Jon, List, A few more points: 1. The quotations you cited are from a time when Peirce still thought that a sign of illation was important for deduction. Note that in R670, he says that the EGs have just three syntactic features: a line of identity, a spot for a rheme and a shaded area for negation. The scroll is "equivalent" to a nest of two negations. It is not a primitive feature. In L231 and the later MSS, he did not draw a scroll or use the word. 2. When I wrote (in slide 10) that an inference was required for negation, I meant a "process of inference", not a "special sign for inference". But I admit that I should have been more precise: A negation results from an observation of a difference or distinction between two perceptions or two aspects of a single perception. That observation may be expressed in the form "A is not B". That process is far more primitive than an application of modus ponens. 3. In R270, the word he actually used to compare a scroll to a nest of two negations is "equivalent". Equivalence implies that one can be substituted for the other in any context. Since I wasn't looking at the MS at the moment, I said that a scroll is "nothing but" a nest of two negations. Equivalence implies that point. It also implies that a nest of two negations is "nothing but" a scroll. In any case, he did not draw a scroll or mention the word in L231 or the later MSS. 4. The word 'analytical' means "pertaining to analysis". It's not at all obvious what the phrase "more analytical" would mean. Although Peirce stated his "permissions" in different ways over the years, every proof from 1897 to the end took exactly the same number of steps. 5. Notice the proof of the Praeclarum Theorema in egintro.pdf. That proof took exactly 7 steps from a blank to the conclusion. In the Principia Mathematica, Whitehead & Russell took 43 steps, starting from 5 non-obvious axioms. That length does not make their method "more analytical" . More appropriate adjectives would be inefficient, inelegant, awkward, clumsy, not recommended... (And by the way, one of the 5 axioms in the 1910 edition was redundant, But nobody noticed that fact until 1926. 6. The most efficient proof procedures used today, do not depend on a special sign of illation or the rule of modus ponens. In dumping the scroll, Peirce was, as usual, ahead of his time. John
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