/pdf/a-course-in-universal-algebra-3xkp961xgs.pdf

A Course in Universal Algebra

From the Publisher:
Among the many approaches to formal reasoning about programs, Dynamic Logic enjoys the singular advantage of being strongly related to classical logic. Its variants constitute natural generalizations and extensions of classical formalisms. For example, Propositional Dynamic Logic (PDL) can be described as a blend of three complementary classical ingredients: propositional calculus, modal logic, and the algebra of regular events. In First-Order Dynamic Logic (DL), the propositional calculus is replaced by classical first-order predicate calculus. Dynamic Logic is a system of remarkable unity that is theoretically rich as well as of practical value. It can be used for formalizing correctness specifications and proving rigorously that those specifications are met by a particular program. Other uses include determining the equivalence of programs, comparing the expressive power of various programming constructs, and synthesizing programs from specifications. 
This book provides the first comprehensive introduction to Dynamic Logic. It is divided into three parts. The first part reviews the appropriate fundamental concepts of logic and computability theory and can stand alone as an introduction to these topics. The second part discusses PDL and its variants, and the third part discusses DL and its variants. Examples are provided throughout, and exercises and a short historical section are included at the end of each chapter.

Dynamic Logic

Parts of recognition (A)

Parts of recognition

13 EADERS of Mr. Bertrand Russell's philosophical £v works know that one of his pupils before the outbreak of the war, an Austrian, Mr. Ludwig Wittgenstein, caused him to change his views in some important particulars. Curiosity can now be satisfied. The “Tractatus Logico-Philosophicus “which Mr. Ogden has included in his new library of philosophy is a remarkable and strikingly original work. It is published in German and English in parallel pages. It is difficult to appreciate the reason for this, seeing that the author is evidently familiar with our language and has himself carefully revised the proofs of the translation. Also we should have liked to have the Tractatus without Mr. Russell's Introduction, not, we hasten to add, on account of any fault or shortcoming in that introduction, which is highly appreciative and in part a defence of himself, in part explanatory of the author, but for the reason that good wine needs no bush and that Mr. Russell's bush has the unfortunate effect of dulling the palate instead of whetting the appetite. In his penultimate sentence Mr. Russell says; “To have constructed a theory of logic which is not at any point obviously wrong is to have achieved a work of extraordinary difficulty and importance.” We agree, but how uninspiring when compared with Mr. Wittgenstein's own statement of aim: “What can be said at all can be said clearly, and whereof one cannot speak, thereof one must be silent.” Tractatus Logico-Philosophicus. By Ludwig Wittgenstein. (International Library of Psychology, Philosophy and Scientific Method.) Pp. 189. (London: Kegan Paul and Co., Ltd.; New York: Harcourt, Brace and Co., Inc., 1922.) 10s. 6d. net.

Tractatus Logico-Philosophicus

1. Beginning Mathematical Logic. 2. First-Order Logic. 3. First-Order Logic (continued). 4. Boolean Algebras. 5. Model Theory. 6. Recursion Theory. 7. Logic - Limitative Results. 8. Recursion Theory (continued). 9. Intuitionistic First-Order Logic. 10. Axiomatic Set Theory. 11. Nonstandard Analysis. Bibliography.

A course in mathematical logic

Models and ultraproducts

Introduction 1. Basic features of smooth worlds 2. Basic differential calculus 3. First applications of the differential calculus 4. Applications to physics 5. Multivariable calculus and applications 6. The definite integral: Higher order infinitesimals 7. Synthetic geometry 8. Smooth infinitesimal analysis as an axiomatic system Appendix Models for smooth infinitesimal analysis.

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A primer of infinitesimal analysis

THE CLOSE CONNECTION BETWEEN mathematics and philosophy has long been recognized by practitioners of both disciplines. The apparent timelessness of mathematical truth, the exactness and objective nature of its concepts, its applicability to the phenomena of the empirical world—explicating such facts presents philosophy with some of its subtlest problems. In this final chapter we discuss some of the attempts made by philosophers and mathematicians to explain the nature of mathematics. We begin with a brief presentation of the views of four major classical philosophers: Plato, Aristotle, Leibniz, and Kant. We conclude with a more detailed discussion of the three “schools” of mathematical philosophy which have emerged in the twentieth century: Logic ism, Formalism, and Intuitionism.

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The Philosophy of Mathematics

Part 1 Elements of category theory: including categories, functors, adjunctions, uniqueness of adjoints, Cartesian closed categories, reflective subcategories, Galois connections Part 2 Introducing toposes: including geometric morphisms, power objects - the concept of topos Part 3 Local set theories: including local languages and local set theories, interpreting a local language in a topos - the Soundness Theorem, the Completeness Theorem, the Equivalence Theorem, adjoining indeterminates, introduction of function values Part 4 Fundamental properties of toposes: including slicing a topos, Beth-Kripke-Joyal semantics Part 5 From logic to sheaves: including truth sets, modalities and universal closure operations, the Sheafification functor, modalized toposes, sheaves over locales and topological spaces Part 6 Locale-valued sets: including the topos of sheaves over a topological space, decidable, subconstant and fuzzy sets, Boolean extensions as toposes Part 7 Natural numbers and real numbers: including natural and real numbers in local set theories, the free topos Part 8 Epilogue - the wider significance of topos theory: from set theory to topos theory, some analogies with the Theory of Relativity, the negation of constancy Appendix - geometric theories and classifying toposes Historical and bibliographical notes References Index of symbols Index of terms

John L. Bell

Papers

A course in mathematical logic

Models and ultraproducts

A primer of infinitesimal analysis

The Philosophy of Mathematics

Toposes and Local Set Theories: An Introduction