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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] 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.


• 
  George Boole's algebraic reformulation of logic, his system of Boolean logic;


• The first-order logic found in Gottlob Frege's Begriffsschrift.

Generalized semantics

With the advent of algebraic logic it became apparent 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

• 
  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.


• 
  Many-valued logic rejects bivalence, allowing for truth values other than true and false. The most popular forms are three-valued logic, as initially developed by Jan Łukasiewicz, and infinitely-valued logics such as fuzzy logic, which permits any real number between 0 and 1 as a truth value.


• 
  Intuitionistic logic rejects the law of the excluded middle, double negation elimination, and part of De Morgan's laws;

• 
  Linear logic rejects idempotency of entailment as well;


• 
  Modal logic extends classical logic with non-truth-functional ("modal") operators.


• 
  Paraconsistent logic (e.g., relevance logic) rejects the principle of explosion, and has a close relation to dialetheism;

• Quantum logic


• 
  Relevance logic, linear logic, and non-monotonic logic reject monotonicity of entailment;


• 
  Non-reflexive logic (also known as "Schrödinger logics") rejects or restricts the law of identity;^[6]
  

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

References

Further reading

• 
  Graham Priest, An Introduction to Non-Classical Logic: From If to Is, 2nd Edition, CUP, 2008,
  ISBN 978-0-521-67026-5
  


• Warren Goldfard, "Deductive Logic", 1st edition, 2003,
  ISBN 0-87220-660-2