Jon S, all, You say, "however, it seems to me that Peirce's three cosmologies are derived from elliptical, parabolic, and hyperbolic lines, not surfaces--continua of one dimension, not two (or more), isomorphic to time and the inferential process of reasoning." It helps to think of the lines as cuts through surfaces. In the NEM, Peirce points out that the early mathematical development of the elliptical, parabolic, and hyperbolic lines was as planar cuts through conics. As such, the lines are on a planar surface, which function as a kind of mapping. Typically, I find Peirce moving between different dimensions in his discussions (one, two, three, etc.) of topology, projective geometry and metrical geometries. I think he is doing the same in the application of the EG as a system of mathematical logic, to his philosophical study of inquiry.
I am interested in the tinctured graphs, as such I'd like to stick with what he says about the Gamma system for now. Peirce argues that ampliative inferences by abduction and induction have correlative expressions as deductive inferences. This much should be pretty straightforward: take the premisses of the inductive argument and the inferential rule as the antecedent of a conditional and the conclusion as a consequent, and then assert premisses, and then the conclusion will follow deductively. In doing so, we will need to properly color the various parts of the propositions, and the inference rule, to clarify the modal character of each. Doing so will shed some light on the character of what is being asserted in each of the propositions. You ask: "how we might represent plausibility, probability, and associated conceptions such as degrees of confidence, margins of error, and the like" in EGs. John Sowa shows how easy it is to insert mathematical arguments, such as the proof of a theorem in Euclid's Elements, into the EGs. Doing so yields heterogeneous diagrams. We can do the same with assertions and diagrams expressed in the language of probability theory, statistics, etc. They can functions as the premisses and conclusions of arguments in the EGs. --Jeff ________________________________ From: [email protected] <[email protected]> on behalf of Jon Alan Schmidt <[email protected]> Sent: Friday, August 15, 2025 2:45 PM To: Peirce-L <[email protected]> Subject: Re: [PEIRCE-L] Topology, Cosmology, and Existential Graphs (was Semiosic Ontology and Evolution) Jeff, List: Your point about projective and metrical geometry being specializations of topology is well taken. However, it seems to me that Peirce's three cosmologies are derived from elliptical, parabolic, and hyperbolic lines, not surfaces--continua of one dimension, not two (or more), isomorphic to time and the inferential process of reasoning. I share your interest in how EGs might shed further light on this, but I remain puzzled about what exactly you have in mind. Peirce's transformation rules for EGs correspond to inferences that are strictly deductive, not abductive nor inductive--they ensure that a false EG is never derived from a true EG, rather than facilitating ampliative reasoning. As I noted before, he never completed the Gamma part, and after abandoning cuts (including broken cuts) in favor of shading, he recognized the need for a new Delta part to represent and reason about modal propositions involving possibility and necessity. My two recent papers on this subject propose his Logic Notebook entry of 1909 Jan 7 as a potential candidate for that, with heavy lines of compossibility (my term) representing possible states of things in which propositions denoted by letters attached to them would be true (https://doi.org/10.2979/csp.00026, https://doi.org/10.23925/2316-5278.2025v26i1:e60449). Personally, I am at a loss as to "how we might represent" plausibility, probability, and associated conceptions such as "degrees of confidence, margins of error, and the like" in EGs. Do you have any specific suggestions? 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> On Tue, Aug 12, 2025 at 11:46 PM Jeffrey Brian Downard <[email protected]<mailto:[email protected]>> wrote: Jon S, Gary R, Gary F, all, It is worth noting that Peirce argues topology supplies the key principles (i.e., postulates and theorem) that are foundational for projective and metrical geometries. See the discussion of topology and theses geometries in the NEM. The distinction between elliptical, parabolic and hyperbolic surfaces and higher dimensional spaces is rooted in topology. See the helpful discussion in Jeffrey Weeks, The Shape of Space, for instance. The reason I raised the topological questions about Peirce's distinction between three kinds of philosophies is that they are, at root, different accounts of the beginning and ending points of inquiry, considered as some kind of limit. Those beginning and ending points are taken to mirror an analogous account of the beginning and ending points of cosmological evolution, also considered as some kind of limit. Jon S claims that Peirce's claims should be understood in terms of projective geometry and not topology. I am suggesting that the claims can be understood more deeply if we consider both, and especially the way the postulates of topology provide a foundation for projective and metrical systems of geometry. I take Peirce seriously when he claims that the EG's were designed to have the richness needed to represent any kind of proposition, and any kind of inference, regardless of whether it is deductive, inductive or abductive. As such, I am interested in various ways we might draw on the existential graphs, especially in their gamma form, to clarify the claims he is making about elliptical, parabolic and hyperbolic philosophies. Without getting into the details, the differences in the assumptions will likely show up in the different ways philosophers might interpret the different universes of discourse, and the manner in which the conceptions of truth and reality are represented in a logical system such as the EG's. The differences will show up prominently when we consider the ways in which abductive and inductive inferences might be represented--especially when we consider the ways the modal features of are treated--such as a the modal features of the claims that a hypothesis is plausible or that the conclusion of an inductive inference is probable. The claims of probability involved in induction need to be interpreted in terms of metrics for degrees of confidence, margins of error, and the like. I'm curious about how we might represent these conceptions in propositions expressed in terms of the gamma graphs. --Jeff
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