We give formal semantics in terms of a hybrid algebraic coalgebraic scheme, namely course-of-value iteration.
2.4 Courses-of-Values, Extensions, and Proposed Mathematical Foundations 2.4.1 Courses-of-Values and Extensions Frege's ontology consisted of two fundamentally different types of entities, namely, functions and objects (1891, 1892b, 1904).
Though we shall briefly discuss Frege's notation for courses-of-values, we'll subsequently switch to simpler notation for naming the extensions of concepts.
However, both systems imply the existence of objects which are not extensions (or courses-of-values), and indeed, they imply an infinite number of such objects.
However, recent work by Wehmeier (1999) suggests that, in §10, Frege was not attempting to restrict the quantifiers of his system to extensions (nor, more generally, to courses-of-values).
The principle Frege used to systematize courses-of-values is Basic Law V (1893/§20;): The course-of-values of the concept ƒ is identical to the course-of-values of the concept g if and only if ƒ and g agree on the value of every argument (i.e., if and only if for every object x, ƒ(x) = g(x)).
Readers whose main goal is to understand Frege's Theorem can now skip directly to Section 3. Though the present section is not required for understanding the proof of Frege's Theorem, we include it so that the reader can get some sense of how second-order logic (with comprehension) gives rise to Russell's paradox when one adds Frege's theory of courses-of-values and extensions.
Frege called the course-of-values of a concept F its extension.
The course-of-values of a function is a record of the value of the function for each argument.
