Classical logic

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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[1]; non-classical logics are those that lack one or more of these properties, which are:

  1. Law of the excluded middle and Double negative elimination;
  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.

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.[2][3]

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.

Non-classical logics

In Deviant Logic, Fuzzy Logic: Beyond the Formalism, Susan Haack divided non-classical logics into deviant, quasi-deviant, and extended logics.[3]

References

  1. 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.
  2. Shapiro, Stewart (2000). Classical Logic. In Stanford Encyclopedia of Philosophy [Web]. Stanford: The Metaphysics Research Lab. Retrieved October 28, 2006, from http://plato.stanford.edu/entries/logic-classical/
  3. 3.0 3.1 Haack, Susan, (1996). Deviant Logic, Fuzzy Logic: Beyond the Formalism. Chicago: The University of Chicago Press.

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