Frege: how are functions defined in concept script?

Question

Does anyone have an example of defining a functions both untyped and typed arguments using concept script? Defining functions with single, multiple arguments and varying arguments? My background is CS and perhaps guidance on a simple function like OR or NAND would be helpful.

I had a theory that a function was an inference string and its arguments the first inference. To get the value of a function, one would deduct the argument. For instance $\phi \implies (B \implies (A))$ and invoking $\phi(B) \equiv A$. For multiple arguments $\phi \implies ((C \land D) \implies (A))$. Perhaps my thinking's there is incorrect.

Any guidance appreciated.

Thanks

Answer 1

The ontology of Frege's logic is : there are objects and there are functions.

Objects are "argument" of functions but not vice-versa.

The concept-script has symbols for both, that means that functions in Frege's Begriffsschrift are expressed through function symbols; see §10 :

$\Phi(A)$ and $\Psi(A,B)$.

Some further examples from §23-on, withe the General Theory of Sequences.

See also Frege's Logic :

It is important to mention here that the predicate calculus formulable in Frege's logic is a ‘second-order’ predicate calculus. This means it allows quantification over functions as well as quantification over objects; i.e., statements of the form ‘Every function ƒ is such that …’ and ‘Some function ƒ is such that …’ are allowed. Thus, the statement ‘objects a and b fall under the same concepts’ would be written as follows in Frege' notation:

$(\mathfrak f) \mathfrak fa \equiv \mathfrak f b$

and in the modern second-order predicate calculus, we write this as:

$∀F(Fa ≡ Fb)$.

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See also the complete English translation of Frege's Grundgesetze der Arithmetik (1893/1903) with the "specifications" for the language.

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Very useful : Gregory Landini, Frege’s Notations : What They Are and How They Mean (Palgrave, 2012).