Article
History of Classical Logic
Classical logic is a 19th and 20th-century
innovation. The name does not refer to classical antiquity, which used
the term logic of Aristotle. Classical logic was the reconciliation of
Aristotle's logic, which dominated most of the last 2000 years, with the
propositional Stoic logic. The two were sometimes seen as
irreconcilable.
Leibniz's calculus ratiocinator can be seen as
foreshadowing classical logic. Bernard Bolzano has the understanding of
existential import found in classical logic and not in Aristotle. Though
he never questioned Aristotle, George Boole's algebraic reformulation
of logic, so-called Boolean logic, was a predecessor of modern
mathematical logic and classical logic. William Stanley Jevons and John
Venn, who also had the modern understanding of existential import,
expanded Boole's system.
The original first-order, classical logic is
found in Gottlob Frege's Begriffsschrift. It has a wider application
than Aristotle's logic and is capable of expressing Aristotle's logic as
a special case. It explains the quantifiers in terms of mathematical
functions. It was also the first logic capable of dealing with the
problem 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
make arithmetic rigorous as David Hilbert had done for geometry, the
doctrine is known as logicism in the foundations of mathematics. The
notation Frege used never much caught on. Hugh MacColl published a
variant of propositional logic two years prior.
The writings of Augustus De Morgan and Charles
Sanders Peirce also pioneered classical logic with the logic of
relations. Peirce influenced Giuseppe Peano and Ernst Schröder.
Classical logic reached fruition in Bertrand
Russell and A. N. Whitehead's Principia Mathematica, and Ludwig
Wittgenstein's Tractatus 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 the Tractatus to have solved
all problems of philosophy.
Willard Van Orman Quine insisted on classical,
first-order logic as the true logic, saying higher-order logic was "set
theory in disguise".
Jan Łukasiewicz pioneered non-classical logic.