I have been reading Kevin C. Klement's article on Gottlob Frege in the Internet Encyclopedia of Philosophy. I am confused by the term "value-range", which is used numerous times throughout the article. Here is what is said about it:

"value-ranges" (Werthverläufe) of functions, understood as objects corresponding to the complete argument-value mappings generated by functions.

I could use some help parsing this concept.

  • At the time of Frege's writing logicians did not yet clearly distinguish between extension, set of pre-defined objects falling under a concept, and intensional class of objects, possibly not yet known or defined, that might conceivably fall under the concept at some point. Frege's "value-ranges" were a conflation of the two, which is why he got in trouble with his Basic Law V. "Value-range of a concept not included in itself" is ambiguous as to its extension (the value-range itself is not pre-defined as an object), if it is treated as a set it leads to Russell's paradox.
    – Conifold
    Sep 12 '17 at 20:10
  • @Conifold can one create an axiom to restrict "the set that does not include itself" as a predicate and be free of the paradox?
    – Nick
    Oct 23 at 14:41
  • @Nick That's what Zermelo did by restricting comprehension to predefined sets, it is the specification axiom schema of ZF.
    – Conifold
    Oct 24 at 4:48

Consider a numerical function, defined by:

f(1)=2, f(2)=4, ...

The Wertverlauf is the "graph" of the function, i.e. the collection of couples:

(1,2), (2,4),...

For concepts, i.e. functions from objects to truth-values, like e.g. Philosopher, where we have:

Phil(Plato)=True and Phil(Napoleon)=False, etc.

we have similarly for the Wertverlauf:

(Plato,True), (Napoleon, False), ...

Thus, strictly speaking, the Wertverlauf of a concept is not (but quite similar to) its extension: the collection of objects that fall under the concept (i.e. that satisfy it). [See Frege: "In the case of concepts, their value-ranges were identified with their extensions."]

See Function and Concept (1891): "Wertverlauf" has been translated as "value-range" and as "course-of-values".

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