Extensions of First Order Logic

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Cambridge University Press, 1996 - History - 388 pages
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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

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