23 Apr 2016

Frege (§10) Begriffsschrift, Chapter 1 (Geach transl.), “[untitled. notating functions]", summary

by Corry Shores

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[The following is summary. Bracketed commentary is my own. Please forgive my typos, as proofreading is incomplete.]

Summary of
Gottlob Frege
Begriffsschrift, Chapter 1
(Geach transl.)

Brief Summary: 
We express an indeterminate function of argument A as: Φ(A) and with the judgment stroke as
begriff 10 a
which is read, “A has the property Φ”. An indeterminate function for two arguments may be written Ψ(A,B) and with the judgment stroke as
begriff 10 b
which is read, “B stands in the Ψ-relation to A” or “B is a result of applying the operation Ψ to the object A.” We can also vary the function letter as well.
[Recall our discussion of functions from the last section. We discussed indeterminate arguments and indeterminate functions. The idea seemed to be that an indeterminate argument or function would be similar to a letter-variable in algebra where it can be filled in with a variety of options, but we have not specified them.]
In order to express an indeterminate function of the argument A, we put A in brackets after a letter, as in
means a function (not further determined) of the two arguments A and B. Here the places of A and B within the brackets represent the places occupied by A and B in the function (whether A and B each occupy one place in it or more). Accordingly in general
Ψ(A,B) and Ψ(B,A)
are different.
(Frege 15)
Frege then shows how to render functions using his notation system. [He will use the judgment stroke. See section 2.]

Indeterminate functions of several arguments are expressed similarly.

begriff 10 a

may be read as ‘A has the property Φ’.

begriff 10 b

may be read as ‘B stands in the Ψ-relation to A’ or as ‘B is a result of applying the operation Ψ to the object A.’

(Frege 15)


[The next point I do not understand. Let me quote it

In the expression
p. 19] the symbol Φ occurs in one place; and we may imagine it replaced by other symbols Ψ, Χ, so as to express different functions of the argument A; we may thus regard Φ(A) as a function of the argument Φ. This makes it specially clear that the concept of function in Analysis, which in general I have followed, is far more restricted than the one developed here.
[Normally we can make a number of substitutions for A. So Φ is a function of argument A. Now we think of Φ also being substituted with other functions. I do not understand this conclusion: “we may thus regard Φ(A) as a function of the argument Φ”. Does it mean that we have: (Φ(A))(Φ) or something like that? I do not think so, because as the argument changes, so too should the function Φ of Φ(A). But I am not sure how else to formulate and understand what Frege has written here. We get the point that we have the flexibility to change the function with other functions, however.]
Frege, Gottlob. “Begriffsschrift (Chapter 1)”. Transl. P.T. Geach. In Translations from the Philosophical Writings of Gottlob Frege. Eds. P.T. Geach and Max Black. Oxford: Basil Blackwell, 1960, second edition (1952 first edition).


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