List, All:
For the benefit of those on the cc: line who are not List members, here is
a link to my entire post to which John Sowa was replying.
https://list.iupui.edu/sympa/arc/peirce-l/2021-01/msg2.html
In addition, here are links to an earlier exchange between us, in which I
provided several relevant excerpts from a 2016 paper by Bellucci and
Pietarinen (
https://www.cambridge.org/core/journals/review-of-symbolic-logic/article/existential-graphs-as-an-instrument-of-logical-analysis-part-i-alpha/9C4689940BDC5B17F739C34A87C2B77F
).
https://list.iupui.edu/sympa/arc/peirce-l/2020-12/msg00086.html
https://list.iupui.edu/sympa/arc/peirce-l/2020-12/msg00095.html
As they rightly observe, "Taking the idea of negation as primary is
philosophically inaccurate ... . 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" (pp. 220-221). I sincerely hope that Parts
II and III on Beta and Gamma are still forthcoming from them, and I would
welcome any feedback from them and/or others on what follows.
JFS: I realized that Peirce's insight on 2 June 1911 was that the adjective
'illative' is irrelevant and misleading for all three permissions (rules of
inference).
This is obviously false. Peirce does state in R 669 that although the
double cut rule "ought to be reckoned as a permission," it nevertheless "is
not an illative permission, i.e. a permission authorizing a species of
inference." However, he then goes on to present deletion/insertion as the
"First Illative Permission" and iteration/deiteration as the "Second
Illative Permission." In other words, only these two permissions are rules
of *inference*, such that together they "will suffice to enable any valid
deduction to be performed."
JFS: I believe that R669 is the *last* MS in which he wrote the words
'illative' or 'illation'. I have not read all his extant MSS, but I very
strongly doubt that he would continue using a word he had rejected.
Omission is not rejection. I am not aware of any textual evidence that
Peirce ever explicitly *rejects *the words "illation" and "illative," let
alone the associated concept.
JFS: Any comment about modal issues in 1913 should be evaluated in terms of
the Delta graphs, for which we don't have any MSS.
I agree, with one major exception--there is an important modal aspect to
shading since it corresponds to a universe of possibility rather than
actuality. In fact, Peirce ultimately considers Beta EGs plus shading to be
the Gamma part (CP 4.576-581, 1906) as a welcome simplification of the
"nonsensical tinctures" that he later regrets introducing in "Prolegomena"
(RL 477, 1913). That is why a new Delta part would be necessary to include
and extend the additional modal considerations that he had explored
previously in the Gamma part of 1903.
JFS: In R670, negation is a primitive. The scroll is nothing but a way of
drawing a nest of two negations without raising the pen.
I agree, and I continue to acknowledge that the same is true in RL 231, RL
378, and RL 376. However, I disagree strongly about Peirce's *reason *for
taking this approach. In my view, he deliberately *simplifies *his
presentation of EGs for the uninitiated, accepting the tradeoff of making
it *less analytical* by not explaining how shading for negation is derived
from the scroll for consequence. The latter is the true third primitive in
addition to the blank sheet for coexistence and the heavy line for identity.
CSP: We have seen that there are three relations which subsist between the
parts of graphs. The first is the relation expressed by the scroll. This is
the most important of all, since this is the relation of premiss and
conclusion; that is, if it be true that if A is true B is true, then should
A occur as a premiss we have a right to conclude B. The second relation is
that expressed by writing two graphs side by side AB, that is to say, the
relation of coexistence, and the third is the relation of individual
identity expressed by the heavy line. (R 466:18-19, 1903)
I am not aware of any textual evidence that Peirce ever explicitly *rejects
*this analysis.
JFS: Oostra's choice of the scroll as a marker for intuitionistic rules has
no similarity to Peirce's use for any version of EGs.
On the contrary, Peirce on multiple occasions uses the scroll to derive the
cut for negation from a consequence with falsity as its consequent (CP
4.402, 1903; CP 4.454-456, 1903; CP 4.564n, c. 1906; R 669:18-20[16-18],
1911). This is precisely how negation is defined in intuitionistic logic.
Moreover, he calls it an "error" and an "inaccuracy" to analyze "if A then
B," which is represented by a scroll, as strictly equivalent to "not-(A and
not-B)," which is represented by nested cuts (R 300:48-49[47-48], 1908).
The latter can be inferred from the former in intuitionistic logic, but not
