Gary F., List:
Thanks for the link to your current draft, which I think is an excellent
overview. My only suggestion, besides the obvious one of finishing and
posting it, is to add links at the very end to the 1973 book by Roberts (
https://www.felsemiotica.com/descargas/Roberts-Don-D.-The-Existential-Graphs-of-Charles-S.-Peirce.pdf)
that you reference, as well as his 1992 paper that provides a nice summary (
https://core.ac.uk/download/pdf/82124291.pdf) and perhaps also Zeman's 1964
dissertation that remains the most comprehensive treatment of Gamma with
broken cuts for modal logic (https://isidore.co/calibre/get/pdf/4481).
Personally, I have found it challenging to present Peirce's ideas in a way
that is digestible for the novice, because I have to overcome the "curse of
knowledge" that results in assuming too much of the reader; i.e., taking
more for granted than is warranted. With that in mind, what follows is my
own attempt at a brief introduction to EG, which also happens to touch on
the previous thread topic. It will appear in a chapter, "Peirce on
Abduction and Diagrams in Mathematical Reasoning," on which Gary R. and I
recently collaborated with mathematical historian and Peirce scholar Joseph
Dauben for the forthcoming Springer book, *Handbook of Cognitive
Mathematics*. As always, I would likewise welcome any feedback.
Peirce ultimately considered his most important contribution to logic to be
the development of a "diagrammatic syntax" for propositions and a set of
transformation rules for carrying out deductive inferences from them, a
system that he dubbed "Existential Graphs." He had three objectives in mind
for it: "to afford a method (1) as *simple *as possible (that is to say,
with as small a number of arbitrary conventions as possible), for
representing propositions (2) as *iconically*, or diagrammatically and (3)
as *analytically *as possible" (CP 4.561n, 1908).
A blank sheet stands for the continuum of all true propositions, any of
which may be explicitly "scribed" on it as a graph-instance consisting of a
single letter in the "Alpha" version for propositional logic, or of names
denoting abstract general concepts and heavy lines denoting concrete
indefinite individuals ("something") in the "Beta" version for first-order
predicate logic. Each name has one, two, or three "pegs" where a heavy line
may be attached, signifying the attribution of the concept to that
individual. The number of pegs associated with the name corresponds to the
"valency" of the concept as monadic ("redness"), dyadic ("killing"), or
triadic ("giving").
Existential graphs for "some apple is red," "Cain killed Abel," and "Bob
gives a ball to Larry":
[image: image.png]
There is no limitation on the number of graph-instances that may be scribed
on the sheet, which signifies the primitive relation of *coexistence *such
that juxtaposing multiple graph-instances expresses the conjunction of the
propositions that they represent. There is also no limitation on the number
of branches that may be added to a heavy line, which corresponds to the
primitive relation of *identity *such that each branch attached to a name
attributes another concept to the same individual. Coexistence and identity
are thus *continuous *relations, and they are also symmetrical: "A and B"
is logically equivalent to "B and A," while "some S is P" is logically
equivalent to "some P is S."
A third primitive relation is required, and for Peirce it is the most
fundamental of all: "The first relation of logic, that of antecedent and
consequent, is unsymmetrical. Now an unsymmetrical relation cannot result
from any combination of symmetrical relations alone" (NEM 3:821, 1905).
This is what he usually called "consequence," now typically referred to as
"implication." It is represented in Existential Graphs by a "scroll," which
is "a curved line without contrary flexure and returning into itself after
once crossing itself, and thus forming an outer and an inner 'close.' … In
the outer I scribed the Antecedent [A], in the inner the Consequent [C]"
(CP 4.564, 1906).
Existential graph for "if A then C":
[image: image.png]
The continuity of the scroll itself thus reflects the continuity of the
inference from the antecedent to the consequent, and disjunction is derived
from it by simply adding more loops with inner closes. Any graph-instance,
including the blank, is always interchangeable with a scroll that has an
empty outer close and that same graph-instance in its inner close.
On the other hand, the scroll also serves as a discontinuity or topical
singularity that interrupts the sheet. With that in mind, Peirce eventually
realized that shading oddly enclosed areas is the best way to distinguish
them from unenclosed and evenly enclosed areas, which are unshaded: simpler
than counting lines, more iconic for conveying that they are different
surfaces, and more analytical because shaded areas correspond to a universe
of possibility rather than actuality (CP 4.576-581, 1906). The permissible
transformations of graph-instances, which correspond to rules of inference,
are then summarized as follows:
1. Erasure - In an unshaded area, any graph-instance or portion of a line
may be deleted.
2. Insertion - In a shaded area, any graph-instance may be added and any
lines may be joined.
3. Iteration - Any graph-instance already scribed may be reproduced
identically in the same area or in a more enclosed area, and any unattached
end of a line may be extended into a more enclosed area.
4. Deiteration - Any graph-instance that could have resulted from iteration
may be deleted.
For classical logic, a continuous scroll is logically equivalent to nested
"cuts," which are simple oval lines or shaded/unshaded areas that represent
negation. Peirce derived this from a scroll whose consequent is "a
proposition implying that every proposition is true," resulting in "a black
spot entirely filling the close in which it is," which "may be drawn
invisibly small" (CP 4.454-456, 1906). However, at one point he retracted
this last provision and instead advocated retaining a small blackened inner
close attached to the cut as a reminder of its theoretical basis (CP
4.564n, c. 1906). Moreover, he called it an "error" and an "inaccuracy" to
analyze "if A then C" as no different from "not both A and not-C."
Existential graph for "not both A and not-C":
[image: image.png]
CSP: For in reasoning, at least, when we first affirm, or affirmatively
judge, the conjugate of premisses, the judgment of the conclusion has not
yet been performed. There then follows a real movement of thought in the
mind, in which that judgment of the conclusion comes to pass. Now surely,
speaking of the same A and B as above, it were absurd to say that a real
change of A into a sequent B consists in a state of things that should
consist in there not being an A without a B. For in such a state of things
there would be no change at all. (R 300:49[48], 1908)
It is interesting to note that in intuitionistic logic, negation is
likewise defined as the implication of a contradiction, and although "if A
then C" implies "not both A and not-C," the inference in the other
direction is invalid. In fact, simply by explicitly distinguishing a scroll
for implication from nested cuts for double negation, Existential Graphs
can be employed in accordance with intuitionistic logic rather than
classical logic (Oostra 2010; Oostra 2011). Either way, implementing a
series of diagrammatic transformations to draw a deductive conclusion from
a set of premisses serves as "a moving-picture of Thought" (CP 4.11, 1906).
For a more detailed exposition of Existential Graphs, see (Roberts 1992).
Regards,
Jon Alan Schmidt - Olathe, Kansas, USA
Structural Engineer, Synechist Philosopher, Lutheran Christian
www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt
On Wed, May 19, 2021 at 8:14 AM <[email protected]> wrote:
> Gary R and list,
>
> Being a non-mathematician myself, I’ve been drafting an introduction to
> Peirce’s EGs for the likes of us. I have the current draft online now here: EG
> introduction (gnusystems.ca) <https://gnusystems.ca/TS/draftEGs.htm> It
> includes many links both to Peirce’s own introduction to EGs (Lowell
> Lecture 2 of 1903) and to other (later) Peirce quotations.
>
>
>
> Of course I’d welcome comments and suggestions (even from mathematicians!).
>
>
>
> Gary f.
>
>
>
> *From:* [email protected] <[email protected]> *On
> Behalf Of *Gary Richmond
> *Sent:* 18-May-21 23:58
>
> John, Jon, List,
>
> JAS: Peirce anticipates aspects of the formal system that is now known as
> intuitionistic logic--e.g., defining negation as the implication of
> falsity...
>
> JFS: Nobody knows what Peirce would have said about the less dogmatic
> treatment of intuitionistic logic by Heyting and others, including Oostra.
> But it's doubtful that he would have approved of the drastic reduction in
> acceptable mathematical theories. He was always highly suspicious of any
> attempt to block the way of inquiry.
>
> "Dogmatic treatment of intuitionistic logic, etc."?? How in the world is
> "intuitionistic logic--e.g., defining negation as the *implication* of
> falsity..." (emphasis added) a "drastic reduction in acceptable
> mathematical theories"? Acceptable to whom? You? Who else?
>
> And, really, it seems fairly over the top to describe it as "an attempt to
> block the way of inquiry?" Certainly your saying so does not make it so.
> Please explain how this "blocks the way of inquiry" for folk like me who
> are apparently radically deficient in mathematics and logic so simply can't
> see it as such.
>
> As JAS wrote:
>
> JAS: . . . Peirce seeks to account for *objective *or *ontological
> *indeterminacy--there
> are some propositions that *are *neither true nor false because *reality *is
> indeterminate. Years later, he briefly explores the logical implications of
> this in his "Logic Notebook."
>
> CSP: Triadic Logic is that logic which, though not rejecting entirely the
> Principle of Excluded Middle, nevertheless recognizes that every
> proposition, S is P, is either true, or false, or else S has a lower mode
> of being such that it can neither be determinately P, nor determinately
> not-P, but is at the limit between P and not P. (R 339:515[344r], 1909)
>
> JAS: Intuitionistic logic also recognizes this limit but does not assign a
> third truth value to it, which is what Peirce goes on to propose.
> Accordingly, in today's terminology, *what he [Peirce] is primarily
> rejecting in these passages is not the (so-called) law of excluded middle
> ("either a proposition or its negation is true"), but rather the principle
> of bivalence ("every proposition is either true or false") *[emphasis
> added].
>
> In a word: Who is actually being 'dogmatic' here?
>
> Best,
> Gary R
> “Let everything happen to you
> Beauty and terror
> Just keep going
> No feeling is final”
> ― Rainer Maria Rilke
> *Gary Richmond*
>
> *Philosophy and Critical Thinking*
>
> *Communication Studies*
>
> *LaGuardia College of the City University of New York*
>
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