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: 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
Cf: Inquiry Into Inquiry • Flash Back
| The fault, dear Brutus, is not in our stars,
| But in ourselves …
| Julius Caesar • 1.2.141–142
Signs have a power to inform, to lead our thoughts and thus our actions
in accord with reality, to make reality our friend. And signs have
a power to misinform, to corrupt our thoughts and thus our actions
and lead us to despair of all our ends.
For now I'll just post a clean copy of a text for later discussion.
Excerpt from Bertrand Russell • “The Philosophy of Logical Atomism” (1918)
4. Propositions and Facts with More than One Verb: Beliefs, Etc.
4.3. How shall we describe the logical form of a belief?
I want to try to get an account of the way that a belief is made up.
That is not an easy question at all. You cannot make what I should
call a map-in-space of a belief. You can make a map of an atomic fact
but not of a belief, for the simple reason that space-relations always
are of the atomic sort or complications of the atomic sort. I will try
to illustrate what I mean.
The point is in connexion with there being two verbs in the judgment
and with the fact that both verbs have got to occur as verbs, because
if a thing is a verb it cannot occur otherwise than as a verb.
Suppose I take ‘A believes that B loves C’. ‘Othello believes that
Desdemona loves Cassio’. There you have a false belief. You have this
odd state of affairs that the verb ‘loves’ occurs in that proposition and
seems to occur as relating Desdemona to Cassio whereas in fact it does not
do so, but yet it does occur as a verb, it does occur in the sort of way
that a verb should do.
I mean that when A believes that B loves C, you have to have a verb
in the place where ‘loves’ occurs. You cannot put a substantive in
its place. Therefore it is clear that the subordinate verb (i.e. the
verb other than believing) is functioning as a verb, and seems to be
relating two terms, but as a matter of fact does not when a judgment
happens to be false. That is what constitutes the puzzle about the
nature of belief.
You will notice that whenever one gets to really close quarters
with the theory of error one has the puzzle of how to deal with
error without assuming the existence of the non-existent.
I mean that every theory of error sooner or later wrecks itself
by assuming the existence of the non-existent. As when I say
‘Desdemona loves Cassio’, it seems as if you have a non-existent
love between Desdemona and Cassio, but that is just as wrong as
a non-existent unicorn. So you have to explain the whole theory
of judgment in some other way.
I come now to this question of a map. Suppose you try such a map as this:
[Figure 1.] Othello Believes Desdemona Loves Cassio
This question of making a map is not so strange as you might suppose because
it is part of the whole theory of symbolism. It is important to realize where
and how a symbolism of that sort would be wrong: Where and how it is wrong is
that in the symbol you have this relationship relating these two things and in
the fact it doesn’t really relate them. You cannot get in space any occurrence
which is logically of the same form as belief.
When I say ‘logically of the same form’ I mean that one can be obtained
from the other by replacing the constituents of the one by the new terms.
If I say ‘Desdemona loves Cassio’ that is of the same form as ‘A is to the
right of B’. Those are of the same form, and I say that nothing that occurs
in space is of the same form as belief.
I have got on here to a new sort of thing, a new beast for our zoo, not another
member of our former species but a new species. The discovery of this fact is
due to Mr. Wittgenstein.
(Russell, POLA, 89–91).
Bertrand Russell, “The Philosophy of Logical Atomism”, pp. 35–155
in The Philosophy of Logical Atomism, edited with an introduction by
David Pears, Open Court, La Salle, IL, 1985. First published 1918.
Notes on Russell’s “Philosophy of Logical Atomism” • Note 25
Cf: Inquiry Into Inquiry • In Medias Res
Re: Dan Everett
I am trying to represent two readings of the three juxtaposed sentences in English. The first reading is that the judge
and the jury both know that Malcolm is guilty. The second is that the judge knows that the jury thinks that Malcolm is
Do these purported EGs of mine seem correct to you?
Apologies for the delay in responding … I won't have much of use to say about those particular graphs as I've long been
following a different fork in Peirce's work about how to get from Alpha to Beta, from propositional to quantificational
logic via graphical syntax.
But the examples raise one of the oldest issues I've bothered about over the years, going back to the days when I read
PQR (Peirce, Quine, Russell) in tandem and many long discussions with my undergrad phil advisor. That is the question
of intentional contexts and ”referential opacity”. The thing is Peirce's pragmatic standpoint yields a radically
distinct analysis of belief, knowledge, and indeed truth from the way things have been handled down the line from
logical atomism and empiricism to analytic philosophy in general. As it happens, there was a critical branch point in
time when Russell almost got a clue but Wittgenstein bullied him into dropping it, at least so far as I could tell from
a scattered sample of texts.
At any rate, I fell down the Wayback Machine rabbit hole looking for things I wrote about all this on the Peirce List
and other places around the web at the turn of the millennium …
I'd almost be tempted to start a blog series on this, probably simulcast on the Facebook Peirce Matters page if you're
into discussing it online … I have enough off the cuff to start an anchor post or two, but it might be the middle of
August before I could do much more.
Cf: Sign Relations • Anthesis
Thus, if a sunflower, in turning towards the sun, becomes by that
very act fully capable, without further condition, of reproducing
a sunflower which turns in precisely corresponding ways toward the
sun, and of doing so with the same reproductive power, the sunflower
would become a Representamen of the sun.
— C.S. Peirce, Collected Papers, CP 2.274
In his picturesque illustration of a sign relation, along with his tracing
of a corresponding sign process, or “semiosis”, Peirce uses the technical term
“representamen” for his concept of a sign, but the shorter word is precise enough,
so long as one recognizes its meaning in a particular theory of signs is given by
a specific definition of what it means to be a sign.
• Semeiotic ( https://oeis.org/wiki/Semeiotic )
• Logic Syllabus ( https://inquiryintoinquiry.com/logic-syllabus/ )
• Sign Relations ( https://oeis.org/wiki/Sign_relation )
• Triadic Relations ( https://oeis.org/wiki/Triadic_relation )
• Relation Theory ( https://oeis.org/wiki/Relation_theory )
cc: Conceptual Graphs • Cybernetics • Laws of Form • Ontolog Forum
cc: FB | Semeiotics • Structural Modeling • Systems Science
Cf: Theme One Program • Discussion 7
Re: Ontolog Forum
::: Alex Shkotin
As we both like digraphs and looking at your way of rendering, let me
share my lazy way of using Graphviz ( https://graphviz.org/ ) on one of
the last pictures produced ( https://photos.app.goo.gl/pJEGBnNqJRBE7JUT9 ).
This is a picture of a derivation tree (aka AST) for the text of four
statements of context-free grammar of some kind. It is important that
this is a digraph with ordered children, and nodes have some attributes.
In your case attributes are “sign”, “code”. In my case attributes are:
* node id,
** for syntactic nonterminal: rule id used for derivation,
** for lexical nonterminal: value taken from text.
Many thanks, the Graphviz suite looks very nice and I will
spend some time looking through the docs. I kept a few samples
of my old ASCII graphics, mostly from a sense of nostalgia, but
I've reached a point in reworking my Theme One Exposition where
I need to upgrade the graphics. My original aim was to have the
program display its own visuals, but it doesn't look like I'll
be the one doing that. Visualizing proofs requires animation —
I used to have an app for that bundled with CorelDraw but it
quit working in a previous platform change and I haven't gotten
around to hunting up a new one. At any rate, there's a sampler
of animated proofs in logical graphs on the following page.
* Proof Animations
( https://oeis.org/wiki/User:Jon_Awbrey/ANIMATION#Proof_Animations )