Extensions of First Order Logic

Cambridge University Press, 1996 - History - 388 pages
Classical logic has proved inadequate in various areas of computer science, artificial intelligence, mathematics, philosopy and linguistics. This is an introduction to extensions of first-order logic, based on the principle that many-sorted logic (MSL) provides a unifying framework in which to place, for example, second-order logic, type theory, modal and dynamic logics and MSL itself. The aim is two fold: only one theorem-prover is needed; proofs of the metaproperties of the different existing calculi can be avoided by borrowing them from MSL. To make the book accessible to readers from different disciplines, whilst maintaining precision, the author has supplied detailed step-by-step proofs, avoiding difficult arguments, and continually motivating the material with examples. Consequently this can be used as a reference, for self-teaching or for first-year graduate courses.

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Contents

 STANDARD SECOND ORDER LOGIC 1 Standard structures 22 Standard semantics 30 Semantic theorems 62 DEDUCTIVE CALCULI 69 Sequent calculi 75 Soundness theorem in standard semantics 90 Incompleteness in standard structures 94
 Algebraic definition of relational general structures 197 A functional theory of types 205 Equational presentation of the functional theory of finite types 214 MANYSORTED LOGIC 220 Structures 227 Substitution of a term for a variable 236 Reduction to onesorted logic 257 APPLYING MANYSORTED LOGIC 236 263

 CATEGORICITY OF SECOND ORDER PEANO ARITHMETIC 115 Categoricity of Peano axioms 122 Peano models and primitive recursion 129 Induction models 135 Induction models and primitive recursion in induction models 140 Second order frames 154 Algebraic definition of general structures 171 TYPE THEORY 180 A relational theory of finite types 187
 Applying manysorted logic to higher order logic 277 Applying manysorted logic to modal logic 291 Prepositional modal logic as manysorted logic 312 First order modal logic as manysorted logic 327 Applying manysorted logic to dynamic logic 335 Bibliography 352 Index 369 Copyright