John, List: Again, as we have been advised repeatedly, since the post to which I am replying lacks "an exact quotation by Peirce," everything in it is someone else's opinion and cannot be claimed as "what Peirce meant" or "what Peirce intended."
JFS: In R670, CSP explicitly stated that the scroll is equivalent to a nest of two negations (12 June 1911). Peirce "explicitly stated" no such thing, since neither the word "scroll" nor the word "nest" appears *even once* in R 670. However, he does provide three relevant EGs, all of which express "if A then either B or C" (see attached R670Fig9-11.jpg). Fig. 9 shows a cut with two nested cuts, Fig. 10 shows a scroll with two loops, and Fig. 11 shows a shaded area with two nested unshaded areas. Peirce's only mention of Fig. 10 is not about *scrolls *vs. cuts, but about *shading *vs. cuts--"It is a help to shade the oddly-enclosed areas and omit the lines that represent the cuts, as in Fig. 11, which is equivalent to Fig. 10" (R 670:17[16], 1911 June 12). Notice that Peirce *does not* say that Fig. 10 is equivalent to Fig. 9, although this would likewise be accurate for classical logic. More importantly, he also *does not* say, in this manuscript or any other as far as I know, that he *rejects *his extensive previous writings about negation being properly derived from the implication of falsity. JFS: On the next day, he wrote that it's so easy to mistake the cellar door of hell for the cellar door of heaven. This is a good example of why we should provide exact quotations wherever possible, including as much of the context as necessary to convey the significance of the key portion. Here is the entire passage that concludes with his remark to this effect, in which everything from "consists" to the end is on a discarded page of the manuscript. CSP: In the first place, if the reader can put up with another technical term, it must be remarked that the line of identity, that heavy kind of line that when evenly enclosed affirms that that individual object that is denoted by one of its extremities *is identically the same* as the individual denoted by its other extremity [*sic*]. Since no concept stands less in need of analysis in the process of deduction than does that of identity, it ought to be regarded as a *simple graph* expressing that “Something (denoted by one of its extremities) *is *the *same individually as*” (that which its other extremity denotes); and consequently it must be scribed, if at all, upon some single area within the border of the phemic sheet. Any instance of it is, however, scribed as a continuous line, which is divisible into and consists of as many parts as one may like; so that it might be urged that what it actually expresses [is] something more like this: "Something (its one end) is something that is something that is something that is something that is something that is something that is something that is something that is something that is (its other end.)" Only instead of a finite series of "somethings," there is, not an infinite number, (for that would not suffice) but a continuum of somethings! This might be urged, but it would not be correct; and in this case, as in many others, the difference which amounts, quite absolutely, to nothing at all of substance, is of high importance in logic and will sometimes make the difference practically of falsehood in place of truth,--especially, in that long and dark corridor of metaphysical speculation it might be so easy to mistake the cellar door of hell for that of heaven (R 670:23&37[22-23], 1911 June 13) I frankly do not see the relevance of this for the issue at hand. Peirce is discussing the line of identity, not scrolls vs. cuts vs. shading or implication vs. negation, and his remark about cellar doors to hell and heaven pertains quite specifically to the dangers of "metaphysical speculation." Consistent with what he writes later to replace the discarded content, it seems to me that he is highlighting the error of treating a line as if it consisted of discrete individuals (i.e., points). Instead, the line of identity iconically signifies the continuity of that relation itself, as he explains elsewhere. JFS: That is the day when he consigned that ridiculous story to the netherworld. What allegedly "ridiculous story" is being referenced here? Surely not Peirce's discussion of "paradisaical logic," since he invokes it again six months later (RL 376, 1911 Dec 6). JFS: Re intuitionistic logic: it is incompatible with Peirce rules of inference, which are the foundation for every EG proof in every version of EGs. On the contrary, intuitionistic logic is *perfectly *compatible with Peirce's rules of inference for EG proofs--namely, erasure/insertion and iteration/deiteration--as amply demonstrated a decade ago by Oostra, and more recently by Ma and Pietarinen. The *only *difference from classical logic is that the primitive sign, besides the blank sheet, *must *be the scroll for implication; i.e., it *cannot *be the cut for negation. Of course, this is *perfectly *compatible with Peirce's derivation of the latter from the former on multiple occasions. Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt On Thu, May 20, 2021 at 5:00 PM John F. Sowa <[email protected]> wrote: > Jon AS, > > I'm tied up with other deadlines, and I'll post a longer article that goes > into all the details in a few more days. > > Re rejecting R669: In R670, CSP explicitly stated that the scroll is > equivalent to a nest of two negations (12 June 1911). On the next day, he > wrote that it's so easy to mistake the cellar door of hell for the cellar > door of heaven. That is the day when he consigned that ridiculous story to > the netherworld.. > > Re intuitionistic logic: it is incompatible with Peirce rules of > inference, which are the foundation for every EG proof in every version of > EGs. > > John >
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