Thanks for that response, Jon. After tweaking my draft a bit and inserting a link to a list of the sources you mentioned, I’ve now put up the whole chapter which includes the section on EGs (Turning Signs 13: Meaning Spaces (gnusystems.ca) <https://gnusystems.ca/TS/mns.htm#lgspc> and taken down the draft.
I found your introduction both concise and readable, but then I’m familiar with the material you’ve posted to the list on the subject, so maybe mine is not an “innocent eye.” It’s good to see that it will be part of a “Handbook of Cognitive Mathematics” (a field I hadn’t heard of before). It gives a more complete sketch of EGs than mine does, but then mine is embedded in a book that deals more with the positive cognitive sciences (as Peirce would call them)(philosophy, semiotic/logic, biology, psychology, linguistics etc.) than with mathematics, so it’s adapted to that purpose. Gary f. From: peirce-l-requ...@list.iupui.edu <peirce-l-requ...@list.iupui.edu> On Behalf Of Jon Alan Schmidt Sent: 19-May-21 12:24 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": 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": 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": 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 <http://www.LinkedIn.com/in/JonAlanSchmidt> - twitter.com/JonAlanSchmidt <http://twitter.com/JonAlanSchmidt>
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