Non-classical logic

Non-classical logics are formal systems that differ significantly from classical logic systems such as propositional logic and predicate logic . There are several ways this can be the case - e.g. B. by varying certain basic laws of classical logic or by changing or extending them. The aim of all such deviations is to show different possibilities of logical inference and logical truth and to vary principles that are taken for granted and immovable within the classical systems.

Examples

Paraconsistent logics are formal systems,

"[...] in which the logical principle ex contradictione sequitur quodlibet ( Latin for 'anything follows from a contradiction') does not apply, in which it is not possible to derive from two contradicting statements A, ¬A or from a contradiction A∧¬ A to derive any statement. "

- Paraconsistent logic

The intuitionistic logic is based on a different concept of truth than the classical logic:

“While in classical logic the statement is interpreted truth-functionally (see truth value ) as' A applies or B applies', the same statement is interpreted in intuitionist logic as' There is a proof for A, or there is a proof for B '. ${\displaystyle A\lor B}$

This divergent interpretation of connectives (connective) shows that certain theorems of classical logic is not valid in the intuitionistic. One example is the law of the excluded , . The classic interpretation is 'A applies or A does not apply' and is easily recognized as valid. The intuitionist interpretation is 'A is proven or A is refuted'. Under this interpretation, the principle of excluded third parties is obviously not valid, on the one hand because there are statements that are neither proven nor refuted, on the other hand because there are statements that are neither provable nor refutable at all. " ${\displaystyle A\lor \neg A}$

- Intuitionist logic

Extensions of the classical logic

A special type of non-classical logic is the extension of classical logic. In a non-classical extension, additional logical operators are added, e.g. B. “ ” in modal logic ; this new symbol stands for "It is necessary that ...". - The following applies to extensions of classical logic: ${\displaystyle \Box }$

• The set of well-formulated formulas (expressions) is a real superset of the set of expressions generated by classical logic.
• The set of provable theorems is a real superset of the set of theorems that apply in classical logic - but only in the sense that the “new” theorems of extended logic are based on the formation of the new expressions.

Important classes of non-classical logics

• Multi-valued logic
• Fuzzy logic
• Intuitionist logic
• Quantum logic
• Modal logic
• Intensional logics
• Deontic logic
• Paraconsistent logic
• Linear logic , a refinement of classical and intuitionist logic.
• Relevance logic
• Non-monotonous logic
• Temporal logic
• Interrogative logic

See also : Category: Non-Classical Logic

literature

• Dov M. Gabbay: Classical vs. non-classical logic . In: DM Gabbay, CJ Hogger, JA Robinson (Eds.): Handbook of Logic in Artificial Intelligence and Logic Programming . Volume 2. Oxford University Press, 1994, chap. 2.6.
• That. / F. Günthner (Ed.): Handbook of Philosophical Logic . Volume 3: Alternative to Classical logic (= synthesis library, 166). Kluwer Publishing Group, 1986.
• Wolfgang Rautenberg : Classical and non-classical propositional logic . Vieweg, Wiesbaden 1979, ISBN 3-528-08385-9 .

Individual evidence

1. ↑ after: Susan Haack : Deviant Logic, Fuzzy Logic: Beyond the Formalism . Cambridge University Press, 1975, ISBN 0-521-20500-X