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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:
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.
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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.
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Non-classical logics
Computability logic is a semantically constructed formal theory of computability, as opposed to classical logic, which is a formal
Fuzzy logic 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;
theory of truth; integrates and extends classical, linear and intuitionistic logics.
In Deviant Logic, Fuzzy Logic: Beyond the Formalism, Susan Haack divided non-classical logics into deviant, quasi-deviant, and extended logics.
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