# EXTENSIONS OF FIRST ORDER LOGIC

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1 EXTENSIONS OF FIRST ORDER LOGIC Maria Manzano University of Barcelona CAMBRIDGE UNIVERSITY PRESS

2 Table of contents PREFACE xv CHAPTER I: STANDARD SECOND ORDER LOGIC Introduction General idea Expressive power Model-theoretic counterparts of expressiveness Incompleteness Second order grammar Definition (signature and alphabet) Expressions: terms, predicates and formulas Remarks on notation Induction Free and bound variables. > Substitution Standard structures Definition of standard structures Relations between standard structures established without the formal language Standard semantics Assignment Interpretation Consequence and validity. 33

3 Vlll 6.- Induction models and primitive recursion in induction models Addition and multiplication in induction models Exponential operation on induction models Universal operations. 144 CHAPTER IV: FRAMES AND GENERAL STRUCTURES Introduction Frames and general structures Standard/nonstandard view The concept of subset Summary Second order frames Definition of frames Semantics on frames Soundness and completeness in frames Undefinability of identity in frames Frames and lambdas Definable sets and relations in a given frame General structures Definition of general structures Semantics based on general structures General structures and lambdas Soundness and completeness in general structures Algebraic definition of general structures Fundamental relations of a structure Algebraic definition of general structures Logics obtained by weakening the schema of comprehension Weak second order logic General idea. 174

4 IX 6.2. Metaproperties of weak second order logic. 175 CHAPTER V: TYPE THEORY Introduction General idea Paradoxes and their solution in type theory Three presentations of type theory A relational theory of finite types Definition (signature and alphabet) Expressions Equality Free variables and substitution Deductive calculus The relational standard structure and the relational standard hierarchy of types RTT with lambda Incompleteness of standard type theory Relational general structures and relational frames. 193 r 3.- Algebraic definition of relational general structures Fundamental relations Definition of relational general structure by algebraic closure of the domains Theorem Some parametrically definable relations also included ' in the universe of relational general structures defined by algebraic closure Theorem A functional theory of types Definition (signature and alphabet) Expressions. 206

5 4.3. Functional frames, functional general structures and functional standard structures From RTT to FTT Equational presentation of the functional theory of finite types Main features of ETT Connectors and quantifiers in ETT The selector operator in ETT A calculus for ETT. 218 CHAPTER VI: MANY-SORTED LOGIC Introduction Examples Reduction to and comparison with first order logic Uses of many-sorted logic Many-sorted logic as a unifier logic Structures Definition (signature) Definition (structure) Formal many-sorted language Alphabet Expressions: formulas and terms Remarks on notation Abbreviations Induction Free and bound variables Semantics Definitions Satisfiability, validity, consequence and logical equivalence. 236

6 XI 5.- Substitution of a term for a variable Semantic theorems Coincidence lemma Substitution lemma Equals substitution lemma Isomorphism theorem The completeness of many-sorted logic Deductive calculus Syntactic notions Soundness Completeness theorem (countable language) Compactness theorem Lowenheim-Skolem theorem Reduction to one-sorted logic The syntactical translation (relativization of quantifiers) Conversion of structures. 258 CHAPTER VH: APPLYING MANY-SORTED LOGIC General plan Aims Representation theorem Main theorem Testing a given calculus for XL. 272 Applying many-sorted logic to higher order logic. 2.- Higher order logic as many-sorted logic Preliminaries The formal many-sorted language MSL The syntactical translation Structures. 283

7 Xll 2.4. The equivalence SOL-MSL D 288 Applying many-sorted logic to modal logic. 3.- Modal logic Some history A formal language for PML Modal propositional logics Normal modal logics Consistency of all normal modal logics contained in S Kripke models A formal language for FOML Semantics A deductive calculus for FOML(S5) Propositional modal logic as many-sorted logic The formal many-sorted language MSL" Translating function General structures and frames built on PM-structures The MODO theory Reverse conversion Testing the calculus First order modal logic as many-sorted logic The formal many-sorted language MSL Translating function Theorems on semantic equivalence FOML-MSL Metaproperties of FOML-S5: compactness, and Loweheim-Skolem Soundness and completeness of S5. 333

8 Xlll Applying many-sorted logic to dynamic logic. 6.- Dynamic logic General idea A formal language for PDL Semantics The logic PDL Propositional dynamic logic as many-sorted logic The formal many-sorted language MSL Translating function Structures and frames built on PD-structures The SOLO 2 theory. 347 Bibliography 352 List of notation 364 Index 369