Mathematical logic

Mathematical logic studies formal languages, proofs, models, and computability using mathematical methods. It investigates the structure and limits of formal reasoning and mathematical foundations.

Formal language: A language defined by a specified alphabet and rules for forming well-formed expressions. Formal languages let logic represent mathematical statements with precise syntax.
First-order logic: A formal logic with variables, predicates, quantifiers, and rules for reasoning about objects. It is a central framework for expressing mathematical theories and proving statements about structures.
Gottlob Frege: A German philosopher and logician who developed a formal notation for quantified logic and arithmetic. His Begriffsschrift helped establish modern formal logic.
Classical logic: A family of logics that includes the law of excluded middle and classical rules of negation. It supplies the standard background logic for much of mathematical formalization.
Automated theorem proving: The use of computer programs to search for or construct formal proofs. It applies formal proof systems to make mathematical reasoning partly mechanizable.
Axiom: A sentence adopted as a starting assumption within a formal theory. Axioms supply the premises from which formal proofs derive theorems.
Natural deduction: A proof system whose rules model the introduction and elimination of logical connectives and quantifiers. Its rules make the structure of formal reasoning explicit.
Bertrand Russell: A British philosopher and logician whose work with Alfred North Whitehead developed logicism in Principia Mathematica. His paradox and foundational program shaped early twentieth-century debates about logic and mathematics.
Intuitionistic logic: A logic in which a proof of a statement's existence or disjunction must provide constructive evidence. It rejects unrestricted use of classical principles that intuitionists regard as nonconstructive.
SAT solving: The computational task of determining whether a propositional formula has a satisfying assignment. It turns logical satisfiability into a practical method used in verification and optimization.

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