We give formal semantics in terms of a hybrid algebraic coalgebraic scheme, namely course-of-value iteration.
In this entry, we provide an account of the class of recursive functions, with particular emphasis on six basic kinds of recursion: iteration, primitive recursion, primitive recursion with parameters, course-of-value recursion, and double recursion.
Frege called the course-of-values of a concept F its extension.
This principle asserts: the course-of-values of the function ƒ is identical to the course-of-values of the function g if and only if ƒ and g map every object to the same value.
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)).
The course-of-values of a function is a record of the value of the function for each argument.
Course-of-values recursion relaxes this condition, and it allows the use of any number of values for previous arguments.
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).
When a function ƒ is a concept, Frege called the course-of-values for that concept its extension.
Frege used the a Greek epsilon with a smooth breathing mark above it as part of the notation for signifying the course-of-values of the function ƒ: ε'ƒ where the first occurrence of the Greek ε (with the smooth breathing mark above it) is a 'variable-binding operator' which we might read as 'the course-of-values of'.
Notice the use of the two values ƒ(n) and ƒ(n + 1) in the definition of ƒ(n + 2), which makes this a course-of-values recursion.
