Frege (§3) Begriffsschrift, Chapter 1 (Geach transl.), 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.)

§3

Brief Summary:

Often in logic we might distinguish the subject and what predicates it. Frege does not make this distinction in his own system of concept notation. This is because he recognizes a conceptual content which can be expressed with different subjects, as by inverting to a passive voice. For example, “the Greeks defeated the Persians at Plataea” and “the Persians were defeated by the Greeks at Plataea” have the same conceptual content but opposing subjects. The only real subject and predicate in his system are the conceptual content (in the form of a proposition) and the predicate, ‘… is a fact’, for example: ‘the violent death of Archimedes at the capture of Syracuse is a fact.’ This predicate is what his ⊢ symbol stands for.

Summary

[Frege is providing a notation system for describing conceptual relations.] In Frege’s notation system, we do not make the classical distinction in logic between a subject and a predicate. Frege provides his justification for this interesting decision. [I do not understand Frege’s formulation in the following, but it seems he is saying that given two propositions with different content, that content may either have the same power of inferential productivity as the other or it may have different inferential productivity (or maybe, the same or different inferential life), meaning that what each content implies is functionally isomorphic on the level of their inferential behavior. “the Greeks defeated the Persians at Plataea” and “the Persians were defeated by the Greeks at Plataea” literally have different contents and perhaps connotationally have slightly different contents. However, when we look at how inferences made by each interact with other judgments we see that they are inferentially similar. A possible counter-example might be “the Persians defeated the Greeks at Plataea” and “the Greeks defeated the Trojans at Troy” also have distinguishable contents, but in terms of their implications they are very different. But please interpret the following for yourself:]

A distinction of subject and predicate finds no place in my way of representing a judgment. In order to justify this, let me observe that there are two ways in which the content of two judgments may differ; it may, or it may not, be the case that all inferences that can be drawn from the first judgment when combined with | p. 3] certain other ones can always also be drawn from the second when combined with the same other judgments. The two propositions ‘the Greeks defeated the Persians at Plataea’ and ‘the Persians were defeated by the Greeks at Plataea’ differ in the former way; even if a slight difference of sense is discernible.
(2-3)

[The difference between “the Greeks defeated the Persians at Plataea” and “the Persians were defeated by the Greeks at Plataea” might be something like a connotational difference but not something conceptual. They both conceptually have the same content, and this was established by Frege as being a matter of their sharing the same inferential life.]

Now I call the part of the content that is the same in both the conceptual content. Only this has significance for our symbolic language; we need therefore make no distinction between propositions that have the same conceptual content.
(3)

[Perhaps what Frege is doing with the prior examples is showing how the same conceptual content can be expressed in sentences with different subjects.]

When people say ‘the subject is the concept with which the judgment is concerned,’ this applies equally well to the object. Thus all that can be said is: ‘the subject is the concept with which the judgment is chiefly concerned.’
(3)

In language, word order might matter, but conceptually it need not.

In my formalized language there is nothing that corresponds; only that part of judgments which affects the possible inferences is taken into consideration. Whatever is needed for a valid inference is fully expressed; what is not needed is for the most part not indicated either; no scope is left for conjecture. 
(3)

In Frege’s system, there is one basic predicate all conceptual content take in propositions, and that is ‘… is a fact’. This is the meaning of the ⊢ in his system.

In this I follow absolutely the example of the formalized language of mathematics; here too, subject and predicate can be distinguished only by doing violence to the thought. We may imagine a language in which the proposition ‘Archimedes perished at the capture of Syracuse’ would be expressed in the following way: ‘the violent death of Archimedes at the capture of Syracuse is a fact.’ You may if you like distinguish subject and predicate even here; but the subject contains the whole p. 4] content, and the only purpose of the predicate is to present this in the form of a judgment. Such a language would have only a single predicate for all judgments, viz. ‘is a fact.’ We see that there is no question here of subject and predicate in the ordinary sense. | Our symbolic language is a language of this sort; the symbol ⊢ ­ is the common predicate of all judgments.
(3-4)

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).