Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well. They are characterised by a number of properties:
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.
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.
^Gabbay, Dov, (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.