John, All:
JFS: I sent a complete analysis of these issues to you and others on the
CC list.
Any analysis of these issues that treats cuts/shading as primitive in EGs,
rather than derived from the scroll, is *incomplete*. Peirce himself never
claims in R 670 or in RL 231 to be giving a *complete *analysis or
explanation of EGs.
JFS: In response to the other comments in your recent note, I'll reply
with a copy of Peirce's comments about scrolls in L231: " ", AKA
silence.
An argument from silence is always logically weak, in this case especially
so since Peirce elsewhere *explicitly* denies that a consequence is a
composite of two negations and *explicitly *derives the cut from the scroll
with a blackened inner close. Again, I am not at all questioning the value
of shading as a simpler and more iconic improvement over thin lines for
*representing* these relations. In fact, according to what seems to be
Peirce's very first introduction of shading in EGs ("blue tint"), written
five years earlier than R 670 and RL 231, it is precisely what revealed to
him that "if A then B" is not strictly equivalent to "not (A and not-B)."
CSP: But I had better tell you that practically, I content myself with
performing these cuts in my imagination, merely drawing a light line to
represent the cut. The *blue *tint, however, of the area within the cut is
a great aid to the understanding. How great I have only recently
discovered. ...
The new discovery, which sheds such a light is simply that, as the main
part of the sheet represents existence or actuality, *so* the area within a
cut, that is, the *verso *of the sheet, represents a kind of possibility.
>From thence I immediately infer several things that I did not understand
before, as follows: (R 490:12-15, includes CP 4.577-578, 1906)
Peirce now perceives that any oddly enclosed area "represents a kind of
possibility," rather than merely a denial of actuality. He proceeds to
describe three specific ramifications of this, the last of which is what I
have been emphasizing.
CSP: Thirdly, my previous account of Existential Graphs was marred by a
certain rule which, from the point of view from which I thought the system
ought to be regarded, seemed quite out of place and unacceptable, and yet
which I found myself unable to dispute. I will just illustrate this matter
by an example. (R 490:19-20, CP 4.580)
As Ahti-Veikko Pietarinen points out in the introduction to his own
transcription of the manuscript (
https://www.researchgate.net/publication/271419583_Two_Papers_on_Existential_Graphs_by_Charles_Peirce),
its presentation in CP is "seriously incomplete," and the EGs that serve as
Peirce's illustrative example "are all erroneous." Among the unfortunate
omissions is the text at the ellipsis in CP 4.580, which is his statement
of the widely accepted rule that he now deems to be "quite out of place and
unacceptable."
CSP: A conditional proposition is false only if the condition of it is
satisfied, while the consequent is falsified. For the proposition asserts
nothing at all in case the condition is *not *satisfied. So then it is only
if the condition *is *satisfied, while the consequent is falsified, that
the conditional proposition is false. But a proposition that is not false
is true. (R 490:23-24)
There is an assumption underlying this approach that Peirce had only come
to recognize (and abandon) in conjunction with his renewed interest in
pragmatism.
CSP: This reasoning is irrefragable as long as a mere possibility is
treated as an absolute nullity. Some years ago, however, when in
consequence of an invitation to deliver a course of lectures in Harvard
University upon Pragmatism, I was led to revise that doctrine, in which I
had already found difficulties, I soon discovered, upon a critical
analysis, that it was absolutely necessary to insist upon and bring to the
front, the truth that a mere possibility may be quite real. That admitted,
it can no longer be granted that every conditional proposition whose
antecedent does not happen to be realized is true, and the whole reasoning
just given breaks down. (R 490:25-26, CP 4.580)
When A is realized, both "if A then B" and "not (A and not-B)" are true
when B is realized and false when B is not realized. However, when A is
*not* realized, "if A then B" is not *necessarily *true, even though "not
(A and not-B)" is *always *true. Yet "if A then B" is still not *false *in
such cases, either. The usually innocuous but ultimately faulty
supposition here is that "a proposition that is not false is true," which
is the principle of excluded middle--precisely the aspect of classical
logic that intuitionistic logic rejects. Peirce even acknowledges (many
years earlier) that it is not *strictly *true.
CSP: The two principles of contradiction and excluded middle do not stand
at all upon the same plane. ... [C]ertain rudimentary forms of reasoning,
embracing all those that the traditional logic has handed down to us,
depend only upon the impossibility of a fact's being both true and false,
and remain equally sound arguments, if we suppose that some things are
neither true nor false. (NEM 3:753, 1881; cf. NEM 3:751-752, 1881)
CSP: A consequence *ut nunc* is one in which the range of possibility is
limited to the actual state of things. To speak of *the *actual state of
things implies a great assumption, namely that there is a perfectly
definite body of propositions which, if we could only find them out, are
the truth, and that everything is really either true or in positive
conflict with the truth. This assumption, called the principle of excluded
middle, I consider utterly unwarranted, and do not believe it. Still, I
hold that there is reason for thinking it to be very nearly true. (NEM
3:758, 1893; cf. NEM 3:758-760, 1893)
In other words, Peirce denies that excluded middle is an absolutely
exceptionless *law *(NEM 4:xiii, no date), which is presumably why he
typically prefers to call it a *principle *instead. Nevertheless, it is
only after accepting "the truth that a mere possibility may be quite real"
that he "discovers" how a scroll in EGs can represent a consequence whose
range of possibility is *not *limited to the actual state of things, such
that excluded middle *does not* hold for it. Peirce has more to say about
this, but that seems like enough for now.
Regards,
Jon Alan Schmidt - Olathe, Kansas, USA
Structural Engineer, Synechist Philosopher, Lutheran Christian
www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt
On Tue, Aug 4, 2020 at 9:46 AM John F. Sowa <[email protected]> wrote:
> Jon AS,
>
> I sent a complete analysis of these issues to you and others on the CC
> list. For a copy, see the attached eg1911.txt. For the unsolved research
> problem from 1988 and an outline of the solution, see the attached
> ppe65.png. For more detail, see the slides "Peirce, Polya, and Euclid" --
> URL in the eg1911.txt. For even more detail, see the 76-page article from
> the Journal of Applied Logics (URL in a footnote on p. 2 of ppe.pdf).
>
> Re the word 'scroll': In terms of the semantics (endoporeutic) and
> permissions (rules of inference) of eg1911, a scroll is *indistinguishable*
> from a shaded area with a nested unshaded area. Anything that Peirce wrote
> about scrolls prior to 22 June 1911 is useful only for understanding the
> development of his thought. After that date. the word 'scroll' could only
> create confusion -- some readers might be misled by Peirce's earlier
> writings to think that there is some "deeper" meaning that is not expressed
> by a nest of two ovals.
>
> In response to the other comments in your recent note, I'll reply with a
> copy of Peirce's comments about scrolls in L231: " ", AKA silence.
>
> John
>
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