 Nonclassical logic

Nonclassical logics (and sometimes alternative logics) is the name given to formal systems which 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, especially in theoretical computer science, is understood to encompass and focus on nonclassical logics, although the term has other meanings as well.^{[2]}
Contents
Examples of nonclassical logics
 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.
 Fuzzy logic rejects the law of the excluded middle and allows as a truth value any real number between 0 and 1.
 Intuitionistic logic rejects the law of the excluded middle, double negative elimination, and the 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., dialetheism and relevance logic) rejects the law of noncontradiction;
 Relevance logic, linear logic, and nonmonotonic logic reject monotonicity of entailment;
Classification of nonclassical logics
In Deviant Logic (1974) Susan Haack divided nonclassical logics into deviant, quasideviant, and extended logics.^{[3]} The proposed classification is nonexclusive; a logic may be both a deviation and an extension of classical logic.^{[4]} A few other authors have adopted the main distinction between deviation and extension in nonclassical logics.^{[5]}^{[6]}^{[7]} John P. Burgess uses a similar classification but calls the two main classes anticlassical and extraclassical.^{[8]}
In an extension, new and different logical constants are added, for instance the "" in modal logic which stands for "necessarily."^{[5]} 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]}^{[7]}
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.^{[5]}
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.^{[9]}
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.^{[10]}
Attitudes toward nonclassical logics
In a recent academic survey, 51.5% of philosophers polled expressed belief in classical logic; 15.3% nonclassical logic; and 33% 'other.' See: http://philpapers.org/surveys/results.pl
References
 ^ Logic for philosophy, Theodore Sider
 ^ John P. Burgess (2009). Philosophical logic. Princeton University Press. pp. viiviii. ISBN 9780691137896. http://books.google.com/books?id=k32w3_wjBoYC&pg=PR7.
 ^ Susan Haack (1974). Deviant logic: some philosophical issues. CUP Archive. p. 4. ISBN 9780521205009. http://books.google.com/books?id=ANg8AAAAIAAJ&pg=PA4.
 ^ Susan Haack (1978). Philosophy of logics. Cambridge University Press. pp. 204. ISBN 9780521293297. http://books.google.com/books?id=0GsZ8SBQrUcC&pg=PA204.
 ^ ^{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. http://books.google.com/books?id=Z0KhywkpolMC&pg=PA156.
 ^ Seiki Akama (1997). Logic, language, and computation. Springer. p. 3. ISBN 9780792343769. http://books.google.com/books?id=QyksEA5i1QC&pg=PA3.
 ^ ^{a} ^{b} Robert Hanna (2006). Rationality and logic. MIT Press. pp. 40–41. ISBN 9780262083492. http://books.google.com/books?id=ka9BhOL1ev8C&pg=PA40.
 ^ ^{a} ^{b} John P. Burgess (2009). Philosophical logic. Princeton University Press. pp. 1–2. ISBN 9780691137896. http://books.google.com/books?id=k32w3_wjBoYC&pg=PA1.
 ^ Dov M. Gabbay; Larisa Maksimova (2005). Interpolation and definability: modal and intuitionistic logics. Clarendon Press. p. 61. ISBN 9780198511748. http://books.google.com/books?id=v6sDNSaW5wAC&pg=PA61.
 ^ D. Pigozzi (2001). "Abstract algebraic logic". In M. Hazewinkel. Encyclopaedia of mathematics: Supplement Volume III. Springer. pp. 2–13. ISBN 1402001983. Also online.
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.
External links
Categories:
Wikimedia Foundation. 2010.
Look at other dictionaries:
NonAristotelian logic — The term non Aristotelian logic, sometimes shortened to null A, means any non classical system of logic which rejects one of Aristotle s premises (see term logic). Contents 1 History 2 Use in science fiction 3 See also 4 … Wikipedia
Nonmonotonic logic — A non monotonic logic is a formal logic whose consequence relation is not monotonic. Most studied formal logics have a monotonic consequence relation, meaning that adding a formula to a theory never produces a reduction of its set of consequences … Wikipedia
Classical logic — identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well.[1][2] They are characterised by a number of properties:[3] Law of the excluded middle and… … Wikipedia
classical logic — noun A kind of logic based on the principle that each assertion has a truth value of either true or false , but not both. The Lindenbaum Tarski algebra of propositional classical logic is a Boolean algebra. Ant: non classical logic,… … Wiktionary
Nonclassical analysis — In mathematics, non classical analysis is any system of analysis, other than classical real analysis, and complex, vector, tensor, etc., analysis based upon it. Such systems include: Abstract Stone duality,[1] a programme to re axiomatise general … Wikipedia
Laws of classical logic — The laws of classical logic are a small collection of fundamental sentences of propositional logic and Boolean algebra, from which may be derived all true sentences in both of these elementary formal systems.The syntax of the laws of classical… … Wikipedia
Logic — For other uses, see Logic (disambiguation). Philosophy … Wikipedia
Nonstandard model — See also Interpretation (logic) In model theory, a discipline within mathematical logic, a non standard model is a model of a theory that is not isomorphic to the intended model (or standard model). If the intended model is infinite and the… … Wikipedia
Logic programming — is, in its broadest sense, the use of mathematical logic for computer programming. In this view of logic programming, which can be traced at least as far back as John McCarthy s [1958] advice taker proposal, logic is used as a purely declarative… … Wikipedia
Logic in Islamic philosophy — Logic (Arabic: Mantiq ) played an important role in early Islamic philosophy. Islamic law placed importance on formulating standards of argument, which gave rise to a novel approach to logic in Kalam, as seen in the method of qiyas . This… … Wikipedia