Begriffsschrift (German for, roughly, "concept-script") is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book. Begriffsschrift is usually translated as concept writing or concept notation; the full title of the book identifies it as "a formula language, modeled on that of arithmetic, for pure thought." Frege's motivation for developing his formal approach to logic resembled Leibniz's motivation for his calculus ratiocinator (despite that, in the foreword Frege clearly denies that he achieved this aim, and also that his main aim would be constructing an ideal language like Leibniz's, which Frege declares to be a quite hard and idealistic—though not impossible—task). 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 future British and Anglo philosophers such as Bertrand Russell further developed. Notation and the system The calculus contains the first appearance of quantified variables, and is essentially classical bivalent second-order logic with identity. It is bivalent in that sentences or formulas denote either True or False; second order because it includes relation variables in addition to object variables and allows quantification over both. The modifier "with identity" specifies that the language includes the identity relation, =. Frege stated that his book was his version of a characteristica universalis, a Leibnizian concept that would be applied in mathematics. Frege presents his calculus using idiosyncratic two-dimensional notation: connectives and quantifiers are written using lines connecting formulas, rather than the symbols ¬, ∧, and ∀ in use today. For example, that judgement B materially implies judgement A, i.e. is written as . 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. In chapter 1, §5, Frege defines the conditional as follows: "Let A and B refer to judgeable contents, then the four possibilities are: A is asserted, B is asserted; A is asserted, B is negated; A is negated, B is asserted; A is negated, B is negated. Let signify that the third of those possibilities does not obtain, but one of the three others does. So if we