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Classical logic (or standard logic) is an intensively studied and widely used class of formal logics. Each logical system in this class shares characteristic properties:
While not entailed by the preceding conditions, contemporary discussions of classical logic normally only include propositional and first-order logics. In other words, the overwhelming majority of time spent studying classical logic has been spent studying specifically propositional and first-order logic, as opposed to the other more obscure variations of classical logic.
Most semantics of classical logic are bivalent, meaning all of the possible denotations of propositions can be categorised as either true or false.
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.
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.