## 16 Apr 2016

### Frege (§8) Begriffsschrift, Chapter 1 (Geach transl.), “Equality of Content", summary

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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.)

§8 Equality of Content

Brief Summary:
A name has contents, which are what it stands for. And the same contents can have two names. This may be expressed by using an equals symbol between the two names. But not every such equality of names implies that we can dispense with one of the two names, thinking that only one is needed for one set of contents. For, one set of contents can be determined in different ways, and on account of these various ways, we might be inclined to distinguish the names, even though they are equalized. For example, we can have two separate points that under a certain operation come later to coincide. We can then say that when the two points are coinciding, they (that is, both their contents and their names) are equal, since they refer to the same location. However, this does not mean we can dispense with one of the names. For, under a certain mode of determination, that second point is distinct from the other one. Another reason we equalize names is for the purpose of substitution, especially when it allows us to abbreviate longer expressions.

Summary

[The terminology in this section is complicated, and I might misconstrue it. For that reason, all text of this section will be quoted for your interpretation. There are a number of terminological and conceptual distinctions we might need to make. The most basic distinction I can find is between contents and their symbols (the contents’ “signs” or “names”). A content it seems is the conceptual data that is encompassed or evoked by a symbol. Now let us try to conceptually distinguish what seems to be at least a terminological distinction between “signs” and “names.” I cannot tell with certainty, but perhaps a sign is any symbol whose value is merely its capacity to stand for its contents. So when we see a sign, we can call to mind the contents and not place any other significance on the actual symbol itself. In Suppes’ Introduction to Logic, for example, he has the formula AP, which he illustrates by having it mean, “All Americans are Philosophers”. Perhaps Frege would say that the A and the P are signs, because they are merely proxies for the contents A and P, but I am not sure. However, Frege will say that as soon as we join names by an equality function – as when we use the equals symbol in A = P, meaning that they have the same contents – the names appear as themselves. I think what he means is that in this case, we are not just saying something about their contents. We are also saying something about their names too, namely, perhaps, that the one name can be regarded as the other with no misrepresentation of meaning. It is not clear to me from the wording if something is a name only if it has been equated with something else, and it is a sign otherwise. Another possibility is that “sign” and “name” are synonymous. You will have to judge that for yourself. Let me quote:]

Equality of content differs from conditionality and negation by relating to names, not to contents. Elsewhere, signs are mere proxies for their content, and thus any phrase they occur in just expresses a relation between their various contents; but names at once appear in propria persona so soon as they are joined p. 14] together by the symbol for equality of content; for this signifies the circumstance of two names’ having the same content.
(10)

[Frege then says that when we equate two symbols, they have a double meaning. On the one hand, they still stand for their contents. But they also now stand for themselves. I am not exactly sure what is meant by them standing for themselves. I think it might be something like the following. Suppose we take Frege’s classic example, “the morning star is the evening star”, or let us write it: M=E. Because we are using names, we already have an equality relation of sorts you might say, where M is (or equals) Venus in the morning and E is (or equals) Venus in the evening. Since we equate the two names, we are also saying that the contents of each name, in this case, I am guessing the planet itself (or perhaps the concept of the planet, I am not sure), are identically one and the same. Yet we are still doing more. We are saying that one symbol can be substituted for the other. On this level of meaning, we are dealing not with the contents but rather with the symbols themselves, and we are claiming that these symbols themselves have a certain property, namely, mutual substitutability. So in the equation M=E, the E stands both for Venus in the evening and for itself as the symbol that is substitutable with M. Let me quote so you can judge:]

Thus, along with the introduction of a symbol for equality | of content, all symbols are necessarily given a double meaning – the same symbols stand now for their own content, now for themselves.
(10-11)

[The next point is a bit difficult, and he will give an illustration to help convey it. I am supposing that he means the following. We might at this point say that when we equate two symbols as in our M=E, we are doing no more than speaking of the symbols as expressions. And thus, we need not be concerned with the fact that there are two names, because there is no way to conceptualize a difference between M and E once we have equated them. Frege will now demonstrate that on the level of thought, we do in fact in certain cases draw a conceptual distinction between the two symbols’ contents, although that distinction might collapse under certain circumstances or under certain considerations, thus warranting both the need to keep the two symbols while also claiming that they are identical.]

At first sight this makes it appear as though it were here a matter of something pertaining only to expression, not to thought; as though we had no need of two symbols for the same content, and therefore no need of a symbol for equality of content either.
(11)

[Frege then has us picture a geometrical situation.

There is a fixed point A, which is the endpoint of a line that spans the diameter of a circle, extending to (and past) a point B on the opposite side. We move the line from horizontal to vertical. While the line is pivoting, point B continues to mark the point of the line’s intersection with the circle’s circumference, opposite to A. As such, B revolves along the edge of the circle, moving closer and closer to A. Once the line attains a vertical position, B and A overlap.]

In order to show the unreality of this appearance, I choose the following example from geometry. Let a fixed point A lie on the circumference of a circle, and let a straight line rotate around this. When this straight line forms a diameter, let us call the opposite end to A the point B corresponding to this position. Then let us go on to call the point of intersection of the straight line and the circumference, the point B corresponding to the position of the straight line at any given time; this point is given by the rule that to continuous changes in the position of the straight line there must always correspond continuous changes in the position of B.
(11)

[Frege’s point with this seems to be that the name “point B” normally has its own content, which is different from that of “point A”. However, there is one instance where they have the same content, and thus the names become equalized under that circumstance. Even though they do become equalized, that does not mean we can just use one name for both of them. For, under many other circumstances their contents are not identical.]

Thus the name B has an indeterminate meaning until the corresponding position of the straight line is given. We may now ask: What point corresponds to the position of the straight line in which it is perpendicular to the diameter? The answer will be: The point A. The name B thus has in this case the same content as the name A; and yet we could not antecedently use just one name, for only the answer to the question justified our doing so.
(11)

[The next ideas I will not be able to explain very well. Let us work through them, then I will quote. Frege says that there are two ways that we determine that same point. I think here he must mean the point where A and B coincide. The first way that the point is determined is that it is given directly by experience. I am not sure what this means. I suppose he is saying that we are given instructions for how to manipulate the geometrical arrangement, and by actually performing the activity, we experience the points arriving upon a status where they coincide. The second way that we determine the point is that it is given as point B corresponding to the straight line’s being perpendicular to the diameter. I am not sure how this is different from the experiential case. I wonder if he means that in the experiential presentation, we see or imagine it coinciding. But in this other way, we logically infer that the two points coincide. That cannot be right. You will have to interpret this part for yourself. But let me continue first, and I will quote this whole section. The next point makes the situation even more complicated and difficult to grasp. He says that for each of these two ways of determining the point, there is a separate name. I am not sure if he means that when we experience the movement, we need there to be the second point B, but when we think abstractly about it, we would only need one name, perhaps A. I cannot be sure. The next point is potentially very important and interesting. Here he might be returning to this notion that in these situations, it is not enough to use one name for both things, and rather we need two names with an equality operation. He says that the same content (the point where A and B coincide in the example) is given by two ways to determine it (by experiencing that coinciding or by inferring it, but I am not sure). I am also not certain about the next point, but he seems to be saying that when in this case we say A = B, this involves yet another judgment. This judgment might be saying “in this case, there are two ways to determine the same content, namely, by means of (x) and by means of (y)”. And in order to make this judgment, it requires not one but two symbols along with the equality operation. Now I quote so that you can interpret it better.]

The same point is determined in a double way:

(1) It is directly given in experience;
(2) It is given as the point B corresponding to the straight line’s being perpendicular to the diameter.

To each of these two ways of determining it there answers a separate name. The need of a symbol for equality of content thus rests on the following fact: The same content can be fully determined in different ways; and that, in a particular case, the same content actually is given by two ways of determining it, is the content of a judgment. Before this judgment is made, we must supply, corresponding to the two ways of determination, | two different names for the thing thus determined. The judgment p. 15] needs to be expressed by means of a symbol for equality of content, joining the two names together. It is clear from this that different names for the same content are not always just a trivial matter of formulation; if they go along with different ways of determining the content, they are relevant to the essential nature of the case. In these circumstances the judgment as to equality of content is, in Kant’s sense, synthetic.
(11-12)

[Frege’s next point is that there is another more “superficial” reason for symbolizing the equality of contents, namely, we may want to substitute an abbreviation for a longer expression, and the equality allows us to do that.]

A more superficial reason for introducing a symbol for equality of content is that sometimes it is convenient to introduce an abbreviation in place of a lengthy expression; we then have to express equality of content between the abbreviation and the original formula.

is to mean: the symbol A and the symbol B have the same conceptual content, so that A can always be replaced by B and conversely.
(12)

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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