Classical logic identifies a class of formal logic

Formal logic

Classical or traditional system of determining the validity or invalidity of a conclusion deduced from two or more statements...

s that have been most intensively studied and most widely used. The class is sometimes called standard logic as well. They are characterised by a number of properties:

1. Law of the excluded middle and Double negative elimination
   Double negative elimination
   In propositional logic, the inference rules double negative elimination allow deriving the double negative equivalent by adding or removing a pair of negation signs...
   
   ;
2. Law of noncontradiction
   Law of noncontradiction
   In classical logic, the law of non-contradiction is the second of the so-called three classic laws of thought. It states that contradictory statements cannot both at the same time be true, e.g...
   
   , and the principle of explosion
   Principle of explosion
   The principle of explosion, or the principle of Pseudo-Scotus, is the law of classical logic and intuitionistic and similar systems of logic, according to which any statement can be proven from a contradiction...
   
   ;
3. Monotonicity of entailment
   Monotonicity of entailment
   Monotonicity of entailment is a property of many logical systems that states that the hypotheses of any derived fact may be freely extended with additional assumptions. In sequent calculi this property can be captured by an inference rule called weakening, or sometimes thinning, and in such...
   
    and Idempotency of entailment
   Idempotency of entailment
   Idempotency of entailment is a property of logical systems that states that one may derive the same consequences from many instances of a hypothesis as from just one...
   
   ;
4. Commutativity of conjunction
   Commutativity of conjunction
   In logic, the commutativity of conjunction demonstrates that predicates on both sides of a logical conjunction operator are interchangeable. This logical law is a part of classical logic.For any propositions H1, H2, .....
   
   ;
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

First-order logic

First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...

 logics.

The intended semantics of classical logic is bivalent. With the advent of algebraic logic

Algebraic logic

In mathematical logic, algebraic logic is the study of logic presented in an algebraic style.What is now usually called classical algebraic logic focuses on the identification and algebraic description of models appropriate for the study of various logics and connected problems...

 it became apparent however that classical propositional calculus

Propositional calculus

In mathematical logic, a propositional calculus or logic is a formal system in which formulas of a formal language may be interpreted as representing propositions. A system of inference rules and axioms allows certain formulas to be derived, called theorems; which may be interpreted as true...

 admits other semantics

Semantics

Semantics is the study of meaning. It focuses on the relation between signifiers, such as words, phrases, signs and symbols, and what they stand for, their denotata....

. 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.

Examples of classical logics

• Aristotle
  Aristotle
  Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
  
  's Organon
  Organon
  The Organon is the name given by Aristotle's followers, the Peripatetics, to the standard collection of his six works on logic:* Categories* On Interpretation* Prior Analytics* Posterior Analytics...
  
   introduces his theory of syllogism
  Syllogism
  A syllogism is a kind of logical argument in which one proposition is inferred from two or more others of a certain form...
  
  s, 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 opposition
  Square of opposition
  In the system of Aristotelian logic, the square of opposition is a diagram representing the different ways in which each of the four propositions of the system are logically related to each of the others...
  
  s. 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
  George Boole
  George Boole was an English mathematician and philosopher.As the inventor of Boolean logic—the basis of modern digital computer logic—Boole is regarded in hindsight as a founder of the field of computer science. Boole said,...
  
  's algebraic reformulation of logic, his system of Boolean logic
  Boolean logic
  Boolean algebra is a logical calculus of truth values, developed by George Boole in the 1840s. It resembles the algebra of real numbers, but with the numeric operations of multiplication xy, addition x + y, and negation −x replaced by the respective logical operations of...
  
  ;
• The first-order logic found in Gottlob Frege
  Gottlob Frege
  Friedrich Ludwig Gottlob Frege was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic, and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on...
  
  's Begriffsschrift
  Begriffsschrift
  Begriffsschrift is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book...
  
  .

Non-classical logics

• Computability logic
  Computability logic
  Introduced by Giorgi Japaridze in 2003, computability logic is a research programme and mathematical framework for redeveloping logic as a systematic formal theory of computability, as opposed to classical logic which is a formal theory of truth...
  
   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, including fuzzy logic
  Fuzzy logic
  Fuzzy logic is a form of many-valued logic; it deals with reasoning that is approximate rather than fixed and exact. In contrast with traditional logic theory, where binary sets have two-valued logic: true or false, fuzzy logic variables may have a truth value that ranges in degree between 0 and 1...
  
  , which rejects the law of the excluded middle and allows as a truth value any real number between 0 and 1.
• Intuitionistic logic
  Intuitionistic logic
  Intuitionistic logic, or constructive logic, is a symbolic logic system differing from classical logic in its definition of the meaning of a statement being true. In classical logic, all well-formed statements are assumed to be either true or false, even if we do not have a proof of either...
  
   rejects the law of the excluded middle, double negative elimination, and the De Morgan's laws;
• Linear logic
  Linear logic
  Linear logic is a substructural logic proposed by Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the dualities of the former with many of the constructive properties of the latter...
  
   rejects idempotency of entailment
  Entailment
  In logic, entailment is a relation between a set of sentences and a sentence. Let Γ be a set of one or more sentences; let S1 be the conjunction of the elements of Γ, and let S2 be a sentence: then, Γ entails S2 if and only if S1 and not-S2 are logically inconsistent...
  
   as well;
• Modal logic
  Modal logic
  Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
  
   extends classical logic with non-truth-functional ("modal") operators.
• Paraconsistent logic
  Paraconsistent logic
  A paraconsistent logic is a logical system that attempts to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing paraconsistent systems of logic.Inconsistency-tolerant logics have been...
  
   (e.g., dialetheism
  Dialetheism
  Dialetheism is the view that some statements can be both true and false simultaneously. More precisely, it is the belief that there can be a true statement whose negation is also true...
  
   and relevance logic
  Relevance logic
  Relevance logic, also called relevant logic, is a kind of non-classical logic requiring the antecedent and consequent of implications be relevantly related. They may be viewed as a family of substructural or modal logics...
  
  ) rejects the law of noncontradiction;
• Relevance logic
  Relevance logic
  Relevance logic, also called relevant logic, is a kind of non-classical logic requiring the antecedent and consequent of implications be relevantly related. They may be viewed as a family of substructural or modal logics...
  
  , linear logic
  Linear logic
  Linear logic is a substructural logic proposed by Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the dualities of the former with many of the constructive properties of the latter...
  
  , and non-monotonic logic
  Non-monotonic logic
  A non-monotonic logic is a formal logic whose consequence relation is not monotonic. Most studied formal logics have a monotonic consequence relation, meaning that adding a formula to a theory never produces a reduction of its set of consequences. Intuitively, monotonicity indicates that learning a...
  
   reject monotonicity of entailment;

In Deviant Logic, Fuzzy Logic: Beyond the Formalism, Susan Haack

Susan Haack

Susan Haack is an English professor of philosophy and law at the University of Miami in the United States. She has written on logic, the philosophy of language, epistemology, and metaphysics. Her pragmatism follows that of Charles Sanders Peirce.-Career:Haack is a graduate of the University of...

 divided non-classical logics into deviant

Deviant logic

Philosopher Susan Haack uses the term "deviant logic" to describe certain non-classical systems of logic. In these logics,* the set of well-formed formulas generated equals the set of well-formed formulas generated by classical logic....

, quasi-deviant, and extended logics.

Further reading

• Graham Priest
  Graham Priest
  Graham Priest is Boyce Gibson Professor of Philosophy at the University of Melbourne and Distinguished Professor of Philosophy at the CUNY Graduate Center, as well as a regular visitor at St. Andrews University. Priest is a fellow in residence at Ormond College. He was educated at the University...
  
  , An Introduction to Non-Classical Logic: From If to Is, 2nd Edition, CUP, 2008, ISBN 9780521670265
• Warren Goldfard, "Deductive Logic", 1st edition, 2003, ISBN 0-87220-660-2