Introduction to the Theory of Logic

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Westview Press, 2000 - Philosophy - 330 pages
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This book provides a rigorous introduction to the basic concepts and results of contemporary logic. It also presents, in two unhurried chapters, the mathematical tools (mainly from set theory) that are needed to master the technical aspects of the subject. Methods of definition and proof are also discussed at length, with special emphasis on inductive definitions and proofs and recursive definitions. The book is ideally suited for readers who want to undertake a serious study of logic but lack the mathematical background that other texts at this level presuppose. It can be used as a textbook in graduate and advanced undergraduate courses in logic. Hundreds of exercises are provided. Topics covered include basic set theory, propositional and first-order syntax and semantics, a sequent calculus-style deductive system, the soundness and completeness theorems, cardinality, the expressive limitations of first-order logic, with especial attention to the Loewenheim-Skolem theorems and non-standard models of arithmetic, decidability, complete theories, categoricity and quantifier elimination.
  

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Contents

Chapter
36
Chapter
76
Chapter
117
Chapter
149
FirstOrder Languages without Identity
162
FirstOrder Languages with Identity
166
A WellRounded Set of Sentences Is the Theory of its Canonical Structure
170
Negation Completeness
174
Exponentiation
201
Cardinality
204
Contextual Definition
206
Continuities
209
Denumerable Sets
215
Diagonals
218
ZF
224
11 Choice
228

Henkin Constants and Henkin Axioms
178
Consistent NegationComplete Henkin Sets Are Well Rounded
183
Compactness
187
Chapter
189
Criteria for Size Claims
190
Counting
191
Functional Criteria
195
Big and Small
196
Addition
197
Multiplication
199
12 Zorns Lemma
236
Completeness for Uncountable Languages
245
Chapter
250
Isomorphic Structures
252
Isomorphic Structures and Contextual Definitions
259
Representing Finite Structures
261
The Downward LowenheimSkolem Theorem
263
Chapter
292
Symbols and Notation
321
Copyright

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Page 73 - ... f ...f •••t •••t ...f ...f ...t ...f ...f •••t •••t •••t •••t •••t •••t •••t •••t ...t •••t •••: ...t...
Page 238 - B be any two sets. The set of all ordered pairs such that the first member of the ordered pair is an element of A and the second member is an element of B is called the cartesian product of A and B and is written as AX B.
Page 8 - The intersection of sets A and B, written A n B, is the set of all elements common to sets A and B.
Page 117 - An argument will be said to be valid if and only if the conjunction of the premises implies the conclusion, ie, if the premises are all true, the conclusion must also be true. It...
Page xii - I have tried to present the material in such a way as to be intelligible and even helpful to graduate students and, perhaps, undergraduates.
Page 175 - Prove that there exists a one-to-one correspondence between the set of positive integers and the set of all positive rational numbers.
Page 8 - The union of sets A and B, written A u B, is the set whose elements are all the elements of A and all the elements of B.
Page 187 - ... then it is not the case that for every set F of sentences of intensional logic, (3) if every finite subset of F is satisfiable, then F is satisfiable. This is obvious in view of the reduction, at which we hinted earlier, of ordinary second-order logic to intensional logic, together with the wellknown failure of the compactness theorem for second-order logic. On the other hand, let us call $ a predicative sentence if...

About the author (2000)

Jose L. Zalabardo was born in Madrid in 1964, and educated at the Universidad Aut243noma de Madrid, the University of St. Andrews and the University of Michigan, where he obtained a PhD in 1994. He was a lecturer at the University of Birmingham from 1994 to 2000. In 2000 he joined the University College London Philosophy Department, where he is now a reader. He is associate editor of Mind.

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