# Deviant logic

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## Graham priest and maureen eckert deviant logic

Philosopher Susan Haack uses the term "deviant logic" to describe certain non-classical systems of logic. In these logics,

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• the set of well-formed formulas generated equals the set of well-formed formulas generated by classical logic.
• the set of theorems 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, superset, or fully exclusive set. A notable example of this is the trivalent logic developed by Polish logician and mathematician Jan Ćukasiewicz. Under this system, any theorem necessarily dependent on classical logic's principle of bivalence would fail to be valid. The term first appears in Chapter 6 of Willard Van Orman Quine'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.