John, List,
Wow, it all is much more difficult than one thinks! I like that the world cannot be explained so easily. It reminds me of the harmony-topic: Even harmony itself is not harmonic: Either merely the octaves fit, or merely the quints, or the quarts, or one or the other terzes. They all fit to each other nearly, but not exactly. And in the graphs-thing, you can turn things around, and it is close to opposites, but not completely. Great. And we are here, because the symmetry between matter and antimatter has not been a complete symmetry. Thank you for showing, that in this graphs-case too there is such a symmetry-break! But it is annoying, somehow. Is that God telling us, that we can never understand Him, so we can never be like Him? I would apreciate that, if a "close-to-but-not exactly"-argument could be sufficient to keep people like the Kurds and the Turks from shooting at each other. I guess it has to, and can. Education just has to be something else than teachers boasting with catchy parables.
Best,
Helmut
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
-----------------------------
PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON PEIRCE-L to this message. PEIRCE-L posts should go to [email protected] . To UNSUBSCRIBE, send a message not to PEIRCE-L but to [email protected] with the line "UNSubscribe PEIRCE-L" in the BODY of the message. More at http://www.cspeirce.com/peirce-l/peirce-l.htm .
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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