Jon,
That's true:
JAS> I am admittedly curious about the content of your new article. As you
know, there is only one place in Peirce's entire vast corpus of writings where
he mentions Delta.
But note the following excerpt from R514, which also contains a rough draft of
the EGs in L231:
"Since my paper of 1906, I have improved the [EG] system slightly (at least),
and the manner of exposition of it greatly, by first stating the force of the
different signs without going into their deeper significance in the
Since my paper of 1906, I have improved the [EG] system slightly (at least),
and the manner of exposition of it greatly, by first stating the force of the
different signs without going into their deeper significance in the least...
One of my possibly slight improvements, is that I begin by drawing (preferably
with a red pencil), a line all round my sheet at a little distance from the
edge; and in the margin outside the red line, whatever is scribed is merely
asserted to be possible. Thus, if the subject were geometry, I could write in
that margin the postulates, and any pertinent problems stated in the form of
postulates such as, that "if on a plane, there be circle with a ray cutting it,
and two be marked [end of R514]
That operation is the way L376 represents multiple parts of the phemic sheet.
And it is a way of saying the conditions for the nested graph to be possible.
That doesn't say much more. But that operation when combined with a notation
for first-order logic is a method for representing modality in various logics
in the late 20th and early 21st C.
There are also other hints that suggest ways of extending FOL. They don't
prove that Peirce intended exactly the same kinds of applications. But it
shows that his ways of thinking could lead in promising directions. Following
is the abstract of the article I'm writing.
Abstract. In December 1911, Peirce wrote an intriguing claim about existential
graphs: “I shall now have to add a Delta part in order to deal with modals.”
Although his unfinished draft does not specify the details, it explains how an
utterer and an interpreter may use Delta graphs in an investigation. Further
hints may be found in several manuscripts he wrote in the previous six months.
As another hint, the intended recipient of the letter was Allan Risteen. When
that letter is combined with information about Risteen’s expertise and Peirce’s
work on a proof of pragmaticism, it suggests that the phemic sheet of a Delta
graph consists of multiple “papers”, each of which represents a different time,
aspect, or modality of some universe of discourse. Although Peirce did not
specify the details of Delta graphs, a combination of features mentioned in
several 1911 manuscripts would satisfy the hints about Delta graphs. The result
would be similar or perhaps equivalent to a logic for modality that was
invented in 2006.
John
From: "Jon Alan Schmidt"
Sent: 2/18/24 8:08 PM
To: Peirce-L
Subject: [PEIRCE-L] Delta Existential Graphs (was The Proper Way in Logic)
John, List:
JFS: I am now writing the article on Delta Graphs. That is an example where
Peirce was on solid ground with his deep understanding of logic and
mathematics. Next week, I'll send the abstract and preview of the new article,
which shows how Peirce anticipated a version of logic that was developed in the
21st century (2006 to be exact).
(https://list.iupui.edu/sympa/arc/peirce-l/2024-02/msg00038.html)
JFS: I'm moving on to the the article on Delta graphs. I'll send a note with a
preview of that article later this week.
(https://list.iupui.edu/sympa/arc/peirce-l/2024-02/msg00104.html)
I am admittedly curious about the content of your new article. As you know,
there is only one place in Peirce's entire vast corpus of writings where he
mentions Delta.
CSP: In this ["Prolegomena to an Apology for Pragmaticism," CP 4.530-572, 1906]
I made an attempt to make the syntax [of Existential Graphs] cover Modals; but
it has not satisfied me. The description was, on the whole, as bad as it well
could be, in great contrast to the one Dr. Carus rejected [in 1897]. For
although the system itself is marked by extreme simplicity, the description
fills 55 pages, and defines over a hundred technical terms applying to it. The
necessity for these was chiefly due to the lines called "cuts" which simply
appear in the present description as the boundaries of shadings, or shaded
parts of the sheet. The better exposition of 1903 divided the system into three
parts, distinguished as the Alpha, the Beta, and the Gamma, parts; a division I
shall here adhere to, although I shall now have to add a Delta part in order to
deal with modals. (R L376, R 500:2-3, 1911 Dec 6)
For EGs as described in "the better exposition of 1903," modal logic is
implemented with broken cuts in Gamma. However, by the time Peirce wrote this
letter to Allan Douglas Rist
