26 Jun 2016

Frege (§11) Begriffsschrift, Chapter 1 (Geach transl.), “Generality", 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.)

§11 Generality

Brief Summary:
In Frege’s notation system, we denote universal quantification by making an indent into the judgment stroke, above which goes the universally quantified variable, and that variable is also placed in parentheses next to the predicate symbol.
The scope is determined by which stroke (and which part of the stroke) the indent is located on. For example:

Summary

[Recall from section 10 that we can formulate a function using a judgment stroke. For example, we can express an indeterminate function of argument A, that is, Φ(A) with the judgment stroke as
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
I am not entirely certain, but Frege’s point now seems to be that in these prior cases, A and B were determinate arguments. In other words, they seem to be like constants that are being predicated. We can instead think of them as being like algebraic variables. Since he says that the notation goes back to what we see above when we substitute something for the letter, that makes me think it is something like a variable. He will give a separate notation for this other situation. The Gothic letter seems like it is the predicate’s variable, and the concave indent into the judgment stroke above which the variable is written again seems to indicate that it is universal quantification. Many details are unclear to me, so let me quote for now:]
In the expression for a judgment, the complex symbol to the right of ⊢ may always be regarded as a function of one of the symbols that occur in it. Let us replace this argument with a Gothic letter, and insert a concavity in the content-stroke, and make this same Gothic letter stand over the concavity: e.g.:
This signifies the judgment that the function is a fact whatever we take its argument to be. A letter used as a functional symbol, like Φ in Φ(A), may itself be regarded as the argument of a function; accordingly, it may be replaced by a Gothic letter, used in the sense I have just specified. The only restrictions imposed on the meaning of a Gothic letter are the obvious ones: (i) that the complex of symbols following a content-stroke must still remain a possible content of judgment (§2); (ii) that if the Gothic letter occurs as a functional symbol, account must be taken of this circumstance. All further conditions imposed upon the allowable substitutions for a Gothic letter must be made part of the judgment. From such a judgment, therefore, we can always deduce any number we like of judgments with less general content, by substituting something different each time for the Gothic letter; when this is done, the concavity in the content-stroke vanishes again. The horizontal stroke that occurs to the left of the concavity in
is the content-stroke for [the proposition] that Φ(ɑ) holds good whatever is substituted for ɑ; the stroke occurring to the right of p. 2o] the concavity is the content-stroke of Φ(ɑ) – we must here imagine something definite substituted for ɑ.
(Frege 16)
[Frege’s next point will be about how we can understand the meaning of these formulations when we remove the judgment stroke. See section 2. We would just have the content of the judgment, but not the claim that it is the case, I think.]
By what was said before about the meaning of the judgment-stroke, it is easy to see what an expression like
| means.
(16-17)
[For the next point, let us recall from section 5 how the conditionals are structured. When we have
This would be read, “If B, then A.” Let me quote the following, then discuss it.]
This expression may occur as part of a judgment, as in
(17)
[Here I think he is simply saying that the universally quantified expressions can be found in more complex structures.]
[I do not quite get the next point. He might be saying the following. Suppose we say that for all x, x is P. Now let us negate that. It is not the case that for all x, x is P. This can still mean that there some substitution, like Pa, that is true. The point is that not all the substitutions are true. So begin again with the simple case of: for all x, x is P. Here, we can infer that any possible substitution will be true. However, from ‘it is not the case that for all x, x is P’, we cannot infer anything, because we do not know which if any of the constants is P. This idea also holds in the conditional structure. Suppose we have ‘if for all x, x is P, then A.’ From this we likewise cannot infer that ‘if Pa, then A’. The reasoning for this instance I am less certain about. One possibility is that the whole conditional can be true if the antecedent is false. So supposing that the antecedent is false in our example, we cannot infer that ‘if Pa, then A’ because perhaps Pa makes the antecedent true while A is false, and thus the whole conditional is false.
It is obvious that from these judgments we cannot infer less general judgments by substituting something definite for ɑ, as we can from
[seems to end the sentence here.]
serves to deny that Χ(ɑ) is always a fact whatever we substitute for ɑ. But this does not in any way deny the possibility of giving ɑ some meaning Δ such that X(Δ) is a fact.
means the case in which
is affirmed and A denied does not occur. But this does not in any way deny the occurrence of the case in which X(Δ) is affirmed and A denied; for, as we have just seen, X(Δ) may be affirmed and nevertheless
denied. Thus here likewise, we cannot make an arbitrary substitution for a without prejudice to the truth of the judgment.
(17)
[Frege will say that this is why we need to indicate the scope of the quantifier. I am not sure how that would change the prior cases. Before we gave the example, ‘if for all x, x is P, then A.’ Perhaps he is saying that the situation would be different if we instead said, ‘for all x, if x is P, then A.’ I am not sure. But maybe in this case we can derive X(Δ). I do not see how, but I also do not yet understand how the example we just saw demonstrates the need for indicating scope. At any rate, he further shows how quantifier scope works in his system.]
This explains why we need the concavity with the Gothic letter written on it; it delimits the scope of the generality signified by the letter. A Gothic letter retains a fixed meaning only within its scope; the same Gothic letter may occur within various scopes in the same judgment, and the meaning we may ascribe to it in one scope does not extend to any other scope. The scope of one Gothic letter may include that of another, as is shown in p. 21]
In this case different letters must be chosen; we could not replace e by a. It is naturally legitimate to replace a Gothic letter everywhere in its scope by some other definite letter, provided that there are still different letters standing where different letters | stood before. This has no effect on the content. Other substitutions are permissible only if the concavity directly follows the judgment stroke, so that the scope of the Gothic letter is constituted by the content of the whole judgment. Since this is a specially important case, I shall introduce the following abbreviation: an italic letter is always to have as its scope the content of the whole judgment, and thus scope is not marked out by a concavity in the content stroke. If an italic letter occurs in an expression not preceded by a judgment stroke, the expression is senseless.
(Frege 18)

[I may not be following the ideas of italic and Gothic letters very well. In the German version the differences might be a little more apparent. Here are some examples.
(Begriffsschrift und andere Aufsätze p.21)
So we have lower case italic (Roman) letters, lowercase Gothic, and uppercase Greek. There is also the possibility that there are uppercase Roman letters, like A, B, and P, but I am inclined to think they are uppercase Greek, as I do not know why predicates would be distinguished. My guess is that predicates take uppercase Greek letters; predicate constants take lowercase Roman letters; and predicate variables take Gothic letters. It seems that what he says next is something like universal introduction. See Agler Symbolic Logic section 8.1.3.]
An italic letter may always be replaced by a Gothic letter that does not yet occur in the judgment; in this case the concavity must be inserted immediately after the judgment-stroke. E.g. for
we may put
since a occurs only in the argument-position within X(a).
Likewise it is obvious that from
we may deduce
if A is an expression in which a does not occur, and a occupies only argument-positions in Φ(a). If
is denied, we must be able to specify a meaning for a such that Φ(a) is denied. Thus if
were denied and A affirmed, we should have to be able to specify a meaning for a such that A was affirmed and Φ(a) denied.
(18)
[I may be missing his next point. It might be that since we have a conditional, our substitution cannot make it that the antecedent is affirmed and the consequent is denied. Let me quote.]
But since we have
we cannot do so; for this formula means that whatever a may be the case in which Φ(a) would be denied and A affirmed does not | occur. Hence we likewise cannot both deny
and affirm A: i.e.
... Similarly when we have several conditional strokes.
(Frege 18-19)

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).

An image comes from:
Frege, Gottlob. Begriffsschrift und andere Aufsätze. 2nd edn.  Hildesheim/Zürich: Geor Olms, 1993.
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