Cactus Language

[Jon Awbrey]

Cactus Language • Overview 1
• https://inquiryintoinquiry.com/2025/03/01/cactus-language-overview-1/

❝Thus, what looks to us like a sphere of scientific knowledge more accurately
should be represented as the inside of a highly irregular and spiky object,
like a pincushion or porcupine, with very sharp extensions in certain
directions, and virtually no knowledge in immediately adjacent areas.
If our intellectual gaze could shift slightly, it would alter each
quill’s direction, and suddenly our entire reality would change.❞

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

The following report describes a calculus for representing propositions
as sentences, that is, as syntactically defined sequences of signs, and
for working with those sentences in light of their semantically defined
contents as logical propositions. In their computational representation
the expressions of the calculus parse into a class of graph‑theoretic
data structures whose underlying graphs are called “painted cacti”.

Painted cacti are a specialization of what graph‑theorists call “cacti”,
which are in turn a generalization of what they call “trees”. The data
structures corresponding to painted cacti have especially nice properties,
not only useful in computational terms but interesting from a theoretical
standpoint. The remainder of the present Overview is devoted to motivating
the development of the indicated family of formal languages, going under the
generic name of Cactus Language.

Resource —

For readers interested and intrepid enough to read ahead,
here's an outline of my work in progress on the OEIS Wiki,
which I'll be revising and serializing to my Inquiry blog.

Part 1
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1

Cactus Language • Syntax
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Syntax

Part 2
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_2

Generalities About Formal Grammars
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_2#Generalities

Part 3
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3

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

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

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

Stretching Exercises
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Stretching_Exercises

References
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_References

Document History
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Document_History

Regards,

Jon

cc: https://www.academia.edu/community/VXZjbz
cc: https://mathstodon.xyz/@Inquiry/114100591406422570
cc: https://www.researchgate.net/post/Cactus_Language_for_Propositional_Calculus

[Jon Awbrey]

Cactus Language • Overview 2
• https://inquiryintoinquiry.com/2025/03/06/cactus-language-overview-2/

In order to facilitate the use of propositions as indicator functions
it helps to acquire a flexible notation for referring to propositions
in that light, for interpreting sentences in a corresponding role, and
for negotiating the requirements of mutual sense between the two domains.
If none of the formalisms readily available or in common use meet all the
design requirements coming to mind then it is necessary to contemplate the
design of a new language especially tailored to the purpose.

In the present application, there is a pressing need to devise a general
calculus for composing propositions, computing their values on particular
arguments, and inverting their indications to arrive at the sets of things
in the universe which are indicated by them.

For computational purposes it is convenient to have a middle ground or
an intermediate language for negotiating between the “koine” of sentences
regarded as strings of literal characters and the realm of propositions
regarded as objects of logical value, even if that makes it necessary
to introduce an artificial medium of exchange between the two domains.

If the necessary computations are to be carried out in an organized fashion,
and ultimately or partially by familiar classes of machines, then the strings
expressing logical propositions are likely to find themselves parsed into
tree‑like data structures at some stage of the game. As far as their abstract
structures as graphs are concerned, there are several species of graph‑theoretic
data structures fitting the task in a reasonably effective and efficient way.

Resources —

Cactus Language • Overview
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Overview

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/

Survey of Theme One Program
• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/

Regards,

Jon

[Jon Awbrey]

Cactus Language • Overview 3
• https://inquiryintoinquiry.com/2025/03/07/cactus-language-overview-3/

In the development of Cactus Language to date
the following two species of graphs have been
instrumental.

• Painted And Rooted Cacti (PARCAI).
• Painted And Rooted Conifers (PARCOI).

It suffices to begin with the first class of data structures,
developing their properties and uses in full, leaving discussion
of the latter class to a part of the project where their distinctive
features are key to developments at that stage. Partly because the
two species are so closely related and partly for the sake of brevity,
we'll always use the genus name “PARC” to denote the corresponding cacti.

To provide a computational middle ground between sentences seen as
syntactic strings and propositions seen as indicator functions the
language designer must not only supply a medium for the expression
of propositions but also link the assertion of sentences to a means
for inverting the indicator functions, that is, for computing the
“fibers” or “inverse images” of the propositions.

Given a body of conceivable propositions we need a way to follow
the threads of their indications from their object domain to their
values for the mind and a way to follow those same threads back again.
Moreover, we need to implement both ways of proceeding in computational
form. Thus we need programs for tracing the clues sentences provide from
the universe of their objects to the signs of their values and, in turn,
from signs to objects. Ultimately, we need to render propositions so
functional as indicators of sets and so essential for examining the
equality of sets as to give a rule for the practical conceivability
of sets. Tackling that task requires us to introduce a number of
new definitions and a collection of additional notational devices,
to which we now turn.

Resources —

Cactus Language • Overview
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Overview

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/

Survey of Theme One Program
• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/

Regards,

Jon

cc: https://www.academia.edu/community/L2XRbj

[Jon Awbrey]

Cactus Language • Overview 4
• https://inquiryintoinquiry.com/2025/03/12/cactus-language-overview-4/

Depending on whether a formal language is called by the type of sign
it enlists or the type of object its signs denote, a cactus language
may be called a “sentential calculus” or a “propositional calculus”,
respectively.

When the syntactic definition of a language is well enough understood
the language can begin to acquire a semantic function. In natural
circumstances the syntax and the semantics are likely to be engaged
in a process of co‑evolution, whether in ontogeny or in phylogeny,
which is to say the two developments tend to form parallel sides of
a single bootstrap. But that is not always the easiest way, at least
not at first, to formally comprehend the nature of their action or
the power of their interaction.

According to the customary modes of formal reconstruction, a language
of the type we are considering is first presented in terms of its syntax,
in other words, as a formal language of strings called “sentences”, and
thus amounting to a particular subset of the possible strings which can be
formed on a finite alphabet of signs. A syntactic definition of a specific
cactus language which proceeds along purely formal lines is carried out in
Cactus Language • Syntax. After that, the development of the language's
more concrete aspects can be seen as a matter of defining the following
two functions.

• The first is a function which takes each sentence of the language
into a computational data structure, namely, a generalized tree‑like
parse graph called a “painted cactus”.

• The second is a function which takes each sentence of the language
or its interpolated parse graph into a logical proposition, ending
with an indicator function as the object denoted by the sentence.

The discussion of syntax brings up a number of associated issues which
need to be clarified before going on. They may be thought of as questions
of “style”, in other words, the manner of description, grammar, or theory one
finds available or chooses as preferable for a given language. Those issues
are discussed in Cactus Language • Stylistics.

There is an aspect of syntax so schematic in its basic character that it can
be conveyed by computational data structures, so algorithmic in its uses that
it can be automated by routine mechanisms, and so fixed in its nature that its
practical exploitation can be served by the usual devices of computation.
Because it involves the transformation of signs it can be recognized as an
aspect of semiotics. Since it can be carried out in abstraction from meaning
it is not up to the level of semantics, much less a complete pragmatics, though
it does incline to the pragmatic aspects of computation which are auxiliary to
and incidental to the human use of language. That aspect of formal language
use may be described as the “algorithmics” or “mechanics” of language processing.
A mechanical conversion of cactus languages into their associated data structures
is discussed in Cactus Language • Mechanics.

In the usual way of proceeding on formal grounds, meaning is added by
giving each grammatical sentence, or each syntactically distinguished
string, an interpretation as a logically meaningful sentence, in effect,
equipping or providing each abstractly well‑formed sentence with a logical
proposition for it to denote. A semantic interpretation of cactus language
is carried out in Cactus Language • Semantics.

Resources —

Cactus Language • Overview
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Overview

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/

Survey of Theme One Program
• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/

Regards,

Jon

cc: https://www.academia.edu/community/lJvgBA
cc: https://www.researchgate.net/post/Cactus_Language_for_Propositional_Calculus