Hello Gary F, John S, List,

Gary F has raised the following question about a remark John made concerning 
the SA in the EG:


"Having said that, I have to say also that some of the statements in your post 
are even more confusing than Peirce’s presentation in Lowell 2. You wrote, 
“Peirce said that a blank sheet of assertion is a graph.  Since it's a graph, 
you can draw a double negation around it.”  — Eh? How can you draw anything 
around the sheet of assertion, which (by Peirce’s definition) is unbounded?? 
Can you show us a replica?


It would be usual for those of us who learned Euclidean geometry in middle and 
high school to think of the SA as a surface that is akin to the Euclidean 
plane. Under the postulates that govern this system, parallel lines never meet, 
so we picture the plane as stretching out in all directions endlessly.


In topology, we think of an unbounded surface differently. After all, the 
figures constructed in a 2-dimensional topological surface can be stretched and 
twisted indefinitely without changing any of the continuous connections between 
the parts of such figures. Leaving aside the homoloidal character of lines 
taken to be straight and the metrical properties of such a surface, the 
underlying topology of the Euclidean plane is that of a parabolic surface. Such 
a surface is unbounded, but lines return to themselves. The reason is that the 
parabola surface has the global structure of a torus.


It is clear that Peirce is reflecting on the topological character of the SA 
itself as he explains the starting assumptions for the alpha and beta system of 
graphs. Such reflections are prominent in the NEM, the 9th Lecture in Reasoning 
and the Logic of Things, etc. The global character of the SA will be determined 
by the assumptions that govern the construction of figures in this 
2-dimensional surface. We can study this surface the same that that we would 
study any 2-dimensional surface in topology using the Euler characteristic, and 
we can study its global properties more carefully by reflecting on the 
additional features that Listing and Peirce added to Euler's version of the 
equation.


For a classification of types of surfaces based on the Euler characteristic, 
see:  https://en.wikipedia.org/wiki/Euler_characteristic

For richer explanations of Peirce's understanding and development of this 
formula, see:  Havenel, Jérôme. "Peirce’s topological concepts." New essays on 
Peirce’s mathematical philosophy (2010): 283-322.


Let's consider an example taken from the table that classifies the different 
kinds of surfaces. The projective plane is an non-orientable unbounded surface, 
and it has an Euler characteristic of 1. The global properties of this surface 
are quite interesting. Within the system of postulates that govern the 
generation of the surface, all parallel lines converge. This is something we 
can picture in a more familiar way by considering a perspective drawing in 
which all parallel lines converge on the infinitely distant horizon. In effect, 
the projective space is a generalization on this idea from perspective 
geometry. What we should note is that the absolute in a projective surface is 
effectively a generalization of the infinitely distant horizon within the 
perspective geometry.


My reason for picking this example of a topological surface is that it provides 
us with an example of a 2 dimensional space in which a path can be drawn all of 
the way "around" the surface--i.e., as the line that serves as the absolute. Is 
there any restriction on doing the same kind of thing in other sorts of 
topological surfaces? That is can we draw a path all of the way around a 
spherical (i.e., elliptical) or toroidal surface?


If such a path can be drawn "all the way around" these sorts of unbounded 
surfaces, is there a restriction on making a cut "all the way around" the SA? 
We shouldn't lose sight of the fact that, for the SA, a cut is not simply a 
path. Rather, the cut takes what is inside the boundary and moves the that part 
of the surface to a different surface--one that represents what is negated. As 
such, already in the Alpha graphs, the SA is not a simple 2 dimensional 
surface. Rather, the SA can be used to represent is all that can be positively 
asserted, and this surface appears to be related--in some fashion--to another 
surface that represents all that can be denied.


Spending some effort on the question of how those surfaces are related within 
the Alpha and Beta systems might be worth our time. My hunch is that there is a 
significant difference between the way they are related in these two systems, 
and and even more significant difference when we consider, as John S suggests, 
what they represent when we are using these systems--especially gamma--to 
analyze synthetic forms of inference such as induction.


The first step in approaching this sort of question here in the context of our 
discussion of the 2nd Lowell Lecture is to consider how the part of the surface 
that is inside the inner portion of the scroll is related to the part of the 
surface that is inside the outer part of the scroll. With a clearer idea of 
what that figure represents as a relation between those two parts of the 
surface, we can consider the import of the drawing the pseudograph as a cut 
that is entirely filled in as black. With this diagram, we see Peirce exploring 
a way to represent the relation between what is possible as a positive 
assertion and what is impossible--at least within the context of the 
development of the alpha system that he is explaining here.


--Jeff







Jeffrey Downard
Associate Professor
Department of Philosophy
Northern Arizona University
(o) 928 523-8354
________________________________
From: John F Sowa <s...@bestweb.net>
Sent: Monday, October 30, 2017 7:20:48 AM
To: peirce-l@list.iupui.edu
Cc: Dau, Frithjof
Subject: Re: [PEIRCE-L] Lowell Lecture 2.6

Gary F,

The issues are far deeper than notation or computer processing.
1903 was a critical year in which Peirce began his correspondence
with Lady Welby.  That led him to address fundamental semiotic issues.

> I’ll have to confess at this point that I have no interest in learning
> EGs for the sake of learning a new notation system, or for the sake
> of knowledge representation in automated systems.

Last week, Ontolog Forum sponsored a telecon, in which I presented
slides on "Context in Language and Logic".  It addressed complex
semiotic issues, and I mentioned Peirce at various points.
Following are the slides.  Slide 2 also has the URL for the audio:
http://jfsowa.com/ikl/contexts/contexts.pdf

> the elementary phenomena of reasoning, that I’d like to understand better.

I agree that's important, and I also agree that Peirce was seeking
the most fundamental methods he could discover.  But I also believe
that he abandoned the recto/verso system because (a) the questions
raised by Lady Welby led him to more significant problems, and
(b) those low-level ideas paled in comparison to his goal of
representing "a moving picture of the action of the mind in thought."

> The three pairs of rules you attached (from NEM) are essentially
> the same as the three pairs he gives later on in Lowell 2, except
> for the shading and the absence of lines of identity.

For his EGs of 1903, they are logically equivalent.  In fact, that
is why his recto/verso description and his "magic blot" have no real
meaning:  they have no implications on the use of the graphs in
perception, learning, reasoning, or action.  But the 1911 system
can be generalized to modal logic, 3-valued logic, and probability.

And by the way, that letter of 1911 was addressed to Mr. Kehler,
one of Lady Welby's correspondents, and the main topic was
probability and induction.  That's also significant.

Implications of his 1911 system:

  1. The rules come in 3 symmetric pairs, and each pair consists
     of an insertion rule (i) and an erasure rule (e), each of
     which is the inverse of the other.  This feature supports
     some important theorems, which are difficult or impossible
     to prove with other rules of inference.

  2. The rules are *notation independent*:  with minor adaptations
     to the syntax, they can be used for reasoning in a very wide
     range of notations:  the algebraic notation for predicate
     calculus (Peirce, Peano, or Polish notations); Kamp's discourse
     representation structures; many kinds of diagrams and networks,
     and even natural languages.

  3. They can be adapted to theorem proving with arbitrary icons
     inside an EG.  I demonstrated that with Euclid's diagrams inside
     the ovals of EGs.  But they can also be used with icons of any
     complexity -- far beyond Euclidean-style diagrams.

  4. The psycholinguist Philip Johnson-Laird observed that Peirce's
     notation and rules are sufficiently simple to make them a
     promising candidate for a logic that could be supported by
     the neural mechanisms of the human brain.  That is true of
     his later system, but not the recto/verso system.

For an overview of these issues, see my slides on visualization:
http://jfsowa.com/talks/visual.pdf

To show that Kamp's DRS notation is isomorphic to a subset of EGs,
see slides 20 to 27 of visual.pdf.  To see the application to English,
see slides 28 to 30.  (But this is true only for that subset of English
or other NLs that can be translated to or from Kamp's DRS notation.)

For the option of including icons inside the areas of EGs, see slides
31 to 42 of visual.pdf.  For more detail about Euclid, see slides
19 to 39 of http://www.jfsowa.com/talks/ppe.pdf

Note:  There is considerable overlap between visual.pdf and ppe.pdf,
but slides 19 to 39 of ppe.pdf go into more detail about Euclid.

For theoretical issues, see slides 43 to 53 of visual.pdf.
For the theoretical details, see http://jfsowa.com/pubs/egtut.pdf

I'm working on another paper that goes into more detail about Peirce's
"magic lantern of thought".  The 1911 system can support it.  But the
recto/verso system cannot.

John
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