Classical logic

Classical logic (or standard logic^[1]^[2]) is an intensively studied and widely used class of formal logics. Each logical system in this class shares characteristic properties:^[3]

Law of excluded middle and double negative elimination
Law of noncontradiction, and the principle of explosion
Monotonicity of entailment and idempotency of entailment
Commutativity of conjunction
De Morgan duality: every logical operator is dual to another

While not entailed by the preceding conditions, contemporary discussions of classical logic normally only include propositional and first-order logics.^[4]^[5]

Classical logic was originally devised as a two-level (bivalent) logical system, with simple semantics for the levels representing "true" and "false".

Examples of classical logics

Aristotle's Organon introduces his theory 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 themselves if two pairs of two dual operators, and each operator is the negation of another, relationships that Aristotle summarised with his square of oppositions. Aristotle explicitly formulated the law of the excluded middle and law of non-contradiction in justifying his system, although these laws cannot be expressed as judgments within the syllogistic framework.
George Boole's algebraic reformulation of logic, his system of Boolean logic;
The first-order logic found in Gottlob Frege's Begriffsschrift.

Generalized semantics

With the advent of algebraic logic it became apparent however that classical propositional calculus admits other semantics. In Boolean-valued semantics (for classical propositional logic), the truth values are the elements of an arbitrary Boolean algebra; "true" corresponds to the maximal element of the algebra, and "false" corresponds to the minimal element. Intermediate elements of the algebra correspond to truth values other than "true" and "false". The principle of bivalence holds only when the Boolean algebra is taken to be the two-element algebra, which has no intermediate elements.

Non-classical logics

Main article: Non-classical logic

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.
Many-valued logic, including fuzzy logic, which 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 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;

In Deviant Logic, Fuzzy Logic: Beyond the Formalism, Susan Haack divided non-classical logics into deviant, quasi-deviant, and extended logics.^[5]

References

↑ Nicholas Bunnin; Jiyuan Yu (2004). The Blackwell dictionary of Western philosophy. Wiley-Blackwell. p. 266. ISBN 978-1-4051-0679-5.
↑ L. T. F. Gamut (1991). Logic, language, and meaning, Volume 1: Introduction to Logic. University of Chicago Press. pp. 156–157. ISBN 978-0-226-28085-1.
↑ Gabbay, Dov, (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/
1 2 Haack, Susan, (1996). Deviant Logic, Fuzzy Logic: Beyond the Formalism. Chicago: The University of Chicago Press.

Mathematical logic

General	Formal language Formation rule Formal system Deductive system Formal proof Formal semantics Well-formed formula Set Element Class Classical logic Axiom Natural deduction Rule of inference Relation Theorem Logical consequence Axiomatic system Type theory Symbol Syntax Theory

Traditional logic	Proposition Inference Argument Validity Cogency Syllogism Square of opposition Venn diagram

Propositional calculus Boolean logic	Boolean functions Propositional calculus Propositional formula Logical connectives Truth tables Many-valued logic

Predicate logic	First-order Quantifiers Predicate Second-order Monadic predicate calculus

Naive set theory	Set Empty set Element Enumeration Extensionality Finite set Infinite set Subset Power set Countable set Uncountable set Recursive set Domain Codomain Image Map Function Relation Ordered pair

Set theory	Foundations of mathematics Zermelo–Fraenkel set theory Axiom of choice General set theory Kripke–Platek set theory Von Neumann–Bernays–Gödel set theory Morse–Kelley set theory Tarski–Grothendieck set theory

Model theory	Model Interpretation Non-standard model Finite model theory Truth value Validity

Proof theory	Formal proof Deductive system Formal system Theorem Logical consequence Rule of inference Syntax

Computability theory	Recursion Recursive set Recursively enumerable set Decision problem Church–Turing thesis Computable function Primitive recursive function

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Classical logic

Law of excluded middle Double negative elimination Law of noncontradiction Principle of explosion Monotonicity of entailment Idempotency of entailment Commutativity of conjunction De Morgan's laws Principle of bivalence Propositional logic Predicate logic

Classical logic

Examples of classical logics

Generalized semantics

Non-classical logics

References

Further reading