Deviant logic is a type of logic incompatible with

classical logic Classical logic (or standard logic or Frege-Russell logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this class ...

. Philosopher

Susan Haack Susan Haack (born 1945) is a distinguished professor in the humanities, Cooper Senior Scholar in Arts and Sciences, professor of philosophy, and professor of law at the University of Miami in Coral Gables, Florida. Haack has written on logic, ...

uses the term ''deviant logic'' to describe certain non-classical systems of logic. In these logics: * the

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

well-formed formula In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language. A formal language can be ...

s generated equals the set of well-formed formulas generated by classical logic. * the set of

theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...

s generated is different from the set of theorems generated by classical logic. The set of theorems of a deviant logic can differ in any possible way from classical logic's set of theorems: as a proper

subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...

, superset, or fully exclusive set. A notable example of this is the trivalent logic developed by

Polish Polish may refer to: * Anything from or related to Poland, a country in Europe * Polish language * Poles Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, w ...

logician Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...

and

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

Jan Łukasiewicz Jan Łukasiewicz (; 21 December 1878 – 13 February 1956) was a Polish logician and philosopher who is best known for Polish notation and Łukasiewicz logic His work centred on philosophical logic, mathematical logic and history of logic. He ...

. Under this system, any theorem necessarily dependent on classical logic's

principle of bivalence In logic, the semantic principle (or law) of bivalence states that every declarative sentence expressing a proposition (of a theory under inspection) has exactly one truth value, either true or false. A logic satisfying this principle is called ...

would fail to be valid. The term ''deviant logic'' first appears in Chapter 6 of

Willard Van Orman Quine Willard Van Orman Quine (; known to his friends as "Van"; June 25, 1908 – December 25, 2000) was an American philosopher and logician in the analytic tradition, recognized as "one of the most influential philosophers of the twentieth century". ...

's ''Philosophy of Logic'', New Jersey: Prentice Hall (1970), which is cited by Haack on p. 15 of her book.

Quasi-deviant and extended logics

Haack also described what she calls a ''quasi''-deviant logic. These logics are different from pure deviant logics in that: * the set of well-formed formulas generated is a proper superset of the set of well-formed formulas generated by classical logic. * the set of theorems generated is a proper superset of the set of theorems generated by classical logic, both in that the quasi-deviant logic generates novel theorems using well-formed formulas held in common with classical logic, as well as novel theorems using novel well-formed formulas. Finally, Haack defined a class of merely ''extended'' logics. In these, * the set of well-formed formulas generated is a proper superset of the set of well-formed formulas generated by classical logic. * the set of theorems 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. Some systems of

modal logic Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend other ...

meet this definition. In such systems, any novel theorem would not parse in classical logic due to modal operators. While deviant and quasi-deviant logics are typically proposed as rivals to classical logic, the impetus behind extended logics is normally only to provide a supplement to it.

Two decades later

Achille Varzi Achille Varzi (8 August 1904 – 1 July 1948) was an Italian Grand Prix driver. Career Born in Galliate, province of Novara (Piedmont), Achille Varzi was the son of a textile manufacturer. As a young man, he was a successful motorcycle ra ...

in his review of the 1996 edition of Haack's book writes that the survey did not stand well the test of time, particularly with the "extraordinary proliferation of nonclassical logics in the past two decades—

paraconsistent logic A paraconsistent logic is an attempt at a logical system to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing "inconsistency-tolerant" syste ...

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. Although the logic has also be ...

substructural logic In logic, a substructural logic is a logic lacking one of the usual structural rules (e.g. of classical and intuitionistic logic), such as weakening, contraction, exchange or associativity. Two of the more significant substructural logics are r ...

nonmonotonic logic A non-monotonic logic is a formal logic whose conclusion relation is not monotonic. In other words, non-monotonic logics are devised to capture and represent defeasible inferences (cf. defeasible reasoning), i.e., a kind of inference in which ...

s, innumerable other logics for AI and computer science." He also finds that Haack's account of

vagueness In linguistics and philosophy, a vague predicate is one which gives rise to borderline cases. For example, the English adjective "tall" is vague since it is not clearly true or false for someone of middling height. By contrast, the word "prime" is ...

"is now seriously defective." He concedes however that "as a defense of a philosophical position, ''Deviant Logic'' retains its significance."

References

{{Reflist, colwidth=30em Non-classical logic