Cactus Language • Pragmatics
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Jon Awbrey
Jul 23, 2025, 12:00:44 PMJul 23
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Pragmatics 1
• https://inquiryintoinquiry.com/2025/07/22/cactus-language-pragmatics-1/
Expanding our perspective on the options for formal grammar style
brings us to questions about the manner in which the abstract theory
of formal languages and the pragmatic theory of sign relations interact
with each other.
Formal language theory can seem like an awfully picky subject at times,
treating every symbol as a thing in itself the way it does, sorting out
the nominal types of symbols as objects in themselves, and singling out
the passing tokens of symbols as distinct entities in their own rights.
It has to continue doing that, if not for any better reason than to aid
in clarifying the kinds of languages people are accustomed to use, to
assist in writing computer programs capable of parsing real sentences,
and to serve in designing programming languages people would like to
become accustomed to use.
As it happens, the only time formal language theory becomes too picky,
or a bit too myopic in its focus, is when it leads one to think one is
dealing with the thing itself and not just the sign of it, in other words,
when the people who use the tools of formal language theory forget they are
dealing with the mere signs of more interesting objects and not the objects
of ultimate interest in and of themselves.
There are then a number of deleterious effects at risk of arising from
the extreme pickiness of formal language theory, arising, as often the case,
when theorists forget the practical context of theorization. The exacting
task of defining the membership of a formal language leads one to think that
object and that object alone is the justifiable end of the whole exercise.
The distractions of the mediate objective render one liable to forget one's
penultimate interest lies always with various equivalence classes of signs,
not entirely or exclusively with their more meticulous representatives.
When that happens, one typically goes on working oblivious to the circumstance
that many details about what transpires in the meantime do not matter at all in
the end, and one is likely to remain in blissful ignorance of the fact that many
special details of language membership are bound, destined, and pre‑determined
to be glossed over with some measure of indifference, especially when it comes
to the final constitution of those equivalence classes of signs which answer
for the genuine objects of the whole enterprise of language.
Whenever a form of theory, against its initial and its best intentions, permits
an absence of mind no longer beneficial in its main effects, an antidotal form
of theory is called for to restore the presence of mind all forms of theory
are meant to advance.
Resources —
Cactus Language • Pragmatics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Pragmatics
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/LxjBy3
cc: https://www.researchgate.net/post/Cactus_Language_Pragmatics
• https://inquiryintoinquiry.com/2025/07/22/cactus-language-pragmatics-1/
Expanding our perspective on the options for formal grammar style
brings us to questions about the manner in which the abstract theory
of formal languages and the pragmatic theory of sign relations interact
with each other.
Formal language theory can seem like an awfully picky subject at times,
treating every symbol as a thing in itself the way it does, sorting out
the nominal types of symbols as objects in themselves, and singling out
the passing tokens of symbols as distinct entities in their own rights.
It has to continue doing that, if not for any better reason than to aid
in clarifying the kinds of languages people are accustomed to use, to
assist in writing computer programs capable of parsing real sentences,
and to serve in designing programming languages people would like to
become accustomed to use.
As it happens, the only time formal language theory becomes too picky,
or a bit too myopic in its focus, is when it leads one to think one is
dealing with the thing itself and not just the sign of it, in other words,
when the people who use the tools of formal language theory forget they are
dealing with the mere signs of more interesting objects and not the objects
of ultimate interest in and of themselves.
There are then a number of deleterious effects at risk of arising from
the extreme pickiness of formal language theory, arising, as often the case,
when theorists forget the practical context of theorization. The exacting
task of defining the membership of a formal language leads one to think that
object and that object alone is the justifiable end of the whole exercise.
The distractions of the mediate objective render one liable to forget one's
penultimate interest lies always with various equivalence classes of signs,
not entirely or exclusively with their more meticulous representatives.
When that happens, one typically goes on working oblivious to the circumstance
that many details about what transpires in the meantime do not matter at all in
the end, and one is likely to remain in blissful ignorance of the fact that many
special details of language membership are bound, destined, and pre‑determined
to be glossed over with some measure of indifference, especially when it comes
to the final constitution of those equivalence classes of signs which answer
for the genuine objects of the whole enterprise of language.
Whenever a form of theory, against its initial and its best intentions, permits
an absence of mind no longer beneficial in its main effects, an antidotal form
of theory is called for to restore the presence of mind all forms of theory
are meant to advance.
Resources —
Cactus Language • Pragmatics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Pragmatics
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/LxjBy3
cc: https://www.researchgate.net/post/Cactus_Language_Pragmatics
Jon Awbrey
Jul 26, 2025, 1:15:29 PMJul 26
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Pragmatics 2
• https://inquiryintoinquiry.com/2025/07/26/cactus-language-pragmatics-2/
The pragmatic theory of sign relations is called for in settings
where everything that can be named has any number of other names,
that is to say, the usual case. Of course we'd like to replace
the multiplicity of signs with an organized system of canonical
signs, one for each object that needs to be named, but reducing
the redundancy too far, beyond what is necessary to eliminate the
factor of “noise” in the language and thus clear up its effectively
useless distractions, can destroy the utility of natural languages
and bespoke formal systems, which are evolved to provide a ready
means for expressing present situations, clear or not, and to
describe ongoing conditions of experience in just the way they
present themselves.
Within a fully fleshed out framework of language, moreover, the process
of transforming the manifestations of a sign from its ordinary appearance
to its canonical aspect is the whole problem of computation in a nutshell.
It's a well‑known fact but an often forgotten truth that no one computes
with numbers, but solely with numerals in respect of numbers, and numerals
themselves are symbols. Among other things, that renders all discussion
of numeric versus symbolic computation a bit beside the point, since it's
only a question of what types of symbols are required for one's immediate
application or for one's selection of ongoing objectives.
The numerals everyone knows best are just the canonical symbols, the standard
signs or the normal forms for numbers, and the process of computation is a matter
of getting from the obscure signs a situation impresses on us in the form of data
to the indications of the situation's character which can be rendered clear enough
to motivate action.
Resources —
Cactus Language • Pragmatics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Pragmatics
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/V9NRYO
cc: https://www.researchgate.net/post/Cactus_Language_Pragmatics
• https://inquiryintoinquiry.com/2025/07/26/cactus-language-pragmatics-2/
The pragmatic theory of sign relations is called for in settings
where everything that can be named has any number of other names,
that is to say, the usual case. Of course we'd like to replace
the multiplicity of signs with an organized system of canonical
signs, one for each object that needs to be named, but reducing
the redundancy too far, beyond what is necessary to eliminate the
factor of “noise” in the language and thus clear up its effectively
useless distractions, can destroy the utility of natural languages
and bespoke formal systems, which are evolved to provide a ready
means for expressing present situations, clear or not, and to
describe ongoing conditions of experience in just the way they
present themselves.
Within a fully fleshed out framework of language, moreover, the process
of transforming the manifestations of a sign from its ordinary appearance
to its canonical aspect is the whole problem of computation in a nutshell.
It's a well‑known fact but an often forgotten truth that no one computes
with numbers, but solely with numerals in respect of numbers, and numerals
themselves are symbols. Among other things, that renders all discussion
of numeric versus symbolic computation a bit beside the point, since it's
only a question of what types of symbols are required for one's immediate
application or for one's selection of ongoing objectives.
The numerals everyone knows best are just the canonical symbols, the standard
signs or the normal forms for numbers, and the process of computation is a matter
of getting from the obscure signs a situation impresses on us in the form of data
to the indications of the situation's character which can be rendered clear enough
to motivate action.
Resources —
Cactus Language • Pragmatics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Pragmatics
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.researchgate.net/post/Cactus_Language_Pragmatics
Jon Awbrey
Jul 27, 2025, 12:48:46 PMJul 27
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Pragmatics 3
• https://inquiryintoinquiry.com/2025/07/27/cactus-language-pragmatics-3/
Having broached the distinction between objective propositions
and syntactic sentences, its analogy to the distinction between
numbers and numerals becomes clear. What are the implications of
that distinction for the realm of reasoning about propositions and
its representation in sentential logic?
If the purpose of a sentence is precisely to denote a proposition
then the proposition is simply the object of whatever sign is taken
for the sentence. The computational manifestation of a piece of
reasoning about propositions thus amounts to a process taking place
entirely within a language of sentences, being a procedure which can
rationalize its account by referring to the denominations of sentences
among propositions.
As far as it bears on our current context of problems, the upshot is this:
Do not worry too much about what roles the empty string ε = “” and the
blank symbol m₁ = “ ” are supposed to play in a given species of formal
language. As it happens, it is far less important to wonder whether those
types of formal tokens actually constitute genuine sentences than it is to
decide what equivalence classes it makes sense to form over all the sentences
in the resulting language, and only then to bother about what equivalence
classes those limiting cases of sentences are most conveniently taken to
represent.
Resources —
Cactus Language • Pragmatics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Pragmatics
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/VBvWNo
cc: https://www.researchgate.net/post/Cactus_Language_Pragmatics
• https://inquiryintoinquiry.com/2025/07/27/cactus-language-pragmatics-3/
Having broached the distinction between objective propositions
and syntactic sentences, its analogy to the distinction between
numbers and numerals becomes clear. What are the implications of
that distinction for the realm of reasoning about propositions and
its representation in sentential logic?
If the purpose of a sentence is precisely to denote a proposition
then the proposition is simply the object of whatever sign is taken
for the sentence. The computational manifestation of a piece of
reasoning about propositions thus amounts to a process taking place
entirely within a language of sentences, being a procedure which can
rationalize its account by referring to the denominations of sentences
among propositions.
As far as it bears on our current context of problems, the upshot is this:
Do not worry too much about what roles the empty string ε = “” and the
blank symbol m₁ = “ ” are supposed to play in a given species of formal
language. As it happens, it is far less important to wonder whether those
types of formal tokens actually constitute genuine sentences than it is to
decide what equivalence classes it makes sense to form over all the sentences
in the resulting language, and only then to bother about what equivalence
classes those limiting cases of sentences are most conveniently taken to
represent.
Resources —
Cactus Language • Pragmatics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Pragmatics
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.researchgate.net/post/Cactus_Language_Pragmatics
Jon Awbrey
Jul 28, 2025, 3:00:42 PMJul 28
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Pragmatics 4
• https://inquiryintoinquiry.com/2025/07/28/cactus-language-pragmatics-4/
The questions about boundary conditions we keep encountering betray
a more general issue. Already by this point in the discussion the
limits of a purely syntactic approach to language are becoming visible.
It is not that one cannot go a long way by that road in the analysis of
a particular language and the study of languages in general but when it
comes to understanding the purpose of a language, extending its use in
a chosen direction, or designing a language for a particular set of uses,
what matters above all are the “pragmatic equivalence classes” of signs
demanded by the application and intended by the designer and not so much
the peculiar characters of signs representing the classes of practical
meaning.
Any description of a language is bound to have alternative descriptions.
In particular, a formally circumscribed description of a formal language,
as any effectively finite description is bound to be, is certain to suggest
the equally likely existence and possible utility of other descriptions.
A single formal grammar describes but a single formal language, but any
formal language is described by many formal grammars, not all of which
afford the same grasp of its structure, provide equivalent comprehensions
of its character, or yield interchangeable views of its aspects. Even with
respect to the same formal language, different formal grammars are typically
better for different purposes.
Resources —
Cactus Language • Pragmatics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Pragmatics
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/5NrBrx
cc: https://www.researchgate.net/post/Cactus_Language_Pragmatics
• https://inquiryintoinquiry.com/2025/07/28/cactus-language-pragmatics-4/
The questions about boundary conditions we keep encountering betray
a more general issue. Already by this point in the discussion the
limits of a purely syntactic approach to language are becoming visible.
It is not that one cannot go a long way by that road in the analysis of
a particular language and the study of languages in general but when it
comes to understanding the purpose of a language, extending its use in
a chosen direction, or designing a language for a particular set of uses,
what matters above all are the “pragmatic equivalence classes” of signs
demanded by the application and intended by the designer and not so much
the peculiar characters of signs representing the classes of practical
meaning.
Any description of a language is bound to have alternative descriptions.
In particular, a formally circumscribed description of a formal language,
as any effectively finite description is bound to be, is certain to suggest
the equally likely existence and possible utility of other descriptions.
A single formal grammar describes but a single formal language, but any
formal language is described by many formal grammars, not all of which
afford the same grasp of its structure, provide equivalent comprehensions
of its character, or yield interchangeable views of its aspects. Even with
respect to the same formal language, different formal grammars are typically
better for different purposes.
Resources —
Cactus Language • Pragmatics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Pragmatics
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.researchgate.net/post/Cactus_Language_Pragmatics
Jon Awbrey
Jul 30, 2025, 6:12:44 AMJul 30
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Pragmatics 5
• https://inquiryintoinquiry.com/2025/07/29/cactus-language-pragmatics-5/
Along with the distinctions we see evolving among different
styles of grammar and the preferences different observers
display toward them, there naturally arises the question:
What is the root of that evolution?
One dimension of variation in formal grammar style can be seen by
treating a union of languages, and especially a disjoint union of
languages, as a sum (∑), by treating a concatenation of languages
as a product (∏), and then by distinguishing the styles of analysis
favoring sums of products (∑∏) from those favoring products of sums (∏∑)
as their canonical forms of description.
If one examines the relationship between grammars and languages
closely enough to detect the influence of the above two styles
and comes to appreciate how different grammar styles may be used
with different degrees of success for different purposes then one
begins to see the possibility that alternative styles of description
might be based on altogether different linguistic and logical operations.
Resources —
Cactus Language • Pragmatics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Pragmatics
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/L68dQj
cc: https://www.researchgate.net/post/Cactus_Language_Pragmatics
• https://inquiryintoinquiry.com/2025/07/29/cactus-language-pragmatics-5/
Along with the distinctions we see evolving among different
styles of grammar and the preferences different observers
display toward them, there naturally arises the question:
What is the root of that evolution?
One dimension of variation in formal grammar style can be seen by
treating a union of languages, and especially a disjoint union of
languages, as a sum (∑), by treating a concatenation of languages
as a product (∏), and then by distinguishing the styles of analysis
favoring sums of products (∑∏) from those favoring products of sums (∏∑)
as their canonical forms of description.
If one examines the relationship between grammars and languages
closely enough to detect the influence of the above two styles
and comes to appreciate how different grammar styles may be used
with different degrees of success for different purposes then one
begins to see the possibility that alternative styles of description
might be based on altogether different linguistic and logical operations.
Resources —
Cactus Language • Pragmatics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Pragmatics
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.researchgate.net/post/Cactus_Language_Pragmatics
Jon Awbrey
Jul 31, 2025, 9:32:27 AMJul 31
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Pragmatics 6
• https://inquiryintoinquiry.com/2025/07/30/cactus-language-pragmatics-6/
It is possible to trace the divergence of formal grammar styles
to an even more primitive division, distinguishing between the
“additive” or “parallel” styles and the “multiplicative” or
“serial” styles.
The issue is somewhat confused by the fact that an additive
analysis is typically expressed in the form of a series, in
other words, a disjoint union of sets or a linear sum of their
independent effects. But it is easy enough to sort things out
if one observes the more telling connection between “parallel”
and “independent”. Another way to keep the right associations
straight is to use the term “sequential” in preference to the
more misleading term “serial”. Whatever one calls the broad
division of styles, the scope and sweep of their dimensions
of variation can be delineated in the following way.
“Additive” or “parallel” styles favor sums of products (∑∏) as
canonical forms of expression, pulling sums, unions, co‑products,
and logical disjunctions to the outermost layers of analysis and
synthesis, while pushing products, intersections, concatenations,
and logical conjunctions to the innermost levels of articulation
and generation. The analogous style in propositional logic leads
to the “disjunctive normal form” (DNF).
“Multiplicative” or “serial” styles favor products of sums (∏∑)
as canonical forms of expression, pulling products, intersections,
concatenations, and logical conjunctions to the outermost layers
of analysis and synthesis, while pushing sums, unions, co‑products,
and logical disjunctions to the innermost levels of articulation
and generation. The analogous style in propositional logic leads
to the “conjunctive normal form” (CNF).
Resources —
Cactus Language • Pragmatics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Pragmatics
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/lzXxvn
cc: https://www.researchgate.net/post/Cactus_Language_Pragmatics
• https://inquiryintoinquiry.com/2025/07/30/cactus-language-pragmatics-6/
It is possible to trace the divergence of formal grammar styles
to an even more primitive division, distinguishing between the
“additive” or “parallel” styles and the “multiplicative” or
“serial” styles.
The issue is somewhat confused by the fact that an additive
analysis is typically expressed in the form of a series, in
other words, a disjoint union of sets or a linear sum of their
independent effects. But it is easy enough to sort things out
if one observes the more telling connection between “parallel”
and “independent”. Another way to keep the right associations
straight is to use the term “sequential” in preference to the
more misleading term “serial”. Whatever one calls the broad
division of styles, the scope and sweep of their dimensions
of variation can be delineated in the following way.
“Additive” or “parallel” styles favor sums of products (∑∏) as
canonical forms of expression, pulling sums, unions, co‑products,
and logical disjunctions to the outermost layers of analysis and
synthesis, while pushing products, intersections, concatenations,
and logical conjunctions to the innermost levels of articulation
and generation. The analogous style in propositional logic leads
to the “disjunctive normal form” (DNF).
“Multiplicative” or “serial” styles favor products of sums (∏∑)
as canonical forms of expression, pulling products, intersections,
concatenations, and logical conjunctions to the outermost layers
of analysis and synthesis, while pushing sums, unions, co‑products,
and logical disjunctions to the innermost levels of articulation
and generation. The analogous style in propositional logic leads
to the “conjunctive normal form” (CNF).
Resources —
Cactus Language • Pragmatics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Pragmatics
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.researchgate.net/post/Cactus_Language_Pragmatics
Jon Awbrey
Aug 2, 2025, 7:00:46 AMAug 2
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Pragmatics 7
• https://inquiryintoinquiry.com/2025/08/01/cactus-language-pragmatics-7/
There is a curious sort of diagnostic clue which often serves
to reveal the dominance of one mode or the other within an
individual thinker's cognitive style. Examined on the question
of what constitutes the natural numbers, an “additive thinker”
tends to start the sequence at 0 while a “multiplicative thinker”
tends to regard it as beginning at 1.
In any style of description, grammar, or theory of a language, it is
usually possible to tease out the influence of the contrasting traits,
namely, the additive attitude versus the multiplicative tendency which
go to make up the style in question, and even to determine the dominant
inclination or point of view which establishes its perspective on the
target domain.
In each style of formal grammar, the multiplicative aspect is present
in the sequential concatenation of signs, in both the augmented strings
and the terminal strings. In settings where the non‑terminal symbols
classify types of strings, the concatenation of the non-terminal symbols
signifies the cartesian product over the corresponding sets of strings.
In the context‑free style of formal grammar, the additive aspect is easy
to spot. It is signaled by the parallel covering of many augmented strings
or sentential forms by the same non‑terminal symbol. In active terms, it
calls for the independent rewriting of that non‑terminal symbol by a number
of different successors, as in the following scheme.
• q :> W₁
• … … …
• q :> Wₖ
Resources —
Cactus Language • Pragmatics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Pragmatics
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/leRQyw
cc: https://www.researchgate.net/post/Cactus_Language_Pragmatics
• https://inquiryintoinquiry.com/2025/08/01/cactus-language-pragmatics-7/
There is a curious sort of diagnostic clue which often serves
to reveal the dominance of one mode or the other within an
individual thinker's cognitive style. Examined on the question
of what constitutes the natural numbers, an “additive thinker”
tends to start the sequence at 0 while a “multiplicative thinker”
tends to regard it as beginning at 1.
In any style of description, grammar, or theory of a language, it is
usually possible to tease out the influence of the contrasting traits,
namely, the additive attitude versus the multiplicative tendency which
go to make up the style in question, and even to determine the dominant
inclination or point of view which establishes its perspective on the
target domain.
In each style of formal grammar, the multiplicative aspect is present
in the sequential concatenation of signs, in both the augmented strings
and the terminal strings. In settings where the non‑terminal symbols
classify types of strings, the concatenation of the non-terminal symbols
signifies the cartesian product over the corresponding sets of strings.
In the context‑free style of formal grammar, the additive aspect is easy
to spot. It is signaled by the parallel covering of many augmented strings
or sentential forms by the same non‑terminal symbol. In active terms, it
calls for the independent rewriting of that non‑terminal symbol by a number
of different successors, as in the following scheme.
• q :> W₁
• … … …
• q :> Wₖ
Resources —
Cactus Language • Pragmatics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Pragmatics
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.researchgate.net/post/Cactus_Language_Pragmatics
Jon Awbrey
Aug 3, 2025, 8:00:48 AMAug 3
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Pragmatics 8
• https://inquiryintoinquiry.com/2025/08/02/cactus-language-pragmatics-8/
It is useful to examine the relation between syntactic production (:>)
and logical implication (⇒) with one eye to what they have in common
and another eye to how they differ.
The production q :> W says the appearance of the symbol q in
a sentential form implies the possibility of replacing q with W.
Although that sounds like a “possible implication”, to the extent
that “q implies a possible W” or that “q possibly implies W”, the
qualifiers possible and possibly are essential to the meaning of
what is actually implied. In effect, those qualifications reverse
the direction of implication, making “q ⇐ W” the best analogue for
the sense of the production.
One way to understand a production of the form q :> W is to realize
non‑terminal symbols have the significance of hypotheses. The terminal
strings form the empirical matter of the language in question while the
non‑terminal symbols mark the patterns or types of substrings which may
be recognized in the linguistic corpus. If one observes a portion of
a terminal string which fits the pattern of a sentential form W then
it is an admissible hypothesis, according to the theory of the language
afforded by the formal grammar, that the piece of string not only fits
the type q but even comes to be generated under the auspices of the
non‑terminal symbol “q”.
Resources —
Cactus Language • Pragmatics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Pragmatics
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/V37RDZ
cc: https://www.researchgate.net/post/Cactus_Language_Pragmatics
• https://inquiryintoinquiry.com/2025/08/02/cactus-language-pragmatics-8/
It is useful to examine the relation between syntactic production (:>)
and logical implication (⇒) with one eye to what they have in common
and another eye to how they differ.
The production q :> W says the appearance of the symbol q in
a sentential form implies the possibility of replacing q with W.
Although that sounds like a “possible implication”, to the extent
that “q implies a possible W” or that “q possibly implies W”, the
qualifiers possible and possibly are essential to the meaning of
what is actually implied. In effect, those qualifications reverse
the direction of implication, making “q ⇐ W” the best analogue for
the sense of the production.
One way to understand a production of the form q :> W is to realize
non‑terminal symbols have the significance of hypotheses. The terminal
strings form the empirical matter of the language in question while the
non‑terminal symbols mark the patterns or types of substrings which may
be recognized in the linguistic corpus. If one observes a portion of
a terminal string which fits the pattern of a sentential form W then
it is an admissible hypothesis, according to the theory of the language
afforded by the formal grammar, that the piece of string not only fits
the type q but even comes to be generated under the auspices of the
non‑terminal symbol “q”.
Resources —
Cactus Language • Pragmatics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Pragmatics
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.researchgate.net/post/Cactus_Language_Pragmatics
Jon Awbrey
Aug 4, 2025, 12:32:38 PMAug 4
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Pragmatics 9
• https://inquiryintoinquiry.com/2025/08/04/cactus-language-pragmatics-9/
A moment's reflection on the issue of style, giving due consideration
to the received array of stylistic choices, ought to inspire at least
the question: “Are those the only choices there are?”
There are abundant indications that other options, more differentiated varieties
of description and more integrated ways of approaching individual languages, are
likely to be conceivable, feasible, and even more ultimately viable.
If a suitably generic style, one that incorporates the full scope of logical
combinations and operations, is broadly available, then it would no longer be
necessary, or even apt, to argue in universal terms about which style is best,
but more useful to investigate how we might adapt the local styles to the local
requirements. The medium of a fully generic style would yield a viable compromise
between additive and multiplicative canons and render the choice between parallel
and serial a false alternative, at least, when expressed in the globally exclusive
terms which are currently and most commonly adopted to pose it.
One set of indications comes from the study of machines, languages, and computation,
including theories of their structures and relations. The forms of composition and
decomposition known as parallel and serial are merely the limiting special cases in
two directions of specialization of a more generic form, commonly known as the “cascade”
form of combination. That is a well‑known fact in the theories dealing with automata
and their associated formal languages but its implications do not seem to be widely
appreciated outside those fields. In particular, the availability of that option
dispells the need to choose one extreme or the other, since most of the natural
cases are likely to exist somewhere in between.
Another set of indications appears in algebra and category theory, where forms
of composition and decomposition related to the cascade combination, namely, the
“semi‑direct product” and its special case, the “wreath product”, are encountered
at higher levels of generality than the cartesian products of sets or the direct
products of spaces.
In those domains of operation, one finds it necessary to consider also the “co‑product”
of sets and spaces, a construction which artificially creates a disjoint union of sets,
that is, a union of spaces which are being treated as independent. It does that, in
effect, by indexing, coloring, or preparing the otherwise possibly overlapping domains
which are being combined. What renders that a chimera or a hybrid form of combination
is the fact that the indexing is tantamount to a cartesian product of a singleton set,
namely, the conventional index, color, or affix in question, with the individual domain
which is entering as a factor, a term, or a participant in the final result.
Resources —
Cactus Language • Pragmatics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Pragmatics
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/V1pnAp
cc: https://www.researchgate.net/post/Cactus_Language_Pragmatics
• https://inquiryintoinquiry.com/2025/08/04/cactus-language-pragmatics-9/
A moment's reflection on the issue of style, giving due consideration
to the received array of stylistic choices, ought to inspire at least
the question: “Are those the only choices there are?”
There are abundant indications that other options, more differentiated varieties
of description and more integrated ways of approaching individual languages, are
likely to be conceivable, feasible, and even more ultimately viable.
If a suitably generic style, one that incorporates the full scope of logical
combinations and operations, is broadly available, then it would no longer be
necessary, or even apt, to argue in universal terms about which style is best,
but more useful to investigate how we might adapt the local styles to the local
requirements. The medium of a fully generic style would yield a viable compromise
between additive and multiplicative canons and render the choice between parallel
and serial a false alternative, at least, when expressed in the globally exclusive
terms which are currently and most commonly adopted to pose it.
One set of indications comes from the study of machines, languages, and computation,
including theories of their structures and relations. The forms of composition and
decomposition known as parallel and serial are merely the limiting special cases in
two directions of specialization of a more generic form, commonly known as the “cascade”
form of combination. That is a well‑known fact in the theories dealing with automata
and their associated formal languages but its implications do not seem to be widely
appreciated outside those fields. In particular, the availability of that option
dispells the need to choose one extreme or the other, since most of the natural
cases are likely to exist somewhere in between.
Another set of indications appears in algebra and category theory, where forms
of composition and decomposition related to the cascade combination, namely, the
“semi‑direct product” and its special case, the “wreath product”, are encountered
at higher levels of generality than the cartesian products of sets or the direct
products of spaces.
In those domains of operation, one finds it necessary to consider also the “co‑product”
of sets and spaces, a construction which artificially creates a disjoint union of sets,
that is, a union of spaces which are being treated as independent. It does that, in
effect, by indexing, coloring, or preparing the otherwise possibly overlapping domains
which are being combined. What renders that a chimera or a hybrid form of combination
is the fact that the indexing is tantamount to a cartesian product of a singleton set,
namely, the conventional index, color, or affix in question, with the individual domain
which is entering as a factor, a term, or a participant in the final result.
Resources —
Cactus Language • Pragmatics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Pragmatics
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.researchgate.net/post/Cactus_Language_Pragmatics
Jon Awbrey
Aug 6, 2025, 12:00:22 PMAug 6
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Pragmatics 10
• https://inquiryintoinquiry.com/2025/08/05/cactus-language-pragmatics-10/
One insight arising from Peirce's work on the mathematics
underlying logic is that the operations on sets known as
complementation, intersection, and union, along with the
corresponding logical operations of negation, conjunction,
and disjunction, are not as fundamental as they first appear.
That is because all of them can be constructed or derived from
a smaller set of operations, in fact, taking the logical side
of things, from either one of two “sole sufficient operators”
called “amphecks” by Peirce, “strokes” by those who re‑discovered
them later, and known in computer science as the operators “nand”
and “nnor”. Thus by virtue of their precedence in the orders of
construction and derivation, the sole sufficient operators have
to be regarded as the simplest and most primitive in principle,
even if they are scarcely recognized as lying among the more
familiar elements of logic.
Resources —
Cactus Language • Pragmatics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Pragmatics
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/L68Wnd
cc: https://www.researchgate.net/post/Cactus_Language_Pragmatics
• https://inquiryintoinquiry.com/2025/08/05/cactus-language-pragmatics-10/
One insight arising from Peirce's work on the mathematics
underlying logic is that the operations on sets known as
complementation, intersection, and union, along with the
corresponding logical operations of negation, conjunction,
and disjunction, are not as fundamental as they first appear.
That is because all of them can be constructed or derived from
a smaller set of operations, in fact, taking the logical side
of things, from either one of two “sole sufficient operators”
called “amphecks” by Peirce, “strokes” by those who re‑discovered
them later, and known in computer science as the operators “nand”
and “nnor”. Thus by virtue of their precedence in the orders of
construction and derivation, the sole sufficient operators have
to be regarded as the simplest and most primitive in principle,
even if they are scarcely recognized as lying among the more
familiar elements of logic.
Resources —
Cactus Language • Pragmatics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Pragmatics
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.researchgate.net/post/Cactus_Language_Pragmatics
Jon Awbrey
Aug 9, 2025, 3:04:35 PMAug 9
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Pragmatics 11
• https://inquiryintoinquiry.com/2025/08/09/cactus-language-pragmatics-11/
I am throwing together a wide variety of different operations into
the bins labeled “additive” and “multiplicative” but it's easy to
observe a natural organization and even some relations approaching
isomorphisms among and between the members of each class.
The relation between logical disjunction and the union of sets and the
relation between logical conjunction and the intersection of sets ought
to be clear enough for present purposes. But the relation of set‑theoretic
union to category‑theoretic co‑product and the relation of set‑theoretic
intersection to syntactic concatenation deserve a closer look at this point.
The effect of a co‑product as a “disjointed union”, in other words, that
creates an object tantamount to a disjoint union of sets in the resulting
co‑product even if some of those sets intersect non‑trivially and even if
some of them are identical “in reality”, can be achieved in several ways.
The usual conception is that of making a separate copy, for each
part of the intended co‑product, of the set assigned to that part.
One imagines the set assigned to a particular part of the co‑product
as being distinguished by a particular color, in other words, by the
attachment of a distinct index, label, or tag, any sort of marker
inherited by and passed on to every element of the set in that part.
A concrete image of the construction can be achieved by imagining each
set and each element of each set is placed in an ordered pair with the
sign of its color, index, label, or tag. One describes that as the
“injection” of each set into the corresponding part of the co‑product.
For example, given the sets P and Q, overlapping or not, one defines the
indexed or marked sets P₍₁₎ and Q₍₂₎, amounting to the copy of P into the
first part of the co‑product and the copy of Q into the second part of the
co‑product, in the following manner.
• P₍₁₎ = (P, 1) = {(x, 1) : x ∈ P}
• Q₍₂₎ = (Q, 2) = {(x, 2) : x ∈ Q}
Using the co‑product operator (∐) for the construction, the “sum”, the
“co‑product”, or the “disjointed union” of P and Q in that order can be
represented as the ordinary union of P₍₁₎ and Q₍₂₎.
• P ∐ Q = P₍₁₎ ∪ Q₍₂₎
Resources —
Cactus Language • Pragmatics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Pragmatics
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/Vjon4W
cc: https://www.researchgate.net/post/Cactus_Language_Pragmatics
• https://inquiryintoinquiry.com/2025/08/09/cactus-language-pragmatics-11/
I am throwing together a wide variety of different operations into
the bins labeled “additive” and “multiplicative” but it's easy to
observe a natural organization and even some relations approaching
isomorphisms among and between the members of each class.
The relation between logical disjunction and the union of sets and the
relation between logical conjunction and the intersection of sets ought
to be clear enough for present purposes. But the relation of set‑theoretic
union to category‑theoretic co‑product and the relation of set‑theoretic
intersection to syntactic concatenation deserve a closer look at this point.
The effect of a co‑product as a “disjointed union”, in other words, that
creates an object tantamount to a disjoint union of sets in the resulting
co‑product even if some of those sets intersect non‑trivially and even if
some of them are identical “in reality”, can be achieved in several ways.
The usual conception is that of making a separate copy, for each
part of the intended co‑product, of the set assigned to that part.
One imagines the set assigned to a particular part of the co‑product
as being distinguished by a particular color, in other words, by the
attachment of a distinct index, label, or tag, any sort of marker
inherited by and passed on to every element of the set in that part.
A concrete image of the construction can be achieved by imagining each
set and each element of each set is placed in an ordered pair with the
sign of its color, index, label, or tag. One describes that as the
“injection” of each set into the corresponding part of the co‑product.
For example, given the sets P and Q, overlapping or not, one defines the
indexed or marked sets P₍₁₎ and Q₍₂₎, amounting to the copy of P into the
first part of the co‑product and the copy of Q into the second part of the
co‑product, in the following manner.
• P₍₁₎ = (P, 1) = {(x, 1) : x ∈ P}
• Q₍₂₎ = (Q, 2) = {(x, 2) : x ∈ Q}
Using the co‑product operator (∐) for the construction, the “sum”, the
“co‑product”, or the “disjointed union” of P and Q in that order can be
represented as the ordinary union of P₍₁₎ and Q₍₂₎.
• P ∐ Q = P₍₁₎ ∪ Q₍₂₎
Resources —
Cactus Language • Pragmatics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Pragmatics
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.researchgate.net/post/Cactus_Language_Pragmatics
Jon Awbrey
Aug 11, 2025, 9:45:37 AMAug 11
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Pragmatics 12
• https://inquiryintoinquiry.com/2025/08/10/cactus-language-pragmatics-12/
The concatenation ‡L‡₁ ∙ ‡L‡₂ of the formal languages ‡L‡₁ and ‡L‡₂ is
just a cartesian product ‡L‡₁ × ‡L‡₂ of the sets ‡L‡₁ and ‡L‡₂ but the
relation of cartesian products to set‑theoretic intersections and thus
to logical conjunctions is not immediately clear.
One way of seeing a type of relation is to focus on the information needed
to specify each construction and thus to reflect on the signs used to bear
the information. As a first approach to the topic of information I introduce
the following set of ideas, intended to be taken in a very provisional way.
A “stricture” is a specification of a certain set in a certain place,
relative to a number of other sets yet to be specified. It is assumed
one knows enough to tell if two strictures are equivalent as pieces of
information but any more determinate indications, for instance, names
for the places mentioned in the stricture or bounds on the number of
places involved, are regarded as extraneous impositions, outside the
proper concern of the definition, no matter how convenient they happen
to be for a particular discussion. As a schematic form of illustration,
a stricture can be pictured in the following shape.
• “… × X × Q × X × …”
A “strait” is the object specified by a stricture, in other words,
a certain set in a certain place of an otherwise yet to be specified
relation. Somewhat sketchily, the strait corresponding to the stricture
just given can be pictured in the following shape.
• … × X × Q × X × …
In that picture Q is a certain set and X is the universe of discourse relevant
to a given discussion. As a stricture does not contain a sufficient amount of
information to specify the number of sets it intends to set in place, or even to
pin down the absolute location of the set it does set in place, it appears to place
an unspecified number of unspecified sets in a vague and uncertain state of affairs.
Taken out of its interpretive context the residual information a stricture is able
to bear makes all of the following potentially equivalent as strictures.
• “Q” ,
• “X × Q × X” ,
• “X × X × Q × X × X” ,
• …
With respect to what those strictures specify, that leaves all of the following
equivalent as straits.
• Q =
• X × Q × X =
• X × X × Q × X × X =
• …
Resources —
Cactus Language • Pragmatics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Pragmatics
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/LYxNEm
cc: https://www.researchgate.net/post/Cactus_Language_Pragmatics
• https://inquiryintoinquiry.com/2025/08/10/cactus-language-pragmatics-12/
The concatenation ‡L‡₁ ∙ ‡L‡₂ of the formal languages ‡L‡₁ and ‡L‡₂ is
just a cartesian product ‡L‡₁ × ‡L‡₂ of the sets ‡L‡₁ and ‡L‡₂ but the
relation of cartesian products to set‑theoretic intersections and thus
to logical conjunctions is not immediately clear.
One way of seeing a type of relation is to focus on the information needed
to specify each construction and thus to reflect on the signs used to bear
the information. As a first approach to the topic of information I introduce
the following set of ideas, intended to be taken in a very provisional way.
A “stricture” is a specification of a certain set in a certain place,
relative to a number of other sets yet to be specified. It is assumed
one knows enough to tell if two strictures are equivalent as pieces of
information but any more determinate indications, for instance, names
for the places mentioned in the stricture or bounds on the number of
places involved, are regarded as extraneous impositions, outside the
proper concern of the definition, no matter how convenient they happen
to be for a particular discussion. As a schematic form of illustration,
a stricture can be pictured in the following shape.
• “… × X × Q × X × …”
A “strait” is the object specified by a stricture, in other words,
a certain set in a certain place of an otherwise yet to be specified
relation. Somewhat sketchily, the strait corresponding to the stricture
just given can be pictured in the following shape.
• … × X × Q × X × …
In that picture Q is a certain set and X is the universe of discourse relevant
to a given discussion. As a stricture does not contain a sufficient amount of
information to specify the number of sets it intends to set in place, or even to
pin down the absolute location of the set it does set in place, it appears to place
an unspecified number of unspecified sets in a vague and uncertain state of affairs.
Taken out of its interpretive context the residual information a stricture is able
to bear makes all of the following potentially equivalent as strictures.
• “Q” ,
• “X × Q × X” ,
• “X × X × Q × X × X” ,
• …
With respect to what those strictures specify, that leaves all of the following
equivalent as straits.
• Q =
• X × Q × X =
• X × X × Q × X × X =
• …
Resources —
Cactus Language • Pragmatics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Pragmatics
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.researchgate.net/post/Cactus_Language_Pragmatics
Jon Awbrey
Aug 12, 2025, 11:01:11 AMAug 12
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Pragmatics 13
• https://inquiryintoinquiry.com/2025/08/12/cactus-language-pragmatics-13/
Stricture, Strait, Constraint, Information, Complexity —
Within the framework of a particular discussion, it is customary
to set a bound on the number of places and to limit the variety of
sets regarded as being under active consideration and it is further
convenient to index the places of the indicated relations and their
encompassing cartesian products in some fixed way.
But the whole idea of a stricture is to specify a strait capable of extending
through and beyond fixed frames of discussion. In other words, a stricture
is conceived to constrain a strait at a certain point and then to leave it
literally embedded, if tacitly expressed, in a yet to be fully specified
relation, one involving an unspecified number of unspecified domains.
A quantity of information is a measure of constraint. In that respect,
a set of comparable strictures is ordered on account of the information
each one conveys and a system of comparable straits is ordered in accord
with the amount of information it takes to pin each one down.
Strictures which are more constraining and straits which are more constrained
are placed at higher levels of information than those which are less so and
entities involving more information are said to have greater “complexity”
than entities involving less information, which are said to have greater
“simplicity”.
Resources —
Cactus Language • Pragmatics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Pragmatics
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/LZ3KOg
cc: https://www.researchgate.net/post/Cactus_Language_Pragmatics
• https://inquiryintoinquiry.com/2025/08/12/cactus-language-pragmatics-13/
Stricture, Strait, Constraint, Information, Complexity —
Within the framework of a particular discussion, it is customary
to set a bound on the number of places and to limit the variety of
sets regarded as being under active consideration and it is further
convenient to index the places of the indicated relations and their
encompassing cartesian products in some fixed way.
But the whole idea of a stricture is to specify a strait capable of extending
through and beyond fixed frames of discussion. In other words, a stricture
is conceived to constrain a strait at a certain point and then to leave it
literally embedded, if tacitly expressed, in a yet to be fully specified
relation, one involving an unspecified number of unspecified domains.
A quantity of information is a measure of constraint. In that respect,
a set of comparable strictures is ordered on account of the information
each one conveys and a system of comparable straits is ordered in accord
with the amount of information it takes to pin each one down.
Strictures which are more constraining and straits which are more constrained
are placed at higher levels of information than those which are less so and
entities involving more information are said to have greater “complexity”
than entities involving less information, which are said to have greater
“simplicity”.
Resources —
Cactus Language • Pragmatics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Pragmatics
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.researchgate.net/post/Cactus_Language_Pragmatics
Jon Awbrey
Aug 19, 2025, 12:34:36 PMAug 19
to CYBCOM
Cactus Language • Pragmatics 14
• https://inquiryintoinquiry.com/2025/08/14/cactus-language-pragmatics-14/
Stricture, Strait, Constraint, Information, Complexity —
To give a concrete example of strictures and straits in action, let us institute a frame of discussion where the number of places in a relation is bounded at two and the variety of sets under active consideration is limited to the subsets P and Q of a universe X. Under those conditions one may use the following sorts of expression as schematic strictures.
Stricture Table 1
• https://inquiryintoinquiry.com/wp-content/uploads/2025/08/stricture-table-1.png
The above strictures and their corresponding straits are stratified according to the amounts of information they contain, or the levels of constraint they impose, as shown in the following table.
Stricture Table 2
• https://inquiryintoinquiry.com/wp-content/uploads/2025/08/stricture-table-2.png
In that framework, the complex strait P×Q can be defined in terms of the simpler straits P×X and X×Q as the following set‑theoretic intersection.
• P×Q = P×X ∩ X×Q
Resources —
Cactus Language • Pragmatics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Pragmatics
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/
• https://inquiryintoinquiry.com/2025/08/14/cactus-language-pragmatics-14/
Stricture, Strait, Constraint, Information, Complexity —
Stricture Table 1
• https://inquiryintoinquiry.com/wp-content/uploads/2025/08/stricture-table-1.png
The above strictures and their corresponding straits are stratified according to the amounts of information they contain, or the levels of constraint they impose, as shown in the following table.
Stricture Table 2
• https://inquiryintoinquiry.com/wp-content/uploads/2025/08/stricture-table-2.png
In that framework, the complex strait P×Q can be defined in terms of the simpler straits P×X and X×Q as the following set‑theoretic intersection.
• P×Q = P×X ∩ X×Q
Resources —
Cactus Language • Pragmatics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Pragmatics
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
Jon Awbrey
Aug 24, 2025, 11:00:27 AM (13 days ago) Aug 24
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Pragmatics 14
• https://inquiryintoinquiry.com/2025/08/14/cactus-language-pragmatics-14/
• https://inquiryintoinquiry.com/2025/08/14/cactus-language-pragmatics-14/
Stricture, Strait, Constraint, Information, Complexity —
To give a concrete example of strictures and straits in action,
let us institute a frame of discussion where the number of places
in a relation is bounded at two and the variety of sets under active
consideration is limited to the subsets P and Q of a universe X.
Under those conditions one may use the following sorts of expression
as schematic strictures.
Stricture Table 1
• https://inquiryintoinquiry.com/wp-content/uploads/2025/08/stricture-table-1.png
The above strictures and their corresponding straits are stratified
according to the amounts of information they contain, or the levels
of constraint they impose, as shown in the following table.
Stricture Table 2
• https://inquiryintoinquiry.com/wp-content/uploads/2025/08/stricture-table-2.png
In that framework, the complex strait P×Q can be defined in terms of the
simpler straits P×X and X×Q as the following set‑theoretic intersection.
• P×Q = P×X ∩ X×Q
let us institute a frame of discussion where the number of places
in a relation is bounded at two and the variety of sets under active
consideration is limited to the subsets P and Q of a universe X.
Under those conditions one may use the following sorts of expression
as schematic strictures.
Stricture Table 1
• https://inquiryintoinquiry.com/wp-content/uploads/2025/08/stricture-table-1.png
The above strictures and their corresponding straits are stratified
according to the amounts of information they contain, or the levels
of constraint they impose, as shown in the following table.
Stricture Table 2
• https://inquiryintoinquiry.com/wp-content/uploads/2025/08/stricture-table-2.png
In that framework, the complex strait P×Q can be defined in terms of the
simpler straits P×X and X×Q as the following set‑theoretic intersection.
• P×Q = P×X ∩ X×Q
Resources —
Cactus Language • Pragmatics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Pragmatics
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/L2jmYr
Cactus Language • Pragmatics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Pragmatics
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.researchgate.net/post/Cactus_Language_Pragmatics
Jon Awbrey
Aug 25, 2025, 12:30:31 PM (12 days ago) Aug 25
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Pragmatics 15
• https://inquiryintoinquiry.com/2025/08/18/cactus-language-pragmatics-15/
Stricture, Strait, Constraint, Information, Complexity —
From here it is easy to see how the concatenation of languages is
related to the intersection of sets and thus to the conjunction of
logical propositions. In the upshot a cartesian product P × Q is
described by a logical proposition P₍₁₎ ∧ Q₍₂₎ subject to the
following interpretation.
• P₍₁₎ says there is an element from the set P
in the 1st place of the product P × Q.
• Q₍₂₎ says there is an element from the set Q
in the 2nd place of the product P × Q.
The integration of those two pieces of information can be taken
to specify a yet to be fully determined relation.
In a corresponding fashion at the level of elements, the ordered
pair (p, q) is described by a conjunction of propositions, namely
p₍₁₎ ∧ q₍₂₎, subject to the following interpretation.
• p₍₁₎ says that p occupies the 1st place of
the product element under construction.
• q₍₂₎ says that q occupies the 2nd place of
the product element under construction.
Taking the cartesian product of P and Q or the concatenation of ‡L‡₁ and ‡L‡₂
in the above manner shifts the level of active construction from the tupling
of elements in P and Q or the concatenation of strings in ‡L‡₁ and ‡L‡₂ to
the concatenation of external signs describing those sets or languages.
Thus we pass to a conjunction of indexed propositions P₍₁₎ and Q₍₂₎ or a
conjunction of assertions (‡L‡₁)₍₁₎ and (‡L‡₂)₍₂₎ which mark the indicated
sets or languages for insertion in the indicated places of a product set or
product language, respectively. On closer examination, we can recognize the
subscripting by the indices “(1)” and “(2)” as a type of concatenation, in
this case accomplished through the posting of editorial remarks from an
external “mark‑up” language.
Resources —
Cactus Language • Pragmatics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Pragmatics
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/laZjRW
cc: https://www.researchgate.net/post/Cactus_Language_Pragmatics
• https://inquiryintoinquiry.com/2025/08/18/cactus-language-pragmatics-15/
Stricture, Strait, Constraint, Information, Complexity —
related to the intersection of sets and thus to the conjunction of
logical propositions. In the upshot a cartesian product P × Q is
described by a logical proposition P₍₁₎ ∧ Q₍₂₎ subject to the
following interpretation.
• P₍₁₎ says there is an element from the set P
in the 1st place of the product P × Q.
• Q₍₂₎ says there is an element from the set Q
in the 2nd place of the product P × Q.
The integration of those two pieces of information can be taken
to specify a yet to be fully determined relation.
In a corresponding fashion at the level of elements, the ordered
pair (p, q) is described by a conjunction of propositions, namely
p₍₁₎ ∧ q₍₂₎, subject to the following interpretation.
• p₍₁₎ says that p occupies the 1st place of
the product element under construction.
• q₍₂₎ says that q occupies the 2nd place of
the product element under construction.
Taking the cartesian product of P and Q or the concatenation of ‡L‡₁ and ‡L‡₂
in the above manner shifts the level of active construction from the tupling
of elements in P and Q or the concatenation of strings in ‡L‡₁ and ‡L‡₂ to
the concatenation of external signs describing those sets or languages.
Thus we pass to a conjunction of indexed propositions P₍₁₎ and Q₍₂₎ or a
conjunction of assertions (‡L‡₁)₍₁₎ and (‡L‡₂)₍₂₎ which mark the indicated
sets or languages for insertion in the indicated places of a product set or
product language, respectively. On closer examination, we can recognize the
subscripting by the indices “(1)” and “(2)” as a type of concatenation, in
this case accomplished through the posting of editorial remarks from an
external “mark‑up” language.
Resources —
Cactus Language • Pragmatics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Pragmatics
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.researchgate.net/post/Cactus_Language_Pragmatics
Jon Awbrey
Aug 26, 2025, 10:40:28 AM (11 days ago) Aug 26
to Conceptual Graphs, Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Pragmatics 16
• https://inquiryintoinquiry.com/2025/08/20/cactus-language-pragmatics-16/
Stricture, Strait, Constraint, Information, Complexity —
The ways in which strictures and straits at different levels
of complexity relate to one another can be given systematic
treatment by introducing the following pair of definitions.
Excerpt of a Stricture —
• The n‑th “excerpt” of a stricture “S₁×…×Sₖ”, regarded in a frame
of discussion where the number of places is bounded by k, is
a stricture of the form “X×…×Sₙ×…×X”.
• The n‑th excerpt can be written more briefly in context
as the stricture “(Sₙ)₍ₙ₎”, an assertion which places
the n‑th set in the n‑th place of the product.
Extract of a Strait —
• The n‑th “extract” of a strait S₁×…×Sₖ, regarded in a frame
of discussion where the number of places is bounded by k,
is a strait of the form X×…×Sₙ×…×X.
• The n‑th extract can be denoted more briefly in context
by the stricture “(Sₙ)₍ₙ₎”, an assertion which places
the n‑th set in the n‑th place of the product.
Using the above definitions, a stricture of the form “S₁×…×Sₖ”
can be expressed in terms of simpler strictures, namely, as
the following conjunction of its individual excerpts.
• “S₁ × … × Sₖ” = “(S₁)₍₁₎” ∧ … ∧ “(Sₖ)₍ₖ₎”
In a similar vein, a strait of the form S₁×…×Sₖ can be
expressed in terms of simpler straits, namely, as the
following intersection of its individual extracts.
• S₁ × … × Sₖ = (S₁)₍₁₎ ∩ … ∩ (Sₖ)₍ₖ₎
Resources —
Cactus Language • Pragmatics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Pragmatics
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/LmaNXm
cc: https://www.researchgate.net/post/Cactus_Language_Pragmatics
• https://inquiryintoinquiry.com/2025/08/20/cactus-language-pragmatics-16/
Stricture, Strait, Constraint, Information, Complexity —
of complexity relate to one another can be given systematic
treatment by introducing the following pair of definitions.
Excerpt of a Stricture —
• The n‑th “excerpt” of a stricture “S₁×…×Sₖ”, regarded in a frame
of discussion where the number of places is bounded by k, is
a stricture of the form “X×…×Sₙ×…×X”.
• The n‑th excerpt can be written more briefly in context
as the stricture “(Sₙ)₍ₙ₎”, an assertion which places
the n‑th set in the n‑th place of the product.
Extract of a Strait —
• The n‑th “extract” of a strait S₁×…×Sₖ, regarded in a frame
of discussion where the number of places is bounded by k,
is a strait of the form X×…×Sₙ×…×X.
• The n‑th extract can be denoted more briefly in context
by the stricture “(Sₙ)₍ₙ₎”, an assertion which places
the n‑th set in the n‑th place of the product.
Using the above definitions, a stricture of the form “S₁×…×Sₖ”
can be expressed in terms of simpler strictures, namely, as
the following conjunction of its individual excerpts.
• “S₁ × … × Sₖ” = “(S₁)₍₁₎” ∧ … ∧ “(Sₖ)₍ₖ₎”
In a similar vein, a strait of the form S₁×…×Sₖ can be
expressed in terms of simpler straits, namely, as the
following intersection of its individual extracts.
• S₁ × … × Sₖ = (S₁)₍₁₎ ∩ … ∩ (Sₖ)₍ₖ₎
Resources —
Cactus Language • Pragmatics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_.E2.80.A2_Pragmatics
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.researchgate.net/post/Cactus_Language_Pragmatics
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