The purpose of this entry is to survey those modern logics that are often called "non-classical," classical logic being the theory of validity concerning truth functions and first-order quantifiers likely to be found in introductory textbooks of formal logic at the end of the twentieth century.

For the sake of uniformity I will give a model-theoretic account of the logics. All of the logics also have proof-theoretic characterizations, and in some cases (such as linear logic) these characterizations are somewhat more natural. I will not discuss combinatory logic, which is not so much a non-classical logic as it is a way of expressing inferences that may be deployed for both classical and non-classical logics. I will use A, B, … for arbitrary sentences; ∧, ∨, ¬, and →, for the standard conjunction, disjunction, negation, and conditional operators for whichever logic is at issue. "Iff" means "if and only if." For references...