# Classical logic

Classical logic (or standard logic) is the intensively studied and most widely used class of logics. Classical logic_sentence_0

Classical logic has had much influence on analytic philosophy, the type of philosophy most often found in the English-speaking world. Classical logic_sentence_1

## Characteristics Classical logic_section_0

Each logical system in this class shares characteristic properties: Classical logic_sentence_2

Classical logic_ordered_list_0

1. Law of excluded middle and double negation eliminationClassical logic_item_0_0
2. Law of noncontradiction, and the principle of explosionClassical logic_item_0_1
3. Monotonicity of entailment and idempotency of entailmentClassical logic_item_0_2
4. Commutativity of conjunctionClassical logic_item_0_3
5. De Morgan duality: every logical operator is dual to anotherClassical logic_item_0_4

While not entailed by the preceding conditions, contemporary discussions of classical logic normally only include propositional and first-order logics. Classical logic_sentence_3

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. Classical logic_sentence_4

Most semantics of classical logic are bivalent, meaning all of the possible denotations of propositions can be categorised as either true or false. Classical logic_sentence_5

## History Classical logic_section_1

Main article: History of logic Classical logic_sentence_6

Classical logic is a 19th and 20th century innovation. Classical logic_sentence_7

The name does not refer to classical antiquity, which used the term logic of Aristotle. Classical logic_sentence_8

In fact, classical logic was the reconciliation of Aristotle's logic, which dominated most of the last 2000 years, with the propositional Stoic logic. Classical logic_sentence_9

The two were sometimes seen as irreconcilable. Classical logic_sentence_10

Leibniz's calculus ratiocinator can be seen as foreshadowing classical logic. Classical logic_sentence_11

Bernard Bolzano has the understanding of existential import found in classical logic and not in Aristotle. Classical logic_sentence_12

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. Classical logic_sentence_13

William Stanley Jevons and John Venn, who also had the modern understanding of existential import, expanded Boole's system. Classical logic_sentence_14

The original first-order, classical logic is found in Gottlob Frege's Begriffsschrift. Classical logic_sentence_15

It has a wider application than Aristotle's logic, and is capable of expressing Aristotle's logic as a special case. Classical logic_sentence_16

It explains the quantifiers in terms of mathematical functions. Classical logic_sentence_17

It was also the first logic capable of dealing with the problem of multiple generality, for which Aristotle's system was impotent. Classical logic_sentence_18

Frege, who is considered the founder of analytic philosophy, invented it so as to show all of mathematics was derivable from logic, and make arithmetic rigorous as David Hilbert had done for geometry, the doctrine known as logicism in the foundations of mathematics. Classical logic_sentence_19

The notation Frege used never much caught on. Classical logic_sentence_20

Hugh MacColl published a variant of propositional logic two years prior. Classical logic_sentence_21

The writings of Augustus De Morgan and Charles Sanders Peirce also pioneered classical logic with the logic of relations. Classical logic_sentence_22

Peirce influenced Giuseppe Peano and Ernst Schröder. Classical logic_sentence_23

Classical logic reached fruition in Bertrand Russell and A. Classical logic_sentence_24 N. Whitehead's Principia Mathematica, and Ludwig Wittgenstein's Tractatus Logico Philosophicus. Classical logic_sentence_25

Russell and Whitehead were influenced by Peano (it uses his notation) and Frege, and sought to show mathematics was derived from logic. Classical logic_sentence_26

Wittgenstein was influenced by Frege and Russell, and initially considered the Tractatus to have solved all problems of philosophy. Classical logic_sentence_27

Willard Van Orman Quine insisted on classical, first-order logic as the true logic, saying higher-order logic was "set theory in disguise". Classical logic_sentence_28

Jan Łukasiewicz pioneered non-classical logic. Classical logic_sentence_29

## Generalized semantics Classical logic_section_2

With the advent of algebraic logic it became apparent that classical propositional calculus admits other semantics. Classical logic_sentence_30

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. Classical logic_sentence_31

Intermediate elements of the algebra correspond to truth values other than "true" and "false". Classical logic_sentence_32

The principle of bivalence holds only when the Boolean algebra is taken to be the two-element algebra, which has no intermediate elements. Classical logic_sentence_33