Non-classical logics is the name given to formal system

Formal system

In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive an expression from one or more other premises that are antecedently supposed or derived . The axioms and rules may be called a deductive apparatus...

s which differ in a significant way from standard logical systems

Classical logic

Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well...

 such as propositional and predicate

Predicate logic

In mathematical logic, predicate logic is the generic term for symbolic formal systems like first-order logic, second-order logic, many-sorted logic or infinitary logic. This formal system is distinguished from other systems in that its formulae contain variables which can be quantified...

 logic. There are several ways in which this is done, including by way of extensions, deviations, and variations. The aim of these departures is to make it possible to construct different models of logical consequence and logical truth

Logical truth

Logical truth is one of the most fundamental concepts in logic, and there are different theories on its nature. A logical truth is a statement which is true and remains true under all reinterpretations of its components other than its logical constants. It is a type of analytic statement.Logical...

.

Philosophical logic

Philosophical logic

Philosophical logic is a term introduced by Bertrand Russell to represent his idea that the workings of natural language and thought can only be adequately represented by an artificial language; essentially it was his formalization program for the natural language...

, especially in theoretical computer science

Theoretical computer science

Theoretical computer science is a division or subset of general computer science and mathematics which focuses on more abstract or mathematical aspects of computing....

, is understood to encompass and focus on non-classical logics, although the term has other meanings as well.

Examples of 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.
• 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...
  
   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;

Classification of non-classical logics

In Deviant Logic (1974) 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. The proposed classification is non-exclusive; a logic may be both a deviation and an extension of classical logic. A few other authors have adopted the main distinction between deviation and extension in non-classical logics. John P. Burgess

John P. Burgess

John Burgess is a John N. Woodhull Professor of Philosophy at Princeton University. He received his Ph.D. from UC Berkeley's Group in Logic and Methodology of Science. His interests include logic, philosophy of mathematics and metaethics...

 uses a similar classification but calls the two main classes anti-classical and extra-classical.

In an extension, new and different logical constants are added, for instance the "" in 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...

 which stands for "necessarily." In extensions of a logic,

• the set of well-formed formula
  Well-formed formula
  In mathematical logic, a well-formed formula, shortly wff, often simply formula, is a word which is part of a formal language...
  
  s generated is a proper superset of the set of well-formed formulas generated by classical logic
  Classical logic
  Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well...
  
  .
• the set of theorem
  Theorem
  In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
  
  s generated is a proper superset of the set of theorems generated by classical logic, but only in that the novel theorems generated by the extended logic are only a result of novel well-formed formulas.

(See also Conservative extension

Conservative extension

In mathematical logic, a logical theory T_2 is a conservative extension of a theory T_1 if the language of T_2 extends the language of T_1; every theorem of T_1 is a theorem of T_2; and any theorem of T_2 which is in the language of T_1 is already a theorem of T_1.More generally, if Γ is a set of...

.)

In a deviation, the usual logical constants are used, but are given a different meaning than usual. Only a subset of the theorems from the classical logic hold. A typical example is intuitionistic logic, where the law of excluded middle

Law of excluded middle

In logic, the law of excluded middle is the third of the so-called three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is....

 does not hold.

Additionally, one can identify a variations (or variants), where the content of the system remains the same, while the notation may change substantially. For instance many-sorted

Many-sorted logic

Many-sorted logic can reflect formally our intention, not to handle the universe as a homogeneous collection of objects, but to partition it in a way that is similar to types in typeful programming...

 predicate logic is considered a just variation of predicate logic.

This classification ignores however semantic equivalences. For instance, Gödel

Godel

Godel or similar can mean:*Kurt Gödel , an Austrian logician, mathematician and philosopher*Gödel...

 showed that all theorems from intuitionistic logic have an equivalent theorem in the classical modal logic S4. The result has been generalized to superintuitionistic logics and extensions of S4.

The theory of abstract algebraic logic

Abstract Algebraic Logic

In mathematical logic, abstract algebraic logic is the study of the algebraization of deductive systemsarising as an abstraction of the well-known Lindenbaum-Tarski algebra, and how the resulting algebras are related to logical systems.-Overview:...

 has also provided means to classify logics, with most results having been obtained for propositional logics. The current algebraic hierarchy of propositional logics has five levels, defined in terms of properties of their Leibniz operator: protoalgebraic, (finitely) equivalential, and (finitely) algebraizable.

Attitudes toward non-classical logics

In a recent academic survey, 51.5% of philosophers polled expressed belief in classical logic; 15.3% non-classical logic; and 33% 'other.' See: http://philpapers.org/surveys/results.pl

Further reading

A revised version was published as Brief introduction to non-classical logics, with a primer on the classical one. Chapters 7-16 cover the main non-classical logics of broad interest today.

External links

• Logic & the Unconscious Mind (international research project on classical/non-classical logics and the unconscious mind)