Mathematical Logic

Mathematical Logic explores the application of formal logic to mathematics. It covers various sub-disciplines that intersect with both philosophical logic and the foundations of mathematics.

Sub-categories:

· Propositional Logic · Predicate Logic · Computability Theory · Model Theory · Proof Theory · Set Theory · Modal Logic · Intuitionistic Logic · Temporal Logic · Program Semantics · Non-classical Logic · Descriptive Complexity · Mathematical Induction · Reverse Mathematics · Higher-order Logic · Categorical Logic

Sub-categories:

Propositional Logic

Propositional Logic, also known as sentential logic, deals with logical relationships between propositions devoid of quantifiers or predicates.

Predicate Logic

Predicate Logic extends propositional logic by including quantified variables that can represent objects or entities and allows for the expression of predicates.

Computability Theory

Computability Theory, or recursion theory, studies the capabilities and limitations of algorithms and whether problems can be solved by computers.

Model Theory

Model Theory examines the relationships between formal languages and their interpretations, or models, and applies abstract algebra to logical frameworks.

Proof Theory

Proof Theory analyzes the structure of mathematical proofs, exploring the nature of formal systems and proving consistency.

Set Theory

Set Theory deals with the study of sets, collections of objects, and is the foundational theory for much of modern mathematics.

Modal Logic

Modal Logic investigates logical systems that introduce modality, including necessity and possibility, and is widely applicable in philosophy, linguistics, and computer science.

Intuitionistic Logic

Intuitionistic Logic focuses on constructive reasoning, favoring proofs that can be constructed explicitly, and rejects the Law of the Excluded Middle as a general principle.

Temporal Logic

Temporal Logic encompasses logical systems that account for the temporal aspects of truth, often used to verify computer or systems properties over time.

Program Semantics

Program Semantics involves the rigorous mathematical study of programming languages and their semantics, ensuring correct application interpretations.

Non-classical Logic

Non-classical Logic includes alternative logical systems like fuzzy logic, which admit more truth values than the standard true-or-false dichotomy.

Descriptive Complexity

Descriptive Complexity characterizes computational problems by the complexity of the logic needed to express them, linking logic and computational complexity theory.

Mathematical Induction

Mathematical Induction is a proof technique that establishes the validity of a property for all natural numbers by first proving a base case and an inductive step.

Reverse Mathematics

Reverse Mathematics seeks to determine which axioms are needed to prove theorems of interest within a logical foundation.

Higher-order Logic

Higher-order Logic extends predicate logic by allowing quantification over predicates and sets, accommodating a broader range of formalized truths.

Categorical Logic

Categorical Logic employs category theory to provide a structural view of logic and mathematics, focusing on the relationships between mathematical structures.