Cactus Language • Semantics
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Jon Awbrey
Oct 7, 2025, 10:54:35 AMOct 7
to Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Semantics 1
• https://inquiryintoinquiry.com/2025/10/06/cactus-language-semantics-1/
❝Alas, and yet what are you, my written and painted thoughts! It is not
long ago that you were still so many‑coloured, young and malicious, so
full of thorns and hidden spices you made me sneeze and laugh — and now?
You have already taken off your novelty and some of you, I fear, are
on the point of becoming truths: they already look so immortal, so
pathetically righteous, so boring!❞
— Nietzsche • Beyond Good and Evil
The discussion to follow describes a particular semantics for painted
cactus languages, showing one way to link logical meanings with the
bare syntactic forms of linguistic expressions. Forging those links
between signs and intents gives the parametric family of formal
languages in question one of its principal “interpretations”.
We'll keep that interpretation in our sights for the time being but
it must be remembered it forms just one of many such interpretations
which may be conceivable and even viable in the long run. Indeed, the
distinction between the sign domain and the object domain can be observed
in the fact that many languages can be deployed to depict the same set of
objects while any language worth its salt is bound to give rise to a host
of salient interpretations.
Resources —
Cactus Language • Semantics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_%E2%80%A2_Semantics
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/V0r38A
cc: https://www.researchgate.net/post/Cactus_Language_Semantics
• https://inquiryintoinquiry.com/2025/10/06/cactus-language-semantics-1/
❝Alas, and yet what are you, my written and painted thoughts! It is not
long ago that you were still so many‑coloured, young and malicious, so
full of thorns and hidden spices you made me sneeze and laugh — and now?
You have already taken off your novelty and some of you, I fear, are
on the point of becoming truths: they already look so immortal, so
pathetically righteous, so boring!❞
— Nietzsche • Beyond Good and Evil
The discussion to follow describes a particular semantics for painted
cactus languages, showing one way to link logical meanings with the
bare syntactic forms of linguistic expressions. Forging those links
between signs and intents gives the parametric family of formal
languages in question one of its principal “interpretations”.
We'll keep that interpretation in our sights for the time being but
it must be remembered it forms just one of many such interpretations
which may be conceivable and even viable in the long run. Indeed, the
distinction between the sign domain and the object domain can be observed
in the fact that many languages can be deployed to depict the same set of
objects while any language worth its salt is bound to give rise to a host
of salient interpretations.
Resources —
Cactus Language • Semantics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_%E2%80%A2_Semantics
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/V0r38A
cc: https://www.researchgate.net/post/Cactus_Language_Semantics
Jon Awbrey
Oct 8, 2025, 12:00:35 PMOct 8
to Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Semantics 2
• https://inquiryintoinquiry.com/2025/10/08/cactus-language-semantics-2/
It is common in formal settings to speak of interpretation as if it created
a direct connection from the signs of a formal language to the objects of
the intended domain, in effect, as if it determined the denotative component
of a sign relation. But closer attention to what goes on reveals that the
process of interpretation is more indirect, that what it does is provide each
sign of a prospectively meaningful source language with a translation into
an already established target language, where “already established” means its
relationship to pragmatic objects is taken for granted at the moment in question.
With that in mind, it is clear interpretation is an affair of signs which
at best respects the objects of all the signs entering into it, and so it is
the connotative aspect of semiotics we find to embody the process. There is
nothing wrong with our saying we interpret expressions of a formal language
as signs referring to functions or propositions or other objects so long as
we understand the reference is generally achieved by way of more familiar
and perhaps less formal signs we already take to denote those objects.
Resources —
Cactus Language • Semantics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_%E2%80%A2_Semantics
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/V9KEBE
cc: https://www.researchgate.net/post/Cactus_Language_Semantics
• https://inquiryintoinquiry.com/2025/10/08/cactus-language-semantics-2/
It is common in formal settings to speak of interpretation as if it created
a direct connection from the signs of a formal language to the objects of
the intended domain, in effect, as if it determined the denotative component
of a sign relation. But closer attention to what goes on reveals that the
process of interpretation is more indirect, that what it does is provide each
sign of a prospectively meaningful source language with a translation into
an already established target language, where “already established” means its
relationship to pragmatic objects is taken for granted at the moment in question.
With that in mind, it is clear interpretation is an affair of signs which
at best respects the objects of all the signs entering into it, and so it is
the connotative aspect of semiotics we find to embody the process. There is
nothing wrong with our saying we interpret expressions of a formal language
as signs referring to functions or propositions or other objects so long as
we understand the reference is generally achieved by way of more familiar
and perhaps less formal signs we already take to denote those objects.
Resources —
Cactus Language • Semantics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_%E2%80%A2_Semantics
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_Semantics
Jon Awbrey
Oct 11, 2025, 11:45:44 AMOct 11
to Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Semantics 3
• https://inquiryintoinquiry.com/2025/10/10/cactus-language-semantics-3/
The task before us is to specify a “semantic function” for
the cactus language ‡L‡ = ‡C‡(‡P‡), in other words, to define
a mapping from the space of syntactic expressions to a space of
logical statements which “interprets” each expression of ‡C‡(‡P‡)
as an expression which says something, an expression which bears
a meaning, in short, an expression which denotes a proposition,
and is in the end a sign of an indicator function.
When the syntactic expressions of a formal language are given
a referent significance in logical terms, for example, as denoting
propositions or indicator functions, then each form of syntactic
combination takes on a corresponding form of logical significance.
A handy way of providing a logical interpretation for the expressions of
any given cactus language is to introduce a family of operators on indicator
functions called “propositional connectives”, to be distinguished from the
associated family of syntactic combinations called “sentential connectives”,
where the relationship between the two realms of connection is exactly that
between objects on the one hand and their signs on the other.
A propositional connective, as an entity of a well‑defined functional and
operational type, can be treated in every way as a logical or mathematical
object and thus as the type of object which can be denoted by the corresponding
form of syntactic entity, namely, the sentential connective appropriate to the
case at hand.
There are two basic types of connectives, called the “blank connectives”
and the “bound connectives”, respectively, with one connective of each type
for each natural number k = 0, 1, 2, 3, … .
Blank Connective —
• The “blank connective” of k places is signified by the concatenation
of the k sentences filling those places.
• For the initial case k = 0, the blank connective is an empty string or
a blank symbol, both of which have the same denotation among propositions.
• For the generic case k > 0, the blank connective takes the form s₁ ⋅ … ⋅ sₖ.
In the type of data called a “text”, the use of the center dot “⋅” is generally
supplanted by whatever number of spaces and line breaks serve to improve the
readability of the resulting text.
Bound Connective —
• The “bound connective” of k places is signified by the surcatenation
of the k sentences filling those places.
• For the initial case k = 0, the bound connective is an empty closure,
an expression taking one of the forms (), ( ), ( ), … with any number
of spaces between the parentheses, all of which have the same denotation
among propositions.
• For the generic case k > 0, the bound connective takes the form (s₁ , … , sₖ).
Resources —
Cactus Language • Semantics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_%E2%80%A2_Semantics
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/5kwO0G
cc: https://www.researchgate.net/post/Cactus_Language_Semantics
• https://inquiryintoinquiry.com/2025/10/10/cactus-language-semantics-3/
The task before us is to specify a “semantic function” for
the cactus language ‡L‡ = ‡C‡(‡P‡), in other words, to define
a mapping from the space of syntactic expressions to a space of
logical statements which “interprets” each expression of ‡C‡(‡P‡)
as an expression which says something, an expression which bears
a meaning, in short, an expression which denotes a proposition,
and is in the end a sign of an indicator function.
When the syntactic expressions of a formal language are given
a referent significance in logical terms, for example, as denoting
propositions or indicator functions, then each form of syntactic
combination takes on a corresponding form of logical significance.
A handy way of providing a logical interpretation for the expressions of
any given cactus language is to introduce a family of operators on indicator
functions called “propositional connectives”, to be distinguished from the
associated family of syntactic combinations called “sentential connectives”,
where the relationship between the two realms of connection is exactly that
between objects on the one hand and their signs on the other.
A propositional connective, as an entity of a well‑defined functional and
operational type, can be treated in every way as a logical or mathematical
object and thus as the type of object which can be denoted by the corresponding
form of syntactic entity, namely, the sentential connective appropriate to the
case at hand.
There are two basic types of connectives, called the “blank connectives”
and the “bound connectives”, respectively, with one connective of each type
for each natural number k = 0, 1, 2, 3, … .
Blank Connective —
• The “blank connective” of k places is signified by the concatenation
of the k sentences filling those places.
• For the initial case k = 0, the blank connective is an empty string or
a blank symbol, both of which have the same denotation among propositions.
• For the generic case k > 0, the blank connective takes the form s₁ ⋅ … ⋅ sₖ.
In the type of data called a “text”, the use of the center dot “⋅” is generally
supplanted by whatever number of spaces and line breaks serve to improve the
readability of the resulting text.
Bound Connective —
• The “bound connective” of k places is signified by the surcatenation
of the k sentences filling those places.
• For the initial case k = 0, the bound connective is an empty closure,
an expression taking one of the forms (), ( ), ( ), … with any number
of spaces between the parentheses, all of which have the same denotation
among propositions.
• For the generic case k > 0, the bound connective takes the form (s₁ , … , sₖ).
Resources —
Cactus Language • Semantics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_%E2%80%A2_Semantics
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_Semantics
Jon Awbrey
Oct 13, 2025, 1:00:50 PMOct 13
to Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Semantics 4
• https://inquiryintoinquiry.com/2025/10/12/cactus-language-semantics-4/
❝Words spoken are symbols or signs (“symbola”) of affections or
impressions (“pathemata”) of the soul (“psyche”); written words
are the signs of words spoken. As writing, so also is speech not
the same for all races of men. But the mental affections themselves,
of which these words are primarily signs (“semeia”), are the same for
the whole of mankind, as are also the objects (“pragmata”) of which
those affections are representations or likenesses, images, copies
(“homoiomata”).❞
— Aristotle • On Interpretation
At this point we have two distinct dialects, scripts, or modes of
presentation for the typical cactus language ‡C‡(‡P‡), each of
which needs to be interpreted, that is to say, equipped with
a semantic function defined on its domain.
PARCE(‡P‡) —
• There is the language of strings in PARCE(‡P‡), the
“painted and rooted cactus expressions” collectively
forming the language ‡L‡ = ‡C‡(‡P‡) ⊆ ‡A‡* = (‡M‡ ∪ ‡P‡)*.
PARC(‡P‡) —
• There is the language of graphs in PARC(‡P‡), the “painted
and rooted cacti” themselves, a family of graphs or species
of data structures formed by parsing the language of strings.
Those two modalities of formal language, like written and spoken
natural languages, are meant to have compatible interpretations,
which means it is generally sufficient to give the meanings of
just one or the other.
All that remains is to provide a “codomain” or “target space” for
the intended semantic function, that is, to supply a suitable range
of logical meanings for the memberships of those languages to map into.
One way to do that proceeds by making the following definitions.
Logical Conjunction —
• The “conjunction” Conjₖ qₖ of a set of propositions, (qₖ : k ∈ K}, is
a proposition which is true if and only if every one of the qₖ is true.
• Conjₖ qₖ is true ⇔ qₖ is true for every k ∈ K.
Logical Surjunction —
• The “surjunction” Surjₖ qₖ of a set of propositions, {qₖ : k ∈ K}, is
a proposition which is true if and only if exactly one of the qₖ is untrue.
• Surjₖ qₖ is true ⇔ qₖ is untrue for unique k ∈ K.
Resources —
Cactus Language • Semantics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_%E2%80%A2_Semantics
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/lydmwZ
cc: https://www.researchgate.net/post/Cactus_Language_Semantics
• https://inquiryintoinquiry.com/2025/10/12/cactus-language-semantics-4/
❝Words spoken are symbols or signs (“symbola”) of affections or
impressions (“pathemata”) of the soul (“psyche”); written words
are the signs of words spoken. As writing, so also is speech not
the same for all races of men. But the mental affections themselves,
of which these words are primarily signs (“semeia”), are the same for
the whole of mankind, as are also the objects (“pragmata”) of which
those affections are representations or likenesses, images, copies
(“homoiomata”).❞
— Aristotle • On Interpretation
At this point we have two distinct dialects, scripts, or modes of
presentation for the typical cactus language ‡C‡(‡P‡), each of
which needs to be interpreted, that is to say, equipped with
a semantic function defined on its domain.
PARCE(‡P‡) —
• There is the language of strings in PARCE(‡P‡), the
“painted and rooted cactus expressions” collectively
forming the language ‡L‡ = ‡C‡(‡P‡) ⊆ ‡A‡* = (‡M‡ ∪ ‡P‡)*.
PARC(‡P‡) —
• There is the language of graphs in PARC(‡P‡), the “painted
and rooted cacti” themselves, a family of graphs or species
of data structures formed by parsing the language of strings.
Those two modalities of formal language, like written and spoken
natural languages, are meant to have compatible interpretations,
which means it is generally sufficient to give the meanings of
just one or the other.
All that remains is to provide a “codomain” or “target space” for
the intended semantic function, that is, to supply a suitable range
of logical meanings for the memberships of those languages to map into.
One way to do that proceeds by making the following definitions.
Logical Conjunction —
• The “conjunction” Conjₖ qₖ of a set of propositions, (qₖ : k ∈ K}, is
a proposition which is true if and only if every one of the qₖ is true.
• Conjₖ qₖ is true ⇔ qₖ is true for every k ∈ K.
Logical Surjunction —
• The “surjunction” Surjₖ qₖ of a set of propositions, {qₖ : k ∈ K}, is
a proposition which is true if and only if exactly one of the qₖ is untrue.
• Surjₖ qₖ is true ⇔ qₖ is untrue for unique k ∈ K.
Resources —
Cactus Language • Semantics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_%E2%80%A2_Semantics
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_Semantics
Jon Awbrey
Oct 15, 2025, 4:00:56 PMOct 15
to Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Semantics 5
• https://inquiryintoinquiry.com/2025/10/14/cactus-language-semantics-5/
Last time we reached the threshold of a potential “codomain”
or “target space” for the kind of semantic function we need
at this point, one able to supply logical meanings for the
syntactic strings and graphs of a given cactus language.
In that pursuit we came to contemplate the following
definitions.
Logical Conjunction —
• The “conjunction” Conjₖ qₖ of a set of propositions {qₖ : k ∈ K}
logical conjunction and logical surjunction can be represented
by means of sentential connectives, incorporating the sentences
which represent the propositions into finite strings of symbols.
If K is finite, for instance, if K consists of the integers in the
interval k = 1 to n, and if each proposition qₖ is represented by
a sentence sₖ, then the following forms of expression are possible.
Logical Conjunction —
• The conjunction Conjₖ qₖ can be represented by a sentence which is
constructed by concatenating the sₖ in the following fashion.
• Conjₖ qₖ ⟿ s₁ s₂ … sₙ.
Logical Surjunction —
• The surjunction Surjₖ qₖ can be represented by a sentence which is
constructed by surcatenating the sₖ in the following fashion.
• Surjₖ qₖ ⟿ (s₁ , s₂ , … , sₙ).
If one opts for a mode of interpretation which moves more directly
from the parse graph of a sentence to the potential logical meaning
of both the PARC and the PARCE then the following specifications are
in order.
A cactus graph rooted at a particular node is taken to represent
what that node represents, namely, its logical denotation.
Denotation of a Node —
• The “logical denotation of a node” is the logical conjunction of
that node's arguments, which are defined as the logical denotations
of that node's attachments.
• The logical denotation of either a blank symbol or empty node
is the boolean value 1 = true.
• The logical denotation of the paint ‡p‡ₖ is the proposition pₖ,
a proposition regarded as “primitive”, at least, with respect to
the level of analysis represented in the current instance of ‡C‡(‡P‡).
Denotation of a Lobe —
• The “logical denotation of a lobe” is the logical surjunction of
that lobe's arguments, which are defined as the logical denotations
of that lobe's appendants.
• As a corollary, the logical denotation of the parse graph of (),
also known as a “needle”, is the boolean value 0 = false.
Resources —
Cactus Language • Semantics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_%E2%80%A2_Semantics
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/V0rDX1
cc: https://www.researchgate.net/post/Cactus_Language_Semantics
• https://inquiryintoinquiry.com/2025/10/14/cactus-language-semantics-5/
Last time we reached the threshold of a potential “codomain”
or “target space” for the kind of semantic function we need
at this point, one able to supply logical meanings for the
syntactic strings and graphs of a given cactus language.
In that pursuit we came to contemplate the following
definitions.
Logical Conjunction —
• The “conjunction” Conjₖ qₖ of a set of propositions {qₖ : k ∈ K}
is a proposition which is true if and only if every one of the qₖ
is true.
• Conjₖ qₖ is true ⇔ qₖ is true for every k ∈ K.
Logical Surjunction —
• The “surjunction” Surjₖ qₖ of a set of propositions {qₖ : k ∈ K}
is true.
• Conjₖ qₖ is true ⇔ qₖ is true for every k ∈ K.
Logical Surjunction —
is a proposition which is true if and only if exactly one of the qₖ
is untrue.
• Surjₖ qₖ is true ⇔ qₖ is untrue for unique k ∈ K.
If the set of propositions {qₖ : k ∈ K} is finite then the
is untrue.
• Surjₖ qₖ is true ⇔ qₖ is untrue for unique k ∈ K.
logical conjunction and logical surjunction can be represented
by means of sentential connectives, incorporating the sentences
which represent the propositions into finite strings of symbols.
If K is finite, for instance, if K consists of the integers in the
interval k = 1 to n, and if each proposition qₖ is represented by
a sentence sₖ, then the following forms of expression are possible.
Logical Conjunction —
• The conjunction Conjₖ qₖ can be represented by a sentence which is
constructed by concatenating the sₖ in the following fashion.
• Conjₖ qₖ ⟿ s₁ s₂ … sₙ.
Logical Surjunction —
• The surjunction Surjₖ qₖ can be represented by a sentence which is
constructed by surcatenating the sₖ in the following fashion.
• Surjₖ qₖ ⟿ (s₁ , s₂ , … , sₙ).
If one opts for a mode of interpretation which moves more directly
from the parse graph of a sentence to the potential logical meaning
of both the PARC and the PARCE then the following specifications are
in order.
A cactus graph rooted at a particular node is taken to represent
what that node represents, namely, its logical denotation.
Denotation of a Node —
• The “logical denotation of a node” is the logical conjunction of
that node's arguments, which are defined as the logical denotations
of that node's attachments.
• The logical denotation of either a blank symbol or empty node
is the boolean value 1 = true.
• The logical denotation of the paint ‡p‡ₖ is the proposition pₖ,
a proposition regarded as “primitive”, at least, with respect to
the level of analysis represented in the current instance of ‡C‡(‡P‡).
Denotation of a Lobe —
• The “logical denotation of a lobe” is the logical surjunction of
that lobe's arguments, which are defined as the logical denotations
of that lobe's appendants.
• As a corollary, the logical denotation of the parse graph of (),
also known as a “needle”, is the boolean value 0 = false.
Resources —
Cactus Language • Semantics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_%E2%80%A2_Semantics
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_Semantics
Jon Awbrey
Oct 28, 2025, 9:36:37 AM (8 days ago) Oct 28
to Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Semantics 6
• https://inquiryintoinquiry.com/2025/10/27/cactus-language-semantics-6/
If one takes the view that PARCEs and PARCs amount to a pair
of intertranslatable representations for the same domain of
objects then denotation brackets of the form ⇃…⇂ can be used
to indicate the logical denotation ⇃sₖ⇂ of a sentence sₖ or
the logical denotation ⇃Cₖ⇂ of a cactus Cₖ.
The relations connecting sentences, graphs, and propositions
are shown in the next two Tables.
Semantic Translation • Functional Form
• https://inquiryintoinquiry.com/wp-content/uploads/2025/10/cactus-language-semantic-translation-functional-form.png
Semantic Translation • Equational Form
• https://inquiryintoinquiry.com/wp-content/uploads/2025/10/cactus-language-semantic-translation-equational-form.png
Between the sentences and the propositions run the graph‑theoretic
data structures which arise in the process of parsing sentences
and catalyze their potential for expressing logical propositions
or indicator functions. The graph‑theoretic medium supplies an
intermediate form of representation between the linguistic sentences
and the indicator functions, not only rendering the possibilities
of connection between them more readily conceivable in fact but
facilitating the necessary translations on a practical basis.
In each Table the passage from the first to the middle column articulates
the mechanics of parsing cactus language sentences into graph‑theoretic
data structures while the passage from the middle to the last column
articulates the semantics of interpreting cactus graphs as logical
propositions or indicator functions.
Aside from their common topic, the two Tables present slightly different ways
of drawing the maps which go to make up the full semantic transformation.
Semantic Translation • Functional Form —
• The first Table shows the functional associations connecting each
domain with the next, taking the triple of a sentence sₖ, a cactus Cₖ,
and a proposition qₖ as basic data, and fixing the rest by recursion on
those ingredients.
Semantic Translation • Equational Form —
• The second Table records the transitions in the form of equations,
treating sentences and graphs as alternative types of signs and
generalizing the denotation bracket to indicate the proposition
denoted by either.
It should be clear at this point that either scheme of translation
puts the triples of sentences, graphs, and propositions roughly in
the roles of signs, interpretants, and objects, respectively, of
a triadic sign relation. Indeed, the “roughly” can be rendered
“exactly” as soon as the domains of a suitable sign relation
are specified precisely.
Resources —
Cactus Language • Semantics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_%E2%80%A2_Semantics
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/VrGMnb
cc: https://www.researchgate.net/post/Cactus_Language_Semantics
• https://inquiryintoinquiry.com/2025/10/27/cactus-language-semantics-6/
If one takes the view that PARCEs and PARCs amount to a pair
of intertranslatable representations for the same domain of
objects then denotation brackets of the form ⇃…⇂ can be used
to indicate the logical denotation ⇃sₖ⇂ of a sentence sₖ or
the logical denotation ⇃Cₖ⇂ of a cactus Cₖ.
The relations connecting sentences, graphs, and propositions
are shown in the next two Tables.
Semantic Translation • Functional Form
• https://inquiryintoinquiry.com/wp-content/uploads/2025/10/cactus-language-semantic-translation-functional-form.png
Semantic Translation • Equational Form
• https://inquiryintoinquiry.com/wp-content/uploads/2025/10/cactus-language-semantic-translation-equational-form.png
Between the sentences and the propositions run the graph‑theoretic
data structures which arise in the process of parsing sentences
and catalyze their potential for expressing logical propositions
or indicator functions. The graph‑theoretic medium supplies an
intermediate form of representation between the linguistic sentences
and the indicator functions, not only rendering the possibilities
of connection between them more readily conceivable in fact but
facilitating the necessary translations on a practical basis.
In each Table the passage from the first to the middle column articulates
the mechanics of parsing cactus language sentences into graph‑theoretic
data structures while the passage from the middle to the last column
articulates the semantics of interpreting cactus graphs as logical
propositions or indicator functions.
Aside from their common topic, the two Tables present slightly different ways
of drawing the maps which go to make up the full semantic transformation.
Semantic Translation • Functional Form —
• The first Table shows the functional associations connecting each
domain with the next, taking the triple of a sentence sₖ, a cactus Cₖ,
and a proposition qₖ as basic data, and fixing the rest by recursion on
those ingredients.
Semantic Translation • Equational Form —
• The second Table records the transitions in the form of equations,
treating sentences and graphs as alternative types of signs and
generalizing the denotation bracket to indicate the proposition
denoted by either.
It should be clear at this point that either scheme of translation
puts the triples of sentences, graphs, and propositions roughly in
the roles of signs, interpretants, and objects, respectively, of
a triadic sign relation. Indeed, the “roughly” can be rendered
“exactly” as soon as the domains of a suitable sign relation
are specified precisely.
Resources —
Cactus Language • Semantics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_%E2%80%A2_Semantics
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_Semantics
Jon Awbrey
Oct 31, 2025, 3:00:53 PM (5 days ago) Oct 31
to Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Semantics 7
• https://inquiryintoinquiry.com/2025/10/30/cactus-language-semantics-7/
A good way to illustrate the action of the conjunction and surjunction
operators is to show how they can be used to construct the boolean
functions on any finite number of variables. Though it's not much
to look at let's start with the case of zero variables, boolean
constants by any other word, partly for completeness and partly
to supply an anchor for the cases in its train.
A boolean function F⁽⁰⁾ on zero variables is just an element of the
boolean domain B = {0, 1}. The following Table shows several ways
of referring to those elements, for the sake of consistency using
the same format we'll use in subsequent Tables, however degenerate
it appears in this case.
Boolean Functions on Zero Variables
• https://inquiryintoinquiry.com/wp-content/uploads/2025/10/boolean-functions-on-zero-variables-e280a2-truth-table.png
• Column 1 lists each boolean element or boolean function under its
ordinary constant name or under a succinct nickname, respectively.
• Column 2 lists each boolean function by means of a function name Fₖ⁽ⁿ⁾
of the following form. The superscript (n) gives the dimension of
the functional domain, in effect, the number of variables, and the
subscript k is a binary string formed from the functional values,
using the obvious coding of boolean values into binary values.
• Column 3 lists the values each function takes for each combination
of its domain values.
• Column 4 lists the ordinary cactus expressions for each boolean function.
Here, as usual, the expression “(())” renders the blank expression for
logical truth more visible in context.
The next Table shows the four boolean functions on one variable, F⁽¹⁾ : B → B.
Boolean Functions on One Variable
• https://inquiryintoinquiry.com/wp-content/uploads/2025/10/boolean-functions-on-one-variable-e280a2-truth-table.png
• Column 1 lists the contents of Column 2 in a more concise form,
converting the lists of boolean values in the subscript strings
to their decimal equivalents. Naturally, the boolean constants
reprise themselves in this new setting as constant functions on
one variable. The constant functions are thus expressible in
the following equivalent ways.
• F₀⁽¹⁾ = F₀₀⁽¹⁾ = 0 : B → B.
• F₃⁽¹⁾ = F₁₁⁽¹⁾ = 1 : B → B.
• The other two functions in the Table are easily recognized as the
one‑place logical connectives or the monadic operators on B. Thus
the function F₁⁽¹⁾ = F₀₁⁽¹⁾ is recognizable as the negation operation
and the function F₂⁽¹⁾ = F₁₀⁽¹⁾ is obviously the identity operation.
Resources —
Cactus Language • Semantics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_%E2%80%A2_Semantics
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/lQkXPn
cc: https://www.researchgate.net/post/Cactus_Language_Semantics
• https://inquiryintoinquiry.com/2025/10/30/cactus-language-semantics-7/
A good way to illustrate the action of the conjunction and surjunction
operators is to show how they can be used to construct the boolean
functions on any finite number of variables. Though it's not much
to look at let's start with the case of zero variables, boolean
constants by any other word, partly for completeness and partly
to supply an anchor for the cases in its train.
A boolean function F⁽⁰⁾ on zero variables is just an element of the
boolean domain B = {0, 1}. The following Table shows several ways
of referring to those elements, for the sake of consistency using
the same format we'll use in subsequent Tables, however degenerate
it appears in this case.
Boolean Functions on Zero Variables
• https://inquiryintoinquiry.com/wp-content/uploads/2025/10/boolean-functions-on-zero-variables-e280a2-truth-table.png
• Column 1 lists each boolean element or boolean function under its
ordinary constant name or under a succinct nickname, respectively.
• Column 2 lists each boolean function by means of a function name Fₖ⁽ⁿ⁾
of the following form. The superscript (n) gives the dimension of
the functional domain, in effect, the number of variables, and the
subscript k is a binary string formed from the functional values,
using the obvious coding of boolean values into binary values.
• Column 3 lists the values each function takes for each combination
of its domain values.
• Column 4 lists the ordinary cactus expressions for each boolean function.
Here, as usual, the expression “(())” renders the blank expression for
logical truth more visible in context.
The next Table shows the four boolean functions on one variable, F⁽¹⁾ : B → B.
Boolean Functions on One Variable
• https://inquiryintoinquiry.com/wp-content/uploads/2025/10/boolean-functions-on-one-variable-e280a2-truth-table.png
• Column 1 lists the contents of Column 2 in a more concise form,
converting the lists of boolean values in the subscript strings
to their decimal equivalents. Naturally, the boolean constants
reprise themselves in this new setting as constant functions on
one variable. The constant functions are thus expressible in
the following equivalent ways.
• F₀⁽¹⁾ = F₀₀⁽¹⁾ = 0 : B → B.
• F₃⁽¹⁾ = F₁₁⁽¹⁾ = 1 : B → B.
• The other two functions in the Table are easily recognized as the
one‑place logical connectives or the monadic operators on B. Thus
the function F₁⁽¹⁾ = F₀₁⁽¹⁾ is recognizable as the negation operation
and the function F₂⁽¹⁾ = F₁₀⁽¹⁾ is obviously the identity operation.
Resources —
Cactus Language • Semantics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_%E2%80%A2_Semantics
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_Semantics
Jon Awbrey
Nov 2, 2025, 10:21:05 AM (3 days ago) Nov 2
to Cybernetic Communications, Structural Modeling, SysSciWG
Cactus Language • Semantics 8
• https://inquiryintoinquiry.com/2025/11/01/cactus-language-semantics-8/
The 16 boolean functions on two variables, F⁽²⁾ : B² → B,
are shown in the following Table.
Boolean Functions on Two Variables
• https://inquiryintoinquiry.com/wp-content/uploads/2025/10/boolean-functions-on-two-variables-e280a2-truth-table.png
As before, all boolean functions on proper subsets of
the current variables are subsumed in the Table at hand.
In particular, we have the following inclusions.
• The constant function 0 : B² → B appears under the name F₀⁽²⁾.
• The constant function 1 : B² → B appears under the name F₁₅⁽²⁾.
• The function expressing the assertion of the first variable is F₁₂⁽²⁾.
• The function expressing the negation of the first variable is F₃⁽²⁾.
• The function expressing the assertion of the second variable is F₁₀⁽²⁾.
• The function expressing the negation of the second variable is F₅⁽²⁾.
Next come the functions on two variables whose output values change
depending on changes in both input variables. Notable among them
are the following examples.
• The logical conjunction is given by the function F₈⁽²⁾(x, y) = x ⋅ y.
• The logical disjunction is given by the function F₁₄⁽²⁾(x, y) = (( x )( y )).
Functions expressing the conditionals, implications,
or if‑then statements appear as follows.
• [x ⇒ y] = F₁₁⁽²⁾(x, y) = ( x ( y )) = [not x without y].
• [x ⇐ y] = F₁₃⁽²⁾(x, y) = (( x ) y ) = [not y without x].
The function expressing the biconditional, equivalence,
or if‑and‑only‑if statement appears in the following form.
• [x ⇔ y] = [x = y] = F₉⁽²⁾(x, y) = (( x , y)).
Finally, the boolean function expressing the exclusive disjunction,
inequivalence, or not equals statement, algebraically associated
with the binary sum operation, and geometrically associated with
the symmetric difference of sets, appears as follows.
• [x ≠ y] = [x + y] = F₆⁽²⁾(x, y) = ( x , y ).
Resources —
Cactus Language • Semantics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_%E2%80%A2_Semantics
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/Vj3BQ6
cc: https://www.researchgate.net/post/Cactus_Language_Semantics
• https://inquiryintoinquiry.com/2025/11/01/cactus-language-semantics-8/
The 16 boolean functions on two variables, F⁽²⁾ : B² → B,
are shown in the following Table.
Boolean Functions on Two Variables
• https://inquiryintoinquiry.com/wp-content/uploads/2025/10/boolean-functions-on-two-variables-e280a2-truth-table.png
As before, all boolean functions on proper subsets of
the current variables are subsumed in the Table at hand.
In particular, we have the following inclusions.
• The constant function 0 : B² → B appears under the name F₀⁽²⁾.
• The constant function 1 : B² → B appears under the name F₁₅⁽²⁾.
• The function expressing the assertion of the first variable is F₁₂⁽²⁾.
• The function expressing the negation of the first variable is F₃⁽²⁾.
• The function expressing the assertion of the second variable is F₁₀⁽²⁾.
• The function expressing the negation of the second variable is F₅⁽²⁾.
Next come the functions on two variables whose output values change
depending on changes in both input variables. Notable among them
are the following examples.
• The logical conjunction is given by the function F₈⁽²⁾(x, y) = x ⋅ y.
• The logical disjunction is given by the function F₁₄⁽²⁾(x, y) = (( x )( y )).
Functions expressing the conditionals, implications,
or if‑then statements appear as follows.
• [x ⇒ y] = F₁₁⁽²⁾(x, y) = ( x ( y )) = [not x without y].
• [x ⇐ y] = F₁₃⁽²⁾(x, y) = (( x ) y ) = [not y without x].
The function expressing the biconditional, equivalence,
or if‑and‑only‑if statement appears in the following form.
• [x ⇔ y] = [x = y] = F₉⁽²⁾(x, y) = (( x , y)).
Finally, the boolean function expressing the exclusive disjunction,
inequivalence, or not equals statement, algebraically associated
with the binary sum operation, and geometrically associated with
the symmetric difference of sets, appears as follows.
• [x ≠ y] = [x + y] = F₆⁽²⁾(x, y) = ( x , y ).
Resources —
Cactus Language • Semantics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Cactus_Language_%E2%80%A2_Semantics
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_Semantics
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