John, List, All:
In this post, I will simply respond to the numbered items below rather than quoting them.
1. I am not aware of any evidence that Peirce ever explicitly denied that illation is essential for deduction or rejected the use of the scroll for that purpose in EGs. The absence of these specific terms in his relatively sparse writings during the last 34 months of his life does not outweigh their abundance in his voluminous output over the preceding decades. Again, my hypothesis is that in June 1911 he merely decided to simplify his presentation of EGs for the uninitiated by omitting the derivation of negation (oval) from consequence (scroll).
2. The psychological process of noticing a difference or distinction in perception does seem to be more primitive than the linguistic process of verbalizing the inference of modus ponens, but as Peirce repeatedly affirmed, the logical relation of negation is absolutely not more primitive than the logical relation of consequence. Moreover, "We do not derive these notions [universal elementary relations of logic] from observation, nor by any sense of being opposed, but from our own reason" (CP 8.352, EP 2:485, 1908).
3. I have repeatedly acknowledged that in classical logic, which is what Peirce obviously had in mind in 1911, a scroll is indeed equivalent to a nest of two ovals. Although he anticipated intuitionistic logic in several remarkable ways, including his explicit statements in R 300 (1908) that analyzing a consequence as a composite of two negations is erroneous, unfortunately he did not take the additional steps that would have been necessary to formalize it. Otherwise, we might today be calling it synechistic logic instead.
4. According to Peirce, "I have a complete theory of this process [logical analysis], including its methodeutic, which I base upon my existential graphs which is my chef d'oeuvre" (NEM 3:885, 1908). Whether it was logical analysis or EGs that he considered to be his masterpiece, it is clear that they are closely linked. In the context of logical analysis, he repeatedly defines "more analytical" as "breaking up inferences into the greatest possible number of steps" (CP 4.373, 1902). In every single classical or intuitionistic proof that involves negation, deriving it from a consequence with "the essence of falsity" as its consequent is technically a necessary additional step. As Bellucci and Pietarinen rightly put it, "Taking the idea of negation as primary is philosophically inaccurate."
5. Peirce was obviously not advocating that we add unnecessary steps, and he generally sought to minimize the number of axioms. For example, he eagerly embraced non-Euclidean geometry as demonstrating that the parallel postulate is not essential for a consistent system. I believe that he likewise would have endorsed intuitionistic logic as demonstrating that excluded middle and its corollaries are not essential for a consistent system, had he managed to work out the details himself--or had he lived to see Brouwer and Heyting do so, despite undoubtedly disagreeing with them about the philosophical motivations. In that sense, non-Euclidean geometry is more analytical than Euclidean geometry and intuitionistic logic is more analytical than classical logic, because in each case the former requires an additional postulate/axiom for certain proofs--effectively, an additional step that is presupposed by the latter.
6. According to his own persistent testimony, Peirce was not terribly interested in improving the efficiency of proof procedures, and the _expression_ "dumping the scroll" is frankly both crude and misleading.
This exchange seems like a good summary of our different positions that result from our different purposes, so hopefully we can leave it at that.
Regards,
Jon Alan Schmidt - Olathe, Kansas, USA
Structural Engineer, Synechist Philosopher, Lutheran Christian
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 R670, 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
