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
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
Cf: Higher Order Sign Relations • 1
When interpreters reflect on their own use of signs they require an
appropriate technical language in which to pursue their reflections.
For this they need signs referring to sign relations, signs referring
to elements and components of sign relations, and signs referring to
properties and classes of sign relations. The orders of signs developing
as reflection evolves can be placed under the description of “higher order
signs” and the extended sign relations involving them can be referred to as
“higher order sign relations”.
Continue Reading at “Inquiry Driven Systems • Higher Order Sign Relations”
I’ve been working apace to format my old dissertation proposal on Inquiry Driven Systems
( https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Overview )for the web but I was
reminded of this part when the subject of “signs about signs” came up recently on the
Cf: Inquiry Into Inquiry • Understanding 1
Another passage from Russell further illustrates what I see as a critical
juncture in his thought. The graph-theoretic figure he uses in analyzing
a complex of logical relationships brings him to the edge of seeing the
limits of dyadic analysis — but he veers off and does not make the leap.
At any rate, that's how it looks from a perspective informed by Peirce.
Here's the first part of the passage.
Excerpt from Bertrand Russell • “Theory of Knowledge” (The 1913 Manuscript)
Part 2. Atomic Propositional Thought
Chapter 1. The Understanding of Propositions
(4). We come now to the last problem which has to be treated in
this chapter, namely: What is the logical structure of the fact
which consists in a given subject understanding a given proposition?
The structure of an understanding varies according to the proposition
understood. At present, we are only concerned with the understanding
of atomic propositions; the understanding of molecular propositions
will be dealt with in Part 3.
Let us again take the proposition “A and B are similar”.
It is plain, to begin with, that the complex “A and B being similar”,
even if it exists, does not enter in, for if it did, we could not
understand false propositions, because in their case there is no
It is plain, also, from what has been said, that we cannot understand
the proposition unless we are acquainted with A and B and similarity
and the form “something and something have some relation”. Apart from
these four objects, there does not appear, so far as we can see, to be
any object with which we need be acquainted in order to understand the
It seems to follow that these four objects, and these only, must be
united with the subject in one complex when the subject understands the
proposition. It cannot be any complex composed of them that enters in,
since they need not form any complex, and if they do, we need not be
acquainted with it. But they themselves must all enter in, since if
they did not, it would be at least theoretically possible to understand
the proposition without being acquainted with them.
In this argument, I appeal to the principle that, when we understand,
those objects with which we must be acquainted when we understand,
and those only, are object-constituents (i.e. constituents other than
understanding itself and the subject) of the understanding-complex.
(Russell, TOK, 116–117).
Bertrand Russell, “Theory of Knowledge : The 1913 Manuscript”,
edited by Elizabeth Ramsden Eames in collaboration with
Kenneth Blackwell, Routledge, London, UK, 1992.
First published, George Allen and Unwin, 1984.
Notes on Russell's “Theory of Knowledge” • Note 1