Begriffsschrift
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, of pure thought." Frege's motivation for developing his formal approach to logic resembled Leibniz's motivation for his calculus ratiocinator (despite that, in his 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, however 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.
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 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 
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