Relevant bibliographies by topics / Inference. Logic, Symbolic and mathematical. Mathematics

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Inference. Logic, Symbolic and mathematical. Mathematics'

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Published: 4 June 2021
Last updated: 16 February 2022

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Journal articles on the topic "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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Dissertations / Theses on the topic "Inference. Logic, Symbolic and mathematical. Mathematics"

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Bishop, Joyce Wolfer Otto Albert D. Lubinski Cheryl Ann. "Middle school students' understanding of mathematical patterns and their symbolic representations." Normal, Ill. Illinois State University, 1997. http://wwwlib.umi.com/cr/ilstu/fullcit?p9803721.

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Thesis (Ph. D.)--Illinois State University, 1997.
Title from title page screen, viewed June 1, 2006. Dissertation Committee: Albert D. Otto, Cheryl A. Lubinski (co-chairs), John A. Dossey, Cynthia W. Langrall, George Padavil. Includes bibliographical references (leaves 119-123) and abstract. Also available in print.
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Lavers, Peter Stanley. "Generating intensional logics : the application of paraconsistent logics to investigate certain areas of the boundaries of mathematics /." Title page, table of contents and summary only, 1985. http://web4.library.adelaide.edu.au/theses/09ARM/09arml399.pdf.

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Akishev, Galym. "Monadic bounded algebras : a thesis submitted to the Victoria University of Wellington in fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics /." ResearchArchive@Victoria e-Thesis, 2009. http://hdl.handle.net/10063/915.

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Brierley, William. "Undecidability of intuitionistic theories." Thesis, McGill University, 1985. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=66016.

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Quine, W. V. "The logic of sequences a generalization of Principia mathematica /." New York : Garland Pub, 1990. http://catalog.hathitrust.org/api/volumes/oclc/20797392.html.

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Eliasson, Jonas. "Ultrasheaves." Doctoral thesis, Uppsala : Matematiska institutionen, Univ. [distributör], 2003. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-3762.

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Schwartzkopff, Robert. "The numbers of the marketplace : commitment to numbers in natural language." Thesis, University of Oxford, 2015. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.711821.

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Brinkerhoff, Jennifer Alder. "Applying Toulmin's Argumentation Framework to Explanations in a Reform Oriented Mathematics Class." Diss., CLICK HERE for online access, 2007. http://contentdm.lib.byu.edu/ETD/image/etd1960.pdf.

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Duff, Karen Malina. "What Are Some of the Common Traits in the Thought Processes of Undergraduate Students Capable of Creating Proof?" Diss., CLICK HERE for online access, 2007. http://contentdm.lib.byu.edu/ETD/image/etd1856.pdf.

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Konecny, Jan. "Isotone fuzzy Galois connections and their applications in formal concept analysis." Diss., Online access via UMI:, 2009.

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Thesis (Ph. D.)--State University of New York at Binghamton, Thomas J. Watson School of Engineering and Applied Science, Department of Systems Science and Industrial Engineering, 2009.
Includes bibliographical references.
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Books on the topic "Inference. Logic, Symbolic and mathematical. Mathematics"

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Stuart, Glennan, ed. Elements of deductive inference: An introduction to symbolic logic. Belmont, CA: Wadsworth Publishing, 2000.

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Admissibility of logical inference rules. Amsterdam: Elsevier, 1997.

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Chandru, Vijay. Optimization methods for logical inference. New York: Wiley, 1999.

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International Colloquium on Grammatical Inference (10th 2010 Valencia, Spain). Grammatical inference: theoretical results and applications: 10th international colloquium ; proceedings. Berlin: Springer, 2010.

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Merrill, Daniel D. Augustus De Morgan and the logic of relations. Dordrecht: Kluwer Academic Publishers, 1990.

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Goodman, Irwin R. Conditional inference and logic for intelligent systems: A theory of measure-free conditioning. Amsterdam, Netherlands: North-Holland, 1991.

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Wittgenstein, Ludwig. Philosophike grammatike. Athe na: Morpho tiko Hidryma Ethnike s Trapeze s, 1994.

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Alexander, Clark, Coste François, and Miclet Laurent, eds. Grammatical inference: Algorithms and applications : 9th international colloquium, ICGI 2008, Saint-Malo, France, September 22-24, 2008 : proceedings. [New York]: Springer-Verlag Berlin Heidelberg, 2008.

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International Colloquium on Grammatical Inference (9th 2008 Saint-Malo, France). Grammatical inference: Algorithms and applications : 9th international colloquium, ICGI 2008, Saint-Malo, France, September 22-24, 2008 : proceedings. [New York]: Springer-Verlag Berlin Heidelberg, 2008.

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International Colloquium on Grammatical Inference (9th 2008 Saint-Malo, France). Grammatical inference: Algorithms and applications : 9th international colloquium, ICGI 2008, Saint-Malo, France, September 22-24, 2008 : proceedings. [New York]: Springer-Verlag Berlin Heidelberg, 2008.

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Book chapters on the topic "Inference. Logic, Symbolic and mathematical. Mathematics"

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Roanes-Lozano, Eugenio, Luis M. Laita, and Eugenio Roanes-Macías. "An inference engine for propositional two-valued logic based on the radical membership problem." In Artificial Intelligence and Symbolic Mathematical Computation, 71–86. Berlin, Heidelberg: Springer Berlin Heidelberg, 1996. http://dx.doi.org/10.1007/3-540-61732-9_51.

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Handelman, Matthew. "The Philosophy of Mathematics: Privation and Representation in Gershom Scholem’s Negative Aesthetics." In The Mathematical Imagination, 65–103. Fordham University Press, 2019. http://dx.doi.org/10.5422/fordham/9780823283835.003.0003.

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Chapter 2 investigates the moment in 1917 when the philosophy of mathematics revealed to Gershom Scholem the symbolic potential of privation. Mathematics—in particular, the translation of logic into the symbols and operations of mathematics known as mathematical logic—produced novel results by discarding the conventional representational and meaning-making functions of language. Drawing on these mathematical insights, Scholem’s theorization of the poetic genre of lament and his translations of the biblical book of Lamentations employed erasure on the level of literary form to symbolize experiences, such as the Jewish diaspora, that exceed the limits of linguistic and historical representation. For Scholem, both poetry and history can mobilize deprivation as a means of retaining in language a symbol of experiences and ideas that remain unsayable in language and inexpressible in history—accounting for the erasure of exile and finding historical continuity in moments of silence, rupture, and catastrophe.
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Cohen, Daniel J. "Reasoning and Belief in Victorian Mathematics." In The Organisation of Knowledge in Victorian Britain. British Academy, 2005. http://dx.doi.org/10.5871/bacad/9780197263266.003.0006.

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This chapter focuses on the progress of mathematics in the nineteenth century. British mathematicians of this period showed interest in the formal aspects of mathematics, particularly in symbolic logic. Bertrand Russell and Alfred North Whitehead were among the most highly influential figures in Victorian mathematical circles due to their wide-ranging thought and institutional positions. Principia Mathematica (1910–1913), in which Russell and Whitehead equated logic and mathematics at the deepest level possible, was a culmination of the innovative mathematical research of the Victorian age.
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O’Donnell, Michael J. "Introduction: Logic and Logic Programming Languages." In Handbook of Logic in Artificial Intelligence and Logic Programming: Volume 5: Logic Programming. Oxford University Press, 1998. http://dx.doi.org/10.1093/oso/9780198537922.003.0004.

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Logic, according to Webster’s dictionary [Webster, 1987], is ‘a science that deals with the principles and criteria of validity of inference and demonstration: the science of the formal principles of reasoning.' Such 'principles and criteria’ are always described in terms of a language in which inference, demonstration, and reasoning may be expressed. One of the most useful accomplishments of logic for mathematics is the design of a particular formal language, the First Order Predicate Calculus (FOPC). FOPC is so successful at expressing the assertions arising in mathematical discourse that mathematicians and computer scientists often identify logic with classical logic expressed in FOPC. In order to explore a range of possible uses of logic in the design of programming languages, we discard the conventional identification of logic with FOPC, and formalize a general schema for a variety of logical systems, based on the dictionary meaning of the word. Then, we show how logic programming languages may be designed systematically for any sufficiently effective logic, and explain how to view Prolog, Datalog, λProlog, Equational Logic Programming, and similar programming languages, as instances of the general schema of logic programming. Other generalizations of logic programming have been proposed independently by Meseguer [Meseguer, 1989], Miller, Nadathur, Pfenning and Scedrov [Miller et al., 1991], Goguen and Burstall [Goguen and Burstall, 1992]. The purpose of this chapter is to introduce a set of basic concepts for understanding logic programming, not in terms of its historical development, but in a systematic way based on retrospective insights. In order to achieve a systematic treatment, we need to review a number of elementary definitions from logic and theoretical computer science and adapt them to the needs of logic programming. The result is a slightly modified logical notation, which should be recognizable to those who know the traditional notation. Conventional logical notation is also extended to new and analogous concepts, designed to make the similarities and differences between logical relations and computational relations as transparent as possible. Computational notation is revised radically to make it look similar to logical notation.
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Stillwell, John. "From Hypothesis Testing to Estimating Functionals." In Statistical Inference via Convex Optimization, 185–259. Princeton University Press, 2019. http://dx.doi.org/10.23943/princeton/9780691197296.003.0003.

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This chapter prepares the reader's mind for reverse mathematics. As its name suggests, reverse mathematics seeks not theorems but the right axioms to prove theorems already known. Reverse mathematics began as a technical field of mathematical logic, but its main ideas have precedents in the ancient field of geometry and the early twentieth-century field of set theory. In geometry, the parallel axiom is the right axiom to prove many theorems of Euclidean geometry, such as the Pythagorean theorem. Set theory offers a more modern example: base theory called ZF, a theorem that ZF cannot prove (the well-ordering theorem) and the “right axiom” for proving it—the axiom of choice. From these and similar examples one can guess at a base theory for analysis, and the “right axioms” for proving some of its well-known theorems.
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Stillwell, John. "Sparse Recovery via ℓ1 Minimization." In Statistical Inference via Convex Optimization, 1–40. Princeton University Press, 2019. http://dx.doi.org/10.23943/princeton/9780691197296.003.0001.

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This chapter prepares the reader's mind for reverse mathematics. As its name suggests, reverse mathematics seeks not theorems but the right axioms to prove theorems already known. Reverse mathematics began as a technical field of mathematical logic, but its main ideas have precedents in the ancient field of geometry and the early twentieth-century field of set theory. In geometry, the parallel axiom is the right axiom to prove many theorems of Euclidean geometry, such as the Pythagorean theorem. Set theory offers a more modern example: base theory called ZF, a theorem that ZF cannot prove (the well-ordering theorem) and the “right axiom” for proving it—the axiom of choice. From these and similar examples one can guess at a base theory for analysis, and the “right axioms” for proving some of its well-known theorems.
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Stillwell, John. "Hypothesis Testing." In Statistical Inference via Convex Optimization, 41–184. Princeton University Press, 2019. http://dx.doi.org/10.23943/princeton/9780691197296.003.0002.

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This chapter prepares the reader's mind for reverse mathematics. As its name suggests, reverse mathematics seeks not theorems but the right axioms to prove theorems already known. Reverse mathematics began as a technical field of mathematical logic, but its main ideas have precedents in the ancient field of geometry and the early twentieth-century field of set theory. In geometry, the parallel axiom is the right axiom to prove many theorems of Euclidean geometry, such as the Pythagorean theorem. Set theory offers a more modern example: base theory called ZF, a theorem that ZF cannot prove (the well-ordering theorem) and the “right axiom” for proving it—the axiom of choice. From these and similar examples one can guess at a base theory for analysis, and the “right axioms” for proving some of its well-known theorems.
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Grattan-Guinness, Ivor. "Turing’s mentor, Max Newman." In The Turing Guide. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198747826.003.0052.

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The interaction between mathematicians and mathematical logicians has always been much slighter than one might imagine. This chapter examines the case of Turing’s mentor, Maxwell Hermann Alexander Newman (1897–1984). The young Turing attended a course of lectures on logical matters that Newman gave at Cambridge University in 1935. After briefly discussing examples of the very limited contact between mathematicians and logicians in the period 1850–1930, I describe the rather surprising origins and development of Newman’s own interest in logic. One might expect that the importance to many mathematicians of means of proving theorems, and their desire in many contexts to improve the level of rigour of proofs, would motivate them to examine and refine the logic that they were using. However, inattention to logic has long been common among mathematicians. A very important source of the cleft between mathematics and logic during the 19th century was the founding, from the late 1810s onwards, of the ‘mathematical analysis’ of real variables, grounded on a theory of limits, by the French mathematician Augustin-Louis Cauchy. He and his followers extolled rigour—most especially, careful definitions of major concepts and detailed proofs of theorems. From the 1850s onwards, this project was enriched by the German mathematician Karl Weierstrass and his many followers, who introduced (for example) multiple limit theory, definitions of irrational numbers, and an increasing use of symbols, and then from the early 1870s by Georg Cantor with his set theory. However, absent from all these developments was explicit attention to any kind of logic. This silence continued among the many set theorists who participated in the inauguration of measure theory, functional analysis, and integral equations. The mathematicians Artur Schoenflies and Felix Hausdorff were particularly hostile to logic, targeting the famous 20th-century logician Bertrand Russell. (Even the extensive dispute over the axiom of choice focused mostly on its legitimacy as an assumption in set theory and its use of higher-order quantification: its ability to state an infinitude of independent choices within finitary logic constituted a special difficulty for ‘logicists’ such as Russell.) Russell, George Boole, and other creators of symbolic logics were exceptional among mathematicians in attending to logic, but they made little impact on their colleagues.
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9

Barwise, Jon, and John Etchemendy. "Visual Information and Valid Reasoning." In Logical Reasoning with Diagrams. Oxford University Press, 1996. http://dx.doi.org/10.1093/oso/9780195104271.003.0005.

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Psychologists have long been interested in the relationship between visualization and the mechanisms of human reasoning. Mathematicians have been aware of the value of diagrams and other visual tools both for teaching and as heuristics for mathematical discovery. As the chapters in this volume show, such tools are gaining even greater value, thanks in large part to the graphical potential of modern computers. But despite the obvious importance of visual images in human cognitive activities, visual representation remains a second-class citizen in both the theory and practice of mathematics. In particular, we are all taught to look askance at proofs that make crucial use of diagrams, graphs, or other nonlinguistic forms of representation, and we pass on this disdain to our students. In this chapter, we claim that visual forms of representation can be important, not just as heuristic and pedagogic tools, but as legitimate elements of mathematical proofs. As logicians, we recognize that this is a heretical claim, running counter to centuries of logical and mathematical tradition. This tradition finds its roots in the use of diagrams in geometry. The modern attitude is that diagrams are at best a heuristic in aid of finding a real, formal proof of a theorem of geometry, and at worst a breeding ground for fallacious inferences. For example, in a recent article, the logician Neil Tennant endorses this standard view: . . . [The diagram] is only an heuristic to prompt certain trains of inference; . . . it is dispensable as a proof-theoretic device; indeed, . . . it has no proper place in the proof as such. For the proof is a syntactic object consisting only of sentences arranged in a finite and inspectable array (Tennant [1984]). . . . It is this dogma that we want to challenge. We are by no means the first to question, directly or indirectly, the logocentricity of mathematics arid logic. The mathematicians Euler and Venn are well known for their development of diagrammatic tools for solving mathematical problems, and the logician C. S. Peirce developed an extensive diagrammatic calculus, which he intended as a general reasoning tool.
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Conference papers on the topic "Inference. Logic, Symbolic and mathematical. Mathematics"

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Finch, William W. "An Overview of Inference Mechanisms for Quantified Relations." In ASME 1998 Design Engineering Technical Conferences. American Society of Mechanical Engineers, 1998. http://dx.doi.org/10.1115/detc98/dtm-5666.

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Abstract This paper temporarily sheds formal mathematical treatment and presents a more intuitive overview of inference mechanisms for quantified relations. These predicate logic expressions are a new class of design constraint among sets of variations affecting the design and performance of engineering systems. Simple examples illustrate the use of quantified relations to infer constraints on the membership of feasible sets. A small design problem from the electronics domain joins the mathematical tools with engineering concepts. A brief comparison demonstrates the advantages of this approach over conventional interval mathematics. This paper’s objective is to illustrate the application of quantified relations and their associated methods to engineering design problems.
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