31 Mar
2025
31 Mar
'25
3:15 p.m.
Cactus Language • Preliminaries 1
• https://inquiryintoinquiry.com/2025/03/30/cactus-language-preliminaries-1/
❝Picture two different configurations of such an irregular shape,
superimposed on each other in space, like a double exposure photograph.
Of the two images, the only part which coincides is the body. The two
different sets of quills stick out into very different regions of space.
The objective reality we see from within the first position, seemingly
so full and spherical, actually agrees with the shifted reality only in
the body of common knowledge. In every direction in which we look at all
deeply, the realm of discovered scientific truth could be quite different.
Yet in each of those two different situations, we would have thought the
world complete, firmly known, and rather round in its penetration of the
space of possible knowledge.❞
— Herbert J. Bernstein • “Idols of Modern Science”
The task before us is to describe the syntax of a family of formal languages
intended for use as a sentential calculus, and thus interpreted for the purpose
of reasoning about propositions and their logical relations.
To carry out our discussion we need a way of referring to signs as if they
were objects like any others, in other words, as the sorts of things which
can be named, indicated, described, discussed, and renamed if necessary,
which can be placed, arranged, and rearranged within a suitable medium of
expression — or else manipulated in the mind — which can be articulated and
decomposed into their elementary signs, and which can be strung together in
sequences to form complex signs.
Signs having signs as their objects are known as “higher order signs”,
a topic which demands an adequate level of formalization, but in due time.
The present discussion needs a quicker way to get into the subject, even if
it settles for informal means which cannot be rendered absolutely precise.
Resources —
Cactus Language • Preliminaries
•
https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.…
Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs…
Survey of Theme One Program
• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/
Regards,
Jon
7 Apr
7 Apr
1:24 p.m.
Cactus Language • Preliminaries 2
• https://inquiryintoinquiry.com/2025/04/05/cactus-language-preliminaries-2/
As a temporary notation, let the relationship between
a particular sign s and a particular object o, namely,
the fact that s denotes o or the fact that o is denoted
by s, be symbolized in one of the following two ways.
1. s → o
2. o ← s
Now consider the following paradigm.
Cactus Language Display 1
•
https://inquiryintoinquiry.files.wordpress.com/2025/04/cactus-language-disp…
1. If “A” → Ann
that is, “A” denotes Ann
then A = Ann and Ann = A
Thus “Ann” → A
that is, “Ann” denotes A
2. If Bob ← “B”
that is, Bob is denoted by “B”
then Bob = B and B = Bob
Thus B ← “Bob”
that is, B is denoted by “Bob”
In the same vein, if we let the sign “blank” denote the sign “ ”
then the string of characters inside the first pair of quotation
marks will serve as another name for the string of characters
inside the second pair of quotation marks.
In other words, “blank” is a higher order sign whose object
is the sign “ ” and the string of five characters inside the
first pair of quotation marks is a sign at a higher level of
signification than the string of one character inside the
second pair of quotation marks. The relation in question
can be abbreviated in either one of the following two ways.
“blank” → “ ”
“ ” ← “blank”
Using the raised dot “∙” as a sign to mark the articulation of
a quoted string into a sequence of possibly shorter quoted strings,
and thus to mark the concatenation of a sequence of quoted strings
into a possibly larger quoted string, one can write the following
equation.
“ ” ← “blank” = “b” ∙ “l” ∙ “a” ∙ “n” ∙ “k”
The above tactic lets us refer to the blank as a type of character and
refer to any blank we choose as a token of that type, denoting either in
a markèd way, but without the use of quotation marks. As a blank is just
what the name “blank” names, it is possible to represent the denoting of
the sign “ ” by the name “blank” in the form of an identity between the
named objects, as follows.
“ ” = blank
Given the above identities it is possible to extend the use of
the “∙” sign to mark the articulation of either named or quoted
strings into both named and quoted strings. For example, we have
the following equations.
“ ” = “ ” ∙ “ ” = blank ∙ blank
“ blank” = “ ” ∙ “blank” = blank ∙ “blank”
“blank ” = “blank” ∙ “ ” = “blank” ∙ blank
Resources —
Cactus Language • Preliminaries
•
https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.…
Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs…
Survey of Theme One Program
• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/
Regards,
Jon
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Jon Awbrey