Non-classical logic

Non-classical logics (and sometimes alternative logics) is the name given to formal systems that differ in a significant way from standard logical systems such as propositional and predicate logic. There are several ways in which this is done, including by way of extensions, deviations, and variations. The aim of these departures is to make it possible to construct different models of logical consequence and logical truth.

Philosophical logic, especially in theoretical computer science, is understood to encompass and focus on non-classical logics, although the term has other meanings as well.

Examples of non-classical logics

There are many kinds of non-classical logic, which include:

• Many-valued logic rejects bivalence, allowing for truth values other than true and false. The most popular forms are three-valued logic, as initially developed by Jan Łukasiewicz, and infinitely-valued logics such as fuzzy logic.
• Intuitionistic logic rejects the law of the excluded middle, double negative elimination, and De Morgan's laws;
• Linear logic rejects idempotency of entailment as well;
• Modal logic extends classical logic with non-truth-functional ("modal") operators.
• Paraconsistent logic (e.g., dialetheism and relevance logic) rejects the law of noncontradiction;
• Relevance logic, linear logic, and non-monotonic logic reject monotonicity of entailment;
• Computability logic is a semantically constructed formal theory of computability, as opposed to classical logic, which is a formal theory of truth; integrates and extends classical, linear and intuitionistic logics.

Classification of non-classical logics

In Deviant Logic (1974) Susan Haack divided non-classical logics into deviant, quasi-deviant, and extended logics. The proposed classification is non-exclusive; a logic may be both a deviation and an extension of classical logic. A few other authors have adopted the main distinction between deviation and extension in non-classical logics. John P. Burgess uses a similar classification but calls the two main classes anti-classical and extra-classical.

In an extension, new and different logical constants are added, for instance the "$\Box$" in modal logic, which stands for "necessarily." In extensions of a logic,

• 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.

(See also Conservative extension.)

In a deviation, the usual logical constants are used, but are given a different meaning than usual. Only a subset of the theorems from the classical logic hold. A typical example is intuitionistic logic, where the law of excluded middle does not hold.

Additionally, one can identify a variations (or variants), where the content of the system remains the same, while the notation may change substantially. For instance many-sorted predicate logic is considered a just variation of predicate logic.

This classification ignores however semantic equivalences. For instance, Gödel showed that all theorems from intuitionistic logic have an equivalent theorem in the classical modal logic S4. The result has been generalized to superintuitionistic logics and extensions of S4.

The theory of abstract algebraic logic has also provided means to classify logics, with most results having been obtained for propositional logics. The current algebraic hierarchy of propositional logics has five levels, defined in terms of properties of their Leibniz operator: protoalgebraic, (finitely) equivalential, and (finitely) algebraizable.

Further reading

• A revised version was published as
• Brief introduction to non-classical logics, with a primer on the classical one.
• Chapters 7-16 cover the main non-classical logics of broad interest today.
• Probably covers more logics than any of the other titles in this section; a large part of this 1500-page monograph is cross-sectional, comparing—as its title implies—the logical connectives in various logics; decidability and complexity aspects are generally omitted though.

External links

• Video of Graham Priest & Maureen Eckert on Deviant Logic