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One of the pioneers of mathematical logic in the twentieth century was Alonzo Church. He introduced such concepts as the lambda calculus, now an essential tool of computer science, and was the founder of the Journal of Symbolic Logic. In Introduction to Mathematical Logic, Church presents a masterful overview of the subject--one which should be read by every researcher and student of logic. The previous edition of this book was in the Princeton Mathematical Series. ... Read more

Customer Reviews (2)

a classic, but mostly useful as a historical reference
I give this book 5 stars out of respect for its enormous contribution to mathematical logic; for no doubt many of the authors of the more modern math-logic texts were greatly influenced by this book. But with that said, all of the material here is a proper subset of other current books which present the material much more clearly and using better notation. Examples include Burris' "Logic for Mathematics and Computer Science", Ebbinhaus' "Intro. To Math Logic", and Gallier's "Logic for Computer Scientists".

One of the classics
This book, which first appeared in print as an issue in Annals of Mathematics in 1944, is now a classic in mathematical logic, and is still worth perusing in spite of the out-dated notation. The author outlines comprehensively the propositional calculus and predicate calculus. Although the book is mostly formal in its style, the author does introduce the reader to some elementary notions in logic, and some brief commentary on what would now be classified as philosophical logic. He defines logic as the analysis of propositions and their proof according to their form and not their content. He notes also that inductive logic and the theory of partial confirmation should also be included as part of mathematical logic. There are exercises throughout the book, and so it could conceivably be used as a textbook, in spite of its publication date. The book could better be used as a historical supplement to a course in mathematical logic or one in the philosophy of logic.

In the introduction to the book the author defines the terms and concepts he will use in the book, with a discussion of proper names, constants and variables, functions, and sentences. He adopts the Fregian point of view that sentences are names of a particular kind. His discussion of this is rather vague however, for he does not give enough clarification of the difference between an "assertive" use of a sentence and its "non-assertive" use. Readers will have to do further reading on Frege in order to understand this distinction more clearly, but essentially what Church is saying here is that sentences are names with truth values. The existential and universal quantifiers are introduced as well. And here the author also introduces the concepts of object language and metalanguage, along with a discussion of the axiomatic method. The author distinguishes between informal and formal axiomatic methods. The modern notions of syntax and semantics are given a nice treatment here, and the di

scussion is more in-depth than one might get in more modern texts on mathematical logic.

Chapter 1 is a detailed overview of propositional logic, being the usual formal system with three symbols, one constant, an infinite number of variables, rules on how to form well-formed formulas, and the rules of inference. The deduction theorem is proved in detail along with a discussion of the decision problem for propositional logic, with the famous truth tables due to W. Quine introduced here. The notions of consistency and completeness are briefly discussed.

The discussion of the propositional calculus is continued in the next chapter where a new system of propositional calculus is obtained by dropping the constants from the first one and adding another symbol (negation). The two systems are shown to be equivalent to each other using a particular well-formed formula in the second one to replace the constant in the first. Other systems of propositional calculus are also introduced here, using the idea of primitive connectives such as disjunction, along with various rules of inference. Church also outlines an interesting propositional calculus due to J.G.P.Nicod, which assumes only one primitive connective, one axiom, and only one rule of inference (besides substitution). The author also introduces partial systems of propositional calculus, with the goal of showing just what must be added to these systems to obtain the full propositional calculus. He discusses the highly interesting and thought-provoking intuitionistic propositional calculus, due to A. Heyting, which is a formalization of the famous mathematical intuitionism of L.E.J. Brouwer. The system he discusses is a variant of Heyting's and he gives references to the positive solution of the decision problem for this system. The author ends the chapter with a brief discussion of how to construct a propositional calculus by employing axiom schemata.

The author then moves on to what he has termed functional calculi of first order beginning in the next chapter. Called predicate calculi in today's parlance, the author first defines the pure functional calculus of first order, and shows that the theorems of the propositional calculus also follow when considered as part of this system. Free and bound variables are defined, and Church proves explicitly the consistency of this system, and the deduction theorem. The important construction of a prenex normal form of a well-formed formula is discussed, and the author shows that every well-formed formula of the functional calculus is equivalent to some well-formed formula in prenex normal form.

In chapter 4, the author gives an alternative formulation of pure functional calculus of first order, wherein rules of substitution are used and axiom schemata are replaced by instances, making the number of axioms finite. The Skolem normal form of a well-formed formula is defined, which sets up a discussion of satisfiability and validity. The author then proves the Godel completeness theorem, which states that every valid well-formed formula is a theorem. This is followed by a very well written discussion of the Skolem-Lowenheim theorem, and an overview of the decision problem in functional (predicate) calculus.

In the last chapter of the book the author considers functional (predicate) calculi of second order, which is distinguished from the first order case by allowing the variables to range over what its predicates and subjects represent. In second-order functional calculus, propositional and predicate variables can have bound occurrences. The author discusses the elimination problem and consistency for second-order predicate calculus, and gives a proof of the (Henkin) completeness theorem. A fairly detailed discussion of a logical system for elementary number theory is given, but the treatment involves notation that is somewhat clumsy and the discussion is difficult to follow. ... Read more

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This volume began as a remembrance of Alonzo Church while he was still with us and is now finally complete. It contains papers by many well-known scholars, most of whom have been directly influenced by Church's own work. Often the emphasis is on foundational issues in logic, mathematics, computation, and philosophy -- as was the case with Church's contributions, now universally recognized as having been of profound fundamental significance in those areas.The volume will be of interest to logicians, computer scientists, philosophers, and linguists. The contributions concern classical first-order logic, higher-order logic, non-classical theories of implication, set theories with universal sets, the logical and semantical paradoxes, the lambda-calculus, especially as it is used in computation, philosophical issues about meaning and ontology in the abstract sciences and in natural language, and much else. The material will be accessible to specialists in these areas and to advanced graduate students in the respective fields. ... Read more

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Originally published in 1895.This volume from the Cornell University Library's print collections was scanned on an APT BookScan and converted to JPG 2000 format by Kirtas Technologies.All titles scanned cover to cover and pages may include marks notations and other marginalia present in the original volume. ... Read more

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This digital document, covering the life and work of Alonzo Church, is an entry from Contemporary Authors, a reference volume published by Thompson Gale. The length of the entry is 1165 words. The page length listed above is based on a typical 300-word page. Although the exact content of each entry from this volume can vary, typical entries include the following information:

• Place and date of birth and death (if deceased)
• Family members
• Education
• Professional associations and honors
• Employment
• Writings, including books and periodicals
• A description of the author's work
• References to further readings about the author

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Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Chapters: Alonzo Church, Charles Sanders Peirce, Raymond Smullyan, Haskell Curry, George Boolos, Clarence Irving Lewis, John Corcoran, Ralph Johnson, Rohit Jivanlal Parikh, Bas Van Fraassen, Paul Benacerraf, Raphael M. Robinson, Hao Wang, Colin Mclarty, Jon Barwise, Richard Montague, Richard Jeffrey, Penelope Maddy, Gila Sher, Edward Nelson, Ronald Jensen, Alan Ross Anderson, Matthew Foreman, Tomek Bartoszyński, Hartley Rogers, Jr., John Etchemendy, Robert M. Solovay, J. Barkley Rosser, Leo Harrington, Nate Ackerman, James Earl Baumgartner, Jack Silver, W. Hugh Woodin, Harvey Friedman, Richard Shore, Yiannis N. Moschovakis, Solomon Feferman, Valentina Harizanov, Theodore Slaman, David Blitz, Peter B. Andrews, Alice Ambrose, Charles Parsons, Gerald Sacks, Irving Copi, Nuel Belnap, Frederic Brenton Fitch, Bob Meyer, Alexander S. Kechris, Donald A. Martin, John R. Steel, Isaac Malitz, John P. Burgess. Source: Wikipedia. Free updates online. Not illustrated. Excerpt: Charles Sanders Peirce (pronounced purse) (September 10, 1839 April 19, 1914) was an American philosopher, logician, mathematician, and scientist, born in Cambridge, Massachusetts. Peirce was educated as a chemist and employed as a scientist for 30 years. It is largely his contributions to logic, mathematics, philosophy, and semiotics (and his founding of pragmatism) that are appreciated today. In 1934, the philosopher Paul Weiss called Peirce "the most original and versatile of American philosophers and America's greatest logician". An innovator in many fields including philosophy of science, epistemology, metaphysics, mathematics, statistics, research methodology, and the design of experiments in astronomy, geophysics, and psychology Peirce considered himself a logician first and foremost. He made major contributions to logic, but...More: http://booksllc.net/?id=6117 ... Read more

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Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Chapters: Bertrand Russell, Alonzo Church, Alan Kay, Fm-2030, Paul Pimsleur, Abraham Robinson, Jared Diamond, Al Gore, Joseph W. Eaton, Emory Tolbert, Wolfgang Smith, Paul Auerbach, Henry Masauko Blasius Chipembere, Gregory Stock, Peter Ladefoged, Ruth Milkman, Julian Schwinger, Kendall Houk, Richard C. Atkinson, Stanford J. Shaw, Lev Manovich, Johanna Drucker, William H. Oldendorf, Benjamin Karney, Terence Tao, Douglas Kellner, Henry Samueli, Alex Cline, Peter Mclaren, John A. Agnew, Larry Pressler, Daniel Dăianu, Eugene Volokh, Paul Terasaki, Marcia J. Bates, Kamyar Kalantar-Zadeh, Leonard Kleinrock, Sheila Greibach, Andrea M. Ghez, Robert Stevenson, James Fraser Stoddart, Chih-Ming Ho, Robert Epp, Helmut Ringsdorf, Stanley Wolpert, Debora Shuger, Louis Ignarro, Greg Graffin, Paul D. Boyer, Robert Bruce Merrifield, Asad Abidi, Utpal Banerjee, Charles E. Young, Eugen Weber, Myron Tribus, N. Katherine Hayles, Murray Leaf, Jennifer Steinkamp, George Rousseau, Ernest Morrell, Julián Marías, Dianne Walker, Jennifer Tour Chayes, Judea Pearl, Val D. Rust, Andrew Benson, Benjamin Anderson, Kenneth Sokoloff, D. Lewis Mingori, Ricardo Azziz, Mazisi Kunene, Lotfi Mansouri, Richard Yarborough, Larry Estrada, Gerald Jay Goldberg, Mike Rose, Gerald J. Popek, Harold Demsetz, Allen J. Scott, Christopher Mott, Sanjay Subrahmanyam, Jack Penn, Harold Garfinkel, Armen Alchian, Evelyn Venable, Johnnetta B. Cole, Jerry Leaf, Richard Popkin, Colin Westerbeck, Peter Wollen, Sue-Ellen Case, Geshe Gyeltsen, Laura E. Gómez, Brenda Marie Osbey, Peter Kollock, Diana Raab, Neil Denari, Anthony Pagden, David Wong Louie, Russell Leong, Page Smith, Murray Jarvik, Vaughan Jones, John Garcia, William Brice, Robert Livingston, Sandra Harding, Charles H. Kraft, Albert Boime, Norman H. Anderson, Paul Habibi, William Ouchi, ...More: http://booksllc.net/?id=5042706 ... Read more

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This digital document is an article from Encyclopedia of Philosophy, brought to you by Gale®, a part of Cengage Learning, a world leader in e-research and educational publishing for libraries, schools and businesses.The length of the article is 1115 words.The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase.You can view it with any web browser.Explores major marketing and advertising campaigns from 1999-2006. Entries profile recent print, radio, television, billboard and Internet campaigns. Each essay discusses the historical context of the campaign, the target market, the competition, marketing strategy, and the outcome. ... Read more

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This digital document is an article from Science and Its Times, brought to you by Gale®, a part of Cengage Learning, a world leader in e-research and educational publishing for libraries, schools and businesses.The length of the article is 515 words.The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase.You can view it with any web browser.The histories of science, technology, and mathematics merge with the study of humanities and social science in this interdisciplinary reference work. Essays on people, theories, discoveries, and concepts are combined with overviews, bibliographies of primary documents, and chronological elements to offer students a fascinating way to understand the impact of science on the course of human history and how science affects everyday life. Entries represent people and developments throughout the world, from about 2000 B.C. through the end of the twentieth century. ... Read more

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Chapters: Kurt Gödel, John Von Neumann, Grover Cleveland, Aaron Burr, Alonzo Church, Eugene Wigner, Jonathan Edwards, Henry Norris Russell, George Gallup, John Witherspoon, B. B. Warfield, David Hunter, David Kellogg Lewis, Charles Hodge, John Tukey, Roger Atkinson Pryor, William Drew Robeson I, Robert F. Stockton, Moses Taylor Pyne, William Willet, George Wildman Ball, Princeton Cemetery, Sylvia Beach, Frances Folsom Cleveland Preston, John Maclean, Jr., Maria Louisa Bustill, James Waddel Alexander, William Cowper Alexander, Edward Parke Custis Lewis, Frank Lewin, Archibald Alexander, Lyman Spitzer, Aaron Burr, Sr., Richard Stockton, Richard Stockton Field, Howard Alexander Smith, John P. Stockton, John Huston Finley, Joseph Addison Alexander, Canvass White, Paul Tulane, John Renshaw Thomson, Peter Charles Harris, Edward Samuel Corwin, Samuel Davies, William G. Thompson, Ruth Cleveland. Source: Wikipedia. Pages: 279. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: Stephen Grover Cleveland (March 18, 1837 June 24, 1908) was the 22nd and 24th President of the United States. Cleveland is the only president to serve two non-consecutive terms (18851889 and 18931897) and therefore is the only individual to be counted twice in the numbering of the presidents. He was the winner of the popular vote for president three timesin 1884, 1888, and 1892and was the only Democrat elected to the presidency in the era of Republican political domination that lasted from 1860 to 1912. Cleveland's admirers praise him for his honesty, independence, integrity, and commitment to the principles of classical liberalism. As a leader of the Bourbon Democrats, he opposed imperialism, taxes, subsidies and inflationary policies. As a reformer he also worked against political corruption, patronage, and bossism. Some of Cle...More: http://booksllc.net/?id=12495 ... Read more

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This digital document is an article from Encyclopedia of Philosophy, brought to you by Gale®, a part of Cengage Learning, a world leader in e-research and educational publishing for libraries, schools and businesses.The length of the article is 928 words.The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase.You can view it with any web browser.Explores major marketing and advertising campaigns from 1999-2006. Entries profile recent print, radio, television, billboard and Internet campaigns. Each essay discusses the historical context of the campaign, the target market, the competition, marketing strategy, and the outcome. ... Read more

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"Like the Gospel, this book challenges us in our depths to receive the gift of God and to live it without compromise in the one 'place' where that is possible - His Church." - Most Rev. Charles Chaput, Archbishop of Denver "In an era of widespread ecclesiological confusion, What Is The Church? is filled with clarity, charity, and discernment. Dr. Martin wears his learning lightly and shares it beguilingly in a book that is at the same time light for the intellect and music for the soul" - Russell B. Shaw, author of Papal Primacy in the Third Millennium ... Read more

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This volume is produced from digital images created through the University of Michigan University Library's large-scale digitization efforts. The Library seeks to preserve the intellectual content of items in a manner that facilitates and promotes a variety of uses. The digital reformatting process results in an electronic version of the original text that can be both accessed online and used to create new print copies. The Library also understands and values the usefulness of print and makes reprints available to the public whenever possible. This book and hundreds of thousands of others can be found in the HathiTrust, an archive of the digitized collections of many great research libraries. For access to the University of Michigan Library's digital collections, please see http://www.lib.umich.edu and for information about the HathiTrust, please visit http://www.hathitrust.org ... Read more