03Bxx General logic
This subtopic studies core logical systems, including propositional and first-order logic, together with modal, substructural, many-valued, and computationally oriented variants. It is used to formalize arguments, test validity, study completeness and decidability, and analyze expressiveness across logical languages. Applications include theorem proving, formal verification, programming language semantics, knowledge representation, and the rigorous specification of reasoning processes.
Specific topics
03B05 Classical propositional logic
Overview
This topic studies classical propositional logic, including truth-functional connectives, tautologies, satisfiability, and formal proof systems. It is the simplest fully worked logical setting and a natural entry point for logical semantics.
Related Wikipedia Page
Propositional calculus (Wikipedia)
Useful Links
Key Ideas
- Truth tables and truth-functional connectives
- Tautology, contradiction, entailment, and satisfiability
- Soundness and completeness for propositional proof systems
Typical Uses
Used to introduce formal semantics, study basic validity, and provide the propositional core of more advanced logical systems.
Applications
- Circuit reasoning and Boolean algebra
- Foundational training for proof systems
- Entry point for automated reasoning
References
Recommended Textbooks
03B10 Classical first-order logic
Overview
This topic studies classical first-order logic, including quantification, interpretation, satisfiability, and the semantics of structures. It is the central logical framework used across mathematics.
Related Wikipedia Page
First-order logic (Wikipedia)
Useful Links
Key Ideas
- Syntax of terms, formulas, and quantifiers
- Structures, interpretations, and truth in a model
- Compactness, completeness, and Löwenheim-Skolem phenomena
Typical Uses
Used to formalize mathematical theories, interpret axioms in structures, and provide the backbone for model theory and proof theory.
Applications
- Foundations of mathematics
- Axiomatization of algebra, geometry, and analysis
- Specification of formal systems in computer science
References
Recommended Textbooks
03B20 Subsystems of classical logic
Overview
This specific topic studies subsystems of classical logic within general logic. It focuses on core definitions, canonical constructions, and key results used to analyze structures and prove classification or existence statements in mathematical logic.
Related Wikipedia Page
Logic (Wikipedia)
Useful Links
Key Ideas
- Formal definitions and equivalent formulations for subsystems of classical logic
- Representative theorems, extremal bounds, or invariants
- Connections to adjacent methods in general logic
Typical Uses
Used to structure proofs, compare examples and counterexamples, and select appropriate tools when working on problems in general logic.
Applications
- Theoretical research and classification questions
- Algorithmic or computational formulations where relevant
- Cross-links to neighboring areas in pure and applied mathematics
References
Recommended Textbooks
03B25 Decidability of theories
Overview
This specific topic studies decidability of theories within general logic. It focuses on core definitions, canonical constructions, and key results used to analyze structures and prove classification or existence statements in mathematical logic.
Related Wikipedia Page
Logic (Wikipedia)
Useful Links
Key Ideas
- Formal definitions and equivalent formulations for decidability of theories
- Representative theorems, extremal bounds, or invariants
- Connections to adjacent methods in general logic
Typical Uses
Used to structure proofs, compare examples and counterexamples, and select appropriate tools when working on problems in general logic.
Applications
- Theoretical research and classification questions
- Algorithmic or computational formulations where relevant
- Cross-links to neighboring areas in pure and applied mathematics
References
Recommended Textbooks
03B35 Mechanization of proofs and logical operations
Overview
This specific topic studies mechanization of proofs and logical operations within general logic. It focuses on core definitions, canonical constructions, and key results used to analyze structures and prove classification or existence statements in mathematical logic.
Related Wikipedia Page
Logic (Wikipedia)
Useful Links
Key Ideas
- Formal definitions and equivalent formulations for mechanization of proofs and logical operations
- Representative theorems, extremal bounds, or invariants
- Connections to adjacent methods in general logic
Typical Uses
Used to structure proofs, compare examples and counterexamples, and select appropriate tools when working on problems in general logic.
Applications
- Theoretical research and classification questions
- Algorithmic or computational formulations where relevant
- Cross-links to neighboring areas in pure and applied mathematics
References
Recommended Textbooks
03B45 Modal logic
Overview
This specific topic studies modal logic within general logic. It focuses on core definitions, canonical constructions, and key results used to analyze structures and prove classification or existence statements in mathematical logic.
Related Wikipedia Page
Logic (Wikipedia)
Useful Links
Key Ideas
- Formal definitions and equivalent formulations for modal logic
- Representative theorems, extremal bounds, or invariants
- Connections to adjacent methods in general logic
Typical Uses
Used to structure proofs, compare examples and counterexamples, and select appropriate tools when working on problems in general logic.
Applications
- Theoretical research and classification questions
- Algorithmic or computational formulations where relevant
- Cross-links to neighboring areas in pure and applied mathematics
References
Recommended Textbooks
03B47 Substructural logics
Overview
This specific topic studies substructural logics within general logic. It focuses on core definitions, canonical constructions, and key results used to analyze structures and prove classification or existence statements in mathematical logic.
Related Wikipedia Page
Logic (Wikipedia)
Useful Links
Key Ideas
- Formal definitions and equivalent formulations for substructural logics
- Representative theorems, extremal bounds, or invariants
- Connections to adjacent methods in general logic
Typical Uses
Used to structure proofs, compare examples and counterexamples, and select appropriate tools when working on problems in general logic.
Applications
- Theoretical research and classification questions
- Algorithmic or computational formulations where relevant
- Cross-links to neighboring areas in pure and applied mathematics
References
Recommended Textbooks
03B50 Many-valued logic
Overview
This specific topic studies many-valued logic within general logic. It focuses on core definitions, canonical constructions, and key results used to analyze structures and prove classification or existence statements in mathematical logic.
Related Wikipedia Page
Logic (Wikipedia)
Useful Links
Key Ideas
- Formal definitions and equivalent formulations for many-valued logic
- Representative theorems, extremal bounds, or invariants
- Connections to adjacent methods in general logic
Typical Uses
Used to structure proofs, compare examples and counterexamples, and select appropriate tools when working on problems in general logic.
Applications
- Theoretical research and classification questions
- Algorithmic or computational formulations where relevant
- Cross-links to neighboring areas in pure and applied mathematics
References
Recommended Textbooks
03B52 Fuzzy logic
Overview
This specific topic studies fuzzy logic within general logic. It focuses on core definitions, canonical constructions, and key results used to analyze structures and prove classification or existence statements in mathematical logic.
Related Wikipedia Page
Logic (Wikipedia)
Useful Links
Key Ideas
- Formal definitions and equivalent formulations for fuzzy logic
- Representative theorems, extremal bounds, or invariants
- Connections to adjacent methods in general logic
Typical Uses
Used to structure proofs, compare examples and counterexamples, and select appropriate tools when working on problems in general logic.
Applications
- Theoretical research and classification questions
- Algorithmic or computational formulations where relevant
- Cross-links to neighboring areas in pure and applied mathematics
References
Recommended Textbooks
03B70 Logic in computer science
Overview
This specific topic studies logic in computer science within general logic. It focuses on core definitions, canonical constructions, and key results used to analyze structures and prove classification or existence statements in mathematical logic.
Related Wikipedia Page
Logic (Wikipedia)
Useful Links
Key Ideas
- Formal definitions and equivalent formulations for logic in computer science
- Representative theorems, extremal bounds, or invariants
- Connections to adjacent methods in general logic
Typical Uses
Used to structure proofs, compare examples and counterexamples, and select appropriate tools when working on problems in general logic.
Applications
- Theoretical research and classification questions
- Algorithmic or computational formulations where relevant
- Cross-links to neighboring areas in pure and applied mathematics
References
Recommended Textbooks