World Library  
Flag as Inappropriate
Email this Article

Classical logic

Article Id: WHEBN0000055868
Reproduction Date:

Title: Classical logic  
Author: World Heritage Encyclopedia
Language: English
Subject: Natural deduction, Logical connective, Foundations of mathematics, Logic, Classical logic
Collection: Classical Logic, History of Logic, Logic, Term Logic
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Classical logic

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]

  1. Law of the excluded middle and double negative elimination;
  2. Law of noncontradiction, and the principle of explosion;
  3. Monotonicity of entailment and idempotency of entailment;
  4. Commutativity of conjunction;
  5. 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]

The intended semantics of classical logic is bivalent. 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.

Contents

  • Examples of classical logics 1
  • Non-classical logics 2
  • References 3
  • Further reading 4

Examples of classical logics

  • 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.
  • Boolean logic;
  • The first-order logic found in Gottlob Frege's Begriffsschrift.

Non-classical logics

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

References

  1. ^ Nicholas Bunnin; Jiyuan Yu (2004). The Blackwell dictionary of Western philosophy. Wiley-Blackwell. p. 266.  
  2. ^  
  3. ^ 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.
  4. ^ 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. ^ a b Haack, Susan, (1996). Deviant Logic, Fuzzy Logic: Beyond the Formalism. Chicago: The University of Chicago Press.

Further 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
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 



Copyright © World Library Foundation. All rights reserved. eBooks from World eBook Library are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.