Class logic

In a broader sense, class logic is a logic whose objects are called classes. In the narrower sense, one speaks of class logic only when classes are described by a property of their elements. This class logic is therefore a generalization of set theory that only allows a restricted class formation.

Class logic in the broader sense

The forerunners of the class logic are the Dihairesis in Plato and above all the syllogistics of Aristotle together with their later modifications. Aristotle mostly works with concepts (termini, terms) without referring to them as classes. Not until 1847 did George Boole designate concepts as classes in his mathematical analysis of the Aristotelian syllogistics; however, he did not describe classes in terms of their elements and their properties, so that he did not yet have class logic in the narrower sense. This also applies to modern Boolean algebra , which Boole's successors developed from his calculus.

The Neumann-Bernays-Gödel set theory NBG from 1937 to 1940 is also a class logic in the broader sense, which is designed as an extension of the Zermelo-Fraenkel set theory ZF, so that in addition to the usual sets it also has so-called real classes as objects that are absent in ZF because they produce antinomies in naive set theory . In today's strict formalization, NBG and ZF are based on predicate logic and officially do not have any class terms typical for class logic , but only use them virtually as spelling on a metalinguistic level. In practice, both set theories are therefore notated according to class. However, they can also be set up formally correctly within the framework of a class logic in the narrower sense. ${\displaystyle \left\{x\mid A(x)\right\}}$

Class logic in the narrower sense

The first class logic in the narrower sense was created by Giuseppe Peano in 1889 as the basis for his arithmetic ( Peano axioms ). He introduced the class term, which describes classes formally correctly through a property of their elements. Today this class term is noted in the form where there is any statement that all class elements satisfy. Peano axiomatized the class term for the first time and used it without restriction. Gottlob Frege also tried in 1893 to justify arithmetic in a logic with class terms; in 1902, however , Bertrand Russell discovered a contradiction in it, which became known as Russell's antinomy . This made it generally known that one cannot use class terms without hesitation. ${\displaystyle \left\{x\mid A(x)\right\}}$ ${\displaystyle A(x)}$ ${\displaystyle x}$

To solve the problem, Russell developed his type theory from 1903 to 1908 , which only allowed a very limited use of class terms. In the long run, however, it did not prevail, but rather the more convenient and efficient set theory initiated by Ernst Zermelo in 1907. However, in its current form (ZF or NBG) it is no longer a class logic in the narrower sense, since it does not axiomatize the class term, but only uses it in practice as a useful notation. In 1937, Willard Van Orman Quine described a set theory based not on Cantor or Zermelo-Fraenkel, but on type theory in New Foundations (NF). 1940 Quine expanded NF to Mathematical Logic (ML). Since the antinomy of Burali-Forti could be derived in the first version of ML , Quine specified ML, retained the widespread use of classes, took up a suggestion by Hao Wang and introduced it into his set theory as a virtual class in 1963 , so that classes are not fully-fledged terms, but partial terms in defined contexts. ${\displaystyle \left\{x\mid A(x)\right\}}$

Starting in 1974 , Arnold Oberschelp developed the first fully functional modern axiomatic class logic based on Quine . It is a consistent extension of predicate logic and allows (like Peano) the unrestricted use of class terms. It also uses all classes that produce antinomies in naive set theory as terms. This is possible because it does not take any axioms of existence for classes. In particular, it does not require set axioms, but it can also include them and formulate them syntactically correct in the traditionally simple representation with class terms; For example, the Oberschelp set theory develops the Zermelo-Fraenkel set theory within the framework of general class logic. A class-logical increase in the ZF language is guaranteed by three principles with which cumbersome ZF formulas can be translated into convenient class formulas; without set axioms, together with the axioms of predicate logic, they form an axiom system for a simple general class logic:

The principle of abstraction states that classes describe their elements through a logical property:

${\displaystyle \forall y\colon (y\in \{x\mid A(x)\}\iff A(y))}$

The extensionality principle describes the equality of classes through the correspondence of their elements and eliminates the need for the axiom of extensionality in ZF, which is restricted to sets :

${\displaystyle A=B\iff \forall x\colon (x\in A\iff x\in B)}$

The Comprehension Principle defines the existence of a class as an element:

${\displaystyle \{x\mid A(x)\}\in B\iff \exists y\colon (y=\{x\mid A(x)\}\land y\in B)}$

Efficiency

"A class logic language corresponds to the mathematical language actually used far better than a predicate logic language."

- Oberschelp : General set theory, 1994, foreword on page 5

Literature (chronological)

• Giuseppe Peano : Arithmetices principia. Nova method exposures. Corso, Torino et al. 1889 (Also in: Giuseppe Peano: Opere scelte. Volume 2. Cremonese, Rome 1958, pp. 20-55).
• G. Frege : Fundamental laws of arithmetic. Derived conceptually. Volume 1. Pohle, Jena 1893.
• Willard Van Orman Quine : New Foundations for Mathematical Logic. In: American Mathematical Monthly 44 (1937), pp. 70-80.
• Willard Van Orman Quine: Set Theory and its Logic. Harvard University Press, Cambridge MA 1963 (German translation: set theory and their logic (= logic and foundations of mathematics. Vol. 10). Vieweg, Braunschweig 1973, ISBN 3-548-03532-9 ).
• Arnold Oberschelp: Elementary logic and set theory (= BI university pocket books 407–408). 2 volumes. Bibliographisches Institut, Mannheim et al. 1974–1978, ISBN 3-411-00407-X (vol. 1), ISBN 3-411-00408-8 (vol. 2).
• Albert Menne outline of formal logic (= Uni-Taschenbücher 59 UTB for science ). Schöningh, Paderborn 1983, ISBN 3-506-99153-1 (from the 5th edition renamed from Grundriß der Logistik - The book shows, in addition to other calculi , a possible calculation of class logic, based on the propositional and predicate calculus, and uses these to introduce basic terms of the formal system of class logic. It also briefly deals with paradoxes and type theory).
• Jürgen-Michael Glubrecht, Arnold Oberschelp, Günter Todt: Class logic. Bibliographisches Institut, Mannheim et al. 1983, ISBN 3-411-01634-5 .
• Arnold Oberschelp: General set theory. BI-Wissenschafts-Verlag, Mannheim et al. 1994, ISBN 3-411-17271-1 .

Individual evidence

1. ↑ John Barkley Rosser : Burali-Forti paradox. In: Journal of Symbolic Logic. Volume 7, 1942, pp. 1-17.
2. ^ Hao Wang: A formal system for logic. In: Journal of Symbolic Logic. Volume 15, 1950, pp. 25-32.
3. ↑ Willard Van Orman Quine: Set theory and its logic. 1973, p. 12.
4. ↑ Arnold Oberschelp: General set theory. 1994, p. 75 f.
5. ↑ The advantages of class logic are shown in a comparison of ZFC in the form of class logic and predicate logic in: Arnold Oberschelp: General set theory. 1994, p. 261.
6. ^ Arnold Oberschelp, p. 262, 41.7. The axiomatic is much more complicated there, but is reduced to the essentials at the end of the book.