Frege's logic: implication question, erratum?

Question

Frege's Begriffsshrift Page 15 English edition details the following form:

It's my understanding that this form translates to:

$\vdash ((B \implies A) \implies \Gamma)$

Frege claims this form is denied in the case when $B$ is affirmed; $A$ and $B$ are denied. Is this statement Erratum?

My understanding of the case when the form is denied is $(B \implies A)$ is affirmed and $\Gamma$ denied. If $(B \implies A)$ must be affirmed, then the case that $B$ is affirmed and $A$ is denied is excluded. This is the very case Frege claims is required for denial of the entire form, which is a contradiction to my analysis above.

This leaves for cases of denial of the entire form: $B$ and $A$ affirmed; $\Gamma$ denied. $B$ denied and $A$ denied; $\Gamma$ denied. $B$ denied and $A$ affirmed; $\Gamma$ denied.

SOLUTION

• The form I posted has been interpreted correctly
• Frege's comments on affirmation and denial in Begriffsshrift Page 15 of the form appear to be Erratum, documented by Schroder (Page 88 Schroder, Zeitschrift our Mathematik und Physik) and a letter from Frege to Bertrand Russell (Frege letter to Russell).
• The source of the mistake may be Frege's miss of a second negation in form ¬Γ∧¬(¬∧), in which Frege accidentally translates the form to ¬Γ∧¬∧.

Thanks!

Answer 1

You're translating the form correctly. I see you have the version from Van Heijenoort's "From Frege to Gödel", which mentions that Ernst Schröder already pointed out there was an oversight there in the footnotes.

If you can read German, you could find Schröder's comments here, on the middle of the page (the three Schema's he talks about are on the previous page)

It roughly says that there is a mistake (the only one Schröder found), of two interpretations that contradict each other. Only the second ("If $A$ is a necessary consequence of $B$, one can infer that $\Gamma$ takes place") is correct. He translates the correct interpretation to Boole's algebraic notation, which would in modern times be written as that the sentence is false whenever $\lnot \Gamma\land \lnot(\lnot A\land B)$.

Frege himself admits the error in a letter to Bertrand Russell, to be found in his original correspondence with Russell, luckily this excerpt contains the relevant page 213. On the bottom the error is explained.

There is an english translation of their letters to be found in Van Heijenoort's book. In particular the second paragraph of Frege's letter on page 127.