Frege went on to employ his logical calculus in his research on the foundations of mathematics, carried out over the next quarter-century.

This is the first work in Analytical Philosophy, a field that later British and Anglo philosophers such as Bertrand Russell further developed.

The calculus contains the first appearance of quantified variables, and is essentially classical bivalent second-order logic with identity.

Frege stated that his book was his version of a characteristica universalis, a Leibnizian concept that would be applied in mathematics.

In the first chapter, Frege defines basic ideas and notation, like proposition ("judgement"), the universal quantifier ("the generality"), the conditional, negation and the "sign for identity of content"

(which he used to indicate both material equivalence and identity proper); in the second chapter he declares nine formalized propositions as axioms.

Frege declared nine of his propositions to be axioms, and justified them by arguing informally that, given their intended meanings, they express self-evident truths.

All other propositions are deduced from (1)–(9) by invoking any of the following inference rules: The main results of the third chapter, titled "Parts from a general series theory," concern what is now called the ancestral of a relation R. "a is an R-ancestor of b" is written "aR*b".

Frege applied the results from the Begriffsschrifft, including those on the ancestral of a relation, in his later work The Foundations of Arithmetic.

For a careful recent study of how the Begriffsschrift was reviewed in the German mathematical literature, see Vilko (1998).

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, "On Sense and Reference," recants some of the conclusions of the Begriffsschrifft about identity (denoted in mathematics by the "=" sign).