deviant logic

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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 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 ...
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 the ...
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 developed by Polish
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 ...
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. H ...
. 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 calle ...
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.