Classical logic

Classical logic (or standard logic[1][2]) is an intensively studied and widely used class of formal logics. Each logical system in this class shares characteristic properties:[3]

  1. Law of excluded middle and double negative elimination
  2. Law of noncontradiction, and the principle of explosion
  3. Monotonicity of entailment and idempotency of entailment
  4. Commutativity of conjunction
  5. De Morgan duality: every logical operator is dual to another

While not entailed by the preceding conditions, contemporary discussions of classical logic normally only include propositional and first-order logics.[4][5]

Classical logic was originally devised as a two-level (bivalent) logical system, with simple semantics for the levels representing "true" and "false".

Examples of classical logics

Generalized semantics

With the advent of algebraic logic it became apparent however that classical propositional calculus admits other semantics. In Boolean-valued semantics (for classical propositional logic), the truth values are the elements of an arbitrary Boolean algebra; "true" corresponds to the maximal element of the algebra, and "false" corresponds to the minimal element. Intermediate elements of the algebra correspond to truth values other than "true" and "false". The principle of bivalence holds only when the Boolean algebra is taken to be the two-element algebra, which has no intermediate elements.

Non-classical logics

Main article: Non-classical logic

In Deviant Logic, Fuzzy Logic: Beyond the Formalism, Susan Haack divided non-classical logics into deviant, quasi-deviant, and extended logics.[5]


  1. Nicholas Bunnin; Jiyuan Yu (2004). The Blackwell dictionary of Western philosophy. Wiley-Blackwell. p. 266. ISBN 978-1-4051-0679-5.
  2. L. T. F. Gamut (1991). Logic, language, and meaning, Volume 1: Introduction to Logic. University of Chicago Press. pp. 156–157. ISBN 978-0-226-28085-1.
  3. 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.
  4. Shapiro, Stewart (2000). Classical Logic. In Stanford Encyclopedia of Philosophy [Web]. Stanford: The Metaphysics Research Lab. Retrieved October 28, 2006, from
  5. 1 2 Haack, Susan, (1996). Deviant Logic, Fuzzy Logic: Beyond the Formalism. Chicago: The University of Chicago Press.

Further reading

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