While intuitionistic logic is obtained from classical logic by dropping the principle of the excluded third, other logics have also been proposed, though none has had a comparable impact on the foundations of mathematics. One may mention many-valued, or multivalued, logics, which admit a finite number of truth-values; fuzzy logic, with an imprecise membership relationship (though, paradoxically, a precise equality relation); and quantum logic, where conjunction may be only partially defined and implication may not be defined at all. Perhaps more important have been various so-called substructural logics in which the usual properties of the deduction symbol are weakened: relevance logic is studied by philosophers, linear logic by computer scientists, and a noncommutative version of the latter by linguists.
Zenos-paradox-illustrated-by-Achilles-racing-a-tortoiseFigure 1: Zeno’s paradox, illustrated by Achilles racing a tortoise.[Credits : Encyclopædia Britannica, Inc.]
Contrasting-triangles-in-Euclidean-elliptic-and-hyperbolic-spacesFigure 2: Contrasting triangles in Euclidean, elliptic, and hyperbolic spaces.[Credits : Encyclopædia Britannica, Inc.]
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