Nonclassical logics (and sometimes alternative logics) are 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.^{[1]}
Philosophical logic is understood to encompass and focus on nonclassical logics, although the term has other meanings as well.^{[2]} In addition, some parts of theoretical computer science can be thought of as using nonclassical reasoning, although this varies according to the subject area. For example, the basic boolean functions (e.g. AND, OR, NOT, etc) in computer science are very much classical in nature, as is clearly the case given that they can be fully described by classical truth tables. However, in contrast, some computerized proof methods may not use classical logic in the reasoning process.
YouTube Encyclopedic

1/5Views:3 8166 3673 4452 699741

NonClassical Logic  Jc Beall

Intuitionistic Logic  Attic Philosophy

3Valued Logic  NonClassical Logic  Attic Philosophy

True, False, Other  NonClassical Logic  Attic Philosophy

Graham Priest: The catuṣkoṭi, the saptabhaṇgī, and 'nonclassical' logic
Transcription
Examples of nonclassical logics
There are many kinds of nonclassical 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.
 Dynamic semantics interprets formulas as update functions, opening the door to a variety of nonclassical behaviours
 Manyvalued logic rejects bivalence, allowing for truth values other than true and false. The most popular forms are threevalued logic, as initially developed by Jan Łukasiewicz, and infinitelyvalued logics such as fuzzy logic, which permit any real number between 0 and 1 as a truth value.
 Intuitionistic logic rejects the law of the excluded middle, double negation elimination, and part of De Morgan's laws;
 Linear logic rejects idempotency of entailment as well;
 Modal logic extends classical logic with nontruthfunctional ("modal") operators.
 Paraconsistent logic (e.g., relevance logic) rejects the principle of explosion, and has a close relation to dialetheism;
 Quantum logic
 Relevance logic, linear logic, and nonmonotonic logic reject monotonicity of entailment;
 Nonreflexive logic (also known as "Schrödinger logics") rejects or restricts the law of identity;^{[3]}
Classification of nonclassical logics according to specific authors
In Deviant Logic (1974) Susan Haack divided nonclassical logics into deviant, quasideviant, and extended logics.^{[4]} The proposed classification is nonexclusive; 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 nonclassical logics.^{[6]}^{[7]}^{[8]} John P. Burgess uses a similar classification but calls the two main classes anticlassical and extraclassical.^{[9]} Although some systems of classification for nonclassical logic have been proposed, such as those of Haack and Burgess as described above for example, many people who study nonclassical logic ignore these classification systems. As such, none of the classification systems in this section should be treated as standard.
In an extension, new and different logical constants are added, for instance the "" in modal logic, which stands for "necessarily."^{[6]} In extensions of a logic,
 the set of wellformed formulas generated is a proper superset of the set of wellformed 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 wellformed 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.^{[8]}^{[9]}
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 manysorted 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 to superintuitionistic logics and extensions of S4.^{[10]}
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.^{[11]}
See also
References
 ^ Logic for philosophy, Theodore Sider
 ^ John P. Burgess (2009). Philosophical logic. Princeton University Press. pp. vii–viii. ISBN 9780691137896.
 ^ Krause, D, Da Costa, N.C.A. and (1994), "Schrödinger logics", Studia Logica, 53 (4): 533, doi:10.1007/BF01057649.
 ^ Haack, Susan (1974). Deviant logic: some philosophical issues. CUP Archive. p. 4. ISBN 9780521205009.
 ^ Haack, Susan (1978). Philosophy of logics. Cambridge University Press. p. 204. ISBN 9780521293297.
 ^ ^{a} ^{b} ^{c} L. T. F. Gamut (1991). Logic, language, and meaning, Volume 1: Introduction to Logic. University of Chicago Press. pp. 156–157. ISBN 9780226280851.
 ^ Seiki Akama (1997). Logic, language, and computation. Springer. p. 3. ISBN 9780792343769.
 ^ ^{a} ^{b} Robert Hanna (2006). Rationality and logic. MIT Press. pp. 40–41. ISBN 9780262083492.
 ^ ^{a} ^{b} John P. Burgess (2009). Philosophical logic. Princeton University Press. pp. 1–2. ISBN 9780691137896.
 ^ Dov M. Gabbay; Larisa Maksimova (2005). Interpolation and definability: modal and intuitionistic logics. Clarendon Press. p. 61. ISBN 9780198511748.
 ^ D. Pigozzi (2001). "Abstract algebraic logic". In M. Hazewinkel (ed.). Encyclopaedia of mathematics: Supplement Volume III. Springer. pp. 2–13. ISBN 9781402001987. Also online: "Abstract algebraic logic", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Further reading
 Graham Priest (2008). An introduction to nonclassical logic: from if to is (2nd ed.). Cambridge University Press. ISBN 9780521854337.
 Dov M. Gabbay (1998). Elementary logics: a procedural perspective. Prentice Hall Europe. ISBN 9780137263653. A revised version was published as D. M. Gabbay (2007). Logic for Artificial Intelligence and Information Technology. College Publications. ISBN 9781904987390.
 John P. Burgess (2009). Philosophical logic. Princeton University Press. ISBN 9780691137896. Brief introduction to nonclassical logics, with a primer on the classical one.
 Lou Goble, ed. (2001). The Blackwell guide to philosophical logic. WileyBlackwell. ISBN 9780631206934. Chapters 716 cover the main nonclassical logics of broad interest today.
 Lloyd Humberstone (2011). The Connectives. MIT Press. ISBN 9780262016544. Probably covers more logics than any of the other titles in this section; a large part of this 1500page monograph is crosssectional, comparing—as its title implies—the logical connectives in various logics; decidability and complexity aspects are generally omitted though.