In logic, the term decidable refers to the decision problem, the question of the existence of an effective method for determining membership in a set of formulas, or, more precisely, an algorithm that can and will return a Boolean true or false value (instead of looping indefinitely). Logical systems such as propositional logic are decidable if membership in their set of logically valid formulas (or theorems) can be effectively determined. A theory (set of formulas closed under logical consequence) in a fixed logical system is decidable if there is an effective method for determining whether arbitrary formulas are included in the theory. Many important problems are undecidable.

As with the concept of a decidable set, the definition of a decidable theory or logical system can be given either in terms of effective methods or in terms of computable functions. These are generally considered equivalent per Church's thesis. Indeed, the proof that a logical system or theory is undecidable will use the formal definition of computability to show that an appropriate set is not a decidable set, and then invoke Church's thesis to show that the theory or logical system is not decidable by any effective method (Enderton 2001, pp. 206ff.).

Logic (from the Greek λογική logikē) is the philosophical study of valid reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science. It examines general forms that arguments may take, which forms are valid, and which are fallacies. In philosophy, the study of logic is applied in most major areas: metaphysics, ontology, epistemology, and ethics. In mathematics, it is the study of valid inferences within some formal language. Logic is also studied in argumentation theory.

Logic was studied in several ancient civilizations, including India,China, and Greece. In the West, logic was established as a formal discipline by Aristotle, who gave it a fundamental place in philosophy. The study of logic was part of the classical trivium, which also included grammar and rhetoric.

Logic is often divided into three parts, inductive reasoning, abductive reasoning, and deductive reasoning.

The concept of logical form is central to logic, it being held that the validity of an argument is determined by its logical form, not by its content. Traditional Aristotelian syllogistic logic and modern symbolic logic are examples of formal logics.

Barry Charles Mazur (born December 19, 1937) is an influential mathematician and professor at Harvard.

Born in New York City, Mazur attended the Bronx High School of Science and MIT, although he did not graduate from the latter on account of failing a then-present ROTC requirement. Regardless, he was accepted for graduate school and received his Ph.D. from Princeton University in 1959, becoming a Junior Fellow at Harvard from 1961 to 1964. He is currently the Gerhard Gade University Professor and a Senior Fellow at Harvard. In 1982 he was elected a member of the National Academy of Sciences. Mazur has received the Veblen Prize in geometry, the Cole Prize in number theory, the Chauvenet Prize for exposition, and the Steele Prize for seminal contribution to research from the American Mathematical Society.

His early work was in geometric topology. In a clever, elementary fashion, he proved the generalized Schoenflies conjecture (his complete proof required an additional result by Marston Morse), around the same time as Morton Brown. Both Brown and Mazur received the Veblen Prize for this achievement. He also discovered the Mazur manifold and the Mazur swindle.