Search results for 'Mathematical notation' (try it on Scholar)

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  1. W. E. Underwood (1980). Symbolic Configurations and Two-Dimensional Mathematical Notation. Semiotics:523-532.score: 150.0
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  2. W. Douglas Maurer (1999). The Influence of the Computer Upon Mathematical Notation. Semiotica 125 (1-3):165-168.score: 150.0
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  3. Yanjie Zhao (1997). What is Mathematical Notation. Semiotics:257-273.score: 150.0
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  4. Ivor Bulmer-Thomas (1985). Boethian Number Theory Michael Masi: Boethian Number Theory: A Translation of the De Institutione Arithmetica (with Introduction and Notes). (Studies in Classical Antiquity, 6.) Pp. 198; 8 Figures with Mathematical Diagrams and Musical Notation in Text. Amsterdam: Editions Rodopi, 1983. Paper, Fl. 60. [REVIEW] The Classical Review 35 (01):86-87.score: 120.0
  5. Toshiyasu Arai (2002). Buchholz Wilfried. Notation Systems for Infinitary Derivations. Archive for Mathematical Logic, Vol. 30 No. 5–6 (1991), Pp. 277–296. Buchholz Wilfried. Explaining Gentzen's Consistency Proof Within Infinitary Proof Theory. Computational Logic and Proof Theory, 5th Kurt Gödel Colloquium, KGC'97, Vienna, Austria, August 25–29, 1997, Proceedings, Edited by Gottlob Georg, Leitsch Alexander, and Mundici Daniele, Lecture Notes in Computer Science, Vol. 1289, Springer, Berlin, Heidelberg, New York, Etc., 1997 ... [REVIEW] Bulletin of Symbolic Logic 8 (3):437-439.score: 120.0
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  6. William James Meyers (1975). A Mathematical Theory of Parenthesis, Free Notations. Państwowe Wydawn. Naukowe.score: 120.0
     
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  7. Robert Feys (1969). Dictionary of Symbols of Mathematical Logic. Amsterdam, North-Holland Pub. Co..score: 90.0
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  8. Michael Kohlhase, Adaptation of Notations in Living Mathematical Documents.score: 86.0
    Notations are central for understanding mathematical discourse. Readers would like to read notations that transport the meaning well and prefer notations that are familiar to them. Therefore, authors optimize the choice of notations with respect to these two criteria, while at the same time trying to remain consistent over the document and their own prior publications. In print media where notations are fixed at publication time, this is an over-constrained problem. In living documents notations can be adapted at reading (...)
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  9. Michael Kohlhase, Notations for Living Mathematical Documents.score: 86.0
    Notations are central for understanding mathematical discourse. Readers would like to read notations that transport the meaning well and prefer notations that are familiar to them. Therefore, authors optimize the choice of notations with respect to these two criteria, while at the same time trying to remain consistent over the document and their own prior publications. In print media where notations are fixed at publication time, this is an over-constrained problem. In living documents notations can be adapted at reading (...)
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  10. Madeline M. Muntersbjorn (1999). Naturalism, Notation, and the Metaphysics of Mathematics. Philosophia Mathematica 7 (2):178-199.score: 84.0
    The instability inherent in the historical inventory of mathematical objects challenges philosophers. Naturalism suggests we can construct enduring answers to ontological questions through an investigation of the processes whereby mathematical objects come into existence. Patterns of historical development suggest that mathematical objects undergo an intelligible process of reification in tandem with notational innovation. Investigating changes in mathematical languages is a necessary first step towards a viable ontology. For this reason, scholars should not modernize historical texts without (...)
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  11. Gregory Landini (2012). Frege's Notations: What They Are and How They Mean. Palgrave Macmillan.score: 70.0
     
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  12. P. B. Andrews (2002). An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof. Kluwer Academic Publishers.score: 66.0
    This introduction to mathematical logic starts with propositional calculus and first-order logic. Topics covered include syntax, semantics, soundness, completeness, independence, normal forms, vertical paths through negation normal formulas, compactness, Smullyan's Unifying Principle, natural deduction, cut-elimination, semantic tableaux, Skolemization, Herbrand's Theorem, unification, duality, interpolation, and definability. The last three chapters of the book provide an introduction to type theory (higher-order logic). It is shown how various mathematical concepts can be formalized in this very expressive formal language. This expressive (...) facilitates proofs of the classical incompleteness and undecidability theorems which are very elegant and easy to understand. The discussion of semantics makes clear the important distinction between standard and nonstandard models which is so important in understanding puzzling phenomena such as the incompleteness theorems and Skolem's Paradox about countable models of set theory. Some of the numerous exercises require giving formal proofs. A computer program called ETPS which is available from the web facilitates doing and checking such exercises. Audience: This volume will be of interest to mathematicians, computer scientists, and philosophers in universities, as well as to computer scientists in industry who wish to use higher-order logic for hardware and software specification and verification. (shrink)
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  13. Eszter Szabó Attila Krajcsi (2012). The Role of Number Notation: Sign-Value Notation Number Processing is Easier Than Place-Value. Frontiers in Psychology 3.score: 66.0
    Number notations can influence the way numbers are handled in computations; however, the role of notation itself in mental processing has not been examined directly. From a mathematical point of view, it is believed that place-value number notation systems, such as the Indo-Arabic numbers, are superior to sign-value systems, such as the Roman numbers. However, sign-value notation might have sufficient efficiency; for example, sign-value notations were common in flourishing cultures, such as in ancient Egypt. Herein we (...)
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  14. Attila Krajcsi & Eszter Szabó (2012). The Role of Number Notation: Sign-Value Notation Number Processing is Easier Than Place-Value. Frontiers in Psychology 3.score: 66.0
    Number notations can influence the way numbers are handled in computations; however, the role of notation itself in mental processing has not been examined directly. From a mathematical point of view, it is believed that place-value number notation systems, such as the Indo-Arabic numbers, are superior to sign-value systems, such as the Roman numbers. However, sign-value notation might have sufficient efficiency; for example, sign-value notations were common in flourishing cultures, such as in ancient Egypt. Herein we (...)
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  15. Jan Von Plato (2007). In the Shadows of the Löwenheim-Skolem Theorem: Early Combinatorial Analyses of Mathematical Proofs. Bulletin of Symbolic Logic 13 (2):189-225.score: 54.0
    The Löwenheim-Skolem theorem was published in Skolem's long paper of 1920, with the first section dedicated to the theorem. The second section of the paper contains a proof-theoretical analysis of derivations in lattice theory. The main result, otherwise believed to have been established in the late 1980s, was a polynomial-time decision algorithm for these derivations. Skolem did not develop any notation for the representation of derivations, which makes the proofs of his results hard to follow. Such a formal (...) is given here by which these proofs become transparent. A third section of Skolem's paper gives an analysis for derivations in plane projective geometry. To clear a gap in Skolem's result, a new conservativity property is shown for projective geometry, to the effect that a proper use of the axiom that gives the uniqueness of connecting lines and intersection points requires a conclusion with proper cases (logically, a disjunction in a positive part) to be proved. The forgotten parts of Skolem's first paper on the Löwenheim-Skolem theorem are the perhaps earliest combinatorial analyses of formal mathematical proofs, and at least the earliest analyses with profound results. (shrink)
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  16. Vladislav A. Shaposhnikov (1999). Mathematical Notational Systems and the Visual Representation of Metaphysical Ideas. Semiotica 125 (1-3):135-142.score: 50.0
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  17. Stefano Mazzanti (2013). Iteration on Notation and Unary Functions. Mathematical Logic Quarterly 59 (6):415-434.score: 48.0
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  18. Harold N. Lee (1931). The Meaning of the Notation of Mathematics and Logic. The Monist 41 (4):594-617.score: 40.0
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  19. Wayne Richter (1996). Putnam Hilary. On Hierarchies and Systems of Notations. Proceedings of the American Mathematical Society, Vol. 15 (1964), Pp. 44–50. [REVIEW] Journal of Symbolic Logic 31 (1):136-137.score: 40.0
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  20. James Robert Brown (1999). Philosophy of Mathematics: An Introduction to the World of Proofs and Pictures. Routledge.score: 38.0
    Philosophy of Mathematics is clear and engaging, and student friendly The book discusses the great philosophers and the importance of mathematics to their thought. Among topics discussed in the book are the mathematical image, platonism, picture-proofs, applied mathematics, Hilbert and Godel, knots and notation definitions, picture-proofs and Wittgenstein, computation, proof and conjecture.
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  21. Axel Gelfert (2011). Mathematical Formalisms in Scientific Practice: From Denotation to Model-Based Representation. Studies in History and Philosophy of Science 42 (2):272-286.score: 38.0
    The present paper argues that ‘mature mathematical formalisms’ play a central role in achieving representation via scientific models. A close discussion of two contemporary accounts of how mathematical models apply—the DDI account (according to which representation depends on the successful interplay of denotation, demonstration and interpretation) and the ‘matching model’ account—reveals shortcomings of each, which, it is argued, suggests that scientific representation may be ineliminably heterogeneous in character. In order to achieve a degree of unification that is compatible (...)
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  22. James Robert Brown (2008). Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures. Routledge.score: 38.0
    1. Introduction : the mathematical image -- 2. Platonism -- 3. Picture-proofs and Platonism -- 4. What is applied mathematics? -- 5. Hilbert and Gödel -- 6. Knots and notation -- 7. What is a definition? -- 8. Constructive approaches -- 9. Proofs, pictures and procedures in Wittgenstein -- 10. Computation, proof and conjecture -- 11. How to refute the continuum hypothesis -- 12. Calling the bluff.
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  23. Stephen Cole Kleene (1967/2002). Mathematical Logic. Dover Publications.score: 38.0
    Undergraduate students with no prior classroom instruction in mathematical logic will benefit from this evenhanded multipart text by one of the centuries greatest authorities on the subject. Part I offers an elementary but thorough overview of mathematical logic of first order. The treatment does not stop with a single method of formulating logic; students receive instruction in a variety of techniques, first learning model theory (truth tables), then Hilbert-type proof theory, and proof theory handled through derived rules. Part (...)
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  24. Douglas S. Bridges (1978). A Note on Morse's Lambda‐Notation in Set Theory. Mathematical Logic Quarterly 24 (8):113-114.score: 36.0
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  25. S. Dennis, M. S. Humphreys & J. Wiles (1996). Mathematical Constraints on a Theory of Human Memory - Response. Behavioral and Brain Sciences 19 (3):559-560.score: 36.0
    Colonius suggests that, in using standard set theory as the language in which to express our computational-level theory of human memory, we would need to violate the axiom of foundation in order to express meaningful memory bindings in which a context is identical to an item in the list. We circumvent Colonius's objection by allowing that a list item may serve as a label for a context without being identical to that context. This debate serves to highlight the value of (...)
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  26. Wilfried Buchholz (1991). Notation Systems for Infinitary Derivations. Archive for Mathematical Logic 30 (5-6):277-296.score: 36.0
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  27. Gerhard Jäger (1984). Ρ-Inaccessible Ordinals, Collapsing Functions and a Recursive Notation System. Archive for Mathematical Logic 24 (1):49-62.score: 36.0
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  28. Noriya Kadota (1993). On Wainer's Notation for a Minimal Subrecursive Inaccessible Ordinal. Mathematical Logic Quarterly 39 (1):217-227.score: 36.0
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  29. Zdzisław Pawlar (1963). Organization of Addressless Computers Working in Parenthesis Notation. Mathematical Logic Quarterly 9 (16‐17):243-249.score: 36.0
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  30. Helmut Pfeiffer & H. Pfeiffer (1992). A Notation System for Ordinal Using Ψ‐Functions on Inaccessible Mahlo Numbers. Mathematical Logic Quarterly 38 (1):431-456.score: 36.0
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  31. Hans Moravec (1995). Roger Penrose's Gravitonic Brains: A Review of Shadows of the Mind by Roger Penrose. [REVIEW] Psyche 2 (1).score: 30.0
    Summarizing a surrounding 200 pages, pages 179 to 190 of Shadows of the Mind contain a future dialog between a human identified as "Albert Imperator" and an advanced robot, the "Mathematically Justified Cybersystem", allegedly Albert's creation. The two have been discussing a Gödel sentence for an algorithm by which a robot society named SMIRC certifies mathematical proofs. The sentence, referred to in mathematical notation as Omega(Q*), is to be precisely constructed from on a definition of SMIRC's algorithm. (...)
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  32. Thomas J. Scheff (2007). A Concept of Social Integration. Philosophical Psychology 20 (5):579 – 593.score: 30.0
    Clear definitions of alienation and solidarity are needed as a step toward an explicit theory of social integration. The idea of alienation has played a key role in the development of sociology, but it's meaning has never been clear. Both theories and empirical studies confound relational-dispositional, cognitive-emotional and/or interpersonal-societal components. This essay proposes definitions that follow from the work of Erving Goffman and others. Goffman's idea of "co-presence" implies a model of solidarity as mutual awareness to the point of merging (...)
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  33. Thomas J. Scheff (2005). The Structure of Context: Deciphering "Frame Analysis". Sociological Theory 23 (4):368-385.score: 30.0
    This article proposes that Goffman's "Frame Analysis" can be interpreted as a step toward unpacking the idea of context. His analysis implies a recursive model involving frames within frames. The key problem is that neither Goffman nor anyone else has clearly defined what is meant by a frame. I propose that it can be represented by a word, phrase, or proposition. A subjective context can be represented as an assembly of these items, joined together by operators such as and, since, (...)
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  34. Pavel Tichy (1986). Constructions. Philosophy of Science 53 (4):514-534.score: 30.0
    The paper deals with the semantics of mathematical notation. In arithmetic, for example, the syntactic shape of a formula represents a particular way of specifying, arriving at, or constructing an arithmetical object (that is, a number, a function, or a truth value). A general definition of this sense of "construction" is proposed and compared with related notions, in particular with Frege's concept of "function" and Carnap's concept of "intensional isomorphism." It is argued that constructions constitute the proper subject (...)
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  35. Front Page, Earliest Uses of Symbols of Set Theory and Logic.score: 30.0
    The study of logic goes back more than two thousand years and in that time many symbols and diagrams have been devised. Around 300 BC Aristotle introduced letters as term-variables, a "new and epoch-making device in logical technique." (W. & M. Kneale The Development of Logic (1962, p. 61). The modern era of mathematical notation in logic began with George Boole (1815- 1864), although none of his notation survives. Set theory came into being in the late 19th (...)
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  36. Chen Chung Chang (1966). Continuous Model Theory. Princeton, Princeton University Press.score: 30.0
    CONTINUOUS MODEL THEORY CHAPTER I TOPOLOGICAL PRELIMINARIES. Notation Throughout the monograph our mathematical notation does not differ drastically from ...
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  37. Fulya Horozal, Florian Rabe & Michael Kohlhase, Extending OpenMath with Sequences.score: 30.0
    Sequences play a great role in mathematical communication. In mathematical notation, we use sequence ellipsis (. . . ) to denote "obvious" sequences like 1, 2, . . . , 7, and in conceptualizations sequence constructors like (i 2+1) i∈N. Furthermore, sequences have a prominent role as argument sequences of flexary functions. While the former cases can adequately be represented and reasoned about as domain objects in Open- Math and MathML, argument sequences are at the language level, (...)
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  38. William M. Farmer & Joshua D. Guttman (2000). A Set Theory with Support for Partial Functions. Studia Logica 66 (1):59-78.score: 28.0
    Partial functions can be easily represented in set theory as certain sets of ordered pairs. However, classical set theory provides no special machinery for reasoning about partial functions. For instance, there is no direct way of handling the application of a function to an argument outside its domain as in partial logic. There is also no utilization of lambda-notation and sorts or types as in type theory. This paper introduces a version of von-Neumann-Bernays-Gödel set theory for reasoning about sets, (...)
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  39. Thomas Mormann (2005). Mathematical Metaphors in Natorp’s Neo-Kantian Epistemology and Philosophy of Science. In Falk Seeger, Johannes Lenard & Michael H. G. Hoffmann (eds.), Activity and Sign. Grounding Mathematical Education. Springer.score: 27.0
    A basic thesis of Neokantian epistemology and philosophy of science contends that the knowing subject and the object to be known are only abstractions. What really exists, is the relation between both. For the elucidation of this “knowledge relation ("Erkenntnisrelation") the Neokantians of the Marburg school used a variety of mathematical metaphors. In this con-tribution I reconsider some of these metaphors proposed by Paul Natorp, who was one of the leading members of the Marburg school. It is shown that (...)
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  40. Michael Kohlhase, Cdmtcs.score: 26.0
    In the last two decades, the World Wide Web has become the universal, and — for many users — main information source. Search engines can efficiently serve daily life information needs due to the enormous redundancy of relevant resources on the web. For educational — and even more so for scientific information needs, the web functions much less efficiently: Scientific publishing is built on a culture of unique reference publications, and moreover abounds with specialized structures, such as technical nomenclature, notational (...)
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  41. Helen De Cruz & Johan De Smedt (2013). Mathematical Symbols as Epistemic Actions. Synthese 190 (1):3-19.score: 24.0
    Recent experimental evidence from developmental psychology and cognitive neuroscience indicates that humans are equipped with unlearned elementary mathematical skills. However, formal mathematics has properties that cannot be reduced to these elementary cognitive capacities. The question then arises how human beings cognitively deal with more advanced mathematical ideas. This paper draws on the extended mind thesis to suggest that mathematical symbols enable us to delegate some mathematical operations to the external environment. In this view, mathematical symbols (...)
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  42. Valeria Giardino (2010). Intuition and Visualization in Mathematical Problem Solving. Topoi 29 (1):29-39.score: 24.0
    In this article, I will discuss the relationship between mathematical intuition and mathematical visualization. I will argue that in order to investigate this relationship, it is necessary to consider mathematical activity as a complex phenomenon, which involves many different cognitive resources. I will focus on two kinds of danger in recurring to visualization and I will show that they are not a good reason to conclude that visualization is not reliable, if we consider its use in (...) practice. Then, I will give an example of mathematical reasoning with a figure, and show that both visualization and intuition are involved. I claim that mathematical intuition depends on background knowledge and expertise, and that it allows to see the generality of the conclusions obtained by means of visualization. (shrink)
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  43. Giuseppe Longo & Arnaud Viarouge (2010). Mathematical Intuition and the Cognitive Roots of Mathematical Concepts. Topoi 29 (1):15-27.score: 24.0
    The foundation of Mathematics is both a logico-formal issue and an epistemological one. By the first, we mean the explicitation and analysis of formal proof principles, which, largely a posteriori, ground proof on general deduction rules and schemata. By the second, we mean the investigation of the constitutive genesis of concepts and structures, the aim of this paper. This “genealogy of concepts”, so dear to Riemann, Poincaré and Enriques among others, is necessary both in order to enrich the foundational analysis (...)
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  44. Helen De Cruz & Johan De Smedt (2010). The Innateness Hypothesis and Mathematical Concepts. Topoi 29 (1):3-13.score: 24.0
    In historical claims for nativism, mathematics is a paradigmatic example of innate knowledge. Claims by contemporary developmental psychologists of elementary mathematical skills in human infants are a legacy of this. However, the connection between these skills and more formal mathematical concepts and methods remains unclear. This paper assesses the current debates surrounding nativism and mathematical knowledge by teasing them apart into two distinct claims. First, in what way does the experimental evidence from infants, nonhuman animals and neuropsychology (...)
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  45. Marianna Antonutti Marfori (2010). Informal Proofs and Mathematical Rigour. Studia Logica 96 (2):261-272.score: 24.0
    The aim of this paper is to provide epistemic reasons for investigating the notions of informal rigour and informal provability. I argue that the standard view of mathematical proof and rigour yields an implausible account of mathematical knowledge, and falls short of explaining the success of mathematical practice. I conclude that careful consideration of mathematical practice urges us to pursue a theory of informal provability.
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  46. José Ferreirós (2009). Hilbert, Logicism, and Mathematical Existence. Synthese 170 (1):33 - 70.score: 24.0
    David Hilbert’s early foundational views, especially those corresponding to the 1890s, are analysed here. I consider strong evidence for the fact that Hilbert was a logicist at that time, following upon Dedekind’s footsteps in his understanding of pure mathematics. This insight makes it possible to throw new light on the evolution of Hilbert’s foundational ideas, including his early contributions to the foundations of geometry and the real number system. The context of Dedekind-style logicism makes it possible to offer a new (...)
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  47. W. V. Quine (1951). Mathematical Logic. Cambridge, Harvard University Press.score: 24.0
    INTRODUCTION MATHEMATICAL logic differs from the traditional formal logic so markedly in method, and so far surpasses it in power and subtlety, ...
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  48. Michał Walicki (2012). Introduction to Mathematical Logic. World Scientific.score: 24.0
    A history of logic -- Patterns of reasoning -- A language and its meaning -- A symbolic language -- 1850-1950 mathematical logic -- Modern symbolic logic -- Elements of set theory -- Sets, functions, relations -- Induction -- Turning machines -- Computability and decidability -- Propositional logic -- Syntax and proof systems -- Semantics of PL -- Soundness and completeness -- First order logic -- Syntax and proof systems of FOL -- Semantics of FOL -- More semantics -- Soundness (...)
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