Cf: Theme One Program • Exposition 5
Lexical, Literal, Logical
Theme One puts cactus graphs to work in three distinct but related ways,
called their “lexical”, “literal”, and “logical” uses. Those three modes
of operation employ three distinct but overlapping subsets of the broader
species of cacti. Accordingly we find ourselves working with graphs, files,
and expressions of lexical, literal, and logical types, depending on the task
The logical class of cacti is the broadest, encompassing the whole species
described above, of which we have already seen a typical example in its
several avatars as abstract graph, pointer data structure, and string
of characters suitable for storage in a text file.
Being a “logical cactus” is not just a matter of syntactic form —
it means being subject to meaningful interpretations as a sign of
a logical proposition. To enter the logical arena cactus expressions
must express something, a proposition true or false of something.
Fully addressing the logical, interpretive, semantic aspect of cactus graphs
normally requires a mind-boggling mass of preliminary work on the details of
their syntactic structure. Practical, pragmatic, and especially computational
considerations will eventually make that unavoidable. For the sake of the
present discussion, however, let’s put a pin in it and fast forward to the
Basal Ingredients Of Society • Prologue
I settled on the acronym BIOS to suggest the vital elements of life in society,
a life in association with others, and not just any association but one whose
flickers of life are sustained for more than a few vicissitudes of history.
Sustainability in that life requires democracy, a society based on
a distinctive form of social compact.
Best Regards on this Day of Contemplation,
Your questions about Peirce's proof of pragmatitiicsm are important.
Jeff> I tend to think the later writings often build on the earlier. As
such, I wonder what the later proof borrows by way of premisses from the
arguments developed in the 1903 Harvard Lectures on Pragmatism. Once that
is clearer, we can then ask what might have been added to the later
argument by way of additional premisses.
Peirce's ideas were constantly evolving up to the very end. He frequently
went back to earlier ideas, but always with some new insights or directions
from his later developments. For pragmaticism, his 1903 Harvard and
Lowell lectures were an important starting point. And the word
'prolegomena' in 1906 is an important clue.
An interesting occurrence in December 1902: Carus published a new English
translation of Kant's "Prolegomena to any future metaphysics" and Peirce
published a notice of it in the Nation in June 1903. -- he must have been
reading (or rereading) it around the same time as he was preparing those
Peirce must have read it (in German) during the time that he and his father
were studying Kant. After Kant finished the first edition of KdrV (or CdrV
as Peirce preferred to refer to it), he wrote the short Prolegomena as an
intro and overview of the questions that he tried to answer in the first
edition of the K(C)drV. Those questions were the prelude to his second
edition, which he finished a few years later. Although Peirce had
criticized some of Kant's fundamental assumptions, he always had a high
regard for Kant, and he cited him frequently throughout CP. And he had a
very high regard for Kant's questions, which are the main topic of his own
Prolegomena. If you (a) read Kant's questions and (b) read Peirce's
writings from 1903 onwards, you can see a strong influence of Kant's
questions on Peirce. In fact, Peirce's 1903 classification of the sciences
seems to be part of Peirce's answers to K's three transcendental questions.
Even stronger evidence for Kant's influence is Peirce's 1906 Apology for
Pragmaticism, which is written as a prelude to a series of articles he
planned for the Monist. Unfortunately, he ran into difficulties around
1909, which led him to the series of ten MSS on "Assurance" (R661 to R670)
from 1910 to 1911.
There's much more to be said about all these issues. I recommend an
article about Peirce's Apology by Max Fisch (1982) and reprinted in a book
by Fisch in 1986. In that article, Max F. wrote that methodeutic is a key
topic that Peirce was addressing in his planned proof. I agree.
And I also believe that there were two reasons why Peirce stopped in 1909:
(1) problems with phaneroscopy as a science egg (R645) and (2) problems
with logic, which were the reason for his ten studies (R661 to R670) from
1910 to 1911. These are the reasons for major revisions that Peirce made
in 1911 and 1912. It's sad that he was converging on important new ideas
just when he had that accident in 1911 followed by the cancer.
There is, of course, much more to say about all these issues. And as Max
Fisch also said, that's why we need all of Peirce's late MSS available in
suitable formats. As Peirce's late letters show -- he was thinking very
hard about all these issues. And he didn't hesitate to make major
revisions when necessary.
Cf: Theme One Program • Jets and Sharks 1
It is easy to spend a long time on the rudiments of learning and logic
before getting down to practical applications — but I think we've
circled square one long enough to expand our scope and see what
the category of programs envisioned in Theme One can do with
more substantial examples and exercises.
During the development of the Theme One program I tested successive
implementations of its Reasoning Module or Logical Modeler on
appropriate examples of logical problems current in the literature
of the day. The PDP Handbook of McClelland and Rumelhart set one
of the wittiest gems ever to whet one's app‑titude so I could hardly
help but take it on. The following text is a light revision of the
way I set it up in the program's User Guide.
Example 5. Jets and Sharks
The propositional calculus based on the minimal negation operator
( https://oeis.org/wiki/Minimal_negation_operator ) can be interpreted
in a way resembling the logic of activation states and competition
constraints in one class of neural network models. One way to do this
is to interpret the blank or unmarked state as the resting state of
a neural pool, the bound or marked state as its activated state, and
to represent a mutually inhibitory pool of neurons A, B, C by the
proposition (A , B , C). The manner of representation may be
illustrated by transcribing a well-known example from the parallel
distributed processing literature (McClelland and Rumelhart 1988)
and working through a couple of the associated exercises as
translated into logical graphs.
Displayed below is the text expression of a traversal string which
Theme One parses into a cactus graph data structure in computer memory.
The cactus graph represents a single logical formula in propositional
calculus and this proposition embodies all the logical constraints
defining the Jets and Sharks data base.
Display. Theme One Guide • Jets and Sharks • Log File
To be continued …
• McClelland, J.L. (2015), Explorations in Parallel Distributed Processing :
A Handbook of Models, Programs, and Exercises, 2nd ed. (draft), Stanford
Parallel Distributed Processing Lab ( https://web.stanford.edu/group/pdplab/ ).
Online ( https://web.stanford.edu/group/pdplab/pdphandbook/ ),
Section 2.3 ( https://web.stanford.edu/group/pdplab/pdphandbook/handbookch3#x7-320002.3 ),
Figure 2.1 ( https://web.stanford.edu/group/pdplab/pdphandbook/jetsandsharkstable.png ).
• McClelland, J.L., and Rumelhart, D.E. (1988), Explorations in Parallel Distributed
Processing : A Handbook of Models, Programs, and Exercises, MIT Press, Cambridge, MA.
“Figure 1. Characteristics of a number of individuals belonging to two gangs, the
Jets and the Sharks”, p. 39, from McClelland (1981).
• McClelland, J.L. (1981), “Retrieving General and Specific Knowledge
From Stored Knowledge of Specifics”, Proceedings of the Third Annual
Conference of the Cognitive Science Society, Berkeley, CA.
• Theme One Program • User Guide
• Example. Jets and Sharks