Deviant logic is a type of logic incompatible with classical logic. Philosopher Susan Haack uses the term ''deviant logic'' to describe certain non-classical

systems of logic A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A fo ...

. In these logics: * the set of

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 ...

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 ''A'' is a subset of a set ''B'' if all 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 unequal, then ''A'' is a proper subset o ...

, superset, or fully exclusive set. A notable example of this is the

trivalent logic In logic, a three-valued logic (also trinary logic, trivalent, ternary, or trilean, sometimes abbreviated 3VL) is any of several many-valued logic systems in which there are three truth values indicating ''true'', ''false'' and some indeterminate ...

developed by Polish

logician Logic is the study of correct reasoning. It includes both Mathematical logic, formal and informal logic. Formal logic is the science of Validity (logic), deductively valid inferences or of logical truths. It is a formal science investigating h ...

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, mathematical structure, structure, space, Mathematica ...

Jan Łukasiewicz. Under this system, any theorem necessarily dependent on classical logic's principle of bivalence 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 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 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" syst ...

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