Many-valued systems

Restrictions of classical logic: deviant logics

One system is a deviation of another if it shares the vocabulary of the first, but has a different set of theorems/valid inferences; a ‘deviant logic’ is a system which is a deviation of classical logic. (A system may involve both an extension and a deviation of classical logic, if it adds new vocabulary and hence new theorems/valid inferences, but at the same time differs from classical logic with respect to theorems/valid inferences involving only shared vocabulary essentially. The system E, considered in ch. 10 §6, would be an example.) Many-valued logics are deviant; sharing its vocabulary, they characteristically lack certain theorems of classical logic, such as the ‘law of excluded middle’, ‘p ∨ −p’. (Some also add new vocabulary and so come in the category of extensions as well.)

The many-valued logics I shall consider in this chapter were devised from two main kinds of motivation: the purely mathematical interest of alternatives to the 2-valued semantics of classical sentence logic; and – of more philosophical interest – dissatisfaction with the classical imposition of an exhaustive dichotomy into the true and the false, and, relatedly, dissatisfaction with certain classical theorems or inferences. The second kind of motivation is characteristic – as I observed in ch. 9 §2 – of proposals for restrictions of classical logic.