Classical logic

Under the classical logic is defined as a logical system that generally contains the statements, the first-or higher -order predicate logic as well as the (logical) concept of identity. A first axiomatization of such a system has developed Gottlob Frege in his Begriffsschrift (1879 ).

Classical logic is characterized by exactly two properties:

• Each statement has one of exactly two truth values, mostly false and true (the principle of bivalence / bivalence ).
• The truth value of each composite statement is uniquely determined by the truth values ​​of its sub-statements (the principle of extensionality ).

The principle of bivalence is to be distinguished from the law of excluded middle:

Represents a set of classical propositional logic, that can be syntactically derived from the axioms and rules of the logical system without the concept of truth explicitly plays a role. In contrast, the principle of the two- value is an indication of the semantics of the logic which associates each message a truth value.

In contrast to classical logic do not arise classical logic systems when you pick up the principle of bivalence, the principle of extensionality or even both principles. Non-classical logics that arise from the abolition of the principle of bivalence, are Multivalued logics. The number of truth values ​​( perhaps better pseudo- truth values) may be finite (eg, three-valued logic ), but is often infinite ( eg, fuzzy logic). Logics that caused by removal of extensionality, however, use connectives ( connectives ) in which the truth value of the composite set can no longer be unambiguously determined from the truth values ​​of its parts. An example of nichtextensionale logic is the modal logic that. Nichtextensionalen the digit operators " it is necessary that " and "it is possible that " introduces Another example, the logic intuitionist, although not introduce new operators, but the existing operators is interpreted differently.

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

Sometimes the term classical logic used as a historical term, that is based on logician of antiquity. But now not only classical logic was operated in antiquity quite; Rather, Aristotle, in the historical sense as exemplary classical logician, issues treated non-classical logic. It is - depending on the context - not always easy to see in what sense a spokesman / a spokeswoman for the term " classical logic " used.

Examples classically valid statements

Some well-known statements that are valid in classical logic are the following: