Gary F,
There are two separate issues here: (1) the isomorphism between Peirce's
1911 system and his earlier presentations; and (2) the relationship
between Peirce's endoporeutic and GTS.
About #1, the issues are clear for first-order logic (Alpha + Beta):
every graph drawn according to the 1903 or 1906 rules can be converted
to one according to the 1911 rules by shading the negative areas.
The rules of inference are also equivalent: a proof by one set of
rules is also a valid proof by the other rules.
There is one point about the scroll, which Peirce does not mention
in 1911 as distinct from a nest of two ovals. But that point has
no effect on any of the graphs or any proof.
Therefore, I regard the 1911 rules as a cleaner, simpler, and more
elegant version of his earlier treatment. But I believe that this
simplicity is a major *improvement* because it makes the rules more
general and more flexible. (As I summarized in my previous note.)
Re graphist and interpreter: Peirce wrote many versions over the
years, in some of them the two parties cooperated and in others
they were more competitive. See the comment by Pietarinen below:
But this is *very* different from Peirce’s own account of the dialog
between graphist and interpreter in the Lowell lectures, in CP 4.431
Peirce wrote many fragmentary remarks about the dialog, most of which
were unpublished. Pietarinen has a summary of the various passages:
http://www.digitalpeirce.fee.unicamp.br/endo.htm
For a more systematic treatment, see the following by Pietarinen:
https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=16&cad=rja&uact=8&ved=0ahUKEwjH9b2dqKHXAhUp8IMKHfP-Bo44ChAWCDQwBQ&url=https%3A%2F%2Fdialnet.unirioja.es%2Fdescarga%2Farticulo%2F4729798.pdf&usg=AOvVaw1DgzAp3gS_pYb_5wiN9gu5
From page 2:
Peirce coined a plethora of names... assertor and critic, concurrent
and antagonist, speaker and hearer, addressor and addressee, scribe and
user, affirmer and denier, compeller and resister, Me and Against-Me...
Pietarinen also said "these names can easily be confused with one
another." That's why I chose the terms proposer and skeptic, which
seem to be clearer and more memorable. The skeptic is willing to be
persuaded, but only after checking all the details.
Summary: Peirce's ideas and terminology were in flux. He didn't
have the advantage of modern computers and the 20th c. techniques
of recursive functions and game-playing programs. But his notion
of a dialog with two parties collaborating and/or competing keeps
recurring in all those discussions.
My specification of the game (URL below) is based on familiarity
with software for playing games like chess. Peirce did not have
that experience, but I believe that he would agree with the method.
John
__________________________________________________________________
From page 18 of http://jfsowa.com/pubs/egtut.pdf
In modern terminology, endoporeutic can be defined as a two-person
zero-sum perfect-information game, of the same genre as board games like
chess, checkers, and tic-tac-toe. Unlike those games, which frequently
end in a draw, every finite EG determines a game that must end in a win
for one of the players in a finite number of moves. In fact, Henkin
(1961), the first modern logician to rediscover the game-theoretical
method, showed that it could evaluate the denotation of some infinitely
long formulas in a finite number of steps. Peirce also considered the
possibility of infinite EGs: “A graph with an endless nest of seps
[ovals] is essentially of doubtful meaning, except in special cases” (CP
4.494). Although Peirce left no record of those special cases, they are
undoubtedly the ones for which endoporeutic terminates in a finite
number of steps. The version of endoporeutic presented here is based on
Peirce’s writings, supplemented with ideas adapted from Hintikka (1973),
Hilpinen (1982), and Pietarinen (2006)...
What distinguishes the game-theoretical method from Tarski’s approach
is its procedural nature. One reason why Peirce had such difficulty
in explaining it is that he and his readers lacked the vocabulary
of the game-playing algorithms of artificial intelligence...
[Follow the URL for the details.]
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