Dear Francesco, Jon, and Dan.
I put Dan Everett on the list
because he is a linguist who
recommended a 600-page book on negation
and is writing a book about
Peirce. I'd like to hear his comments
about a paragraph in R669.
FB> I must say that in the
dispute I agree completely with Jon A.
Schmidt. He has supported his
points with abundant textual evidence
and with several convincing
exegetical arguments.
If we were discussing Peirce's writings
prior to 3 June 1911, I would
agree with you. But that does not
explain why Peirce began R669 with
the 1906 version of EGs and five
days later switched to a new version
for R670.
FB> What
we dispute is whether [Peirce's] 1911 presentations of the
logical
graphs show that he retracted one of his most stable
logico-philosophical ideas.
Peirce's criterion for
significant ideas: "The elements of every
concept enter into
logical thought at the gate of perception and make
their exit at the
gate of purposive action; and whatever cannot show
its passports at
both those two gates is to be arrested as
unauthorized by
reason" (CP 5.212).
Note that this criterion says nothing
about the words used to define
or explain the concept. Two concepts
with different names and verbal
definitions are equivalent if they
produce the same interpretations
and responses at the two gates.
FB> What [Peirce] claimed is that implication ("If A then
B") mirrors
inference; inference ("A, ergo B") is a
transitive, anti-symmetrical
relation, and the only truth-functional
operator that is also
anti-symmetrical and transitive is
implication.
On that issue, it's irrelevant whether the
conditional is primitive
or defined. In MM_AVP.png (attached below),
the only effect of the
1911 EGs is to replace the word
"define" by "it is true that". The
conditional
and any symbol that represents it remain just as iconic,
and all
proof procedures remain valid.
But the 1911 EGs support a
richer, more general set of logico-
philosophical ideas that go
beyond the 1906 ideas. The shaded ovals
for 1911 negations can be
generalized to shaded regions in three
dimensions or even 3D plus
time. But the 1906 recto/verso semantics
is limited to 2D. On that
issue, note what Peirce wrote in L231:
"an algebraist like Boole
plainly thought in algebraic symbols; and
so did I, until, at great
pains, I learned to think in diagrams,
which is a much superior
method. I am convinced there is a far
better one, capable of
wonders; but the great cost of the apparatus
forbids my learning it.
It consists in thinking in stereoscopic
moving pictures. Of course
one might substitute the real objects
moving in solid space; and that
might not be so very unreasonably
costly." (NEM 3:191)
For evidence that Peirce's significant ideas are not affected by the
transition from R669 to R670, I recommend Ahti's 2011 article,
"Existential Graphs: What a Diagrammatic Logic of Cognition
Might
Look Like", History and Philosophy of Logic, 32:3,
265-281. That
article discusses a wide range of philosophical ideas,
including the
quotation from NEM 3:191. But it doesn't mention the
idea of
deriving negation from implication. On the contrary, see
page 14:
"given two nested cuts, implication is ability to
continue a passage
from the area of an outer cut area to the area of
the inner cut."
FB quoting Frege> "By means of
negation the hypothetical mode can
thus be reduced to conjunction and
conjunction to the hypothetical
mode. Looked at from a logical
standpoint both appear equally
primitive. But since the hypothetical
mode is more closely connected
with drawing inferences, it is best to
give it pride of place, and
see it as the primitive form, reducing
conjunction to it." (Frege,
Posthumous Writings, 1979, p.
200)
But Frege assumed *both* negation and implication as
primitives, and
he defined conjunction in terms of them. That is
contrary to Peirce
who assumed that conjunction was so fundamental
that he didn't assign
a symbol to it. Conjunction is implicit when
two or more EGs are
scribed in the same area.
Although
Peirce never mentioned Frege, he did criticize Russell, who
adopted
Frege's primitives for his Principles of Mathematics: "true
in
the main and throughout, however, he betrays insufficient
reflection
on the fundamental conceptions of the subject, with the
primary
difficulty... his not having begun with a thorough
examination of the
elements". (Houghton Library, Series I, no. 12.
Notes
Preparatory to a Criticism of Bertrand Russell's Principles of
Mathematics, 5 February 1912)
That comment shows no sympathy
for Frege's primitives. The tone is
milder, but the comments are
consistent with the sharp criticisms in
L378 and L376 (copy
below).
I also recommend another article by Ahti: "Two
papers on existential
graphs by Charles Peirce", Synthese, July
2014. The two "papers"
actually consist of some MSS that
Ahti grouped in two chapters.
Chapter 1, "Recent
Developments of Existential Graphs and their
Consequences for Logic
(MS 498, 499, 470, S-36, 1906)". In R498 and
R499, Peirce
discussed a variety of ideas, which do not conflict with
any version
of EGs. In R470, he presented the 1906 EGs with
recto/verso
semantics for the cut. But he presented the conditional
as a nest of
two cuts. He did not define a single cut in terms of a
double cut
plus a pseudograph.
Chapter 2, "Assurance Through
Reasoning (MS 669 & MS 670, 1911)."
For this
"paper", Ahti placed R670 first, and treated it as an
introduction to the more detailed R669. He did not comment on any
discrepancies between them.
But on page 27 of R670, Peirce
states "There are but three peculiar
signs that the Syntax of
Existential Graphs absolutely requires. The
first of these is a sign
of identity." On p. 28, "This way of
asserting
identity... [furnishes] the second general sign required
by the
syntax of existential graphs. This is the "Spot of
Teridentity". On p. 29, "the third indispensable peculiar
sign of
the Syntax of Existential Graphs... is one that shall deny
a
Graph-instance, or scribed assertion... But that Conditional
Proposition just written may be expressed by the aid of a sign of
Negation in the form, 'That the Antecedent is true and the Consequent
not true, is not true'. On p. 31, "It is a help to shade the
oddly-enclosed areas and omit the lines that represent the cuts, as
in Fig. 11 [just shading], which is equivalent to Fig. 10 [drawn as a
scroll with no shading]."
Peirce added two non-peculiar
signs to the three peculiar signs:
"according to this syntax any
single word is an assertion" (p. 27)
which may have zero or more
pegs for attaching lines. Conjunction is
another non-peculiar sign,
which is represented by drawing two or
more EGs in the same area.
Altogether, the R670 syntax has five signs: peculiar
(existence,
teridentity, and negation) and non-peculiar (words and
conjunction).
The words represent predicates or rhemes. The
conditional is defined
by two negations and a conjunction.
These definitions, which preserve all the logical ideas, also
preserve philosophical ideas that are violated by the 1906 semantics.
For observation (phaneroscopy), the only logical operators that can
be observed are existence, conjunction and negation: any sensation
is a sign that something exists; any observation of a difference is a
sign of a conjunction (A and B) and a sign of negation (A is not B).
>From those three operaters all others can be derived.
Since
Francesco and Jon claim that the following paragraph is
significant,
I'd like to ask for Dan's opinion: Based on your study
of
linguistics and related issues, what is your opinion about the
following paragraph? Would you consider it a convincing argument for
adopting the conditional as primitive and negation as derivative?
>From R669, p. 43: "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 logicians 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 paradisiacal
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. 24, "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.
25: "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. 26 and finally Fig.
27, in which the idea of flat
falsity is first matured." (R669,
p. 43)
My opinion: this paragraph ie embarrassingly bad. For
years, Peirce
derided the typical reasoning of metaphysicians who
start with a
conclusion they claim is infallible and generate
evidence that makes
it seem plausible. In this paragraph, Peirce is
doing the same
thing. He fabricated a dubious explanation for the
derivation in
Figures 24 to 27 (copy attached in R699figs.png).
But Peirce knew he was fallible. If he had reviewed this
paragraph
(on or about 3 June 1911), he would not have been
satisfied. Five
days later, he wrote R670. On June 22, he began
writing the more
polished L231.
In summary, the 1911
version discards a huge amount of irrelevant
verbiage (see the
quotations below), But it's sufficiently general to
go beyond two
dimensions (see http://jfsowa.com/talks/ppe.pdf ). That
generalization supports a wide rage of new logico-philosophical ideas
without discarding any significant ideas from the earlier versions.
John
____________________________________________________________________
L378 (Sept. 1911) and L376 (Dec. 1911) confirm the fact that R670
and
L231 are a total rejection of R669 and the 1906 version of EGs
on
which it is based. In L378, Peirce wrote: "I use a
diagrammatic
syntax, which I described very badly and at an
intolerable length in
the Monist of October 1906." In L376, he
was more explicit about the
1906 version: "For although the
system itself is marked by extreme
simplicity, the description fills
55 pages, and defines over a
hundred technical terms applying to it.
The necessity for these was
chiefly due to the lines called cuts
which simply appear in the
present description as the boundaries of
shadings, or shaded parts of
the sheet."
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