Classical logic

Classical logic (or standard logic) is an intensively studied and most widely used class of logics. Classical logic has had much influence on analytic philosophy, the type of philosophy most often found in the English-speaking world.

Characteristics

Each logical system in this class shares characteristic properties:

1. Law of excluded middle and double negative elimination
2. Law of noncontradiction, and the principle of explosion
3. Monotonicity of entailment and idempotency of entailment
4. Commutativity of conjunction
5. De Morgan duality: every logical operator is dual to another

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

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

History

Main article: History of logic

Begriffsschrift title page

Classical logic was a 19th and 20th century innovation. The name does not refer to classical antiquity, which used the logic of Aristotle. In fact, classical logic was the departure from Aristotle's logic which reigned supreme for over 2000 years. It arguably shares more with Stoic logic, given its use of propositions.

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 quantifier 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 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. The notation Frege used never much caught on.

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. The results of Kurt Goedel and Alfred Tarski undermined the logicist project.

Generalized semantics

With the advent of algebraic logic it became apparent that classical propositional calculus admits other semantics. 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. 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 the two-element algebra, which has no intermediate elements.

Non-classical logics

Main article: Non-classical logic

• Computability logic is a semantically constructed formal theory of computability—as opposed to classical logic, which is a formal theory of truth—integrates and extends classical, linear and intuitionistic logics.
• Many-valued logic rejects bivalence, allowing for truth values other than true and false. The most popular forms are three-valued logic, as initially developed by Jan Łukasiewicz, and infinitely-valued logics such as fuzzy logic, which permits any real number between 0 and 1 as a truth value.
• Intuitionistic logic rejects the law of the excluded middle, double negation elimination, and part of De Morgan's laws;

• Linear logic rejects idempotency of entailment as well;
• Modal logic extends classical logic with non-truth-functional ("modal") operators.
• Paraconsistent logic (e.g., relevance logic) rejects the principle of explosion, and has a close relation to dialetheism;

• Quantum logic
• Relevance logic, linear logic, and non-monotonic logic reject monotonicity of entailment;
• Non-reflexive logic (also known as "Schrödinger logics") rejects or restricts the law of identity;

In Deviant Logic, Fuzzy Logic: Beyond the Formalism, Susan Haack divided non-classical logics into deviant, quasi-deviant, and extended logics.

Further reading

• Graham Priest, An Introduction to Non-Classical Logic: From If to Is, 2nd Edition, CUP, 2008, ISBN 978-0-521-67026-5
• Warren Goldfard, "Deductive Logic", 1st edition, 2003, ISBN 0-87220-660-2