Classic logic

Under the classical logic one is understood to mean a logical system that the propositional , the predicate logic of first or higher level, and generally the (logical) identity term includes. Gottlob Frege developed the first axiomatization of such a system in his conceptual writing (1879).

Classic logic is characterized by exactly two properties:

Every statement has one of exactly two truth values , mostly false and true ( principle of two- valued / bivalence principle).

The truth value of every compound statement is uniquely determined by the truth values of its partial statements ( principle of extensionality ).

The principle of bivalent is to be distinguished from the principle of the excluded third :

{\ displaystyle p \ lor \ neg p}

(e.g. "It is raining, or it is not the case that it is raining.")

represents a sentence of classical propositional logic, so it can be derived syntactically from the rules and axioms of the logical system without the concept of truth playing an explicit role. In contrast, the principle of bivalent is a statement about the semantics of logic, which assigns a truth value to every statement.

In contrast to classical logic, non-classical logic systems arise when the principle of duality, the principle of extensionality or even both principles are abolished. Non-classical logics that arise from the abolition of the principle of two- valued are multi-valued logics . The number of truth values (perhaps better: pseudo truth values) can be finite (e.g. three-valued logic), but is often also infinite (e.g. fuzzy logic ). However, logics caused by the abolition of extensionality, use logical operators (connectives) in which the truth value of the composite set can no longer be unambiguously determined from the truth value of its parts. An example of non-extension logic is modal logic , which introduces the single-digit non-extension operators “it is necessary that” and “it is possible that”. Another example is intuitionist logic , which does not introduce any new operators but interprets the existing operators differently.

The algebraic structure of classical propositional logic is a two-element Boolean algebra . Formal two-valued logic in the modern sense was developed by Boole, Frege, and others in the second half of the 19th century. The term “classical logic” then emerged in the 20th century to distinguish it from a number of other logics that are described as non-classical.

Sometimes the term classical logic is also used as a historical term; H. based on logicians of antiquity. But in antiquity not only classical logic was practiced; on the contrary, Aristotle , the classic logician who is exemplary in the historical sense, dealt with facts of non-classical logic. Depending on the context, it is not always easy to see in which sense a speaker is using the term “classical logic”.

Examples of classically valid statements

Some well-known statements that are valid in classical logic are as follows:

Theorem of excluded contradiction: ${\ displaystyle \ neg (p \ land \ neg p)}$ (e.g. "It is not the case that it rains (at the same time and in the same place) and does not rain.")
Sentence of the excluded third party: ${\ displaystyle p \ lor \ neg p}$ (e.g. "The earth is round, or it is not the case that the earth is round.")
Verum sequitur ex quodlibet (truth follows from anything): ${\ displaystyle p \ rightarrow (q \ rightarrow p)}$ (e.g. "If it rains, then it rains (also) provided that the earth is flat.")
Ex falso sequitur quodlibet (anything wrong follows anything): ${\ displaystyle \ neg p \ rightarrow (p \ rightarrow q)}$ (e.g. "If it does not rain, then provided that it rains (in the same place and at the same time), the earth is flat.")
Paradox of the material implication: ${\ displaystyle (p \ rightarrow q) \ vee (q \ rightarrow p)}$ (Of any two sentences, at least one is always the sufficient condition for the other.)

literature

Antoine Arnauld , Pierre Nicole : The logic or the art of thinking. Translated from the French and introduced by Christos Axelos. 2nd revised edition with an introduction added. Wissenschaftliche Buchgesellschaft, Darmstadt 1994, ISBN 3-534-03710-3 ( library of classical texts ).
Thank God Frege : Conceptual writing, one of the arithmetic simulated formula language of pure thinking. Nebert, Halle 1879.
Dov Gabbay: Classical vs non-classical logic. In: Don M. Gabbay, Christopher J. Hogger, JA Robinson, J. Siekmann (Eds.): Handbook of Logic in Artificial Intelligence and Logic Programming. Volume 2: Deduction Methodologies. Clarendon Press, Oxford 1994, ISBN 0-19-853746-8 , Chapters 2, 6.
Lothar Kreiser, Siegfried Gottwald , Werner Stelzner (eds.): Non-classical logic. An introduction. 2nd revised edition. Akademie-Verlag, Berlin 1990, ISBN 3-05-000274-3 .
Jan Łukasiewicz : O zasadzie wyłączonego środka. In: Przegląd filozoficzny. 13, 1910, ISSN 1230-1493 , pp. 372-373 (On the proposition of excluded third parties).
Jan Łukasiewicz: Z historii logiki zdań. In: Przegląd filozoficzny. 37, 1934, pp. 417-437 (discovery of the stoic connectivity logic).
Jan Łukasiewicz: On the history of propositional logic. In: Knowledge 5, 1935, ISSN 0165-0106 , pp. 111-131. (German translation of the article: Z historii logiki zdań ).

Web links

Stewart Shapiro: Classical Logic. In: Edward N. Zalta (Ed.): Stanford Encyclopedia of Philosophy .