Mathematical Logic

  1. Propositional Logic
    1. Syntax of Propositional Logic
      1. Alphabet and Symbols
        1. Propositional Variables
          1. Logical Connectives
            1. Auxiliary Symbols
            2. Atomic Propositions
              1. Logical Connectives
                1. Negation
                  1. Conjunction
                    1. Disjunction
                      1. Implication
                        1. Biconditional
                        2. Formation Rules for Well-Formed Formulas
                          1. Parentheses and Precedence Rules
                            1. Parsing Trees for Formulas
                              1. Construction of Parse Trees
                                1. Use in Analyzing Formula Structure
                                2. Unique Readability
                                3. Semantics of Propositional Logic
                                  1. Truth Values and Valuations
                                    1. Assignment of Truth Values
                                      1. Valuation Functions
                                        1. Truth Conditions for Connectives
                                        2. Truth Tables
                                          1. Construction of Truth Tables
                                            1. Evaluating Complex Formulas
                                              1. Truth Table Method for Testing Validity
                                              2. Logical Equivalence
                                                1. Definition and Examples
                                                  1. Laws of Logical Equivalence
                                                    1. Commutativity
                                                      1. Associativity
                                                        1. Distributivity
                                                          1. De Morgan's Laws
                                                            1. Double Negation
                                                              1. Conditional Equivalences
                                                            2. Tautologies, Contradictions, and Contingencies
                                                              1. Identifying Tautologies
                                                                1. Identifying Contradictions
                                                                  1. Identifying Contingent Formulas
                                                                  2. Satisfiability and Validity
                                                                    1. Satisfiable Formulas
                                                                      1. Valid Formulas
                                                                        1. Unsatisfiable Formulas
                                                                        2. Logical Consequence
                                                                          1. Definition of Semantic Entailment
                                                                            1. Use in Argument Analysis
                                                                              1. Relationship to Validity
                                                                            2. Normal Forms
                                                                              1. Disjunctive Normal Form
                                                                                1. Conjunctive Normal Form
                                                                                  1. Conversion Algorithms
                                                                                  2. Formal Proof Systems
                                                                                    1. Axiomatic Systems
                                                                                      1. Structure of Axiomatic Systems
                                                                                        1. Axiom Schemas
                                                                                          1. Rule of Inference
                                                                                            1. Formal Proofs
                                                                                              1. Example Proofs
                                                                                                1. Derived Rules
                                                                                                2. Natural Deduction
                                                                                                  1. Introduction and Elimination Rules
                                                                                                    1. Conjunction Rules
                                                                                                      1. Disjunction Rules
                                                                                                        1. Implication Rules
                                                                                                          1. Negation Rules
                                                                                                            1. Biconditional Rules
                                                                                                            2. Assumptions and Discharge
                                                                                                              1. Proof Trees
                                                                                                                1. Proof Strategies
                                                                                                                2. Sequent Calculus
                                                                                                                  1. Structure of Sequents
                                                                                                                    1. Structural Rules
                                                                                                                      1. Weakening
                                                                                                                        1. Contraction
                                                                                                                          1. Exchange
                                                                                                                          2. Logical Rules
                                                                                                                            1. Left Rules
                                                                                                                              1. Right Rules
                                                                                                                              2. Cut Rule
                                                                                                                            2. Metatheory of Propositional Logic
                                                                                                                              1. The Deduction Theorem
                                                                                                                                1. Statement and Proof
                                                                                                                                2. Soundness Theorem
                                                                                                                                  1. Statement and Proof Outline
                                                                                                                                    1. Implications for Proof Systems
                                                                                                                                    2. Completeness Theorem
                                                                                                                                      1. Statement and Proof Outline
                                                                                                                                        1. Implications for Logic
                                                                                                                                        2. Compactness Theorem
                                                                                                                                          1. Statement and Consequences
                                                                                                                                          2. Decidability of Propositional Logic
                                                                                                                                            1. Decision Procedures
                                                                                                                                              1. Complexity Considerations
                                                                                                                                                1. SAT Problem

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                                                                                                                                            3. First-Order Logic