Jon Schmidt, John Sowa, List,
It might be helpful to make a clearer distinction between what is advantageous for the purposes of developing the EGs as a formal system of mathematical logic and what is advantageous for the purposes of developing theories of philosophical logic. For the sake of illustrating the importance of the distinction, let me take up the following assertion Jon Schmidt: "Hence I continue to maintain that the cut for negation must be derived from the scroll for consequence with a blackened inner close, rather than treated as a primitive, even when shading is employed instead." For the purposes of developing systems of mathematical logic, the logician can adopt various starting points in setting up the logical grammar for a given system. In symbolic systems, the rules determine what does and does not count as a well-formed-formula. The same holds in the case of the EGs. The grammatical rules determine what counts as a well-formed-graph. Given all the work he has done on the symbolic systems of logic, Peirce sees that there are a number of different ways of setting up the grammatical rules that will, when taken together with the rules of inference and transformation, yield consistent results. For the sake of the EGs considered as a formal system, the scroll and two nested circles are logically equivalent. What is more, it makes no difference for the beta graphs whether the scroll (used to represent the conditional) or a shaded area within a boundary (used to represent negation) is taken as "primitive" in one sense or another. Having said that, I do think there is a special philosophical significance that Peirce attaches to the scroll as a representation of the conditional. I do not think that it is mere artifact of his early explorations of the graphs. As Peirce points out, the graphs can be used to express any sort of proposition. As such, they can be put to use in philosophical inquiry for the sake of analyzing the logical relationships between any set of premisses and conclusions. For the sake of giving a deeper philosophical analysis of the different classes of arguments we need to apply the EGs to the problem of analyzing synthetic forms of inference. In doing so, it will be helpful to have a variety of different icons that can be used to study the grounds of the validity of inductive and abductive inference. (MS 296, 499) As far as I can see, the scroll is a special kind of iconic sign because it expresses the continuity in the relationship between antecedent and consequent of the conditional, and this mirrors the continuity in the relationship between premisses and conclusions in an argument. In the case of inductive and abductive inferences, the conditionals may take a variety of forms: epistemic, alethetic, deontic, etc. In each of these cases, the topological character of the relations may vary. Based on my own inquiries using the graphs to analyze these forms of inference, thinking about the relationship between the scroll and the shaded area representing negation has been a fruitful endeavor. It is possible that it has been fruitful given the fact that I am still at an early point in my application of the graphs to these problems of critical logic. Yours, Jeff Jeffrey Downard Associate Professor Department of Philosophy Northern Arizona University (o) 928 523-8354 ________________________________ From: Jon Alan Schmidt <[email protected]> Sent: Monday, August 3, 2020 7:06:34 PM To: [email protected]; [email protected] Cc: [email protected]; [email protected]; [email protected]; [email protected]; Gary Richmond Subject: [PEIRCE-L] Philosophy of Existential Graphs (was Peirce's best and final version of EGs) John, All, List: With your permission given below, I am posting this reply on Peirce-L. Anyone is obviously still free to respond off-List if that is preferred. JFS: The theory of EGs that Peirce presented in L231 (which I have been calling eg1911) is the one he wished Lady Welby and her group to consider his last and best version of EGs. This claim is a plausible interpretative hypothesis based on the circumstances and timing of the letter, but it should be acknowledged that the text itself does not state or imply any such specific intention on Peirce's part. JFS: Some readers might be misled by Peirce's earlier writings to think that there is some "deeper" meaning that is not expressed by a nest of two ovals. Such an impression is not misleading at all, since Peirce explicitly denies that a consequence (scroll) is strictly equivalent to a composite of two negations (nested cuts). I already quoted the following passage in one of my Peirce-L posts, but it is worth repeating. CSP: The second failure of Selectives to be as analytical as possible lies in their encouraging the idea that negation, or denial, is a relatively simple concept, and that the concept of Consequence, is a special composite of two negations, so that to say, “If in the actual state of things A is true, then B is true,” is correctly analyzed as the assertion, “It is false to say that A is true while B is false.” I fully acknowledge that, for most purposes and in a preliminary explanation, the error of this analysis is altogether insignificant. But when we come to the first analysis the inaccuracy must not be passed over. ... Indeed, so far is the concept of Sequence from being a composite of two Negations, that, on the contrary, the concept of the Negation of any state of things, X, is, precisely, a composite of which one element is the concept of Sequence. Namely, it is the concept of a sequence from X of the essence of falsity. (R 300:[47-51], 1908) According to Peirce, it is neither correct nor accurate to analyze "if A then B" as "not (A and not-B)," although "for most purposes ... the error of this analysis is altogether insignificant." Treating negation as a primitive results in a system that is simpler and more iconic, but not "as analytical as possible" because negation is "a composite of which one element is the concept of Sequence," which by contrast is indecomposable and fundamental to logic. CSP: A sequence is a unidimensional form in which there is a difference between the relation of A to B and of B to A. Mathematically considered, in one dimension it is a progress from a point A to a point B, where A and B are different or A and B may coincide, or they may both vanish. Of these three forms of sequence, the first [hyperbolic] is distinctly that of logic since the ultimate antecedent and the ultimate consequent are different in logic. You cannot proceed from antecedent to consequent till you reach again your original antecedent (as in the 3rd kind of sequence, the elliptical), nor do you tend to such a return (as in the second, or parabolic sequence), but the two are distinct. (NEM 4:127, 1897-8) Peirce even generalizes this to the very process of semeiosis, whose sequence is always from the object through the sign to the interpretant. "The object and the interpretant are thus merely the two correlates of the sign; the one being antecedent, the other consequent of the sign" (EP 2:410, 1907). Hence I continue to maintain that the cut for negation must be derived from the scroll for consequence with a blackened inner close, rather than treated as a primitive, even when shading is employed instead. 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 Sat, Aug 1, 2020 at 11:56 PM John F. Sowa <[email protected]<mailto:[email protected]>> wrote: Jon et al., I have no objection to posting any or all of these notes on Peirce-L. I sent my previous note offline to avoid stuffing everybody's inboxes with endless debates about a very straightforward claim: The theory of EGs that Peirce presented in L231 (which I have been calling eg1911) is the one he wished Lady Welby and her group to consider his last and best version of EGs. Re the word 'scroll': In terms of the semantics (endoporeutic) and permissions (rules of inference) of eg1911, a scroll is *indistinguishable* from a shaded area with a nested unshaded area. Anything that Peirce wrote about scrolls prior to 17 June 1911 is useful only for understanding the development of Peirce's thought. After that date. the word 'scroll' can only create confusion. Some readers might be misled by Peirce's earlier writings to think that there is some "deeper" meaning that is not expressed by a nest of two ovals. Re intuitionistic logic: Peirce may have had some vague thoughts along those lines, but he never formulated them precisely. Anybody has a right o develop an intuitionisitic extension to EGs and use whatever notation they prefer. But their choice of syntax and semantics for those EGs is independent of anything that Peirce wrote. John
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