Frege (§1) Begriffsschrift, Chapter 1 (Geach transl.), “Explanation of the Symbols”, summary

by Corry Shores
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[The following is summary. All boldface, underlying and bracketed commentary are my own.]

Gottlob Frege

Begriffsschrift, Chapter 1
(Geach transl.)

§1
Explanation of the Symbols

Brief Summary:

In Frege’s conceptual notation system, letters will stand for variables and express conceptual generality while other symbols will have specific values.

Summary

[Frege begins with the idea of the general theory of magnitude, which seems to refer to mathematics, also known to be a science of quantity]. He says there are two types of symbols in the general theory of magnitude. The first type are letters which represent either indeterminate numbers or indeterminate functions. [If we were only dealing with specific numbers and equations made of them, then we would be finding the validity of or proving just these specific formulations like 2 + 2 = 4. However, if we wanted to articulate more general observations about how quantities relate, then we would need to not refer to specific values.] The indeterminateness of these terms and functions allows us to “express by means of letters the general validity of propositions; e.g.: (a+b)c = ac + bc.” The other kind of symbol includes individual ones each with their own specific meaning; for example: +, –, √ , 0, 1, 2, 3, and so on. (1)

Although these symbols are used normally in mathematics (general theory of magnitude), Frege will use this distinction between types of symbols more generally “in the wider domain of pure thought” (1). [Like the first category of variables and like the second category of specific meanings for symbols:]

Accordingly, I divide all the symbols I use into those that can be taken to mean various things and those that have a fully determinate sense. The first kind are letters, and their main task is to be the expression of generality. For all their indeterminateness, it must be laid down that a letter retains in a given context the meaning once given to it.
(1)

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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