Jon, John, All, JAS: [Peirce's] primary objective in developing both his logical algebras and EGs is *not *"making a calculus which would turn out conclusions by a regular routine." It is "simply and solely the investigation of the theory of logic," which requires "that the system devised for the investigation of logic should be as analytical as possible" (CP 4.373, 1902).
JAS: EGs with shading, rather than cuts, satisfy this criterion as long as the *derivation *of negation from the primitive of consequence, reflecting the fundamental asymmetry of all semeiosis, is kept firmly in mind. Accordingly, I agree with Peirce's "confession" that it is an "error" to assume that "because the blackened Inner Close can be made indefinitely small, therefore it can be struck out entirely like an infinitesimal" (CP 4.564n, c. 1906). Instead, *when a shaded area is intended to represent negation--not the antecedent of a consequence--it should have a darkened circle within it, "however small, to represent iconically, the blackened Inner Close"* (ibid).[emphasis added by GR] QED (more or less), Gary R "Time is not a renewable resource." gnox *Gary Richmond* *Philosophy and Critical Thinking* *Communication Studies* *LaGuardia College of the City University of New York* <http://www.avg.com/email-signature?utm_medium=email&utm_source=link&utm_campaign=sig-email&utm_content=webmail> Virus-free. www.avg.com <http://www.avg.com/email-signature?utm_medium=email&utm_source=link&utm_campaign=sig-email&utm_content=webmail> <#DAB4FAD8-2DD7-40BB-A1B8-4E2AA1F9FDF2> On Wed, Aug 5, 2020 at 8:19 PM Jon Alan Schmidt <[email protected]> wrote: > John, All: > > JFS: The beauty of eg1911, as specified in L231, is its brevity, > simplicity, precision, and bare minimum of verbiage. > > > Again, no one is disputing this. Nevertheless, elsewhere Peirce > explicitly (1) denies that a consequence is a composite of two negations, > (2) derives the cut for negation from the scroll with a blackened inner > close accordingly, and (3) states that shading any oddly enclosed area is > what enabled him to perceive that it "represents a kind of possibility," > not just a denial of actuality. One upshot of this "discovery" is the > breakdown of the reasoning behind the widely accepted rule "that every > conditional proposition whose antecedent does not happen to be realized is > true," because the principle of excluded middle does not hold when that > antecedent is a *real *possibility. What conclusion does Peirce go on to > draw from this? > > CSP: I often think that we logicians are the most obtuse of men, and the > most devoid of common sense. As soon as I saw that this strange rule, so > foreign to the general idea of the System of Existential Graphs, could by > no means be deduced from the other rules, nor from the general idea of the > system, but has to be accepted, if at all, as an arbitrary first > principle,--I ought to have poked myself, and should have asked myself if I > had not been afflicted with the logician’s *bêtise*, What compels the > adoption of this rule? The answer to that must have been that the > *interpretation *requires it; and the inference of common sense from that > answer would have been that the interpretation was too narrow. Yet I did > not think of that until my operose method like that of a hydrographic > surveyor sounding out a harbour, suddenly brought me up to the important > truth that the *verso *of the sheet of Existential Graphs represents a > universe of possibilities. This, taken in connection with other premisses > led me back to the same conclusion to which my studies of Pragmatism had > already brought me, the reality of some possibilities. (R 490:26-28, CP > 4.581,1906) > > > The restriction of the scroll to a consequence *ut nunc*--which, as > Francesco Bellucci has explained ( > http://www.academia.edu/20434982/Charles_S._Peirce_and_the_Medieval_Doctrine_of_consequentiae > ), Peirce seems to conflate with a conditional *de inesse* after 1896--cannot > "be deduced from the other rules" for EGs. Instead, it must be *imposed *as > an additional "arbitrary first principle," like what is now known as > Peirce's Law in the axiomatization of classical logic, which is absent from > intuitionistic logic. Why had he not noticed this previously? For one > thing ... > > JFS: 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. > > JFS: Any EG drawn with a scroll would either be semantically identical > to one with two ovals or it would be meaningless. > > JFS: In the note I just sent, I was talking about the version of EGs in > L231. For that version of logic, there can be no difference in semantics > between a scroll and a nest of two ovals. > > > As Peirce himself ascertains, the relevant distinction is not to be found > in the rules or general idea of EGs, but in the *interpretation *of a > scroll--regardless of whether it appears as a single continuous line that > crosses itself once to form inner and outer loops, as one oval inside > another, or as a shaded area around an unshaded area. The greater > iconicity of the last option is not primarily with respect to negation, but > because an oddly enclosed area is a different *surface* from an evenly > enclosed area, corresponding to a universe of possibility rather than that > of actuality. Moreover ... > > CSP: This is a striking proof of the superiority of the System of > Existential Graphs to either of my algebras of logic. For in both of them > the incongruity of this strange rule is completely hidden behind the > superfluous machinery which is introduced in order to give an appearance of > symmetry to logical law, and in order to facilitate the working of these > algebras considered as reasoning machines. I cannot let this remark pass > without protesting, however, that in the construction of no algebra was the > idea of making a calculus which would turn out conclusions by a regular > routine other than a very secondary purpose. (R 490:28-29, CP 4.581) > > > This has important bearing on "the unsolved research problem from 1988" as > described in John's presentation slide ( > https://list.iupui.edu/sympa/arc/peirce-l/2020-08/msg00017/ppe65.png). > > JFS (off-List): The reason why the problem by Larry Wos (1988) was > unsolved is that Gentzen assumed that an if-then statement was essential > for "illative transformations" (Peirce's term). But an if-then statement > and any proofs that depend on it are not symmetric. Peirce's EG rules, > which depend only on existence, conjunction, and negation, are simpler and > symmetric. > > > For Peirce, symmetry in logical laws is merely "an appearance," the > pursuit of which results in "superfluous machinery" that *obscures *the > non-equivalence of "if A then B" and "not (A and not-B)" for the sake of > facilitating "reasoning machines." By contrast, his primary objective in > developing both his logical algebras and EGs is *not *"making a calculus > which would turn out conclusions by a regular routine." It is "simply > and solely the investigation of the theory of logic," which requires "that > the system devised for the investigation of logic should be as analytical > as possible" (CP 4.373, 1902). > > EGs with shading, rather than cuts, satisfy this criterion as long as the > *derivation *of negation from the primitive of consequence, reflecting > the fundamental asymmetry of all semeiosis, is kept firmly in mind. > Accordingly, I agree with Peirce's "confession" that it is an "error" to > assume that "because the blackened Inner Close can be made indefinitely > small, therefore it can be struck out entirely like an infinitesimal" (CP > 4.564n, c. 1906). Instead, when a shaded area is intended to represent > negation--not the antecedent of a consequence--it should have a darkened > circle within it, "however small, to represent iconically, the blackened > Inner Close" (ibid). > > 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 10:44 PM John F. Sowa <[email protected]> wrote: > >> Jon AS, >> >> This is yet another case where the mathematical structures are precise, >> but the words that describe them leave enough ambiguity to cause confusion. >> >> The beauty of eg1911, as specified in L231, is its brevity, simplicity, >> precision, and bare minimum of verbiage. Every EG that conforms to the >> syntax of eg1911 has a precise translation to and from a logically >> equivalent statement in Peirce's algebra of 1885. >> >> It also has a precise translation to and from a logically equivalent >> statement in every version of classical FOL from Frege's Begriffsschrift >> (1879) to any notation for classical FOL that anyone may publish in the >> future. >> >> Any EG drawn with a scroll would either be semantically identical to one >> with two ovals or it would be meaningless. There is no other option. >> >> End of story. >> John > > <http://www.avg.com/email-signature?utm_medium=email&utm_source=link&utm_campaign=sig-email&utm_content=webmail> Virus-free. www.avg.com <http://www.avg.com/email-signature?utm_medium=email&utm_source=link&utm_campaign=sig-email&utm_content=webmail> <#DAB4FAD8-2DD7-40BB-A1B8-4E2AA1F9FDF2>
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