## 1 Jul 2016

### Frege (§12) Begriffsschrift, Chapter 1 (Geach transl.), [on combinations of structures], 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.)

§12
[on combinations of structures]

Brief Summary:
In Frege’s notation system, we write:

¬(∀z)Xz
“Not everything has the property X”

(∀z)¬Xz
“there is not something with the property X

¬(∀z)¬Λz
“There are Λs”

(∀z)(Xz→Pz)
“If something has the property X, then it has also the property P”
“every X is a P”
“all Xs are Ps”

(∀z)(Xz→¬Pz)
“what has the property Ψ has not the property P
“no Ψ is a P

¬(∀z)(Λz→Pz)
“some Λs are not Ps”

¬(∀z)(Mz→¬Pz)
“some Ms are Ps”
“it is possible for an M to be a P”

Summary

Frege will now explain his notation system for certain combinations of symbols. [Suppose we wanted to write
¬(∀x)Mx
Or, “not everything is movable.” See Agler, Symbolic Logic, section 6.5. This means that the predicate M does not hold for all things.] If we want to notate “find something, say Δ such that X(Δ) is denied. We may thus render it as: “there are some things that have not the property of X” (Frege 19). We would write that as
(Frege 19)

[Now suppose instead that we wanted to write
(∀x)¬Mx
Or, “Nothing is movable.” Again see Agler, Symbolic Logic, section 6.5. This means that the predicate M holds for absolutely none of the objects in the domain.] Suppose instead we wanted to notate “ ‘Whatever a may be, X(a) must always be denied,’ or ‘there is not something with the property X,’ or (calling something that has the property X, a X) ‘there is no X’ ” (Frege 19). We would write it as:

[The next idea might be something like the following. Suppose we have ‘for all x, x is not P.’ This means that none of the x’s are P. Now suppose we have ‘it is not the case that for all x, x is not P.” This would mean that we cannot say that none of the x’s are not P. It also would be making no claim whether or not all of them are either. In other words, it would seem to mean that some x’s are P. That might be something like what Frege does next.]
is denied by
This may be rendered as ‘there are Λ’s.
(Frege 19)

[For the next part, recall first how Agler (Symbolic Logic section 6.5.2) made translations for a formation that might now with respect to this new context be something like ‘all S’s are P’s.’ He gave this sort of a bridging translation:

(3) (∀x)(Zx→Hx)

(3B) For every x, if x is a zombie, then x is happy.

(3B*) Choose any object you please in the domain of discourse; if that object is a zombie, then it will be also be happy.

(3E) Every zombie is happy.

(Agler 271, 272)

So suppose we want to say that all X’s are P’s. We would then say that if  z is an X then z is a P. Given the conditional structure, this means that it cannot be that z is an X and z is not a P. Here are the other possibilities, however:
z is an X and z is a P
z is not an X and z is a P
z is not an X and z is not a P.
All of these situations conform to the notion that all X’s are P’s. As we will see, Frege will render this structure using the conditional notation.]
means: ‘whatever may be substituted for a, the case in which P(a) would have to be denied and X(a) affirmed does not occur.’ It is thus possible that, for some possible meanings of a,
P(a) must be affirmed and X(a) affirmed; for others,
P(a) must be affirmed and X(a) denied; for others again,
P(a) must be denied and X(a) denied.
| We can thus give the rendering: ‘If something has the property X, then it has also the property P,’ or ‘every X is a P,’ or ‘all X’s are P’s.’
(Frege 19-20)

[The next structure Frege says is for causal connexions. I do not know the terminology well enough to understand why this has something to do with causality, as I cannot discern any such meaning. For this formation, recall (again from Agler Symbolic Logic section 6.5.2):

(4) (∀x)(Zx→¬Hx)

(4B) For every x, if x is a zombie, then x is not happy.

(4B*) Choose any object you please in the domain of discourse consisting of human beings (living or dead); if that object is a zombie, then it will not be happy.

(4E) No zombies are happy.

(Agler 271, 272)

Frege will show now how to make this structure.]
This is the way causal connexions are expressed.
means: ‘no meaning can be given to a such that P(a) and Ψ(a) p. 24] could both be affirmed.’ We may thus render it as ‘what has the property Ψ has not the property P’ or ‘no Ψ is a P.’
(Frege 20)
[Now, if we wanted to say that all S’s are P’s, then we would say that for all x, if x is S, then x is P. And if we wanted to say that no S’s are P’s, then we would say, for all x, if x is S then x is not P. But if we want to say that some S’s are not P’s, then we would say that it is not the case that for all x, if x is S then x is P. For, if not all are something (and not none are something), then some are not that something. I think.]
denies
and may be therefore rendered as ‘some Λs are not Ps’
(Frege 20)

[Suppose we said, ‘all M’s are not P’s’. This means there is no M which is a P. Now suppose we said, “it is not the case that all M’s are not P’s’. That would mean that some M’s are P’s.]
denies that no M is a P and thus means ‘some Ms are Ps’ or ‘it is possible for an M to be a P.’
(Frege 20)

From:
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).

Or if otherwise noted:
Agler, David. Symbolic Logic: Syntax, Semantics, and Proof. New York: Rowman & Littlefield, 2013.

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