Classical logic (orstandard logic)[1][2] orFrege–Russell logic[3] is the intensively studied and most widely used class ofdeductive logic.[4] Classical logic has had much influence onanalytic philosophy.
Each logical system in this class shares characteristic properties:[5]
While not entailed by the preceding conditions, contemporary discussions of classical logic normally only includepropositional andfirst-order logics.[4][6] In other words, the overwhelming majority of time spent studying classical logic has been spent studying specifically propositional and first-order logic, as opposed to the other forms of classical logic.
Most semantics of classical logic arebivalent, meaning all of the possible denotations of propositions can be categorized as either true or false.
Classical logic is a 19th and 20th-century innovation. The name does not refer toclassical antiquity, which used theterm logic ofAristotle. Classical logic was the reconciliation of Aristotle's logic, which dominated most of the last 2000 years, with the propositionalStoic logic. The two were sometimes seen as irreconcilable.
Leibniz'scalculus ratiocinator can be seen as foreshadowing classical logic.Bernard Bolzano has the understanding ofexistential import found in classical logic and not in Aristotle. Though he never questioned Aristotle,George Boole's algebraic reformulation of logic, so-calledBoolean logic, was a predecessor of modernmathematical logic and classical logic.William Stanley Jevons andJohn Venn, who also had the modern understanding of existential import, expanded Boole's system.
The originalfirst-order, classical logic is found inGottlob Frege'sBegriffsschrift. It has a wider application than Aristotle's logic and is capable of expressing Aristotle's logic as a special case. It explains thequantifiers in terms of mathematical functions. It was also the first logic capable of dealing with theproblem of multiple generality, for which Aristotle's system was impotent. Frege, who is considered the founder of analytic philosophy, invented it to show all of mathematics was derivable from logic, and makearithmetic rigorous asDavid Hilbert had done forgeometry, the doctrine is known aslogicism in thefoundations of mathematics. The notation Frege used never much caught on.Hugh MacColl published a variant of propositional logic two years prior.
The writings ofAugustus De Morgan andCharles Sanders Peirce also pioneered classical logic with the logic of relations. Peirce influencedGiuseppe Peano andErnst Schröder.
Classical logic reached fruition inBertrand Russell andA. N. Whitehead'sPrincipia Mathematica, andLudwig Wittgenstein'sTractatus Logico Philosophicus. Russell and Whitehead were influenced by Peano (it uses his notation) and Frege and sought to show mathematics was derived from logic. Wittgenstein was influenced by Frege and Russell and initially considered theTractatus to have solved all problems of philosophy.
Willard Van Orman Quine believed that a formal system that allows quantification over predicates (higher-order logic) didn't meet the requirements to be a logic, saying that it was "set theory in disguise".
Classical logic is the standard logic of mathematics. Many mathematical theorems rely on classical rules of inference such asdisjunctive syllogism and thedouble negation elimination. The adjective "classical" in logic is not related to the use of the adjective "classical" in physics, which has another meaning. In logic, "classical" simply means "standard". Classical logic should also not be confused withterm logic, also known as Aristotelian logic.
Jan Łukasiewicz pioneerednon-classical logic.
With the advent ofalgebraic logic, it became apparent that classicalpropositional calculus admits othersemantics. InBoolean-valued semantics (for classicalpropositional logic), the truth values are the elements of an arbitraryBoolean 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 thetwo-element algebra, which has no intermediate elements.
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