List,
Peirce’s most complete draft of “The Bed-Rock Beneath Pragmaticism”
(http://gnusystems.ca/Bedrock.htm) includes a long section beginning on p. {31}
about the problems with Selectives as components of the system of Existential
Graphs. Selectives were introduced in the 1906 Prolegomena at
http://gnusystems.ca/ProlegomPrag.htm#4561 to deal with notational problems
arising from a Line of Identity (or Ligature) crossing a cut (or another
Ligature). But as Peirce explained in “Bedrock”:
[[ Selectives violate the essential idea, or {33} purpose of The System,
Existential Graphs, as it is stated in the Prolegomena; namely, to afford a
method (1) as simple as possible (that is to say, with as small a number of
arbitrary conventions as possible), for representing propositions (2) as
iconically, or diagrammatically and (3) as analytically as possible. … These
three essential aims of the system are, every one of them, missed by
Selectives. ]]
One of the problems Peirce pointed out pertains not only to Selectives but to
the interpretation of the cuts themselves. Up until April of 1906, a cut (a
lightly drawn line around a graph) was to be interpreted as negation. This
interpretation was considered iconic by Peirce because it was derived from the
scroll — the most basic of all diagrams in the EG system — as Peirce explained
in Lowell Lecture 2 (1903), at http://gnusystems.ca/Lowell2.htm#scrl . The gist
of his argument, using the example he gives there, is as follows:
Suppose we want to make a graph-replica asserting a conditional proposition
such as [If it rains, a pear is ripe.] This is intended to be a conditional de
inesse, i.e. no modality is involved in the logical relation. We make a cut on
the sheet of assertion which separates the area inside the cut from whatever is
asserted on the sheet. This area represents a “universe of supposition” as
opposed to the “universe of discourse” represented by the sheet of assertion
itself; and this implies that anything scribed on the area inside the cut is
thereby denied.
Now we scribe on that area [It rains], and another cut inside the first cut but
not including [it rains]. On the area inside this inner cut we scribe [a pear
is ripe]. The inner cut being scribed on the area of the outer cut amounts to a
denial of the denial — and since no modality is involved, a denial of a denial
amounts to an assertion. So we read the whole graph as saying that [a pear is
ripe] can be asserted on the condition that the area in which it is written is
enclosed by the area in which we suppose that [it rains.] By reading both cuts
as signifying negation, we can translate the whole graph into English this way:
“It is not true that it rains and that no pear is ripe.” This is logically
equivalent to saying that “If it rains, a pear is ripe” — assuming that we
treat it that statement as a conditional de inesse (not as a fallible
observation of empirical causation).
The above paragraph is all about Alpha graphs. In Beta graphs we introduce the
icon of the Line of Identity, and this introduces a problem: how do we
interpret a line of identity that crosses a cut, if the cut signifies negation?
Peirce was stumped by this at first, so he made a rule in 1903 that a graph,
such as a line of identity, simply cannot cross a cut. But he changed his mind
in writing up a presentation to the National Academy of Science given in April
1906 (CP 4.573-84, but see Roberts 1973, 88 ff. for the date). Here he began
calling the area inside a single cut the verso, and rethinking what it
represented. This led to a subtly different explanation of why it signified
negation of its contents:
[[ The verso is usually appropriated to imparting information about subjective
possibilities or what may be true for aught we know. To scribe a graph is to
impart an item of information; and this item of information does one of two
things. It either adds to what we know to exist or it cuts off something from
our list of subjective possibilities. Hence, it must be that a graph scribed on
the verso is thereby denied. ] CP 4.574 ]
But it also led Peirce to a “new discovery” about the interpretation of EGs:
[[ 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:
First, the cut may be imagined to extend down to one or another depth into the
paper, so that the overturning of the piece cut out may expose one stratum or
another, these being distinguished by their tints; the different tints
representing different kinds of possibility.
This improvement gives substantially, as far as I can see, nearly the whole of
that Gamma part which I have been endeavoring to discern. ]]
Peirce jumped to the conclusion that he could use various kinds of cuts to
represent various modalities as well as negation. Over the next year or so he
experimented with various ways of doing this, including the “tinctures” of the
1906 Prolegomena and several types of “broken cuts.” This coincided with a
change of plan for his Monist series on Pragmatism: he decided to base his
“proof” of pragmaticism on the Existential Graphs, specifically the Gamma
graphs which he thought capable of analyzing modal logic.
The “Bedrock” draft seems to represent the end of this process, where Peirce
finally had to admit that modalities could not be iconically represented in the
EG system, especially if cuts and the areas inside them had to represent both
modality and negation. The problem with cuts as modalities is the same as the
problem with Selectives, that they were symbolic rather than iconic: their
interpretation had to be guided by explicitly verbalized rules and conventions
rather than the nonverbal iconicity which was supposed to be the main advantage
of of Existential Graphs over other notation systems. From 1908 on, as far as I
can tell, his presentations of EGs used the cuts for negation only, and dropped
the issue of modality altogether.
In my opinion, the part of the Bedrock draft which makes the point most clearly
is the part in which Peirce explains why he regards logical sequence as simpler
than logical negation. This implies that the scroll, read like the one above
(“It is not true that A is true and B is false”), is not analytical enough as a
representation of “If A then B.” Here is the passage from “Bedrock”:
[[ The second failure of Selectives to be as analytical as possible lies in
their encouraging the idea that negation, or denial, is a relatively simple
concept, and that the concept of Consequence, is a special composite of two
negations, so that to say, “If in the actual state of things A is true, then B
is true,” is correctly analyzed as the assertion, “It is false to say that A is
true while B is false.” I fully acknowledge that, for most purposes and in a
preliminary explanation, the error of this analysis is altogether
insignificant. But when we come to the first analysis the inaccuracy must not
be passed over. All my own writings upon formal logic have been based on the
belief {48} that the concept of Sequence, alike in reasonings and in judgments,
whether the latter be conditional or categorical, could in no wise be replaced
by any composition of ideas. For in reasoning, at least, when we first affirm,
or affirmatively judge, the conjugate of premisses, the judgment of the
conclusion has not yet been performed. There then follows a real movement of
thought in the mind, in which that judgment of the conclusion comes to pass.
Now surely, speaking of the same A and B as above, it were absurd to say that a
real change of A into a sequent B consists in a state of things that should
consist in there not being an A without a B. For in such a state of things
there would be no change at all. {49} This judgment is, at first, no more than
a copy, or “generalized” icon (with a symbolic “legend,” or label, indexically
attached to it), of that experience of having been constrained by the
supposition of A to join to that the acknowledgment of the truth (in the case
supposed), of B. Consequently, since the real sequence, as we have seen, cannot
be adequately represented as merely a composite of two negations, no more can
the copy [of] it, which is substantially the concept of the sequence. It is but
very rarely that a proof as satisfactory as this, that a given concept is not
composite, can be obtained. ]]
As you can see, Peirce introduces this argument with a reference to “failure of
Selectives to be as analytical as possible,” but it goes deeper than that: it
shows the failure of the scroll, or cut within a cut, to be as analytical as
possible. But it was from the scroll that the interpretation of cuts as
negations was derived. This implies that the cuts, as a feature of EGs, were
themselves neither as analytical nor as iconic as Peirce had thought they were
in 1903. They are symbolic, in that it has to be stipulated that they should be
read as negations, as this is not visually (or in an optico-muscular way)
evident to anyone who looks at them. It is not surprising, then, that EGs
underwent no further development after this point (although Peirce continued to
recommend them, for instance to Lady Welby, as useful for the purpose of
logical analysis).
I’ll stop here and invite comments, and (finally!) get back to responding to
Jon’s posts in this thread.
Gary f.
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