Non-classical logics (and sometimesalternative logics ornon-Aristotelian logics) areformal systems that differ in a significant way fromstandard logical systems such aspropositional andpredicate logic. There are several ways in which this is commonly the case, including by way of extensions, deviations, and variations. The aim of these departures is to make it possible to construct different models oflogical consequence andlogical truth.^[1]

Philosophical logic is understood to encompass and focus on non-classical logics, although the term has other meanings as well.^[2] In addition, some parts oftheoretical computer science can be thought of as using non-classical reasoning, although this varies according to the subject area. For example, the basicboolean functions (e.g.AND,OR,NOT, etc) incomputer science are very muchclassical in nature, as is clearly the case given that they can be fully described by classicaltruth tables. However, in contrast, somecomputerized proof methods may not use classical logic in the reasoning process.

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

• Computability logic is a semantically constructed formal theory of computability—as opposed to classical logic, which is a formal theory of truth—that integrates and extends classical, linear and intuitionistic logics.
• Dialectical materialism is a materialist theory based upon the writings ofKarl Marx andFriedrich Engels that has found widespread applications in a variety of philosophical disciplines ranging from philosophy of history to philosophy of science and theclass struggle. It also structured theRussian Revolution
• Dynamic semantics interprets formulas as update functions, opening the door to a variety of nonclassical behaviours
• Many-valued logic rejects bivalence, allowing fortruth values other than true and false. The most popular forms arethree-valued logic, as initially developed byJan Łukasiewicz, and infinitely-valued logics such asfuzzy logic, which permit any real number between 0 and 1 as a truth value.
• Intuitionistic logic rejects thelaw of the excluded middle,double negation elimination, and part ofDe Morgan's laws;
• Linear logic rejectsidempotency ofentailment as well;
• Paraconsistent logic (e.g.,relevance logic) rejects theprinciple of explosion, and has a close relation todialetheism;
• Quantum logic
• Relevance logic,linear logic, andnon-monotonic logic reject monotonicity of entailment;
• Non-reflexive logic (also known as"Schrödinger logics") rejects or restricts thelaw of identity;^[3]

InDeviant Logic (1974)Susan Haack divided non-classical logics intodeviant, quasi-deviant, and extended logics.^[4] The proposed classification is non-exclusive; a logic may be both a deviation and an extension of classical logic.^[5] A few other authors have adopted the main distinction between deviation and extension in non-classical logics.^[6][7][8]John P. Burgess uses a similar classification but calls the two main classes anti-classical and extra-classical.^[9] Although some systems of classification for non-classical logic have been proposed, such as those of Haack and Burgess as described above for example, many people who study non-classical logic ignore these classification systems. As such, none of the classification systems in this section should be treated as standard.

In anextension, new and differentlogical constants are added, for instance the "${\displaystyle ◻}$" inmodal logic, which stands for "necessarily".^[6] In extensions of a logic,

• the set ofwell-formed formulas generated is aproper superset of the set of well-formed formulas generated byclassical logic.
• the set oftheorems 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 alsoConservative extension.)

In adeviation, 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 thelaw of excluded middle does not hold.^[8][9]

Additionally, one can identify avariations (orvariants), where the content of the system remains the same, while the notation may change substantially. For instancemany-sorted predicate logic is considered a just variation of predicate logic.^[6]

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 tosuperintuitionistic logics and extensions of S4.^[10]

The theory ofabstract 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 theirLeibniz operator:protoalgebraic, (finitely)equivalential, and (finitely)algebraizable.^[11]

• Logic in Eastern philosophy
  • Logic in China
  • Logic in India

• Priest, Graham (2008).An introduction to non-classical logic: from if to is (2nd ed.). Cambridge University Press.ISBN 978-0-521-85433-7.
• Gabbay, Dov M. (1998).Elementary logics: a procedural perspective. Prentice Hall Europe.ISBN 978-0-13-726365-3. A revised version was published asGabbay, D. M. (2007).Logic for Artificial Intelligence and Information Technology.College Publications.ISBN 978-1-904987-39-0.
• Burgess, John P. (2009).Philosophical logic.Princeton University Press.ISBN 978-0-691-13789-6. Brief introduction to non-classical logics, with a primer on the classical one.
• Goble, Lou, ed. (2001).The Blackwell guide to philosophical logic. Wiley-Blackwell.ISBN 978-0-631-20693-4. Chapters 7-16 cover the main non-classical logics of broad interest today.
• Humberstone, Lloyd (2011).The Connectives. MIT Press.ISBN 978-0-262-01654-4. 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—thelogical connectives in various logics; decidability and complexity aspects are generally omitted though.

• Video of Graham Priest & Maureen Eckert on Deviant Logic

• Non-classical logic
• History of logic
• Formal logic
• Philosophy of logic

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