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;
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.

Examples of classical logics

• Aristotle's Organon introduces his theory of syllogistic, which is a logic with a restricted form of judgements: 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 judgements 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 noncontradiction in justifying his system, although these laws cannot be expressed as judgements within the syllogistic framework.

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

• Gottlob Frege's Begriffsschrift.

• Clarence Irving Lewis's systems S1-S5 of alethic modal logic.

Non-classical logics

• Intuitionistic logic rejects the law of the excluded middle and De Morgan's laws;
• Dialetheism and relevance logic reject the law of noncontradiction;
• Relevance logic and linear logic reject monotonicity of entailment;
• Linear logic rejects idempotency of entailment;
• 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;
• Modal logics extend classical logic with non-truth-functional ("modal") operators;
• Paraconsistent logics do not have the property that contradiction entails everything;
• Non-monotonic logics are used to formalize dynamical or defeasible reasoning, in which the addition of new facts can reduce the set of consequences.

References

• Dov Gabbay, (1994). 'Classical vs non-classical logic'. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, (Eds), Handbook of Logic in Artificial Intelligence and Logic Programming, volume 2, chapter 2.6. Oxford University Press.