From 7a3cda8d95652074657a57a2c9bb9798234557a9 Mon Sep 17 00:00:00 2001
From: Richard Heck <richard_heck@brown.edu>
Date: Fri, 15 Dec 2023 10:34:01 +0100
Subject: [PATCH] Add footnote on Frege contributed by R. Heck; fixes issue
 #342

---
 bib/open-logic.bib                        | 10 ++++++++--
 content/set-theory/story/grundgesetze.tex | 15 +++++++++------
 2 files changed, 17 insertions(+), 8 deletions(-)

diff --git a/bib/open-logic.bib b/bib/open-logic.bib
index 96f87fb2..c24ef86a 100644
--- a/bib/open-logic.bib
+++ b/bib/open-logic.bib
@@ -296,6 +296,14 @@ @article{Grattan-Guinness1971
   year =	 1971,
 }
 
+@book{Heck2012,
+  author = {Richard Kimberly Heck},
+  title = {Reading Frege's Grundgesetze},
+  publisher = {Oxford University Press},
+  year = {2012},
+  address = {Oxford}
+}
+
 @book{Hodges2014,
   title =	 {Alan Turing: The Enigma},
   author =	 {Hodges, Andrew},
@@ -304,8 +312,6 @@ @book{Hodges2014
   address =	 {London}
 }
 
-
-
 @misc{Imitation2014,
   author =	 {Morten Tyldum},
   title =	 {The Imitation Game},
diff --git a/content/set-theory/story/grundgesetze.tex b/content/set-theory/story/grundgesetze.tex
index b5ac3d83..4b4679ba 100644
--- a/content/set-theory/story/grundgesetze.tex
+++ b/content/set-theory/story/grundgesetze.tex
@@ -15,12 +15,15 @@
 Paradox.
 
 Frege's system is \emph{second-order}, and was designed to formulate
-the notion of an \emph{extension of a concept}. Using notation
-inspired by Frege, we will write $\fregeext{x}{F(x)}$ for \emph{the extension
-of the concept $F$}. This is a device which takes a \emph{predicate},
-``$F$'', and turns it into a (first-order) \emph{term},
-``$\fregeext{x}{F(x)}$''. Using this device, Frege offered the following
-\emph{definition} of membership:
+the notion of an \emph{extension of a concept}.\footnote{Strictly
+speaking, Frege attempts to formalize a more general notion: the
+``value-range'' of a function. Extensions of concepts are a special
+case of the more general notion. See \citet[pp.\ 8--9]{Heck2012} for
+the details.} Using notation inspired by Frege, we will write
+$\fregeext{x}{F(x)}$ for \emph{the extension of the concept~$F$}. This
+is a device which takes a \emph{predicate}, ``$F$'', and turns it into
+a (first-order) \emph{term}, ``$\fregeext{x}{F(x)}$''. Using this
+device, Frege offered the following \emph{definition} of membership:
 \[
 	a \in b =_\text{df} \exists G(b = \fregeext{x}{G(x)} \land Ga)
 \]