Lectures on Mathematical Logic, Volume 2

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CRC Press, May 30, 2000 - Mathematics - 330 pages
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In this volume, logic starts from the observation that in everyday arguments, as brought forward by say a lawyer, statements are transformed linguistically, connecting them in formal ways irrespective of their contents. Understanding such arguments as deductive situations, or "sequents" in the technical terminology, the transformations between them can be expressed as logical rules. The book concludes with the algorithms producing the results of Gentzen's midsequent theorem and Herbrand's theorem for prenex formulas.
 

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Contents

Sequent Calculi for Positive Logic
5
Cuts
27
Continuous Cut Elimination
46
Sequent Calculi for Minimal and Intuitionistic Logic
67
Sequent Calculi for Classical Logic
88
Classes of Algebras Associated to a Calculus
126
Calculi of Formulas
168
Modus Ponens Calculi for Positive Logic
174
The Variants cxqt0 of cxqt
240
The Variants cxqt and cxqt2 of cxqt
243
The Calculi cxqs
245
The Deduction Theorem and Other Metarules for the Calculi ccqt
248
Tautologies of Positive Quantifier Logic
250
Tautologies of Minimal Quantifier Logic
252
Tautologies of Classical Quantifier Logic
253
Selected Topics in Sequential Quantifier Logic
256

Modus Ponens Calculi for Minimal and for Intuitionistic Logic
181
Modus Ponens Calculi for Classical Logic
185
Historical Notes to Chapters 17
196
Sequent Calculi for Quantifier Logic
202
Sequent Calculi with Qrules
204
The Replacement Theorem and Cut Elimination for Calculi with rep
212
The Substitution Theorem and Cut Elimination for Calculi with sub
219
The Sets SUB
221
The Substitution Theorem Resumed
226
Cut Elimination Resumed
231
Inversion Rules
233
9 Semantical Consequence Operations and Modus Ponens Calculi
236
Relations between Classical and Intuitionistic Derivability
259
Equality Logic
263
Language Extensions with Predicate Symbols
270
Language Extensions with Function Symbols 1
274
Language Extensions with Function Symbols 2
285
The Midsequent Theorem
288
Herbrands Theorem for Prenex Formulas
298
Tableaux
305
Index of concepts and names
317
Index of symbolic notations
321
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