Jeff, All: JD: 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.
I agree; in fact, it expresses and mirrors not just the *continuity* of that relationship, but also its *asymmetry*. All semeiosis is inferential process, a continuous hyperbolic sequence that always and only has one direction--from premiss to conclusion, from antecedent to consequent, from object to interpretant. I am in general agreement with your other comments, as well. In particular, my interests here are primarily philosophical--as the subject line reflects--which is why I keep insisting on the derivative nature of negation while recognizing that treating it as a primitive does not necessarily affect the *appearance *of the resulting EGs. As I just noted in my other post, it does make an important difference in the *interpretation *of those EGs, which is why I believe that we should follow Peirce's advice to add a small darkened circle to a shaded area when it is intended to represent negation rather than the antecedent of a consequence. Another potential difference in interpretation is where we choose to draw the line between the subjects and predicate of a proposition. I take Peirce seriously when he states that "the proper way in logic is to take as the subject whatever there is of which sufficient knowledge cannot be conveyed in the proposition itself, but collateral experience on the part of its interpreter is requisite ... leaving the *pure *predicate a mere form of connection" (NEM 3:885, 1908 Dec 5). After all, "when we have carried analysis so far as to leave only a continuous predicate, we have carried it to its ultimate elements" (SS 72, 1908 Dec 14). Accordingly, my analysis of an EG is that the names and lines are subjects, respectively denoting abstract general concepts and concrete indefinite individuals, while the syntax of their attachments signifies the pure/continuous predicate. Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt On Tue, Aug 4, 2020 at 10:17 PM Jeffrey Brian Downard < [email protected]> wrote: > 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 > >>
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