Begriffsschrift - Influence On Other Works

Influence On Other Works

For a careful recent study of how the Begriffsschrift was reviewed in the German mathematical literature, see Vilko (1998). Some reviewers, especially Ernst Schröder, were on the whole favorable. All work in formal logic subsequent to the Begriffsschrift is indebted to it, because its second-order logic was the first formal logic capable of representing a fair bit of mathematics and natural language.

Some vestige of Frege's notation survives in the "turnstile" symbol derived from his "Inhaltsstrich"´(i.e. content-dash) ── and "Urteilsstrich" (judging/infering-dash) │. Frege used these symbols in the Begriffsschrift in the unified form ├─ for declaring that a proposition is true. In his later "Grundgesetze" he revises slightly his interpretation of the ├─ symbol.

In "Begriffsschrift" the "Definitionsdoppelstrich" (i.e. definition-double-dash) │├─ indicates that a proposition is a definition. Furthermore, the negation sign can be read as a combination of the horizontal Inhaltsstrich with a vertical negation stroke. This negation symbol was reintroduced by Arend Heyting in 1930 to distinguish intuitionistic from classical negation. It also appears in Gerhard Gentzen's doctoral dissertation.

In the Tractatus Logico Philosophicus, Ludwig Wittgenstein pays homage to Frege by employing the term Begriffsschrift as a synonym for logical formalism.

Frege's 1892 essay, "Sense and reference" recants some of the conclusions of the Begriffsschrifft about identity (denoted in mathematics by the = sign). In particular, he rejects the "Begriffsschrift" view that the identity predicate expresses a relationship between names, in favor of the conclusion that it expresses a relationship between the objects that are denoted by those names.