Double Value-Ranges
PETER M. SIMONS · Oxford University Press eBooks · 2019
From the time of Begriffsschrift onwards, Frege treated functions of two or more places on a par with those of one place. This included the treatment of relations (Beziehungen) as a special case of polyadic functions in the way that concepts (Begriffe) were a special case of monadic functions. By the time of Grundgesetze (and unlike in Begriffsschrift), Frege dealt with relations largely through their extensions, which were what he called “double value-ranges” (Doppelwerthverläufe). This is in some ways a misnomer, since double value-ranges are simply a special case of single or ordinary value-ranges, namely value-ranges of functions derived from the value-ranges of monadic functions with additional saturated places. Frege’s treatment of the extensions of relations (which he came to call simply “Relationen”) thus embodies a move analogous to the treatment of polyadic functions as functions of functions, a device invented in 1920 by Moses Schönfinkel and since (unfairly) known in combinatory logic as “currying”. This paper considers the details of Frege’s Grundgesetze treatment of relations via their extensions, exhibits its grammar, and indicates its formal elegance by comparing it with other possible treatments.