. W. Beth, which shows that the principle of tolerance does not suffice to remove all obstacles to the employment of infinitary rules.

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Fineman, Daniel. "The Anomaly of Anomaly of Anomaly." M/C Journal 23, no. 5 (October 7, 2020). http://dx.doi.org/10.5204/mcj.1649.

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‘Bitzer,’ said Thomas Gradgrind. ‘Your definition of a horse.’‘Quadruped. Graminivorous. Forty teeth, namely twenty-four grinders, four eye-teeth, and twelve incisive. Sheds coat in the spring; in marshy countries, sheds hoofs, too. Hoofs hard, but requiring to be shod with iron. Age known by marks in mouth.’ Thus (and much more) Bitzer.‘Now girl number twenty,’ said Mr. Gradgrind. ‘You know what a horse is.’— Charles Dickens, Hard Times (1854)Dickens’s famous pedant, Thomas Gradgrind, was not an anomaly. He is the pedagogical manifestation of the rise of quantification in modernism that was the necessary adjunct to massive urbanisation and industrialisation. His classroom caricatures the dominant epistemic modality of modern global democracies, our unwavering trust in numbers, “data”, and reproductive predictability. This brief quotation from Hard Times both presents and parodies the 19th century’s displacement of what were previously more commonly living and heterogeneous existential encounters with events and things. The world had not yet been made predictably repetitive through industrialisation, standardisation, law, and ubiquitous codes of construction. Theirs was much more a world of unique events and not the homogenised and orthodox iteration of standardised knowledge. Horses and, by extension, all entities and events gradually were displaced by their rote definitions: individuals of a so-called natural kind were reduced to identicals. Further, these mechanical standardisations were and still are underwritten by mapping them into a numerical and extensive characterisation. On top of standardised objects and procedures appeared assigned numerical equivalents which lent standardisation the seemingly apodictic certainty of deductive demonstrations. The algebraic becomes the socially enforced criterion for the previously more sensory, qualitative, and experiential encounters with becoming that were more likely in pre-industrial life. Here too, we see that the function of this reproductive protocol is not just notational but is the sine qua non for, in Althusser’s famous phrase, the manufacture of citizens as “subject subjects”, those concrete individuals who are educated to understand themselves ideologically in an imaginary relation with their real position in any society’s self-reproduction. Here, however, ideology performs that operation through that nominally least political of cognitive modes, the supposed friend of classical Marxism’s social science, the mathematical. The historical onset of this social and political reproductive hegemony, this uniform supplanting of time’s ineluctable differencing with the parasite of its associated model, can partial be found in the formation of metrics. Before the 19th century, the measures of space and time were local. Units of length and weight varied not just between nations but often by municipality. These parochial standards reflected indigenous traditions, actualities, personalities, and needs. This variation in measurement standards suggested that every exchange or judgment of kind and value relied upon the specificity of that instance. Every evaluation of an instance required perceptual acuity and not the banality of enumeration constituted by commodification and the accounting practices intrinsic to centralised governance. This variability in measure was complicated by similar variability in the currencies of the day. Thus, barter presented the participants with complexities and engagements of skills and discrete observation completely alien to the modern purchase of duplicate consumer objects with stable currencies. Almost nothing of life was iterative: every exchange was, more or less, an anomaly. However, in 1790, immediately following the French Revolution and as a central manifestation of its movement to rational democratisation, Charles Maurice de Talleyrand proposed a metrical system to the French National Assembly. The units of this metric system, based originally on observable features of nature, are now formally codified in all scientific practice by seven physical constants. Further, they are ubiquitous now in almost all public exchanges between individuals, corporations, and states. These units form a coherent and extensible structure whose elements and rules are subject to seemingly lossless symbolic exchange in a mathematic coherence aided by their conformity to decimal representation. From 1960, their basic contemporary form was established as the International System of Units (SI). Since then, all but three of the countries of the world (Myanmar, Liberia, and the United States), regardless of political organisation and individual history, have adopted these standards for commerce and general measurement. The uniformity and rational advantage of this system is easily demonstrable in just the absurd variation in the numeric bases of the Imperial / British system which uses base 16 for ounces/pounds, base 12 for inches/feet, base three for feet/yards, base 180 for degrees between freezing and cooling, 43,560 square feet per acre, eights for division of inches, etc. Even with its abiding antagonism to the French, Britain officially adopted the metric system as was required by its admission to the EU in 1973. The United States is the last great holdout in the public use of the metric system even though SI has long been the standard wanted by the federal government. At first, the move toward U.S. adoption was promising. Following France and rejecting England’s practice, America was founded on a decimal currency system in 1792. In 1793, Jefferson requested a copy of the standard kilogram from France in a first attempt to move to the metric system: however, the ship carrying the copy was captured by pirates. Indeed, The Metric Conversion Act of 1975 expressed a more serious national intention to adopt SI, but after some abortive efforts, the nation fell back into the more archaic measurements dominant since before its revolution. However, the central point remains that while the U.S. is unique in its public measurement standard among dominant powers, it is equally committed to the hegemonic application of a numerical rendition of events.The massive importance of this underlying uniformity is that it supplies the central global mechanism whereby the world’s chaotic variation is continuously parsed and supplanted into comparable, intelligible, and predictable units that understand individuating difference as anomaly. Difference, then, is understood in this method not as qualitative and intensive, which it necessarily is, but quantitative and extensive. Like Gradgrind’s “horse”, the living and unique thing is rendered through the Apollonian dream of standardisation and enumeration. While differencing is the only inherent quality of time’s chaotic flow, accounting and management requite iteration. To order the reproduction of modern society, the unique individuating differences that render an object as “this one”, what the Medieval logicians called haecceities, are only seen as “accidental” and “non-essential” deviations. This is not just odd but illogical since these very differences allow events to be individuated items so to appear as countable at all. As Leibniz’s principle, the indiscernibility of identicals, suggests, the application of the metrical same to different occasions is inherently paradoxical: if each unit were truly the same, there could only be one. As the etymology of “anomaly” suggests, it is that which is unexpected, irregular, out of line, or, going back to the Greek, nomos, at variance with the law. However, as the only “law” that always is at hand is the so-called “Second Law of Thermodynamics”, the inconsistently consistent roiling of entropy, the evident theoretical question might be, “how is anomaly possible when regularity itself is impossible?” The answer lies not in events “themselves” but exactly in the deductive valorisations projected by that most durable invention of the French Revolution adumbrated above, the metric system. This seemingly innocuous system has formed the reproductive and iterative bias of modern post-industrial perceptual homogenisation. Metrical modeling allows – indeed, requires – that one mistake the metrical changeling for the experiential event it replaces. Gilles Deleuze, that most powerful French metaphysician (1925-1995) offers some theories to understand the seminal production (not reproduction) of disparity that is intrinsic to time and to distinguish it from its homogenised representation. For him, and his sometime co-author, Felix Guattari, time’s “chaosmosis” is the host constantly parasitised by its symbolic model. This problem, however, of standardisation in the face of time’s originality, is obscured by its very ubiquity; we must first denaturalise the seemingly self-evident metrical concept of countable and uniform units.A central disagreement in ancient Greece was between the proponents of physis (often translated as “nature” but etymologically indicative of growth and becoming, process and not fixed form) and nomos (law or custom). This is one of the first ethical and so political debates in Western philosophy. For Heraclitus and other pre-Socratics, the emphatic character of nature was change, its differencing dynamism, its processual but not iterative character. In anticipation of Hume, Sophists disparaged nomos (νόμος) as simply the habituated application of synthetic law and custom to the fluidity of natural phenomena. The historical winners of this debate, Plato and the scientific attitudes of regularity and taxonomy characteristic of his best pupil, Aristotle, have dominated ever since, but not without opponents.In the modern era, anti-enlightenment figures such as Hamann, Herder, and the Schlegel brothers gave theoretical voice to romanticism’s repudiation of the paradoxical impulses of the democratic state for regulation and uniformity that Talleyrand’s “revolutionary” metrical proposal personified. They saw the correlationalism (as adumbrated by Meillassoux) between thought and thing based upon their hypothetical equitability as a betrayal of the dynamic physis that experience presented. Variable infinity might come either from the character of God or nature or, as famously in Spinoza’s Ethics, both (“deus sive natura”). In any case, the plenum of nature was never iterative. This rejection of metrical regularity finds its synoptic expression in Nietzsche. As a classicist, Nietzsche supplies the bridge between the pre-Socratics and the “post-structuralists”. His early mobilisation of the Apollonian, the dream of regularity embodied in the sun god, and the Dionysian, the drunken but inarticulate inexpression of the universe’s changing manifold, gives voice to a new resistance to the already dominate metrical system. His is a new spin of the mythic representatives of Nomos and physis. For him, this pair, however, are not – as they are often mischaracterised – in dialectical dialogue. To place them into the thesis / antithesis formulation would be to give them the very binary character that they cannot share and to, tacitly, place both under Apollo’s procedure of analysis. Their modalities are not antithetical but mutually exclusive. To represent the chaotic and non-iterative processes of becoming, of physis, under the rubric of a common metrics, nomos, is to mistake the parasite for the host. In its structural hubris, the ideological placebo of metrical knowing thinks it non-reductively captures the multiplicity it only interpellates. In short, the polyvalent, fluid, and inductive phenomena that empiricists try to render are, in their intrinsic character, unavailable to deductive method except, first, under the reductive equivalence (the Gradgrind pedagogy) of metrical modeling. This incompatibility of physis and nomos was made manifest by David Hume in A Treatise of Human Nature (1739-40) just before the cooptation of the 18th century’s democratic revolutions by “representative” governments. There, Hume displays the Apollonian dream’s inability to accurately and non-reductively capture a phenomenon in the wild, free from the stringent requirements of synthetic reproduction. His argument in Book I is succinct.Now as we call every thing custom, which proceeds from a past repetition, without any new reasoning or conclusion, we may establish it as a certain truth, that all the belief, which follows upon any present impression, is deriv'd solely from that origin. (Part 3, Section 8)There is nothing in any object, consider'd in itself, which can afford us a reason for drawing a conclusion beyond it; ... even after the observation of the frequent or constant conjunction of objects, we have no reason to draw any inference concerning any object beyond those of which we have had experience. (Part 3, Section 12)The rest of mankind ... are nothing but a bundle or collection of different perceptions, which succeed each other with an inconceivable rapidity, and are in a perpetual flux and movement. (Part 4, Section 6)In sum, then, nomos is nothing but habit, a Pavlovian response codified into a symbolic representation and, pragmatically, into a reproductive protocol specifically ordered to exclude anomaly, the inherent chaotic variation that is the hallmark of physis. The Apollonian dream that there can be an adequate metric of unrestricted natural phenomena in their full, open, turbulent, and manifold becoming is just that, a dream. Order, not chaos, is the anomaly. Of course, Kant felt he had overcome this unacceptable challenge to rational application to induction after Hume woke him from his “dogmatic slumber”. But what is perhaps one of the most important assertions of the critiques may be only an evasion of Hume’s radical empiricism: “there are only two ways we can account for the necessary agreement of experience with the concepts of its objects: either experience makes these concepts possible or these concepts make experience possible. The former supposition does not hold of the categories (nor of pure sensible intuition) ... . There remains ... only the second—a system ... of the epigenesis of pure reason” (B167). Unless “necessary agreement” means the dictatorial and unrelenting insistence in a symbolic model of perception of the equivalence of concept and appearance, this assertion appears circular. This “reading” of Kant’s evasion of the very Humean crux, the necessary inequivalence of a metric or concept to the metered or defined, is manifest in Nietzsche.In his early “On Truth and Lies in a Nonmoral Sense” (1873), Nietzsche suggests that there is no possible equivalence between a concept and its objects, or, to use Frege’s vocabulary, between sense or reference. We speak of a "snake" [see “horse” in Dickens]: this designation touches only upon its ability to twist itself and could therefore also fit a worm. What arbitrary differentiations! What one-sided preferences, first for this, then for that property of a thing! The various languages placed side by side show that with words it is never a question of truth, never a question of adequate expression; otherwise, there would not be so many languages. The "thing in itself" (which is precisely what the pure truth, apart from any of its consequences, would be) is likewise something quite incomprehensible to the creator of language and something not in the least worth striving for. This creator only designates the relations of things to men, and for expressing these relations he lays hold of the boldest metaphors.The literal is always already a reductive—as opposed to literature’s sometimes expansive agency—metaphorisation of events as “one of those” (a token of “its” type). The “necessary” equivalence in nomos is uncovered but demanded. The same is reproduced by the habitual projection of certain “essential qualities” at the expense of all those others residing in every experiential multiplicity. Only in this prison of nomos can anomaly appear: otherwise all experience would appear as it is, anomalous. With this paradoxical metaphor of the straight and equal, Nietzsche inverts the paradigm of scientific expression. He reveals as a repressive social and political obligation the symbolic assertion homology where actually none can be. Supposed equality and measurement all transpire within an Apollonian “dream within a dream”. The concept captures not the manifold of chaotic experience but supplies its placebo instead by an analytic tautology worthy of Gradgrind. The equivalence of event and definition is always nothing but a symbolic iteration. Such nominal equivalence is nothing more than shifting events into a symbolic frame where they can be commodified, owned, and controlled in pursuit of that tertiary equivalence which has become the primary repressive modality of modern societies: money. This article has attempted, with absurd rapidity, to hint why some ubiquitous concepts, which are generally considered self-evident and philosophically unassailable, are open not only to metaphysical, political, and ethical challenge, but are existentially unjustified. All this was done to defend the smaller thesis that the concept of anomaly is itself a reflection of a global misrepresentation of the chaos of becoming. This global substitution expresses a conservative model and measure of the world in the place of the world’s intrinsic heterogenesis, a misrepresentation convenient for those who control the representational powers of governance. In conclusion, let us look, again too briefly, at a philosopher who neither accepts this normative world picture of regularity nor surrenders to Nietzschean irony, Gilles Deleuze.Throughout his career, Deleuze uses the word “pure” with senses antithetical to so-called common sense and, even more, Kant. In its traditional concept, pure means an entity or substance whose essence is not mixed or adulterated with any other substance or material, uncontaminated by physical pollution, clean and immaculate. The pure is that which is itself itself. To insure intelligibility, that which is elemental, alphabetic, must be what it is itself and no other. This discrete character forms the necessary, if often tacit, precondition to any analysis and decomposition of beings into their delimited “parts” that are subject to measurement and measured disaggregation. Any entity available for structural decomposition, then, must be pictured as constituted exhaustively by extensive ones, measurable units, its metrically available components. Dualism having established as its primary axiomatic hypothesis the separability of extension and thought must now overcome that very separation with an adequacy, a one to one correspondence, between a supposedly neatly measurable world and ideological hegemony that presents itself as rational governance. Thus, what is needed is not only a purity of substance but a matching purity of reason, and it is this clarification of thought, then, which, as indicated above, is the central concern of Kant’s influential and grand opus, The Critique of Pure Reason.Deleuze heard a repressed alternative to the purity of the measured self-same and equivalent that, as he said about Plato, “rumbled” under the metaphysics of analysis. This was the dark tradition he teased out of the Stoics, Ockham, Gregory of Rimini, Nicholas d’Autrecourt, Spinoza, Meinong, Bergson, Nietzsche, and McLuhan. This is not the purity of identity, A = A, of metrical uniformity and its shadow, anomaly. Rather than repressing, Deleuze revels in the perverse purity of differencing, difference constituted by becoming without the Apollonian imposition of normalcy or definitional identity. One cannot say “difference in itself” because its ontology, its genesis, is not that of anything itself but exactly the impossibility of such a manner of constitution: universal anomaly. No thing or idea can be iterative, separate, or discrete.In his Difference and Repetition, the idea of the purely same is undone: the Ding an sich is a paradox. While the dogmatic image of thought portrays the possibility of the purely self-same, Deleuze never does. His notions of individuation without individuals, of modulation without models, of simulacra without originals, always finds a reflection in his attitudes toward, not language as logical structure, but what necessarily forms the differential making of events, the heterogenesis of ontological symptoms. His theory has none of the categories of Pierce’s triadic construction: not the arbitrary of symbols, the “self-representation” of icons, or even the causal relation of indices. His “signs” are symptoms: the non-representational consequences of the forces that are concurrently producing them. Events, then, are the symptoms of the heterogenetic forces that produce, not reproduce them. To measure them is to export them into a representational modality that is ontologically inapplicable as they are not themselves themselves but the consequences of the ongoing differences of their genesis. Thus, the temperature associated with a fever is neither the body nor the disease.Every event, then, is a diaphora, the pure consequent of the multiplicity of the forces it cannot resemble, an original dynamic anomaly without standard. This term, diaphora, appears at the conclusion of that dialogue some consider Plato’s best, the Theaetetus. There we find perhaps the most important discussion of knowledge in Western metaphysics, which in its final moments attempts to understand how knowledge can be “True Judgement with an Account” (201d-210a). Following this idea leads to a theory, usually known as the “Dream of Socrates”, which posits two kinds of existents, complexes and simples, and proposes that “an account” means “an account of the complexes that analyses them into their simple components … the primary elements (prôta stoikheia)” of which we and everything else are composed (201e2). This—it will be noticed—suggests the ancient heritage of Kant’s own attempted purification of mereological (part/whole relations) nested elementals. He attempts the coordination of pure speculative reason to pure practical reason and, thus, attempts to supply the root of measurement and scientific regularity. However, as adumbrated by the Platonic dialogue, the attempted decompositions, speculative and pragmatic, lead to an impasse, an aporia, as the rational is based upon a correspondence and not the self-synthesis of the diaphorae by their own dynamic disequilibrium. Thus the dialogue ends inconclusively; Socrates rejects the solution, which is the problem itself, and leaves to meet his accusers and quaff his hemlock. The proposal in this article is that the diaphorae are all that exists in Deleuze’s world and indeed any world, including ours. Nor is this production decomposable into pure measured and defined elementals, as such decomposition is indeed exactly opposite what differential production is doing. For Deleuze, what exists is disparate conjunction. But in intensive conjunction the same cannot be the same except in so far as it differs. The diaphorae of events are irremediably asymmetric to their inputs: the actual does not resemble the virtual matrix that is its cause. Indeed, any recourse to those supposedly disaggregate inputs, the supposedly intelligible constituents of the measured image, will always but repeat the problematic of metrical representation at another remove. This is not, however, the traditional postmodern trap of infinite meta-shifting, as the diaphoric always is in each instance the very presentation that is sought. Heterogenesis can never be undone, but it can be affirmed. In a heterogenetic monism, what was the insoluble problem of correspondence in dualism is now its paradoxical solution: the problematic per se. What manifests in becoming is not, nor can be, an object or thought as separate or even separable, measured in units of the self-same. Dogmatic thought habitually translates intensity, the differential medium of chaosmosis, into the nominally same or similar so as to suit the Apollonian illusions of “correlational adequacy”. However, as the measured cannot be other than a calculation’s placebo, the correlation is but the shadow of a shadow. Every diaphora is an event born of an active conjunction of differential forces that give rise to this, their product, an interference pattern. Whatever we know and are is not the correlation of pure entities and thoughts subject to measured analysis but the confused and chaotic confluence of the specific, material, aleatory, differential, and unrepresentable forces under which we subsist not as ourselves but as the always changing product of our milieu. In short, only anomaly without a nominal becomes, and we should view any assertion that maps experience into the “objective” modality of the same, self-evident, and normal as a political prestidigitation motivated, not by “truth”, but by established political interest. ReferencesDella Volpe, Galvano. Logic as a Positive Science. London: NLB, 1980.Deleuze, Gilles. Difference and Repetition. Trans. Paul Patton. New York: Columbia UP, 1994.———. The Logic of Sense. Trans. Mark Lester. New York: Columbia UP, 1990.Guenon, René. The Reign of Quantity. New York: Penguin, 1972.Hawley, K. "Identity and Indiscernibility." Mind 118 (2009): 101-9.Hume, David. A Treatise of Human Nature. Oxford: Clarendon, 2014.Kant, Immanuel. Critique of Pure Reason. Trans. Norman Kemp Smith. London: Palgrave Macmillan, 1929.Meillassoux, Quentin. After Finitude: An Essay on the Necessity of Contingency. Trans. Ray Brassier. New York: Continuum, 2008.Naddaf, Gerard. The Greek Concept of Nature. Albany: SUNY, 2005. Nietzsche, Friedrich. The Birth of Tragedy. Trans. Douglas Smith. Oxford: Oxford UP, 2008.———. “On Truth and Lies in a Nonmoral Sense.” Trans. Walter Kaufmann. The Portable Nietzsche. New York: Viking, 1976.Welch, Kathleen Ethel. "Keywords from Classical Rhetoric: The Example of Physis." Rhetoric Society Quarterly 17.2 (1987): 193–204.

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