Mathematical Logic

Mathematical Logic is a subfield of mathematics that deals with formal systems, proof theory, model theory, set theory, and computability. It provides the foundations for mathematical reasoning and involves the study of the principles of valid reasoning and argument. Mathematical logic uses symbolic notation to represent logical expressions and employs rigorous methods to examine their structures and relationships. It plays a crucial role in various areas such as computer science, philosophy, linguistics, and cognitive science by influencing how mathematical statements are formulated, understood, and manipulated.

1. Formal Systems
   1. Propositional Logic
      1. Syntax
         1. Symbols and Alphabet
            1. Propositional Variables
            2. Logical Connectives
               1. Conjunction
               2. Disjunction
               3. Negation
               4. Implication
               5. Biconditional
         2. Well-formed Formulas
            1. Rules of Formation
            2. Parsing and Bracketing
         3. Expressiveness
            1. Tautologies
            2. Contradictions
            3. Contingencies
      2. Semantics
         1. Truth Tables
            1. Construction of Truth Tables
            2. Evaluation of Logical Statements
         2. Logical Connectives
            1. Truth-functional Connectives
            2. Meaning and Interpretation
            3. Duality Principles
      3. Proof Systems
         1. Natural Deduction
            1. Rules of Inference
               1. Introduction Rules
               2. Elimination Rules
            2. Proof Construction
            3. Examples of Derivations
         2. Sequent Calculus
            1. Structure of Sequents
            2. Inference Rules and Sequents
            3. Cut Rule and Cut Elimination
         3. Resolution
            1. Clause Form Transformation
            2. Unit Resolution
            3. Completeness of Resolution
   2. Predicate Logic
      1. Syntax
         1. Terms and Formulas
            1. Variables and Constants
            2. Function Symbols
            3. Predicate Symbols
         2. Well-formed Formulas
            1. Formation Rules
            2. Bound and Free Variables
      2. Semantics
         1. Interpretations and Models
            1. Domain of Discourse
            2. Assignments of Values
            3. Satisfiability and Validity
      3. Quantifiers
         1. Universal Quantifier
            1. Universal Instantiation
            2. Universal Generalization
         2. Existential Quantifier
            1. Existential Instantiation
            2. Existential Generalization
         3. Scope and Binding
            1. Interaction with Logical Connectives
      4. Proof Techniques
         1. First-order Logic
            1. Soundness and Completeness
            2. Compactness Theorem
            3. Skolemization
         2. Higher-order Logic
            1. Quantification over Functions
            2. Expressiveness and Limitations
            3. Henkin Semantics
   3. Logical Consequence
      1. Definition and Concepts
         1. Formal Consequence
         2. Informal Consequence
      2. Relations Between Sentences
         1. Deductive Systems
         2. Entailment and Derivation
      3. Decision Procedures
         1. Decidability and Undecidability
         2. Gödel’s Completeness Theorem
   4. Formal Languages
      1. Definition and Significance
         1. Alphabet and Vocabulary
         2. Syntax Rules and Grammar
      2. Language Classes
         1. Regular Languages
         2. Context-free Languages
         3. Context-sensitive Languages
      3. Applications of Formal Languages
         1. Parsing Algorithms
         2. Compiler Design
   5. Metalogic
      1. Study of Formal Systems
         1. Syntax versus Semantics
         2. Metatheoretical Concepts
      2. Logical Paradoxes
         1. Semantic Paradoxes
            1. Liar Paradox
         2. Set-theoretical Paradoxes
            1. Russell's Paradox
      3. Properties of Formal Systems
         1. Completeness
         2. Consistency
         3. Soundness
         4. Decidability
   6. Applications of Formal Systems
      1. In Mathematics
         1. Formalization of Mathematical Theorems
         2. Automated Proof Checking
      2. In Computer Science
         1. Logic Programming
         2. Formal Specification and Verification
      3. Interdisciplinary Applications
         1. Contribution to Cognitive Sciences
         2. Influence on Linguistics and Language Processing