Logic for Computer Science

  1. First-Order Logic
    1. Motivation and Limitations of Propositional Logic
      1. Expressiveness Limitations
        1. Need for Quantification
          1. Need for Relations and Functions
            1. Examples Requiring First-Order Logic
            2. Syntax of First-Order Logic
              1. Vocabulary
                1. Constants
                  1. Individual Constants
                    1. Naming Objects
                    2. Variables
                      1. Individual Variables
                        1. Role in Quantification
                        2. Function Symbols
                          1. Function Symbols and Arity
                            1. Function Terms
                            2. Predicate Symbols
                              1. Predicate Symbols and Arity
                                1. Relations
                              2. Terms
                                1. Simple Terms
                                  1. Complex Terms
                                    1. Ground Terms vs Non-Ground Terms
                                    2. Atomic Formulas
                                      1. Predicate Applications
                                        1. Equality
                                        2. Complex Formulas
                                          1. Boolean Combinations
                                            1. Quantified Formulas
                                            2. Quantifiers
                                              1. Universal Quantifier
                                                1. Syntax and Meaning
                                                  1. Scope of Quantification
                                                  2. Existential Quantifier
                                                    1. Syntax and Meaning
                                                      1. Scope of Quantification
                                                    2. Variable Binding
                                                      1. Free Variables
                                                        1. Bound Variables
                                                          1. Scope of Quantifiers
                                                            1. Variable Capture
                                                            2. Well-Formed Formulas in FOL
                                                              1. Formation Rules
                                                                1. Proper Quantifier Usage
                                                              2. Semantics of First-Order Logic
                                                                1. Structures and Interpretations
                                                                  1. Definition of a Structure
                                                                    1. Domain of Discourse
                                                                      1. Interpretation Function
                                                                      2. Variable Assignments
                                                                        1. Assigning Values to Variables
                                                                          1. Modified Assignments
                                                                          2. Truth in a Structure
                                                                            1. Satisfaction Relation
                                                                              1. Truth for Atomic Formulas
                                                                                1. Truth for Complex Formulas
                                                                                  1. Truth for Quantified Formulas
                                                                                  2. Semantic Concepts
                                                                                    1. Satisfiability
                                                                                      1. Validity
                                                                                        1. Logical Consequence
                                                                                          1. Logical Equivalence
                                                                                          2. Substitution
                                                                                            1. Substitution of Terms for Variables
                                                                                              1. Substitution Instances
                                                                                                1. Proper Substitution
                                                                                              2. Proof Systems for First-Order Logic
                                                                                                1. Natural Deduction for FOL
                                                                                                  1. Quantifier Rules
                                                                                                    1. Universal Introduction
                                                                                                      1. Universal Elimination
                                                                                                        1. Existential Introduction
                                                                                                          1. Existential Elimination
                                                                                                          2. Restrictions on Quantifier Rules
                                                                                                            1. Proof Strategies with Quantifiers
                                                                                                            2. Sequent Calculus for FOL
                                                                                                              1. Quantifier Rules in Sequent Calculus
                                                                                                                1. Eigenvariable Conditions
                                                                                                                2. Resolution for FOL
                                                                                                                  1. Skolemization
                                                                                                                    1. Removing Existential Quantifiers
                                                                                                                      1. Skolem Functions
                                                                                                                        1. Skolem Normal Form
                                                                                                                        2. Herbrand Universe
                                                                                                                          1. Definition and Construction
                                                                                                                            1. Herbrand Base
                                                                                                                              1. Herbrand Interpretations
                                                                                                                              2. Unification
                                                                                                                                1. Unification Problem
                                                                                                                                  1. Most General Unifier
                                                                                                                                    1. Unification Algorithm
                                                                                                                                    2. Resolution Rule for FOL
                                                                                                                                      1. First-Order Resolution
                                                                                                                                        1. Factoring
                                                                                                                                        2. Completeness of Resolution
                                                                                                                                      2. Decidability and Undecidability
                                                                                                                                        1. Decidable Fragments
                                                                                                                                          1. Monadic First-Order Logic
                                                                                                                                            1. Two-Variable Logic
                                                                                                                                            2. Undecidability Results
                                                                                                                                              1. Church's Theorem
                                                                                                                                                1. Undecidability of FOL
                                                                                                                                                2. Semi-Decidability
                                                                                                                                                  1. Enumeration of Valid Formulas
                                                                                                                                                    1. Practical Implications

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                                                                                                                                                3. Proof Systems for Propositional Logic

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                                                                                                                                                5. Logic and Computation