Cactus Language • Stylistics

[Jon Awbrey]

Cactus Language • Stylistics 1
• https://inquiryintoinquiry.com/2025/07/06/cactus-language-stylistics-1/

❝As a result, we can hardly conceive of how many possibilities
there are for what we call objective reality. Our sharp quills
of knowledge are so narrow and so concentrated in particular
directions that with science there are myriads of totally
different real worlds, each one accessible from the next
simply by slight alterations — shifts of gaze — of every
particular discipline and subspecialty.❞

— Herbert J. Bernstein • “Idols of Modern Science”

The discussion to follow highlights a question of “style” arising
in describing a formal language. In formal contexts, “style” refers
to a loosely specified family of formal systems, typically ones with
a set of distinctive features in common. For example, a “style” of
proof system dictates one or more rules of inference acknowledged as
conforming to the style in question. When it comes to formal languages,
“style” is a natural choice to characterize the varieties of formal
grammars or other kinds of formal systems contemplated for deriving
the sentences of the language in view.

Resources —

Cactus Language • Stylistics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Stylistics

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/

Survey of Theme One Program
• https://inquiryintoinquiry.com/2025/05/06/survey-of-theme-one-program-7/

Regards,

Jon

cc: https://www.academia.edu/community/lnaBNx
cc: https://www.researchgate.net/post/Cactus_Language_Stylistics

[Jon Awbrey]

Cactus Language • Stylistics 2
• https://inquiryintoinquiry.com/2025/07/09/cactus-language-stylistics-2/

In looking at what seems like an incidental issue, the discussion arrives
at a critical point. The question is: What decides the issue of style?

Taking a given language as the object of discussion, what factors enter
into and determine the choice of a style for its presentation, that is,
a particular way of arranging and selecting the materials involved in
a description, a grammar, or a theory of the language? To what degree
is the determination accidental, empirical, pragmatic, rhetorical, or
stylistic, and to what extent is the choice essential, logical, and
necessary?

For that matter, what determines the order of signs in a word, a sentence,
a text, or a discussion? All the corresponding parallel questions about
the character of the choice can be posed with regard to the constituent
part as well as with regard to the main constitution of the formal language.

Answering the question of choice, at any level of articulation, requires
an inquiry into the type of distinction it invokes, between arrangements
and orders which are essential, logical, and necessary and orders and
arrangements which are accidental, rhetorical, and stylistic.

A “logical order”, if it resides in a subject at all, can be approached by
considering all the ways of saying the same things, in all the languages
capable of saying roughly the same things about the subject in question.

Naturally, the “all” appearing in that rule of thumb has to be interpreted
as a fittingly qualified universal. For all practical purposes, it simply
means “all the ways a person can think of” and “all the languages a person can
conceive of”, with all things being relative to the person and the particular
moment of investigation. For all those reasons, the rule must stand as little
more than a rough idea of how to approach its object.

If it is demonstrated that a given formal language can be presented in any one
of several styles of formal grammar then the choice is accidental, optional, and
stylistic to the very extent it is free. But if it can be shown that a particular
language cannot be successfully presented in a particular style of grammar then
the issue of style is no longer free and rhetorical but becomes to that very
degree essential, necessary, and obligatory, in other words, a question of the
objective logical order which can be found to reside in the object language.

Resources —

Cactus Language • Stylistics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Stylistics

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/

Survey of Theme One Program
• https://inquiryintoinquiry.com/2025/05/06/survey-of-theme-one-program-7/

Regards,

Jon

cc: https://www.academia.edu/community/LYxBxD
cc: https://www.researchgate.net/post/Cactus_Language_Stylistics

[Jon Awbrey]

Cactus Language • Stylistics 3
• https://inquiryintoinquiry.com/2025/07/11/cactus-language-stylistics-3/

As a rough illustration of the difference between rhetorical
and logical orders, consider the contrasting types of order
appearing in the following conjunction of conditionals.

• X ⇒ Y and Y ⇒ Z

The formula exhibits a happy conformity between its rhetorical form and its
logical content, in such a way one hardly notices the difference between them.
The rhetorical form is given by the order of sentences in the conditionals and
the order of conditionals in the conjunction. The logical content is given by
the order of propositions in the following extended implicational sequence.

• X ≤ Y ≤ Z

To see the difference between rhetorical form and logical content, or manner
and matter, it is enough to observe a few ways the expression can be varied
without changing its meaning. For example, the following expression is
logically equivalent to the one at the top.

• Z ⇐ Y and Y ⇐ X

Any style of “declarative programming”, or “logic programming”, depends on
a capacity, as embodied in a programming language or other formal system,
to describe the relation between problems and solutions in logical terms.
A recurring problem in building such a capacity is in bridging the gap
between ostensibly non‑logical orders and the logical orders used to
describe and represent them.

For example, to mention just two of the most pressing cases and the
ones currently proving to be the most resistant to a complete analysis,
one has the orders of dynamic evolution and rhetorical transition which
manifest themselves in the process of inquiry and in the communication
of its results.

Resources —

Cactus Language • Stylistics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Stylistics

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/

Survey of Theme One Program
• https://inquiryintoinquiry.com/2025/05/06/survey-of-theme-one-program-7/

Regards,

Jon

cc: https://www.academia.edu/community/LxjwWa
cc: https://www.researchgate.net/post/Cactus_Language_Stylistics

[Jon Awbrey]

Cactus Language • Stylistics 4
• https://inquiryintoinquiry.com/2025/07/15/cactus-language-stylistics-4/

The present patch of discussion is concerned with describing a family
of formal languages whose typical representative is the painted cactus
language ‡L‡ = ‡C‡(‡P‡). Once we have the abstract forms of cactus
languages well enough in hand to grasp their application, the next
order of business is to interpret them for propositional logic, thus
producing a sentential calculus, an order of reasoning which constitutes
an active ingredient and a significant component of all logical reasoning.

The standard devices of formal grammars and formal language theory
are adequate to describe the language of interest from an external
point of view but the ultimate desire is for something more exacting,
to turn the tables on the order of description and enter on a process
of eversion which evolves to the point of asking: To what extent
can the language capture the essential features and laws of its own
grammar and describe the active principles of its own generation?
In other words: How well can the language be described by using
the language itself to do so?

To address the above questions, we have to express what a grammar
says about a language in terms of what a language can say on its own.
In effect, it is necessary to analyze the kinds of meaningful statements
grammars are capable of making about languages in general and to relate
them to the kinds of meaningful statements the “sentences” of the cactus
language might be interpreted as making about the same topics.

So far in the present discussion, the sentences of the cactus language make
no meaningful statements at all, much less any meaningful statements about
languages and their constitutions. As of yet, those sentences subsist in
the form of purely abstract, formal, and uninterpreted combinatorial
constructions.

Resources —

Cactus Language • Stylistics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Stylistics

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/

Survey of Theme One Program
• https://inquiryintoinquiry.com/2025/05/06/survey-of-theme-one-program-7/

Regards,

Jon

cc: https://www.academia.edu/community/lnaAZv
cc: https://www.researchgate.net/post/Cactus_Language_Stylistics

[Jon Awbrey]

Cactus Language • Stylistics 5
• https://inquiryintoinquiry.com/2025/07/18/cactus-language-stylistics-5/

Before the capacity of a language to describe itself can be evaluated,
the missing link to meaning must be supplied for each of its expressions.
That means opening a dimension of semantics to be navigated by means of
interpretation, topics to be taken up in the case of ‡C‡(‡P‡) at a later
stage of the present inquiry.

The pressing issue at this point is the distinct placements of formal
languages and formal grammars with respect to the question of meaning.
The sentences of a formal language are merely the strings of signs which
happen to belong to a certain set. They do not by themselves make any
meaningful statements at all, not without mounting a separate effort
of interpretation, but the rules of a formal grammar make meaningful
statements about a formal language, to the extent they say what strings
belong to it and what strings do not.

A formal grammar, then, a formalism appearing even more skeletal than
a formal language, still has bits and pieces of meaning attached to it.
In a sense, the question of meaning is factored into two parts, structure
and value, leaving the aspect of value reduced to the simple question of
belonging. Whether that single bit of meaningful value is enough to
encompass all the dimensions of meaning we require, and whether it can
be compounded to cover the complexity which actually exists in the realm
of meaning — those are questions for an extended future inquiry.

Resources —

Cactus Language • Stylistics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Stylistics

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/

Survey of Theme One Program
• https://inquiryintoinquiry.com/2025/05/06/survey-of-theme-one-program-7/

Regards,

Jon

cc: https://www.academia.edu/community/VvkDbE
cc: https://www.researchgate.net/post/Cactus_Language_Stylistics

[Jon Awbrey]

Cactus Language • Stylistics 6
• https://inquiryintoinquiry.com/2025/07/20/cactus-language-stylistics-6/

Perhaps I ought to comment on the differences between the present and
the standard definition of a formal grammar, since I am attempting to
strike a compromise with several alternative conventions of usage, and
thus to leave certain options open for future exploration. All the
changes are minor, in the sense they are not intended to alter the
classes of languages able to be generated but only to clear up the
ambiguities and obscurities affecting their conception.

Perhaps most importantly, the conventional scope of non‑terminal symbols
is expanded to include the sentence symbol, mainly on account of all the
contexts where initial and intermediate symbols are naturally invoked in
the same breath. By way of compensating for the usual exclusion of the
sentence symbol from the non‑terminal class an equivalent distinction is
introduced in the fashion of a distinction between the initial and the
intermediate symbols, and that serves its purpose in all the contexts
where the two kinds of symbols need to be treated separately.

At present I remain a bit worried about the motivations and the justifications
for introducing that distinction in the first place. It is purportedly designed
to guarantee that the process of derivation at least gets started in a definite
direction, while the real question has to do with how it all ends. The excuses
of efficiency and expediency I offered as reasons for distinguishing between
empty and significant sentences are likely to be ephemeral, if not entirely
illusory, since intermediate symbols are still permitted to characterize or
cover themselves, not to mention being allowed to cover the empty string,
and so the very types of traps one exerts oneself to avoid at the outset
are always there to afflict the process at all the intervening times.

If one reflects on the form of grammar being prescribed here, it looks as if
one sought, rather futilely, to avoid the problems of recursion by proscribing
the main program from calling itself, while allowing any subprogram to do so.
But any trouble avoidable in the part is also avoidable in the main, while any
trouble inescapable in the part is also inescapable in the main. Consequently,
I am reserving the right to change my mind at a later stage, perhaps to permit
the initial symbol to characterize, cover, produce, or regenerate itself, if
that turns out to be the best way in the end.

Resources —

Cactus Language • Stylistics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Stylistics

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/

Survey of Theme One Program
• https://inquiryintoinquiry.com/2025/05/06/survey-of-theme-one-program-7/

Regards,

Jon

cc: https://www.academia.edu/community/5MKnag
cc: https://www.researchgate.net/post/Cactus_Language_Stylistics