Frege’s definitions in Part III of Begriffsschrift introduce novel forms of variable-binding and quantification. Frege’s commentary, however, shows that he did not fully grasp the logical significance of his notation, treating the new variables as themselves somehow defined. In Grundgesetze, such issues are avoided by exploiting the value-range notation as a substitute for functional abstraction, relying on the inconsistent Basic Law V. In presenting Frege’s Theorem without appeal to Basic Law V, Richard Heck reinstates a generalized form of the notation of Begriffsschrift by using a higher-order logic employing generalized variable-binding to form names of higher-level functions. Heck and Robert May suggest that by the time of Grundgesetze Frege had achieved the necessary understanding of variable-binding to appreciate Heck’s notation. However, even the later Frege had only a piecemeal characterization of what we now see as falling together as variable-binding. For him, different forms of variable-binding do different logical work, marked by distinct ranges of variables. Eliminating the value-range notation in his definitions after the manner of Heck would require rethinking the role of variable-binding operators. This would not be philosophically cost-free for Frege, though there is some slight evidence that Frege may have begun to move in this direction toward the end of his career.
The role of higher-order modes on the electromagnetic whistler-cyclotron wave fluctuations of thermal and non-thermal plasmas