Wednesday, February 6, 2008

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

Law of the excluded middle and Double negative elimination;
Law of noncontradiction;
Monotonicity of entailment and Idempotency of entailment;
Commutativity of conjunction;
De Morgan duality: every logical operator is dual to another. Non-classical logic 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.