Classical logic

Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. They are characterised by a number of properties; non-classical logics are those that lack one or more of these properties, which are:

1. Law of the excluded middle and Double negative elimination;
2. Law of noncontradiction;
3. Monotonicity of entailment and Idempotency of entailment;
4. Commutativity of conjunction;
5. De Morgan duality: every logical operator is dual to another.

Classical logic is bivalent, i.e. it uses only Boolean-valued functions. And while not entailed by the preceding conditions, contemporary discussions of classical logic normally only include propositional and first-order logics.

Examples of classical logics

• Aristotle's Organon introduces his theory of syllogisms, which is a logic with a restricted form of judgments: assertions take one of four forms, All Ps are Q, Some Ps are Q, No Ps are Q, and Some Ps are not Q. These judgments find themselves if two pairs of two dual operators, and each operator is the negation of another, relationships that Aristotle summarised with his square of oppositions. Aristotle explicitly formulated the law of the excluded middle and law of non-contradiction in justifying his system, although these laws cannot be expressed as judgments within the syllogistic framework.

• George Boole's algebraic reformulation of logic, his system of Boolean logic;

• The first-order logic found in Gottlob Frege's Begriffsschrift.

Non-classical logics

• Computability logic is a semantically constructed formal theory of computability, as opposed to classical logic, which is a formal theory of truth; integrates and extends classical, linear and intuitionistic logics.
• Fuzzy logic rejects the law of the excluded middle and allows as a truth value any real number between 0 and 1.
• Intuitionistic logic rejects the law of the excluded middle, double negative elimination, and the 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., dialetheism and relevance logic) rejects the law of noncontradiction;
• Relevance logic, linear logic, and non-monotonic logic reject monotonicity of entailment;

In Deviant Logic, Fuzzy Logic: Beyond the Formalism, Susan Haack divided non-classical logics into deviant, quasi-deviant, and extended logics.