John, List: JFS: As I said, I'm writing a longer article in which I cover all the details Just a short preview of coming attractions:
I sincerely look forward to reading it, but in the meantime, I respectfully suggest holding off on further posts until time and circumstances permit careful review and accurate citation of the texts. JFS: The story about paradisaical logic is used in R669 to justify the derivation of negation from a scroll. In L376, Peirce happens to use the same term for a logic without negation, but he does not use it to derive negation from the scroll. Indeed, Peirce does not go so far as to derive negation from the implication of falsity on the latter occasion. Nevertheless, his obvious recognition that there *can* be "a logic without negation" is sufficient to establish the underlying philosophical point that there *cannot *be a logic without inference, which corresponds to the primitive formal relation of implication. JFS: On the contrary, he uses it as an excuse for defining the blot as an affirmation, not a denial. Therefore, the blot cannot be used in conjunction with a scroll to derive a negation. Here is what Peirce actually says. CSP: The simplest part of speech which this syntax contemplates, which, as scribed, I shall term *a **blot *... is itself an *assertion*. Ought it to be an affirmation or a denial? A denial is logically the simpler, because it implies merely that the utterer recognizes, however vaguely, *some *discrepancy between the fact and the speech, while an affirmation implies that he has examined all the implications of the latter and finds no discrepancy with the fact. This is a circumstance to be borne in mind; but since the denial implies recognition of the affirmation, while the affirmation is so far from implying recognition of the denial, that one might imagine a paradisaic state of innocence in which men never had the idea of falsity, and yet might reason, we must admit that affirmation is *psychically *the simpler. Now I think that upon this point we must prefer psychical to logical simplicity. I therefore make *the blot* an affirmation. (RL 376, 1911 Dec 6; bold added) The key question is, what exactly does "a blot" or "the blot" affirm? Peirce goes on to discuss "blots" (plural) as if they were the same as what he elsewhere calls "spots"--letters for propositions in Alpha, or names for general concepts in Beta--and we puzzled over this together on-List a couple of years ago. However, in his earlier writings, "blot" (singular) *always *has a very specific reference. CSP: Thus far, we have no means of expressing an absurdity. Let us invent a sign which shall assert that *everything is true*. Nothing could be more illogical than that statement inasmuch as it would render logic false as well as needless. Were every graph asserted to be true, there would be nothing that could be added to that assertion. Accordingly, our expression for it may very appropriately consist in completely filling up the area on which it is asserted. Such filling up of an area may be termed *a blot*. (R 455, 1903; bold added) CSP: Thus, I shall say that the *alpha-signs*, that is to say the signs of this system heretofore described apart from the *graphs *themselves are two, and there are besides two peculiar graphs. Namely, the two peculiar graphs are the blank place which asserts only what is already well-understood between us to be true, and *the **blot *which asserts something well understood to be false. (R 455, 1903; bold added) CSP: Since, the meaning of *the blot* is that everything is true, the predication of the definition of the definitum is that where *the blot* is scribed anything whatever is permitted to be scribed. *The blot*, however, fills its whole area so that it is physically impossible to scribe anything else. But we have already seen that any enclosure having *a blot *on its area can be erased, provided we interpret a cut as precisely denying the truth of the entire graph which it encloses. (R 456(s), 1903; bold added) CSP: Passing to *the blot*, or *pseudograph*, of which, you remember the meaning is that everything is true, the predication of the definition concerning the definitum is that within any even number of cuts where *the blot* is any graph we please may be inserted and within any odd number of cuts where *the blot* is any graph may be erased. *The blot*, it is true, fills its whole area, so as to leave no room for any other graph. But there is an equivalent of it of which this is not true. For since by Permission No. 1 every graph on the sheet of assertion enclosed can be transformed into the blank, it follows, by the principle of contraposition that an enclosure containing nothing but a blank can when evenly enclosed be transformed into anything we please, and consequently into the pseudograph. The vacant enclosure is, therefore, a form of the pseudograph. For evenly enclosed it can be transformed into *the blot*, as *the blot* can be transformed into it. And since these two transformations are the reverse of one another, it follows, by the principle of contraposition that the same is true within any odd number of cuts. (R 456, 1903; bold added) CSP: The idea of falsity, the serpent in their Eden, that could only cause them to eat of the fruit of the Tree of knowledge of Logical Good and Evil, could only come from noticing Words as being different from Things. When that came, they being already in the habit of uttering simple Conditional Propositions, *might conceivably* have formed such a Proposition as, "If A is true, everything is true," and this might have suggested that Not everything is true. At any rate, the Scroll affords me no other means of denying any Graph, say A, than by scribing that if A be true, everything is true. Now since it is impossible by any addition to increase Everything, this I can suitably express by completely filling with *a blot* the Inner Close of a Scroll that carries only A (and the Blank) in its Outer Close, so that there shall be no more room in that Inner Close for anything else. (RS 30, Copy T; c. 1906; bold added) The blot *affirms *that *everything *is true, and since this proposition is an absurdity, it is not a genuine graph but a pseudograph. The empty cut *denies *that *anything *is true, and since this proposition is also an absurdity, it is likewise not a genuine graph but a pseudograph. Either one can thus serve as the basis for defining negation in EG; but as Pietarinen, Bellucci, and three other co-authors summarize in their very recent and aptly named paper, "The Blot" ( https://www.researchgate.net/publication/343688226_The_Blot), "Peirce's preference lies with the former, since it *analyses* falsity and negation without assuming it" (pp. 233-234). They subsequently add the following observations. AvP, FB, et al.: Taking absurdity as "Everything is true" has some other conceptual and formal advantages that we briefly list. (1) It explains *ex falso*: If everything is true then P also is true. There is no need for an axiom or a rule and no need to appeal to proofs by disjunctive syllogisms, which are known to be circular. (2) The Law of Excluded Middle (LEM) and the elimination of double cut are laws not inherent in the nature of negation, which is a desirable feature intuitionistically (Peirce came close to intuitionistic logic in many related senses). (3) The double cut rule is to be derived, if justified, from more primitive, observational considerations. If the cut were defined as reversing its area, then the double cut rule would be immediate by symmetry. But symmetry, though advantageous in calculus, is an unfavorable guideline when the purpose is logical analysis (CP 4.375). On the other hand, although the absurdities "Everything is true" and "There is no truth" are semantically equivalent, the latter is gotten from the former: rules like the elimination of double cut are not eligible at this level of analysis. From "Everything is true" it follows that "It is true that there is no truth." But from "There is no truth" it follows, for example, that "It is wrong that something is wrong," which means that everything is true. However, we need an extra move here. Thus from negative absurdity we cannot directly derive the positive absurdity (its justification would need another rule or an axiom, such as LEM). But from positive absurdity other facets of absurdity follow. (p. 234) Hence, the blot as an absurd *affirmation *is philosophically essential to the derivation of negation from the implication of falsity, as well as philosophically superior to the empty cut as an absurd *denial* in various other ways. JFS: Please reread R270. He does state that the scroll (using that word) is "equivalent" to a nest of negations. The relevant manuscript is R 670, not R 270. I did review it, and again, it *does not* contain the word "scroll" *at all*. If I am somehow repeatedly overlooking its appearance in that text, please provide a quotation and the page number, and I will be grateful for the correction. JFS: Therefore, the loose sheet about heaven and hell has two implications: (a) CSP is rejecting the story that supports the derivation of negation from the scroll, and (b) he doesn't need to include it in the complete version of R670 since R669 will not be included in the final draft. I already quoted the entire passage from that "loose sheet," showing that it is unambiguously about the line of identity and thus has *absolutely nothing* to do with R 669, Peirce's parable of paradisaical logic, or the derivation of negation from the implication of falsity. JFS: But this evidence makes it very hard for anyone to claim the contrary opinion: that CSP preferred R669 to R670. Fortunately, *no one* is arguing that Peirce preferred R 669 to R 670. Instead, my contention (and Francesco's, https://list.iupui.edu/sympa/arc/peirce-l/2021-03/msg00003.html) is that he never *rejected *his longstanding analysis of negation as being derived from the implication of falsity, which extends back several decades to his development of logical algebras long before he even invented EG. Only a direct quotation of an explicit statement by Peirce could warrant attributing such a drastic a change of mind to him, and the *complete absence *of any textual evidence to that effect "makes it very hard for anyone to claim the contrary opinion." 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 10:25 PM John F. Sowa <s...@bestweb.net> wrote: > Jon AS, > > As I said, I'm writing a longer article in which I cover all the details > Just a short preview of coming attractions: > > 1. The story about paradisaical logic is used in R669 to justify the > derivation of negation from a scroll. In L376, Peirce happens to use the > same term for a logic without negation, but he does not use it to derive > negation from the scroll. On the contrary, he uses it as an excuse for > defining the blot as an affirmation, not a denial. Therefore, the blot > cannot be used in conjunction with a scroll to derive a negation. > > 2. Please reread R270. He does state that the scroll (using that word) is > "equivalent" to a nest of negations. He also draws both EGs. The date of > that passage is 12 June 1911. In a loose sheet included with R270 and > dated 13 June 1911, he talks about "the long dark tunnel of metaphysical > speculation" in which it is so easy to mistake the cellar of hell for the > cellar of heaven. > > 3. Since that passage about heaven and hell is only relevant to someone > who had recently read R669, it would be irrelevant if CSP had no intention > of including R669 in a final version of the article he was writing. > Therefore, the loose sheet about heaven and hell has two implications: (a) > CSP is rejecting the story that supports the derivation of negation from > the scroll, and (b) he doesn't need to include it in the complete version > of R670 since R669 will not be included in the final draft. > > I emphasize that this is just my opinion. But this evidence makes it very > hard for anyone to claim the contrary opinion: that CSP preferred R669 to > R670. > > John > >
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