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Encyclopedia > Classical logic

Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. They are characterised by a number of properties; non-classical logics are those that lack one or more of these properties, which are: Logic (from ancient Greek λόγος (logos), meaning reason) is the study of arguments. ...

  1. Law of the excluded middle;
  2. Law of noncontradiction;
  3. Monotonicity of entailment and Idempotency of entailment;
  4. Commutativity of conjunction;
  5. De Morgan duality: every logical operator is dual to another.

The law of excluded middle (tertium non datur in Latin) states that for any proposition P, it is true that (P or ~P). ... In logic, the law of noncontradiction judges as false any proposition P asserting that both proposition Q and its denial, proposition not-Q, are true at the same time and in the same respect. In the words of Aristotle, One cannot say of something that it is and that it... Monotonicity of entailment - Wikipedia /**/ @import /w/skins-1. ... Idempotency of entailment is a property of logical systems that states that one may derive the same consequences from many instances of a hypothesis as from just one. ... In logic, De Morgans laws (or De Morgans theorem) are rules in formal logic relating pairs of dual logical operators in a systematic manner expressed in terms of negation. ... In logical calculus, logical operators or logical connectors serve to connect statements into more complicated compound statements. ...

Examples of classical logics

  • Aristotle's Organon introduces his theory of syllogistic, which is a logic with a restricted form of judgements: 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 judgements 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 noncontradiction in justifying his system, although these laws cannot be expressed as judgements within the syllogistic framework.

Aristotle, marble copy of bronze by Lysippos. ... The Organon is the name given by Aristotles followers, the Peripatetics, for the standard collection of six of his works on logic. ... Aristotelian logic, also known as syllogistic, is the particular type of logic created by Aristotle, primarily in his works Prior Analytics and De Interpretatione. ... George Boole [], (November 2, 1815 Lincoln, Lincolnshire, England - December 8, 1864 Ballintemple, County Cork, Ireland) was a mathematician and philosopher. ... Boolean logic is a system of syllogistic logic invented by 19th-century British mathematician George Boole, which attempts to incorporate the empty set, that is, a class of non-existent entities, such as round squares, without resorting to uncertain truth values. ... Friedrich Ludwig Gottlob Frege Friedrich Ludwig Gottlob Frege (November 8, 1848 – July 26, 1925) was a German mathematician, logician, and philosopher who is regarded as a founder of both modern mathematical logic and analytic philosophy. ... Begriffsschrift is the name of a book on logic by Gottlob Frege published in 1879. ... Clarence Irving Lewis ( April 12, 1883 - February 3, 1964) was a pragmatist philosopher. ...

Non-classical logics

Intuitionistic logic, or constructivist logic, is the logic used in mathematical intuitionism and other forms of mathematical constructivism. ... A paraconsistent logic is a logic which attempts to deal with contradictions. ... Dialetheism is a paraconsistent logic typified by its tolerance of at least some contradictions. ... Relevance logic, also called relevant logic, is any of a family of non-classical substructural logics that impose certain restrictions on implication. ... Relevance logic, also called relevant logic, is any of a family of non-classical substructural logics that impose certain restrictions on implication. ... In mathematical logic, linear logic is a type of substructural logic that denies the structural rules of weakening and contraction. ... A non-monotonic logic is a formal logic whose consequence relation is not monotonic. ... In mathematical logic, linear logic is a type of substructural logic that denies the structural rules of weakening and contraction. ... Computability logic is a formal theory of computability, introduced by Giorgi Japaridze in 2003. ... Modal logic, or (less commonly) intensional logic is the branch of logic that deals with sentences that are qualified by modalities such as can, could, might, may, must, possibly, and necessarily, and others. ...

References

  • 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.

 
 

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