I understand, I think, the general notation for abbreviated arguments in Frege's Begriffsschrift, but I'm a little unclear on the use of one colon versus two, following the invocation of an omitted argument. In §6 Frege writes, in Terrell Bynum's translation:
Now, suppose for example that the judgement [B → A]—or one containing it as a special case—has been labelled with X. Then I can write the inference this way:
$\qquad$ [B]
(X):———
$\qquad$ [A]Here it is left to to the reader to construct the judgement [B → A] from [B] and [A], and to see if it tallies with the cited judgement X.
If, for example, the judgement [B] was labelled by XX, then I also write the same inference this way:
$\qquad$ [B → A]
(XX)::———
$\qquad\quad$ [A].Here the double colon indicates that [B], which is cited only by means of XX, must be formed, by a method other than the one above, from the two judgements which are written out.
(I use square brackets with my interpretations of Frege's notation because of the difficulty of representing the latter in text.)
I find the wording a little confusing, but I think I understand from later examples: In an argument through modus ponens, if one abbreviates through citation of the logical claim (e.g., p → q), one uses one colon; if one abbreviates through citation of the antecedent (e.g., p), or if one abbreviates multiple arguments in a string of applications of modus ponens, one uses two colons.
Am I getting this right?
\vdash. You can also keep the conventional material conditional $\to$ notation (as long as you say so). – Bruno Bentzen Mar 31 '17 at 01:50