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Mathematical Logic A Course with Exercises Propositional Predicate calculus by Rene Cori and Daniel Lascar



Mathematical Logic A Course with Exercises Propositional Predicate calculus, Part I: Propositional calculus, Boolean algebras, Predicate calculus written by Rene Cori and Daniel LascarEquipe de Logique Mathematique, Universite Paris VII, Translated by Donald H. Pelletier, York University, Toronto.

Mathematical Logic A Course with Exercises Propositional Predicate calculus, Part I: Propositional calculus, Boolean algebras, Predicate calculus written by Rene Cori and Daniel Lascar cover the following topics.

  • 1 Propositional calculus
    1.1 Syntax
    1.1.1 Propositional formulas
    1.I .2 Proofs by induction on the set of formulas
    I.1.3 The decomposition tree of a formula
    1.I .4 The unique decomposition theorem
    1.1.5 Definitions by induction on the set of formulas
    1.1.6 Substitutions in a propositional formula
    1.2 Semantics
    1.2.1 Assignments of truth values and truth tables
    1.2.2 Tautologies and logically equivalent formulas
    1.2.3 Some tautologies
    1.3 Normal forms and complete sets of connectives
    1.3.1 Operations on {O, I} and formulas
    1.3.2 Normal forms
    1.3.3 Complete sets of connectives
    1.4 The interpolation lemma
    1.4.1 Interpolation lemma
    1.4.2 The definability theorem
    1.5 The compactness theorem
    1.5.1 Satisfaction of a set of formulas
    1.5.2 The compactness theorem for propositional calculus
    1.6 Exercises for Chapter I

  • 2 Boolean algebras
    2.1 Algebra and topology review
    2.1.1 Algebra
    2.1.2 Topology
    2.1.3 An application to propositional calculus
    2.2 Definition of Boolean algebra
    2.2.1 Properties of Boolean algebras, order relations
    2.2.2 Boolean algebras as ordered sets
    2.3 Atoms in a Boolean algebra
    2.4 Homomorphisms, isomorphisms, subalgebras
    2.4.1 Homomorphisms and isomorphisms
    2.4.2 Boolean subalgebras
    2.5 Ideals and filters
    2.5.1 Properties of ideals
    2.5.2 Maximal ideals
    2.5.3 Filters
    2.5.4 Ultrafilters
    2.5.5 Filterbases
    2.6 Stone's theorem
    2.6.1 The Stone space of a Boolean algebra
    2.6.2 Stone's theorem
    2.6.3 Boolean spaces are Stone spaces
    2.7 Exercises for Chapter 2

  • 3 Predicate calculus
    3.1 Syntax
    3.1.1 First order languages
    3.1.2 Terms of the language
    3.1.3 Substitutions in terms
    3.1.4 Formulas of the language
    3.1.5 Free variables, bound variables, and closed formulas
    3.1.6 Substitutions in formulas
    3.2 Structures
    3.2.1 Models of a language
    3.2.2 Substructures and restrictions
    3.2.3 Homomorphisms and isomorphisms
    3.3 Satisfaction of formulas in structures
    3.3.1 Interpretation in a structure of the terms
    3.3.2 Satisfaction of the formulas in a structure
    3.4 Universal equivalence and semantic consequence
    3.5 Prenex forms and Skolem forms
    3.5.1 Prenex forms
    3.5.2 Skolem forms
    3.6 First steps in model theory
    3.6.1 Satisfaction in a substructure
    3.6.2 Elementary equivalence
    3.6.3 The language associated with a structure and formulas with parameters
    3.6.4 Functions and relations definable in a structure
    3.7 Models that may not respect equality
    3.8 Exercises for Chapter 3

  • 4 The completeness theorems
    4.1 Formal proofs (or derivations)
    4.1.1 Axioms and rules
    4.1.2 Formal proofs
    4.1.3 The finiteness theorem and the deduction theorem
    4.2 Henkin models
    4.2.1 Henkin witnesses
    4.2.2 The completeness theorem
    4.3 Herbrand's method
    4.3.1 Some examples
    4.3.2 The avatars of a formula
    4.4 Proofs using cuts
    4.4.1 The cut rule
    4.4.2 Completeness of the method
    4.5 The method of resolution
    4.5.1 Unification
    4.5.2 Proofs by resolution
    4.6 Exercises for Chapter 4

  • Solutions to the exercises of Part I

  • Bibliography

  • Index

  • Start Now
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