logic

Logic is the branch of philosophy and mathematics that studies the principles of valid reasoning and inference. It seeks to distinguish good from poor arguments by analyzing the forms of reasoning rather than the content of the statements. An argument is valid if the conclusion follows from the premises; it is sound if it is valid and its premises are true.

Formal logic studies the structure of arguments using formal languages. The core domains include propositional logic, which analyzes propositions and their connectives (and, or, not, implies); and predicate logic, which adds quantifiers and variables to express general statements about objects. Beyond classical logic, modal logic, temporal logic, and other non-classical systems study necessity, possibility, time, and other modalities.

Origins in Aristotle's syllogistic, later Boolean algebra by Boole, Frege's predicate calculus, and Russell and Whitehead's Principia Mathematica laid the groundwork for modern logic; Gödel's completeness and incompleteness theorems, model theory (Tarski), proof theory, and computation (Church–Turing) further shaped the field.

Semantics link formal expressions to meanings via models; syntax concerns formal rules of inference. Important properties include validity, soundness, consistency, completeness, and decidability. Truth-functional semantics for propositional logic and model-theoretic semantics for predicate logic are standard frameworks.

Logic underpins mathematics, computer science, formal verification, programming languages, artificial intelligence, linguistics, and philosophy.