From Crispin Sartwell, "Six Names of Beauty" (Routledge 2004)

The first profound advance in logic since Plato's student Aristotle was made by Gottlob Frege in his Begriffschrift or conceptual notation. He introduced predicate logic and quantification, for the first time reaching inside sentences to reveal their logical form, and hence made possible a much greater range of inferences than can be generate in Aristotelian, or "syllogistic" logic. It was Frege's methods of displaying the logical structure of language that led to the possibility of analytic philosophy, to Bertrand Russell, Wittgenstein, Quine, Kripke: a whole century of thought and a whole world of ideas. A remarkable feature of Frege's conceptual notation is that it is a graphic display of logical relations, using horizontal and vertical lines as well as simple curves to represent various logical relations.

This way of presenting logic is, in its own way, extremely clear and economical, indeed conspicuously more clear and economical than most of its competitors, including the notation of Russell and Whitehead's Principia Mathematica, which became canonical. But the basic visual presentation also largely accounts for the fact that Frege's advances were neglected for decades. For the visual language of the Begriffschrift is very difficult to set in type, and its unfamiliarity makes it difficult also, at least at first, to master. The Russell/Whitehead notation makes use of the normal typographical symbols, albeit in unusual configurations.

But the graphical representation of the Begriffschrift is profound. It seems to transport us into the realm of logical relations, to make a visual array that we can enter: to take us into a Platonic abstract realm, a Mondrian of the mind. It emphasizes the distinction of concept from reality: the world of the Begriffschrift is a realm of purity and simplicity and abstraction that Frege made, or as he would have it, discovered. (Frege was a "Platonist": he thought of abstract realms such as that of mathematics as real, as something discovered rather than stipulated.) But it also brings this world to us in a visual array: it makes possible our transportation from the realm of concrete visual experience to the realm of pure abstract relations. Frege's language is, hence, a Platonic illumination, a glance at the true sun.