Mathematics
Logic
Logic in mathematics refers to the systematic study of valid reasoning and inference. It involves the use of formal systems to determine the truth or falsity of mathematical statements. By employing logical principles and rules, mathematicians can construct rigorous proofs and make sound conclusions about the properties and relationships of mathematical objects.
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12 Key excerpts on "Logic"
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter- 4 Logic and Mathematical Logic Logic Logic (from the Greek λογική logikē) is the study of arguments. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathe-matics, and computer science. Logic examines general forms which arguments may take, which forms are valid, and which are fallacies. It is one kind of critical thinking. In philosophy, the study of Logic falls in the area of epistemology, which asks: How do we know what we know? In mathematics, it is the study of valid inferences within some formal language. Logic has origins in several ancient civilizations, including ancient India, China and Greece. Logic was established as a discipline by Aristotle, who established its fun -damental place in philosophy. The study of Logic was part of the classical trivium. Averroes defined Logic as the tool for distinguishing between the true and the false; Richard Whately, the Science, as well as the Art, of reasoning; and Gottlob Frege, the science of the most general laws of truth. The article Definitions of Logic provides citations for these and other definitions. Logic is often divided into two parts, inductive reasoning and deductive reasoning. The first is drawing general conclusions from specific examples, the second drawing Logical conclusions from definitions and axioms. A similar dichotomy, used by Aristotle, is analysis and synthesis. Here the first takes an object of study and examines its component parts. The second considers how parts can be combined to form a whole. Logic is also studied in argumentation theory. Nature The concept of Logical form is central to Logic, it being held that the validity of an argument is determined by its Logical form, not by its content. Traditional Aristotelian syllogistic Logic and modern symbolic Logic are examples of formal Logics.
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- 2014(Publication Date)
- Orange Apple(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter- 1 Mathematical Logic Mathematical Logic (also known as symbolic Logic ) is a subfield of mathematics with close connections to computer science and philosophical Logic. The field includes both the mathematical study of Logic and the applications of formal Logic to other areas of mathematics. The unifying themes in mathematical Logic include the study of the expressive power of formal systems and the deductive power of formal proof systems. Mathematical Logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory. These areas share basic results on Logic, particularly first-order Logic, and definability. Since its inception, mathematical Logic has contributed to, and has been motivated by, the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and analysis. In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory. Contemporary work in the foundations of mathematics often focuses on establishing which parts of mathematics can be formalized in particular formal systems, rather than trying to find theories in which all of mathematics can be developed. History Mathematical Logic emerged in the mid-19th century as a subfield of mathematics independent of the traditional study of Logic (Ferreirós 2001, p. 443). Before this emergence, Logic was studied with rhetoric, through the syllogism, and with philosophy.
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- 2014(Publication Date)
- Orange Apple(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter- 1 Introduction to Mathematical Logic Mathematical Logic (also known as symbolic Logic ) is a subfield of mathematics with close connections to computer science and philosophical Logic. The field includes both the mathematical study of Logic and the applications of formal Logic to other areas of mathematics. The unifying themes in mathematical Logic include the study of the expressive power of formal systems and the deductive power of formal proof systems. Mathematical Logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory. These areas share basic results on Logic, particularly first-order Logic, and definability. Since its inception, mathematical Logic has contributed to, and has been motivated by, the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and analysis. In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory. Contemporary work in the foundations of mathematics often focuses on establishing which parts of mathematics can be formalized in particular formal systems, rather than trying to find theories in which all of mathematics can be developed. History Mathematical Logic emerged in the mid-19th century as a subfield of mathematics independent of the traditional study of Logic (Ferreirós 2001, p. 443). Before this emergence, Logic was studied with rhetoric, through the syllogism, and with philosophy.
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- 2014(Publication Date)
- White Word Publications(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 7 Mathematical Logic Mathematical Logic (also known as symbolic Logic ) is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical Logic. The field includes both the mathematical study of Logic and the applications of formal Logic to other areas of mathematics. The unifying themes in mathematical Logic include the study of the expressive power of formal systems and the deductive power of formal proof systems. Mathematical Logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory. These areas share basic results on Logic, particularly first-order Logic, and definability. Since its inception, mathematical Logic has contributed to, and has been motivated by, the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and analysis. In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory. Contemporary work in the foundations of mathematics often focuses on establishing which parts of mathematics can be formalized in particular formal systems, rather than trying to find theories in which all of mathematics can be developed. History Mathematical Logic emerged in the mid-19th century as a subfield of mathematics independent of the traditional study of Logic (Ferreirós 2001, p. 443). Before this emergence, Logic was studied with rhetoric, through the syllogism, and with philosophy.
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- 2014(Publication Date)
- Orange Apple(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter- 4 Mathematical Logic and Graph Mathematical Logic Mathematical Logic (also known as symbolic Logic ) is a subfield of mathematics with close connections to computer science and philosophical Logic. The field includes both the mathematical study of Logic and the applications of formal Logic to other areas of mathematics. The unifying themes in mathematical Logic include the study of the expressive power of formal systems and the deductive power of formal proof systems. Mathematical Logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory. These areas share basic results on Logic, particularly first-order Logic, and definability. Since its inception, mathematical Logic has contributed to, and has been motivated by, the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and analysis. In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory. Contemporary work in the foundations of mathematics often focuses on establishing which parts of mathematics can be formalized in particular formal systems, rather than trying to find theories in which all of mathematics can be developed. History Mathematical Logic emerged in the mid-19th century as a subfield of mathematics independent of the traditional study of Logic (Ferreirós 2001, p. 443). Before this emergence, Logic was studied with rhetoric, through the syllogism, and with philosophy.
eBook - PDFInexhaustibility
A Non-Exhaustive Treatment
- Torkel Franzén(Author)
- 0(Publication Date)
- Cambridge University Press(Publisher)
119 8 Logic and theories 8.1 Logical reasoning Mathematical proofs, as usually represented in Logic, consist in Logical deduc- tions from axioms stated in a formalized language. We have a set of general logi- cal rules (and in some formulations axioms), applicable in all mathematical rea- soning, and a set of particular mathematical axioms, depending on what mathematics is being represented, from which we draw conclusions using the Logical rules. In using such a representation of mathematical proofs, we are not supposing that mathematicians do or could prove theorems in this way, even though the axiomatic method on a more informal level has been very useful in actual mathematics. (For an elaboration of this point, see the comments on for- mal and actual provability in §1.3.) In this book, the mathematical axioms will be formulas in a first or second order language as defined in chapter 7. The basic mathematical axioms will be pre- sented in chapters 9 and 10. Exactly what Logical rules are used doesn’t matter for the results and questions to be considered here, but we need to give a precise definition of just what “Logically deducible” means for our formal languages, both in order to establish that various relations having to do with Logical deduc- tions can be defined arithmetically, and to prove the completeness theorem for predicate Logic, which will be used at some points in later chapters. The basic relation to be defined in this chapter is Γ⇒φ, which we read as “φ can be Logically deduced from the formulas in Γ”. Here Γ is a set, possibly infinite, of formulas in some first or second order language, and φ a formula in that lan- guage. The formulas in Γ may be axioms in a mathematical theory, but they can also be assumptions used in hypothetical reasoning. The relation ⇒ is defined inductively, and since all the rules defining the relation are finite rules, we will automatically get a concept of Logical deduction in the sense of a finite sequence
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- 2014(Publication Date)
- Learning Press(Publisher)
___________________________ WORLD TECHNOLOGIES ___________________________ Chapter- 11 Mathematical Logic and Analysis Mathematical Logic (also known as symbolic Logic ) is a subfield of mathematics with close connections to computer science and philosophical Logic. The field includes both the mathematical study of Logic and the applications of formal Logic to other areas of mathematics. The unifying themes in mathematical Logic include the study of the expressive power of formal systems and the deductive power of formal proof systems. Mathematical Logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory. These areas share basic results on Logic, particularly first -order Logic, and defi -nability. Since its inception, mathematical Logic has contributed to, and has been motivated by, the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and analysis. In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consis -tency. Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory. Contemporary work in the foundations of mathematics often focuses on establishing which parts of mathematics can be formalized in particular formal systems, rather than trying to find theories in which all of mathematics can be developed. ___________________________ WORLD TECHNOLOGIES ___________________________ History Mathematical Logic emerged in the mid -19th century as a subfield of mathematics independent of the traditional study of Logic (Ferreirós 2001, p. 443).
eBook - PDFPhilosophy
Made Simple
- Richard H. Popkin, Avrum Stroll(Authors)
- 2014(Publication Date)
- Made Simple(Publisher)
Because this is so, Logic has sometimes 274 Philosophy Made Simple been characterized as a discipline which deals with the relations between sentences, or propositions, as the Logician prefers to call them, since he/she is not concerned with interrogative or exclamatory sentences, but only with those that make assertions. Inference is considered to be a process which allows us to establish the truth of a certain proposition, called the conclusion of an argument, from the truth of other propositions which constitute the evidence for the conclusion. On this interpretation, one might define Logic as the branch of philosophy which attempts to determine when a given proposition or a group of propositions permits us correctly to infer some other proposition. Deductive and inductive Logic Philosophers have traditionally divided Logic into two branches. These are called 'deductive Logic' and 'inductive Logic' respectively. Both branches are concerned with the rules for correct reasoning, or correct 'argumentation' as philosophers frequently say. Deductive Logic deals with reasoning which attempts to establish conclusive inferences. To say that an inference is 'conclusive' means that if the reasons given are true, then it will be impossible for the inference based upon these reasons to be false. Such reasoning is called 'valid' reasoning. Deductive Logic is thus concerned with the rules for determining when an argument is valid. Not all reasoning in daily life attempts to provide conclusive evidence for the truth of a given conclusion. Sometimes, by the very nature of the case, conclusive evidence cannot be produced. But very often for practical purposes we do not need conclusive evidence. We merely want the evidence to show that the conclusion we have arrived at is well founded, that it is more probable than some other conclusion we might have reached.
eBook - PDF- Dov M. Gabbay, Paul Thagard, John Woods(Authors)
- 2006(Publication Date)
- North Holland(Publisher)
Logic, like any other subject, depends on philosophical presuppositions. It raises conceptual and especially foundational questions that can only be profitably en-gaged by thinkers with rigorous philosophical training. The fact that so many Logicians have also been philosophers, and that the motivation for so many im-portant innovations in Logic have derived from philosophical concerns about the clarification of concepts, problems and distinctions, means that Logic and philoso-phy of Logic are historically at least if not also thematically inseparably intertwined. It is only by understanding the complex justifications for formal Logical apparatus, for the exact expression of ideas and deductively valid inference of sentences, that we can hope to fathom the explosion of Logical systems that have appeared since the late nineteenth century through the latter half of the twentieth and continuing now into the twenty-first century. With every new addition to the vocabulary and Dale Jacquette Dov M. Gabbay, Paul Thagard and John Woods c Handbook of the Philosophy of Science. Philosophy of Logic General editors: Volume editor: 2007 Elsevier B.V. All rights reserved. 2 Dale Jacquette mechanisms of formal symbolic Logic there arise an exponentially greater num-ber of philosophical questions about the meaning and implications of what Logic introduces to the realms of mathematics and science. IN THE Logic CANDY STORE I want to introduce this handbook of essays in the philosophy of Logic by raising a problem that I think holds a vital key to understanding the nature of Logic in contemporary philosophy and mathematics. Not all of the papers in this volume do not directly address the question, but they provide ample raw material for reflecting on the kinds of answers that might be given. The puzzle in a nutshell is to understand how it is that so many different kinds of formal systems can all deserve to be called Logics.
eBook - PDFLearning to Reason
An Introduction to Logic, Sets, and Relations
- Nancy Rodgers(Author)
- 2011(Publication Date)
- Wiley-Interscience(Publisher)
A proof is a linearly ordered structure of interwoven valid arguments. As we saw in Section 2.1, the rules for the outside construction of a valid argument are essen-tially the rules for how we use the Logical operators and quanti-fiers. The definition of implies gives different ways to set up the structure to derive an implication; the definition also gives the structure for making derivations from an implication. These are two different proof techniques: • how we derive an implication • what we can derive from an implication Similarly, the definition of or gives the rules for setting up the structure to derive an or-sentence; it also gives the structure for making derivations from an or-sentence. In a proof, it is essen-tial to remember that deductions made from an assumption are not stand-alone conclusions. An assumption used in a valid argument must always be included in the final conclusion. For example, if we assume p is true and then derive q, our final conclusion isp=>
- Lev D. Beklemishev(Author)
- 2000(Publication Date)
- Elsevier Science(Publisher)
IV. Philosophy of Logic and Mathematics This page intentionally left blank TOWARDS A FOUNDATION OF A GENERAL PROOF THEORY DAG PRAWITZ University of Oslo, Oslo, Norway Introduction By general proof theory, I understand the study of the notion of proof. The name proof theory was originally given by Hilbert to a constructive study of proofs with certain specific aims. By such a study, he hoped to establish the consistency of mathematics or, more generally, to obtain a reduction of mathematics to a certain constructive part of it. Hence, the study of proofs was here only a tool to obtain this reduction, and it could thus not use principles that were more advanced than those contained in the constructive part of mathematics to which all mathematics was to be reduced. We may call such a study reductive proof theory. In general proof theory, we are-in contrast-interested in understanding the very proofs themselves, i.e., in understanding not only what deductive connections hold but also how they are established, and we do not impose any special restrictions on the means that may be used in the study of these phenomena. Of course, it may also be of interest to analyse the principles used in this theory, and as a by-product, results in this theory may find applications within reductive proof theory.' Some of the more important results in reductive proof theory, in particu-lar Gentzen's well-known results, were indeed such by-products. They were obtained from insights about the general structure of proofs, which insights are in their own right at least as interesting as their applications in reductive proof theory. These insights consists more precisely of the following two connected discoveries: (a) an analysis of first-order inferences into certain atomic 1 For some further remarks concerning general versus reductive proof theory, see PRAWITz (1972).
eBook - PDF- Leslie Burkholder(Author)
- 2019(Publication Date)
- Taylor & Francis(Publisher)
For example, in a recent article, the Logician Neil Tennant endorses this standard view (Tennant, 1986): [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. 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 and 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 160 Visual Information and Valid Reasoning 161 an extensive diagrammatic calculus, which he intended as a general rea-soning tool. Our own challenge is two-pronged. On the one hand, we are developing a computer program, Hyperproof, that follows in the tradition of Euler, Venn, and Peirce. The program will allow students to solve deductive reasoning tasks using an integrated combination of sentences and diagrams. On the other hand, we are developing an information-based theory of deduction rich enough to assess the validity of heterogeneous proofs, proofs that use multiple forms of representation. In this task, we do not want to restrict our attention to any particular dia-grammatic calculus; rather, our aim is to develop a semantic analysis of valid inference that is not inextricably tied to linguistic forms of represen-tation. This volume is not an appropriate forum for presenting the technical details of our theory of heterogeneous inference. Nor can we actually illustrate our program in operation. Consequently, our aim in this paper is just to sow a seed of doubt in the reader's mind about the dogma men-tioned above.
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