John, List:
The diagram in CP 3.469 has been bothering me for while, and I finally
figured out why.
JFS: In CP 3.469, Peirce said that Fig. 1 states the proposition "John
gives John to John."
That is *not *what Peirce actually said; in particular, he did not claim or
imply that Figure 1 *states *the proposition in accordance with the
conventions that he went on to define *later *in the same article for the
system that he eventually called Entitative Graphs. Instead, he was simply
illustrating an *analogy *between the *structure *of a logical relative and
the *structure *of a chemical molecule--"The proposition 'John gives John
to John' corresponds in its constitution, as Figs. 1 and 2 show, precisely
to ammonia" (Cp 3.469; 1897).
JFS: But that reading is derived more directly from the corresponding
existential graph (EG). In this case, the EG can be derived from the
entitative graph (EntG) by erasing the three ovals. But that operation, by
itself, is not sufficient to convert an arbitrary entitative graph (EntG)
to an equivalent EG.
Indeed, as I have already pointed out, Peirce himself stipulated a very
simple *two-step* procedure for converting between Entitative and
Existential Graphs in another manuscript.
CSP: Any entitative graph may be converted into the equivalent existential
graph by, first, enclosing each spot separately and secondly enclosing the
whole graph. (R 485:1)
The table from p. 136 of Roberts is fully consistent with this; carrying
out these *two steps* converts anything in the "Entitative Graphs" column
into its counterpart in the "Existential Graphs" column, *and vice-versa*,
once we erase any resulting double enclosures as we are permitted to do in
both systems. For example, applying the first step to Figure 1, we enclose
the four Spots separately, and then erase those double enclosures.
However, we still must proceed to take the second step--*enclosing the
entire Graph*--in order to obtain the equivalent Existential Graph. Hence
it turns out that Figure 1, interpreted as an Entitative Graph, *does not*
state the proposition "John gives John to John"; instead, it states
the *negation
*of that proposition, "It is *false *that John gives John to John."
Careful consideration of Figure 1 confirms this--keeping in mind that in
Entitative Graphs, juxtaposition means *inclusive disjunction* (or
alternation) rather than conjunction, and a line denotes a
*universally *quantified
subject ("anything") rather than an existentially quantified subject
("something"). From the Roberts table, two enclosed Spots connected by a
line means "No F is G"; i.e., "anything you please is either not F or not
G, or neither F nor G." Likewise, Figure 1 means "(giving is not the
relation) or (John is not the giver) or (John is not the gift) or (John is
not the recipient)"; i.e., *at least one* of these four propositions must
be *true*, such that "It is false that John gives John to John."
On the other hand, if we scribe the Existential Graph for "John gives John
to John," and then follow the same two steps, we get the corresponding
Entitative Graph--which is identical to Figure 1, except *entirely within
an enclosure*.
Regards,
Jon Alan Schmidt - Olathe, Kansas, USA
Professional Engineer, Amateur Philosopher, Lutheran Layman
www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt
On Sun, Jun 30, 2019 at 2:25 PM John F Sowa <[email protected]> wrote:
> Jon AS, Gary F, Helmut, and Jerry LRC,
>
> To illustrate the issues in this thread, see the attached EntG.png
> which shows an entitative graph (Fig. 1) and a chemical graph (Fig. 2).
>
> > CSP, CP 3.469: The proposition "John gives John to John" corresponds
> > in its constitution, as Figs. 1 and 2 show, precisely to ammonia. But
> > beyond this point, the analogy ceases to be striking. In fact, the
> > analogy with the ruling theory of chemical compounds quite breaks down.
> > Yet I cannot resist the temptation to pursue it. After all, any analogy,
> > however fanciful, which serves to focus attention upon matters which
> > might otherwise escape observation is valuable.
>
> GR
> > entitative graphs and existential graphs... seem at first glance
> > to be something of mirror images of each other, especially as I've
> > read that the alpha part of existential graphs is dual to entitative
> > graphs in the sense of an "involution operation" ...
>
> HR
> > perhaps entitative graphs are the inversion of existential graphs
> > and vice versa, like, turned inside out.
>
> JAS, quoting CSP:
> > As a matter of fact, "Existential Graphs, which are merely entitative
> > graphs turned inside out ..." (R 280:21; c. 1904-1905)
>
> The term 'inside out' is a fanciful analogy that can focus attention,
> but more detail is necessary to specify a precise transformation.
> In mathematics and logic, the words 'dual' and 'duality' cover several
> different kinds of transformations. They all transform one structure
> to another and back again. But there are many variations. For an
> overview, see https://en.wikipedia.org/wiki/Duality_(mathematics)
>
> In CP 3.469, Peirce said that Fig. 1 states the proposition "John
> gives John to John." But that reading is derived more directly from
> the corresponding existential graph (EG). In this case, the EG can
> be derived from the entitative graph (EntG) by erasing the three
> ovals. But that operation, by itself, is not sufficient to convert
> an arbitrary entitative graph (EntG) to an equivalent EG.
>
> See the attached EntGEG.png for a copy of the transformation rules
> from p. 136 of Don Roberts' book. The rules for "P and Q" and
> "P or Q" are exact inverses. But the rules for "All F is G" and
> "Some F is G" are not exact inverses.
>
> To show how these rules are applied, consider Fig. 1. In that EntG,
> there are three ovals that contain the monad 'John'. That monad
> corresponds to F in the rule for transforming "All F is G". The oval
> that contains the triad 'gives' corresponds to G.
>
> If you apply that rule to each of the three ovals that contain the
> monad 'John', each one implies an additional oval around the oval that
> contains the triad 'gives'. That would create a double negation around
> 'gives'. Then the inference rule for erasing a double negation would
> produce the EG for the sentence "John gives John to John."
>
> Technically speaking, one could say that the extra inference step
> means that EntGs and EGs are not exact duals. But according to the
> looser criteria summarized in the Wikipedia article, EntGs and EGs
> could be called duals.
>
> GR
> > I have also very recently read in an encyclopedia article that
> > Spencer-Brown's primary algebra as presented in Laws of Form is
> > isomorphic to entitative graphs, while earlier I'd only read of
> > its similarities to existential graphs.
>
> Short answer: It's not isomorphic to either one, but there are
> some similarities. For the details, see "The Mathematics of CSP"
> by Louis Kaufman: http://homepages.math.uic.edu/~kauffman/CHK.pdf
>
> HR
> > in Wikipedia they write, that entitative graphs are "dual" to
> > alpha-existential graphs
>
> Wikipedia cites Kauffman's article, which is more detailed and
> more accurate. Note what Kaufman wrote at the top of p. 23:
>
> > LHK: we have exhibited a version of the Logical Garnet that
> > intermediates between Peirce’s Existential Graphs and the dual
> > approach of Laws of Form... The diagrammatic investigations of
> > Peirce, Venn, Carroll, Nicod and Spencer-Brown are all ways of
> > finding geometry in logic, but in Zellweger’s Logical Garnet
> > classical three-dimensional geometry appears, and this is an
> > indication that one should think again on the relationship of
> > logic and mathematics.
>
> JLRC
> > In light of subsequent developments in graph theory, these assertions
> > of CSP are not consistent with the notion of organic chemistry as
> > the bedrock of pragmatic representations of matter.
>
> That's true. As the quotation in CP 3.469 says, "the analogy with
> the ruling theory of chemical compounds quite breaks down." Peirce
> wrote that in 1896, and many more breaks have been found since then.
>
> JLRC
> > This basic semantic and grammatical difference ... casts a very dark
> > shadow over the meaning and truth content of many many CSP texts.
>
> Note quite. What it shows is that no version of ordinary language
> or logic can accurately represent the details of quantum mechanics.
> But the calculations based on mathematical theories have made
> excellent predictions about the outcome of various experiments.
>
> What that means: Science requires two-levels of definitions.
> (1) An application of logic to specify the axioms and definitions
> of the many theories of mathematics; and (2) an application of
> some of those theories to analyze and make predictions about the
> phenomena of physics and chemistry.
>
> This two-level combination was developed and applied thousands
> of years ago. Euclid's version is the best documented. Whenever
> possible, Euclid used Aristotle's version of logic, but he had to
> add much more fill in the gaps.
>
> Peirce did a lot to clarify those gaps and show where more R & D
> was necessary. That is a sign of strength, not weakness.
>
> John
>
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