Deviant logic

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

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 Łukasiewicz. 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.

Haack also described what she calls a quasi-deviant logic. These logics are different from pure deviant logics in that:

Finally, Haack defined a class of merely extended logics. In these,

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.

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 logics, 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."

...
Wikipedia