John, List, All: JFS: The bottom line (which I added) is a translation of the algebraic notation to EGs. It shows the EG for a single negation defined by an EG with three negations.
That "translation" is blatantly question-begging since it *presupposes *the cut for negation rather than properly *deriving *it from the scroll for implication with a blackened inner close for falsity/absurdity, expressed positively as "every proposition is true." Pietarinen recently wrote another excellent paper with Bellucci, Shafiei, and two other co-authors specifically about the latter ( https://www.researchgate.net/publication/343688226_The_Blot). JFS: It seems strange (dubious? unreasonable? unjustified?) to claim that a triple negation is more fundamental than a single negation. It is strange, dubious, unreasonable, and unjustified to attribute something to Peirce that he never stated, implied, or even so much as suggested. JFS: But his English explanation of Figures 23 to 26 is vague and rambling. He was struggling to make a bad choice seem plausible. On the contrary, the passage is quite lucid and straightforward, as well as perfectly consistent with multiple others in Peirce's writings on the same subject. Here it is in its entirety so that those reading along can judge for themselves. CSP: The Cut came to be thought of because of the immense frequency of occasions on which it was necessary to express the assertion "If X be true, then every assertion is true." It was forced upon the logician’s attention that a certain development of reasoning was possible before, or as if before, the concept of *falsity *had ever been framed, or any recognition of such a thing as a false assertion had ever taken place. Probably every human being passes through such a grade of intellectual life, which may be called the state of paradisaical logic, when reasoning takes place but when the idea of falsity, whether in assertion or in inference, has never been recognized. But it will soon be recognized that not every assertion is true; and that once recognized, as soon as one notices that if a certain thing were true, every assertion would be true, one at once rejects the antecedent that lead to that absurd consequence. Now that conditional proposition "If A is true, every proposition is true," is represented, in the model of Fig. 23, "If A is true, C is true" by blackening the entire inner close, as if there were no room, in reason, for any additional consequence. This gives Fig. 24: "If A be true whatever can be asserted is true," which is as much as to say that A is not true and the inner close being cut very small, we get, first Fig. 25 and finally Fig. 26, in which the idea of flat falsity is first matured. (R 669:18-20[16-18], LF 1:582, 1911 May 31) As Pietarinen emphasizes repeatedly, both in various papers over the years and in the introductory material in *Logic of the Future*, the scroll for implication is a logical primitive for Peirce--just like its algebraic predecessors, the copula of inclusion and the sign of consequence--because it corresponds directly to *inference*, which does not require any prior notion of falsity. By contrast, the cut for negation must be explicitly *defined *as a scroll with an infinitesimally small blackened inner close because otherwise it is a scroll *without *an inner close, thus signifying "if A" with no consequent or "if A, then something is true" where we interpret the inner close as containing the blank, which has been deiterated (CP 4.564n, c. 1906). JFS: And by the way, nothing in the article by Ma and Pietarinen depends on which logical operators are assumed as primitives. With derived rules of inference, the choice of primitives is irrelevant. Peirce has a different opinion, and I agree with him. Bellucci and Pietarinen summarize it as follows. FB & AVP: As an abbreviation, we can interpret the oval as negation and then express the conditional in terms of negation and conjunction (as in EGs) or in terms of negation and disjunction (as in Entitative Graphs). In this way, it is possible to negate *a* without resorting to the more complex conditional form with the blot as in Fig. 4. But Peirce is clear that this is just a derived interpretation, an interpretational corollary (MS 450, 1903). Taking the idea of negation as primary is philosophically inaccurate: All my own writings upon formal logic have been based on the belief that the concept of Sequence, alike in reasonings and in judgments, whether the latter be conditional or categorical, could in no wise be replaced by any composition of ideas [...] 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. (MS 300, pp. 46-48, 1908) Negation is not a primitive idea; rather, it is a conception derived from that of implication. Therefore, the sign of negation ought to be considered as a complication or determination of a more primitive sign, the scroll. The analysis of negation in terms of the conditional is according to Peirce one of the few satisfactory proofs of the indecomposability or uncompoundness of a logical concept that can be obtained. He took that in logical analysis, it is “tolerably easy to demonstrate compoundness, but next to impossible to make sure of elementality, or elementarity” (MS 300, p. 49, 1908). Once all other logical constants of the propositional calculus are defined in terms of the conditional relation, we have carried analysis to its extreme (truth-functional) limit: the conditional is logically unanalyzable, while other logical relations are analyzable through it. ( https://www.cambridge.org/core/services/aop-cambridge-core/content/view/9C4689940BDC5B17F739C34A87C2B77F/S1755020315000362a.pdf/div-class-title-existential-graphs-as-an-instrument-of-logical-analysis-part-i-alpha-div.pdf, pp. 220-221) As Peirce explains at the ellipsis in the quotation, from a strictly theoretical standpoint, implication *must *be a logical primitive and *cannot *be analyzed as a composite of two negations because the latter is missing the "real movement of thought in the mind" from antecedent to consequent. Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt On Sat, Feb 27, 2021 at 11:58 PM John F. Sowa <s...@bestweb.net> wrote: > Jon AS, List, > > To understand Peirce's views prior to 1911, I recommend the detailed > analysis in "Peirce's calculi for classical propositional logic", by > Minghui Ma and Ahti Pietarinen: > https://www.researchgate.net/publication/327235076 > > For a definition of negation in terms of material implication, see > MM_AVP.png (attached below). The top two lines are copied from p. 6 of > that article. The bottom line (which I added) is a translation of the > algebraic notation to EGs. It shows the EG for a single negation defined > by an EG with three negations. It seems strange (dubious? unreasonable? > unjustified?) to claim that a triple negation is more fundamental than a > single negation. > > To state an equivalent definition in R669, Peirce started with a scroll > for (A implies C) and took three steps to derive not-A. (See the attached > R669figs.png.) But his English explanation of Figures 23 to 26 is vague > and rambling. He was struggling to make a bad choice seem plausible. > > After Peirce wrote R669, he could not have been satisfied with those four > EGs and his vague English prose. He probably went to the blackboard to > draw other variations. If he drew the EG at the bottom of MM_AVP.png (or > anything similar), the conclusion would be obvious; negation must be > primitive. > > JAS> Peirce's actual claim was that his primary purpose when developing > both algebraic and graphical systems of logic was "not at all the > construction of a calculus to aid the drawing of inferences," but rather > "that the system devised for the investigation of logic should be as > analytical as possible, breaking up inferences into the greatest possible > number of steps, and exhibiting them under the most general categories > possible" (CP 4.373, 1902). > > Yes, both Peirce and Frege made the sane mistake. As pioneers in logic, > neither of them had any experience with derived rules of inference. To see > how the derived rules enable the same choice of primitives to be used for > both theory and practice, please reread > http://jfsowa.com/peirce/EGcalculus.pdf . > > And by the way, nothing in the article by Ma and Pietarinen depends on > which logical operators are assumed as primitives. With derived rules of > inference, the choice of primitives is irrelevant. The authors could > strengthen their argument by showing that the formal derivations are > independent of the choice of logical primitives. > > JAS> Indeed, treating negation as if it were the third logical primitive > results in a more efficient calculus for classical logic, as well as a > simpler introduction of EG for the previously uninitiated.... > > Yes! And the beauty of the 1911 EGs is that the best foundation for > teaching novices is also the best for professionals in every branch of the > empirical sciences: phaneroscopy, semeiotic, metaphysics, normative > science, logic as semeiotic, physical sciences, and psychical sciences. > For an explanation of that point, see "Natural logic is diagrammatic > reasoning about mental models": http://jfsowa.com/pubs/natlog.pdf . > > Since calculation is the theme of this thread, it's important to note > Peirce's mistake at the end of R669: "the application of the two illative > permissions... can only be made by a living intelligence." In fact, > computer systems today can apply Peirce's permissions and the rules derived > from them much faster and more accurately than any human. > > Today, the enormous volume of data on the www, especially documents > written in natural languages, poses a challenge to the fastest > supercomputers available. In June 2020, I presented a lecture "Language, > ontology, and the Semantic Web" at the European Semantic Web Conference: > http://jfsowa.com/talks/eswc.pdf > > In section 3 (slides 13 to 20 of eswc.pdf), I introduced EGs, and in later > sections I discussed the importance of Peirce's logic and semeiotic for > supporting innovative methods for addressing that challenge. The > generality of the 1911 EGs is essential. > > In summary, the three documents EGcalculus.pdf, natlog.pdf, and eswc.pdf, > which are based on the 1911 EGs, show the importance of Peirce's insight on > or about 3 June 1911. No one has shown any evidence in favor of older > versions of EGs. If anyone has any doubts on that point, I would be happy > to answer them. > > John >
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