Reflective Interpretive Frameworks • Incident 1
• https://inquiryintoinquiry.com/2026/03/26/reflective-interpretive-frameworks-incident-1/

Re: William Waites • The Agent That Doesn't Know Itself
• https://johncarlosbaez.wordpress.com/2026/03/20/the-agent-that-doesnt-know-itself/

WW: ❝Why Has Nobody Done This?❞

People who study C.S. Peirce would say reflective reasoning requires triadic relations at core and there is work being done on that. One of the challenges is clarifying the role of triadic relations in category theory and raising them into higher relief as fundamental operations.

Note. I was looking for a word to describe a random encounter with something that jogs one's memory of a recurring theme — “incident” plays into the “reflection” theme and looked worth trying for now.

Resources —

Inquiry Driven Systems • Inquiry Into Inquiry
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Overview

Reflective Interpretive Frameworks
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_10#Reflective_Interpretive_Frameworks

The Phenomenology of Reflection
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_11#The_Phenomenology_of_Reflection

Higher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_Relations

Notes On Categories
• https://inquiryintoinquiry.com/2013/02/22/notes-on-categories-1/
• https://inquiryintoinquiry.com/2021/07/31/notes-on-categories-2/

#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations

Sign Relations • Semiotic Equivalence Relations 2.3
• https://inquiryintoinquiry.com/2025/12/31/sign-relations-semiotic-equivalence-relations-2-c/

The semiotic equivalence relation for interpreter A yields the following semiotic equations.

• [“A”]_A = [“i”]_A

• [“B”]_A = [“u”]_A

Display 4
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-ser-display-4.png

or

• “A” =_A “i”

• “B” =_A “u”

Display 5
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-ser-display-5.png

In this way the SER for A induces the following semiotic partition.

• {{“A”, “i”}, {“B”, “u”}}.

Display 6
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-ser-display-6.png

The semiotic equivalence relation for interpreter B yields the following semiotic equations.

• [“A”]_B = [“u”]_B

• [“B”]_B = [“i”]_B

Display 7
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-ser-display-7.png

or

• “A” =_B “u”

• “B” =_B “i”

Display 8
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-ser-display-8.png

In this way the SER for B induces the following semiotic partition.

• {{“A”, “u”}, {“B”, “i”}}.

Display 9
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-ser-display-9.png

Taken all together we have the following picture.

Tables 7a and 7b. Semiotic Partitions for Interpreters A and B
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/semiotic-partitions-for-interpreters-a-b-2.0.png

Resources —

Sign Relation
• https://oeis.org/wiki/Sign_relation
• https://mywikibiz.com/Sign_relation
• https://en.wikiversity.org/wiki/Sign_relation

Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2025/05/06/survey-of-semiotics-semiosis-sign-relations-6/

cc: https://www.academia.edu/community/VBAXbj
cc: https://www.researchgate.net/post/Sign_Relations_First_Elements
cc: https://stream.syscoi.com/2026/01/01/sign-relations-semiotic-equivalence-relations-2/

#Peirce #Inquiry #Logic #Mathematics #RelationTheory
#Semiosis #Semiotics #SignRelations #TriadicRelations

Sign Relations • Semiotic Equivalence Relations 2.2
• https://inquiryintoinquiry.com/2025/12/31/sign-relations-semiotic-equivalence-relations-2-c/

In the application to sign relations it is useful to extend the square bracket notation in the following ways. If L is a sign relation whose connotative component L_SI is an equivalence relation on S = I, let [s]_L be the equivalence class of s under L_SI. In short, [s]_L = [s]_{L_{SI}}.

A statement that the signs x and y belong to the same equivalence class under a semiotic equivalence relation L_SI is called a “semiotic equation” (SEQ) and may be written in either of the following forms.

• [x]_L = [y]_L

• x =_L y

Display 3
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-ser-display-3.png

In many situations there is one further adaptation of the square bracket notation for semiotic equivalence classes that can be useful. Namely, when there is known to exist a particular triple (o, s, i) in a sign relation L, it is permissible to let [o]_L be defined as [s]_L. This modifications is designed to make the notation for semiotic equivalence classes harmonize as well as possible with the frequent use of similar devices for the denotations of signs and expressions.

Applying the array of equivalence notations to the sign relations for A and B will serve to illustrate their use and utility.

Tables 6a and 6b. Connotative Components Con(L_A) and Con(L_B)
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/connotative-components-con-la-con-lb-3.0.png

Resources —

Sign Relation
• https://oeis.org/wiki/Sign_relation
• https://mywikibiz.com/Sign_relation
• https://en.wikiversity.org/wiki/Sign_relation

Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2025/05/06/survey-of-semiotics-semiosis-sign-relations-6/

cc: https://www.academia.edu/community/VBAXbj
cc: https://www.researchgate.net/post/Sign_Relations_First_Elements
cc: https://stream.syscoi.com/2026/01/01/sign-relations-semiotic-equivalence-relations-2/

#Peirce #Inquiry #Logic #Mathematics #RelationTheory
#Semiosis #Semiotics #SignRelations #TriadicRelations

Sign Relations • Semiotic Equivalence Relations 2.1
• https://inquiryintoinquiry.com/2025/12/31/sign-relations-semiotic-equivalence-relations-2-c/

A few items of notation are useful in discussing equivalence relations in general and semiotic equivalence relations in particular.

In general, if E is an equivalence relation on a set X then every element x of X belongs to a unique equivalence class under E called “the equivalence class of x under E”. Convention provides the “square bracket notation” for denoting such equivalence classes, in either the form [x]_E or the simpler form [x] when the subscript E is understood.

A statement that the elements x and y are equivalent under E is called an “equation” or an “equivalence” and may be expressed in any of the following ways.

• (x, y) ∈ E

• x ∈ [y]_E

• y ∈ [x]_E

• [x]_E = [y]_E

• x =_E y

Display 1
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-ser-display-1.png

Thus we have the following definitions.

• [x]_E = {y ∈ X : (x, y) ∈ E}

• x =_E y ⇔ (x, y) ∈ E

Display 2
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-ser-display-2.png

Resources —

Sign Relation
• https://oeis.org/wiki/Sign_relation
• https://mywikibiz.com/Sign_relation
• https://en.wikiversity.org/wiki/Sign_relation

Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2025/05/06/survey-of-semiotics-semiosis-sign-relations-6/

cc: https://www.academia.edu/community/VBAXbj
cc: https://www.researchgate.net/post/Sign_Relations_First_Elements
cc: https://stream.syscoi.com/2026/01/01/sign-relations-semiotic-equivalence-relations-2/

#Peirce #Inquiry #Logic #Mathematics #RelationTheory
#Semiosis #Semiotics #SignRelations #TriadicRelations

Sign Relations • Semiotic Equivalence Relations 1.2
• https://inquiryintoinquiry.com/2025/12/30/sign-relations-semiotic-equivalence-relations-1-c/

A nice property of the sign relations L_A and L_B is that their connotative components Con(L_A) and Con(L_B) form a pair of equivalence relations on their common syntactic domain S = I. This type of equivalence relation is called a “semiotic equivalence relation” (SER) because it equates signs having the same meaning to some interpreter.

Each of the semiotic equivalence relations, Con(L_A), Con(L_B) ⊆ S×I ≅ S×S partitions the collection of signs into semiotic equivalence classes. This constitutes a strong form of representation in that the structure of the interpreters' common object domain {A, B} is reflected or reconstructed, part for part, in the structure of each one's semiotic partition of the syntactic domain {“A”, “B”, “i”, “u”}.

It's important to observe the semiotic partitions for interpreters A and B are not identical, indeed, they are “orthogonal” to each other. Thus we may regard the “form” of the partitions as corresponding to an objective structure or invariant reality, but not the literal sets of signs themselves, independent of the individual interpreter's point of view.

Information about the contrasting patterns of semiotic equivalence corresponding to the interpreters A and B is summarized in Tables 7a and 7b. The form of the Tables serves to explain what is meant by saying the SEPs for A and B are “orthogonal” to each other.

Tables 7a and 7b. Semiotic Partitions for Interpreters A and B
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/semiotic-partitions-for-interpreters-a-b-2.0.png

Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2025/05/06/survey-of-semiotics-semiosis-sign-relations-6/

#Peirce #Inquiry #Logic #Mathematics #RelationTheory
#Semiosis #Semiotics #SignRelations #TriadicRelations

Sign Relations • Semiotic Equivalence Relations 1.1
• https://inquiryintoinquiry.com/2025/12/30/sign-relations-semiotic-equivalence-relations-1-c/

A “semiotic equivalence relation” (SER) is a special type of equivalence relation arising in the analysis of sign relations. Generally speaking, any equivalence relation induces a partition of the underlying set of elements, known as the “domain” or “space” of the relation, into a family of equivalence classes. In the case of a SER the equivalence classes are called “semiotic equivalence classes” (SECs) and the partition is called a “semiotic partition” (SEP).

The sign relations L_A and L_B have many interesting properties over and above those possessed by sign relations in general. Some of those properties have to do with the relation between signs and their interpretant signs, as reflected in the projections of L_A and L_B on the SI‑plane, notated as proj_{SI} L_A and proj_{SI} L_B, respectively. The dyadic relations on S×I induced by those projections are also referred to as the “connotative components” of the corresponding sign relations, notated as Con(L_A) and Con(L_B), respectively. Tables 6a and 6b show the corresponding connotative components.

Tables 6a and 6b. Connotative Components Con(L_A) and Con(L_B)
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/connotative-components-con-la-con-lb-3.0.png

Resources —

Sign Relation
• https://oeis.org/wiki/Sign_relation
• https://mywikibiz.com/Sign_relation
• https://en.wikiversity.org/wiki/Sign_relation

Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2025/05/06/survey-of-semiotics-semiosis-sign-relations-6/

cc: https://www.academia.edu/community/Lm48yP
cc: https://www.researchgate.net/post/Sign_Relations_First_Elements
cc: https://stream.syscoi.com/2025/12/30/sign-relations-semiotic-equivalence-relations-1/

#Peirce #Inquiry #Logic #Mathematics #RelationTheory
#Semiosis #Semiotics #SignRelations #TriadicRelations

Sign Relations • Ennotation • Part 2
• https://inquiryintoinquiry.com/2025/12/29/sign-relations-ennotation-c/

As it happens, the sign relations L_A and L_B are fully symmetric with respect to exchanging signs and interpretants, so all the data of proj_{OS} L_A is echoed unchanged in proj_{OI} L_A and all the data of proj_{OS} L_B is echoed unchanged in proj_{OI} L_B.

Tables 5a and 5b show the ennotative components of the sign relations associated with the interpreters A and B, respectively. The rows of each Table list the ordered pairs (o, i) in the corresponding projections, Enn(L_A), Enn(L_B) ⊆ O×I.

• Tables 5a and 5b. Ennotative Components Enn(L_A) and Enn(L_B)
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-twin-tables-enn-la-enn-lb-2.0.png

Resources —

Sign Relation • OEIS • MyWikiBiz • Wikiversity
• https://oeis.org/wiki/Sign_relation
• https://mywikibiz.com/Sign_relation
• https://en.wikiversity.org/wiki/Sign_relation

Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2025/05/06/survey-of-semiotics-semiosis-sign-relations-6/

cc: https://www.academia.edu/community/V0rbOx
cc: https://www.researchgate.net/post/Sign_Relations_First_Elements
cc: https://stream.syscoi.com/2025/12/29/sign-relations-ennotation/

#Peirce #Inquiry #Logic #Mathematics #RelationTheory
#Semiosis #Semiotics #SignRelations #TriadicRelations

Sign Relations • Ennotation • Part 1
• https://inquiryintoinquiry.com/2025/12/29/sign-relations-ennotation-c/

A third aspect of a sign's complete meaning concerns the relation between its objects and its interpretants, which has no standard name in semiotics. It would be called an “induced relation” in graph theory or the result of “relational composition” in relation theory. If an interpretant is recognized as a sign in its own right then its independent reference to an object can be taken as belonging to another moment of denotation, but this neglects the mediational character of the whole transaction in which this occurs. Denotation and connotation have to do with dyadic relations in which the sign plays an active role but here we are dealing with a dyadic relation between objects and interpretants mediated by the sign from an off‑stage position, as it were.

As a relation between objects and interpretants mediated by a sign, this third aspect of meaning may be referred to as the “ennotation” of a sign and the dyadic relation making up the ennotative aspect of a sign relation L may be notated as Enn(L). Information about the ennotative aspect of meaning is obtained from L by taking its projection on the object‑interpretant plane and visualized as the “shadow” L casts on the 2‑dimensional space whose axes are the object domain O and the interpretant domain I. The ennotative component of a sign relation L, variously written as proj_{OI} L, L_OI, proj₁₃ L, or L₁₃, is defined as follows.

• Enn(L) = proj_{OI} L = {(o, i) ∈ O × I : (o, s, i) ∈ L for some s ∈ S}.
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-display-5.png

#Peirce #Inquiry #Logic #Mathematics #RelationTheory
#Semiosis #Semiotics #SignRelations #TriadicRelations

Sign Relations • Connotation • Part 2
• https://inquiryintoinquiry.com/2025/12/28/sign-relations-connotation-c/

Formally speaking, however, the connotative aspect of meaning presents no additional difficulty. The dyadic relation making up the connotative aspect of a sign relation L is notated as Con(L). Information about the connotative aspect of meaning is obtained from L by taking its projection on the sign‑interpretant plane and visualized as the “shadow” L casts on the 2‑dimensional space whose axes are the sign domain S and the interpretant domain I. The connotative component of a sign relation L, variously written as proj_{SI} L, L_SI, proj₂₃ L, or L₂₃, is defined as follows.

• Con(L) = proj_{SI} L = {(s, i) ∈ S × I : (o, s, i) ∈ L for some o ∈ O}.
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-display-4.png

Tables 4a and 4b show the connotative components of the sign relations associated with the interpreters A and B, respectively. The rows of each Table list the ordered pairs (s, i) in the corresponding projections, Con(L_A), Con(L_B) ⊆ S×I.

• Tables 4a and 4b. Connotative Components Con(L_A) and Con(L_B)
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-twin-tables-con-la-con-lb-2.0.png

Resources —

Sign Relation
• https://oeis.org/wiki/Sign_relation
• https://mywikibiz.com/Sign_relation
• https://en.wikiversity.org/wiki/Sign_relation

Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2025/05/06/survey-of-semiotics-semiosis-sign-relations-6/

cc: https://www.academia.edu/community/VqeB0k
cc: https://www.researchgate.net/post/Sign_Relations_First_Elements
cc: https://stream.syscoi.com/2025/12/28/sign-relations-connotation/

#Peirce #Inquiry #Logic #Mathematics #RelationTheory
#Semiosis #Semiotics #SignRelations #TriadicRelations

Sign Relations • Connotation • Part 1
• https://inquiryintoinquiry.com/2025/12/28/sign-relations-connotation-c/

Another aspect of a sign's complete meaning concerns the reference a sign has to its interpretants, which interpretants are collectively known as the “connotation” of the sign. In the pragmatic theory of sign relations, connotative references fall within the projection of the sign relation on the plane spanned by its sign domain and its interpretant domain.

In the full theory of sign relations the connotative aspect of meaning includes the links a sign has to affects, concepts, ideas, impressions, intentions, and the whole realm of an interpretive agent's mental states and allied activities, broadly encompassing intellectual associations, emotional impressions, motivational impulses, and real conduct.

Taken at the full, in the natural setting of semiotic phenomena, this complex system of references is unlikely ever to find itself mapped in much detail, much less completely formalized, but the tangible warp of its accumulated mass is commonly alluded to as the connotative import of language.

Resources —

Sign Relation
• https://oeis.org/wiki/Sign_relation
• https://mywikibiz.com/Sign_relation
• https://en.wikiversity.org/wiki/Sign_relation

Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2025/05/06/survey-of-semiotics-semiosis-sign-relations-6/

cc: https://www.academia.edu/community/VqeB0k
cc: https://www.researchgate.net/post/Sign_Relations_First_Elements
cc: https://stream.syscoi.com/2025/12/28/sign-relations-connotation/

#Peirce #Inquiry #Logic #Mathematics #RelationTheory
#Semiosis #Semiotics #SignRelations #TriadicRelations