Non-classical logics (and sometimes alternative logics) are formal systems that differ in a significant way from standard logical systems such as propositional and predicate logic. There are several ways in which this is done, including by way of extensions, deviations, and variations. The aim of these departures is to make it possible to construct different models of logical consequence and logical truth.
Philosophical logic is understood to encompass and focus on non-classical logics, although the term has other meanings as well. In addition, some parts of theoretical computer science can be thought of as using non-classical reasoning, although this varies according to the subject area. For example, the basic boolean functions (e.g. AND, OR, NOT, etc) in computer science are very much classical in nature, as is clearly the case given that they can be fully described by classical truth tables. However, in contrast, some computerized proof methods may not use classical logic in the reasoning process.
There are many kinds of non-classical logic, which include:
Computability logic is a semantically constructed formal theory of computability—as opposed to classical logic, which is a formal theory of truth—that integrates and extends classical, linear and intuitionistic logics.
Dynamic semantics interprets formulas as update functions, opening the door to a variety of nonclassical behaviours
Many-valued logic rejects bivalence, allowing for truth values other than true and false. The most popular forms are three-valued logic, as initially developed by Jan Łukasiewicz, and infinitely-valued logics such as fuzzy logic, which permit any real number between 0 and 1 as a truth value.
Intuitionistic logic rejects the law of the excluded middle, double negation elimination, and part of De Morgan's laws;
Linear logic rejects idempotency of entailment as well;
Modal logic extends classical logic with non-truth-functional ("modal") operators.
Paraconsistent logic (e.g.
