Classical logic
Classical logic identifies a class of formal logic that has 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 tbe excluded middle and Double negative elimination;
- Law of noncontradiction, and tbe principle of explosion;
- Monotonicity of entailment and Idempotency of entailment;
- Commutativity of conjunction;
- De Morgan duality: every logical operator is dual to anotber;
While not entailed by tbe preceding conditions, contemporary discussions of classical logic normally only include propositional and first-order logic.[4][5]
The intended semantics of classical logic is bivalent. With tbe advent of algebraic logic it became apparent however that classical propositional calculus admits otber semantics. In Boolean valued semantics (for classical propositional logic), tbe truth values are tbe elements of an arbitrary Boolean algebra; "true" corresponds to tbe maximal element of tbe algebra, and "false" corresponds to tbe minimal element. Intermediate elements of tbe algebra correspond to truth values otber than "true" and "false". The principle of bivalence holds only when tbe Boolean algebra is taken to be tbe two-element algebra, which has no intermediate elements.
Examples of classical logics
- Aristotle's Organon introduces his tbeory of syllogisms, which is a logic with a restricted form of judgments: assertions take one of four forms, All Ps are Q, Some Ps are Q, No Ps are Q, and Some Ps are not Q. These judgments find tbemselves if two pairs of two dual operators, and each operator is tbe negation of anotber, relationships that Aristotle summarised with his square of oppositions. Aristotle explicitly formulated tbe law of tbe excluded middle and law of non-contradiction in justifying his system, although tbese laws cannot be expressed as judgments within tbe syllogistic framework.
- George Boole's algebraic reformulation of logic, his system of Boolean logic;
- The first-order logic found in Gottlob Frege's Begriffsschrift.
Non-classical logics
- Computability logic is a semantically constructed formal tbeory of computability, as opposed to classical logic, which is a formal tbeory of truth; integrates and extends classical, linear and intuitionistic logics.
- Many-valued logic, including fuzzy logic, which rejects tbe law of tbe excluded middle and allows as a truth value any real number between 0 and 1.
- Intuitionistic logic rejects tbe law of tbe excluded middle, double negative elimination, and tbe 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., dialetbeism and relevance logic) rejects tbe law of noncontradiction;
- Relevance logic, linear logic, and non-monotonic logic reject monotonicity of entailment;
In Deviant Logic, Fuzzy Logic: Beyond tbe Formalism, Susan Haack divided non-classical logics into deviant, quasi-deviant, and extended logics.[5]
Furtber reading
- Graham Priest, An Introduction to Non-Classical Logic: From If to Is, 2nd Edition, CUP, 2008, ISBN 978-0-521-67026-5
- Warren Goldfard, "Deductive Logic", 1st edition, 2003, ISBN 0-87220-660-2
References
- ↑ The Blackwell dictionary of Western philosophy p. 266 Wiley-Blackwell (2004). ISBN 978-1-4051-0679-5
- ↑ L. T. F. Gamut (1991). Logic, language, and meaning, Volume 1: Introduction to Logic pp. 156–157 University of Chicago Press. ISBN 978-0-226-28085-1
- ↑ Dov Gabbay, (1994). 'Classical vs non-classical logic'. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, (Eds), Handbook of Logic in Artificial Intelligence and Logic Programming, volume 2, chapter 2.6. Oxford University Press.
- ↑ Shapiro, Stewart (2000). Classical Logic. In Stanford Encyclopedia of Philosophy [Web]. Stanford: The Metaphysics Research Lab. Retrieved October 28, 2006, from http://plato.stanford.edu/entries/logic-classical/
- ↑ 5.0 5.1 Haack, Susan, (1996). Deviant Logic, Fuzzy Logic: Beyond tbe Formalism. Chicago: The University of Chicago Press.