#peirce — Public Fediverse posts
Live and recent posts from across the Fediverse tagged #peirce, aggregated by home.social.
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Language Learning And Logical Modeling —
Wrote my first “Language Learning Module”, strictly speaking, a two‑level formal language learner, back in the 80s and it pretty much told me what every conceivable upscale of that ilk would be like. But it did not cross the threshold of logical reasoning, so I used Peirce's logical graphs for that. Et sic deinceps …
#Peirce #Logic #Mathematics #Semiotics #LogicalGraphs
#LanguageLearningAlgorithm #LogicalModelingAlgorithm -
Animated Logical Graphs • 2
• https://inquiryintoinquiry.com/2015/01/14/animated-logical-graphs-2/It's almost 50 years now since I first encountered the volumes of Peirce's “Collected Papers” in the math library at Michigan State, and shortly afterwards a friend called my attention to the entry for Spencer Brown's “Laws of Form” in the Whole Earth Catalog and I sent off for it right away. I would spend the next decade just beginning to figure out what either one of them was talking about in the matter of logical graphs and I would spend another decade after that developing a program, first in Lisp and then in Pascal, that turned graph‑theoretic data structures formed on their ideas to good purpose as the basis of its reasoning engine.
I thought it might contribute to a number of long‑running and ongoing discussions if I could articulate what I think I learned from that experience.
So I'll try to keep focused on that.
Resources —
Logical Graphs • First Impressions
• https://inquiryintoinquiry.com/2024/08/26/logical-graphs-first-impressions-a/Logical Graphs • Formal Development
• https://inquiryintoinquiry.com/2024/09/12/logical-graphs-formal-development-b/Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/#Peirce #Logic #Mathematics #Semiotics #LogicalGraphs #GraphTheory
#SpencerBrown #LawsOfForm #PropositionalCalculus #ProofAnimations -
Animated Logical Graphs • 1
• https://inquiryintoinquiry.com/2015/01/08/animated-logical-graphs-1/For Your Musement …
Here are some animations I made up to illustrate several different styles of proof in an extended topological variant of Peirce's Alpha Graphs for propositional logic.
Proof Animations
• https://oeis.org/wiki/User:Jon_Awbrey/ANIMATION#Proof_AnimationsDouble Negation
• https://inquiryintoinquiry.com/wp-content/uploads/2026/04/proof-animation-e280a2-double-negation-2.0.gifPeirce's Law
• https://inquiryintoinquiry.com/wp-content/uploads/2026/04/proof-animation-e280a2-peirces-law-2.0.gifPraeclarum Theorema
• https://inquiryintoinquiry.com/wp-content/uploads/2026/04/proof-animation-e280a2-praeclarum-theorema-2.0.gifTwo‑Thirds Majority Function
• https://inquiryintoinquiry.com/wp-content/uploads/2026/04/proof-animation-e280a2-two-thirds-majority-function-2.0.gifA full discussion of logical graphs can be found in the following article.
Logical Graphs
• https://oeis.org/wiki/Logical_GraphsResources —
Logical Graphs • First Impressions
• https://inquiryintoinquiry.com/2024/08/26/logical-graphs-first-impressions-a/Logical Graphs • Formal Development
• https://inquiryintoinquiry.com/2024/09/12/logical-graphs-formal-development-b/Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/cc: https://www.academia.edu/community/ldzadj
cc: https://mathstodon.xyz/@Inquiry/116494097283214718
cc: https://www.researchgate.net/post/Animated_Logical_Graphs
cc: https://stream.syscoi.com/2026/04/30/animated-logical-graphs-1/
cc: https://groups.io/g/lawsofform/topic/animated_logical_graphs/119049814#Peirce #Logic #Mathematics #Semiotics #LogicalGraphs #GraphTheory
#SpencerBrown #LawsOfForm #PropositionalCalculus #ProofAnimations -
Animated Logical Graphs • 1
• https://inquiryintoinquiry.com/2015/01/08/animated-logical-graphs-1/For Your Musement …
Here are some animations I made up to illustrate several different styles of proof in an extended topological variant of Peirce’s Alpha Graphs for propositional logic.
Proof Animations
• https://oeis.org/wiki/User:Jon_Awbrey/ANIMATION#Proof_AnimationsSee the following article for a full discussion of this type of logical graph.
Logical Graphs
• https://oeis.org/wiki/Logical_GraphsAdditional Resources —
Logical Graphs • First Impressions
• https://inquiryintoinquiry.com/2024/08/26/logical-graphs-first-impressions-a/Logical Graphs • Formal Development
• https://inquiryintoinquiry.com/2024/09/12/logical-graphs-formal-development-b/#Peirce #Logic #Mathematics #Semiotics #LogicalGraphs #GraphTheory
#SpencerBrown #LawsOfForm #PropositionalCalculus #ProofAnimations -
Reflection On Recursion • Discussion 1
• https://inquiryintoinquiry.com/2026/04/21/reflection-on-recursion-discussion-1/Re: Reflection On Recursion • 1
• https://inquiryintoinquiry.com/2026/04/06/reflection-on-recursion-1/
Re: Laws of Form • John Mingers
• https://groups.io/g/lawsofform/message/4943JM:
❝This is a very important and interesting topic. I think you should consider the relationship to self‑reference, indeed are they really the same thing?❝Also the work of Maturana and Varela on autopoiesis and the neurophysiology of cognition which also has recursion at its heart.❞
Thanks, John. Yes, we certainly find the whole array of self concepts coming into play here — selfhood, autopoiesis or self creation, self reference and self transformation, just to name a few. But one thing I need to emphasize from the start is how radically different such concepts appear when viewed in the x‑ray vision of Peirce’s pragmatic semiotics.
I forget where I first heard it, but it’s fairly common observation that the persistence of a recurring problem is a symptom of how unlikely it is to be solved in the paradigm where it keeps occurring.
After a while, it simply becomes time to change the paradigm …
Just by way of a first example, take the very idea of “self‑reference”. The moment we place it in the medium of triadic sign relations we realize signs do not refer to anything at all except insofar as an interpreter refers them.
And when we ask, “What is this, that we call an interpreter?”, the pragmatic theory of signs tells us we cannot tell when we turn out the light but under the x‑ray of the pragmatic maxim the sum of its effects is effectively modeled by an extended triadic sign relation.
Et sic deinceps …
#Peirce #Logic #Mathematics
#Recursion #Reflection #Semiotics
#SignRelations #TriadicRelations -
Reflection On Recursion • 4
• https://inquiryintoinquiry.com/2026/04/18/reflection-on-recursion-4/A feature worth noting in the recursion diagram is the function traversing the square from one triadic node to the other. It preserves an image of the object n all the while its precedent p(n) is being retrieved and processed — thus it injects a measure of parallel process and a modicum of extra memory over and above that afforded by the serial composition of functions.
Simple Recursion
• https://inquiryintoinquiry.com/wp-content/uploads/2026/03/simple-recursion-fn-mn-fpn.pngResources —
Inquiry Driven Systems • Inquiry Into Inquiry
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_OverviewReflective Interpretive Frameworks
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_10#RIF_1The Phenomenology of Reflection
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_11#The_Phenomenology_of_ReflectionHigher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_Relations#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations -
Reflection On Recursion • 4
• https://inquiryintoinquiry.com/2026/04/18/reflection-on-recursion-4/A feature worth noting in the recursion diagram is the function traversing the square from one triadic node to the other. It preserves an image of the object n all the while its precedent p(n) is being retrieved and processed — thus it injects a measure of parallel process and a modicum of extra memory over and above that afforded by the serial composition of functions.
Simple Recursion
• https://inquiryintoinquiry.com/wp-content/uploads/2026/03/simple-recursion-fn-mn-fpn.pngResources —
Inquiry Driven Systems • Inquiry Into Inquiry
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_OverviewReflective Interpretive Frameworks
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_10#RIF_1The Phenomenology of Reflection
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_11#The_Phenomenology_of_ReflectionHigher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_Relations#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations -
Reflection On Recursion • 4
• https://inquiryintoinquiry.com/2026/04/18/reflection-on-recursion-4/A feature worth noting in the recursion diagram is the function traversing the square from one triadic node to the other. It preserves an image of the object n all the while its precedent p(n) is being retrieved and processed — thus it injects a measure of parallel process and a modicum of extra memory over and above that afforded by the serial composition of functions.
Simple Recursion
• https://inquiryintoinquiry.com/wp-content/uploads/2026/03/simple-recursion-fn-mn-fpn.pngResources —
Inquiry Driven Systems • Inquiry Into Inquiry
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_OverviewReflective Interpretive Frameworks
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_10#RIF_1The Phenomenology of Reflection
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_11#The_Phenomenology_of_ReflectionHigher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_Relations#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations -
Reflection On Recursion • 4
• https://inquiryintoinquiry.com/2026/04/18/reflection-on-recursion-4/A feature worth noting in the recursion diagram is the function traversing the square from one triadic node to the other. It preserves an image of the object n all the while its precedent p(n) is being retrieved and processed — thus it injects a measure of parallel process and a modicum of extra memory over and above that afforded by the serial composition of functions.
Simple Recursion
• https://inquiryintoinquiry.com/wp-content/uploads/2026/03/simple-recursion-fn-mn-fpn.pngResources —
Inquiry Driven Systems • Inquiry Into Inquiry
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_OverviewReflective Interpretive Frameworks
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_10#RIF_1The Phenomenology of Reflection
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_11#The_Phenomenology_of_ReflectionHigher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_Relations#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations -
Reflection On Recursion • 4
• https://inquiryintoinquiry.com/2026/04/18/reflection-on-recursion-4/A feature worth noting in the recursion diagram is the function traversing the square from one triadic node to the other. It preserves an image of the object n all the while its precedent p(n) is being retrieved and processed — thus it injects a measure of parallel process and a modicum of extra memory over and above that afforded by the serial composition of functions.
Simple Recursion
• https://inquiryintoinquiry.com/wp-content/uploads/2026/03/simple-recursion-fn-mn-fpn.pngResources —
Inquiry Driven Systems • Inquiry Into Inquiry
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_OverviewReflective Interpretive Frameworks
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_10#RIF_1The Phenomenology of Reflection
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_11#The_Phenomenology_of_ReflectionHigher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_Relations#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations -
Reflection On Recursion • 3
• https://inquiryintoinquiry.com/2026/04/13/reflection-on-recursion-3/One other feature of syntactic recursion deserves to be brought into higher relief. Evidence of it can be found in the recursion diagram by examining the places where three paths meet. On the descending side there is the point where three paths diverge. On the ascending side there is the point where the middlemost of the three divergent paths joins the upshot arrow in medias res.
Simple Recursion
• https://inquiryintoinquiry.com/wp-content/uploads/2026/03/simple-recursion-fn-mn-fpn.pngThe arrows of the diagram represent functions, a species of dyadic relations, but nodes of degree three signify aspects of triadic relations somewhere in the mix.
• The three arrows from the initial node represent a function F : N → N×N×N such that F(n) = (p(n), n, f(n)).
• The three arrows at the penultimate node represent a function m : N×N → N such that m(j, k) = jk.
For the sake of a first approach, many questions about triadic relations which might arise at this point can be safely left to later discussions, since the current level of generality is comprehensible enough in functional terms.
Resources —
Inquiry Driven Systems • Inquiry Into Inquiry
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_OverviewReflective Interpretive Frameworks
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_10#RIF_1The Phenomenology of Reflection
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_11#The_Phenomenology_of_ReflectionHigher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_Relations#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations -
Reflection On Recursion • 3
• https://inquiryintoinquiry.com/2026/04/13/reflection-on-recursion-3/One other feature of syntactic recursion deserves to be brought into higher relief. Evidence of it can be found in the recursion diagram by examining the places where three paths meet. On the descending side there is the point where three paths diverge. On the ascending side there is the point where the middlemost of the three divergent paths joins the upshot arrow in medias res.
Simple Recursion
• https://inquiryintoinquiry.com/wp-content/uploads/2026/03/simple-recursion-fn-mn-fpn.pngThe arrows of the diagram represent functions, a species of dyadic relations, but nodes of degree three signify aspects of triadic relations somewhere in the mix.
• The three arrows from the initial node represent a function F : N → N×N×N such that F(n) = (p(n), n, f(n)).
• The three arrows at the penultimate node represent a function m : N×N → N such that m(j, k) = jk.
For the sake of a first approach, many questions about triadic relations which might arise at this point can be safely left to later discussions, since the current level of generality is comprehensible enough in functional terms.
Resources —
Inquiry Driven Systems • Inquiry Into Inquiry
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_OverviewReflective Interpretive Frameworks
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_10#RIF_1The Phenomenology of Reflection
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_11#The_Phenomenology_of_ReflectionHigher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_Relations#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations -
Reflection On Recursion • 3
• https://inquiryintoinquiry.com/2026/04/13/reflection-on-recursion-3/One other feature of syntactic recursion deserves to be brought into higher relief. Evidence of it can be found in the recursion diagram by examining the places where three paths meet. On the descending side there is the point where three paths diverge. On the ascending side there is the point where the middlemost of the three divergent paths joins the upshot arrow in medias res.
Simple Recursion
• https://inquiryintoinquiry.com/wp-content/uploads/2026/03/simple-recursion-fn-mn-fpn.pngThe arrows of the diagram represent functions, a species of dyadic relations, but nodes of degree three signify aspects of triadic relations somewhere in the mix.
• The three arrows from the initial node represent a function F : N → N×N×N such that F(n) = (p(n), n, f(n)).
• The three arrows at the penultimate node represent a function m : N×N → N such that m(j, k) = jk.
For the sake of a first approach, many questions about triadic relations which might arise at this point can be safely left to later discussions, since the current level of generality is comprehensible enough in functional terms.
Resources —
Inquiry Driven Systems • Inquiry Into Inquiry
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_OverviewReflective Interpretive Frameworks
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_10#RIF_1The Phenomenology of Reflection
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_11#The_Phenomenology_of_ReflectionHigher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_Relations#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations -
Reflection On Recursion • 3
• https://inquiryintoinquiry.com/2026/04/13/reflection-on-recursion-3/One other feature of syntactic recursion deserves to be brought into higher relief. Evidence of it can be found in the recursion diagram by examining the places where three paths meet. On the descending side there is the point where three paths diverge. On the ascending side there is the point where the middlemost of the three divergent paths joins the upshot arrow in medias res.
Simple Recursion
• https://inquiryintoinquiry.com/wp-content/uploads/2026/03/simple-recursion-fn-mn-fpn.pngThe arrows of the diagram represent functions, a species of dyadic relations, but nodes of degree three signify aspects of triadic relations somewhere in the mix.
• The three arrows from the initial node represent a function F : N → N×N×N such that F(n) = (p(n), n, f(n)).
• The three arrows at the penultimate node represent a function m : N×N → N such that m(j, k) = jk.
For the sake of a first approach, many questions about triadic relations which might arise at this point can be safely left to later discussions, since the current level of generality is comprehensible enough in functional terms.
Resources —
Inquiry Driven Systems • Inquiry Into Inquiry
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_OverviewReflective Interpretive Frameworks
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_10#RIF_1The Phenomenology of Reflection
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_11#The_Phenomenology_of_ReflectionHigher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_Relations#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations -
Reflection On Recursion • 3
• https://inquiryintoinquiry.com/2026/04/13/reflection-on-recursion-3/One other feature of syntactic recursion deserves to be brought into higher relief. Evidence of it can be found in the recursion diagram by examining the places where three paths meet. On the descending side there is the point where three paths diverge. On the ascending side there is the point where the middlemost of the three divergent paths joins the upshot arrow in medias res.
Simple Recursion
• https://inquiryintoinquiry.com/wp-content/uploads/2026/03/simple-recursion-fn-mn-fpn.pngThe arrows of the diagram represent functions, a species of dyadic relations, but nodes of degree three signify aspects of triadic relations somewhere in the mix.
• The three arrows from the initial node represent a function F : N → N×N×N such that F(n) = (p(n), n, f(n)).
• The three arrows at the penultimate node represent a function m : N×N → N such that m(j, k) = jk.
For the sake of a first approach, many questions about triadic relations which might arise at this point can be safely left to later discussions, since the current level of generality is comprehensible enough in functional terms.
Resources —
Inquiry Driven Systems • Inquiry Into Inquiry
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_OverviewReflective Interpretive Frameworks
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_10#RIF_1The Phenomenology of Reflection
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_11#The_Phenomenology_of_ReflectionHigher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_Relations#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations -
Reflection On Recursion • 2
• https://inquiryintoinquiry.com/2026/04/09/reflection-on-recursion-2/Turning to the form of a simple recursive function f(n) = m(n, f(p(n))), the clause we used to define it earns the title of “syntactic recursion” due to the way the function name “f” occurring in the defined phrase “f(n)” re‑occurs in the defining phrase “m(n, f(p(n)))”.
Simple Recursion
• https://inquiryintoinquiry.com/wp-content/uploads/2026/03/simple-recursion-fn-mn-fpn.pngIt needs to be clear there is no circle in the definition — each instance of the type f is defined in terms of an instance one step simpler until the base case is reached and fixed by fiat. Instead of a circle then we have two gyres, the gyre down via the precedent function p and the gyre up via the modifier function m.
cc: https://www.academia.edu/community/L24rvm
cc: https://www.academia.edu/community/LE2mrr
cc: https://www.researchgate.net/post/Reflection_On_Recursion#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations -
Reflection On Recursion • 2
• https://inquiryintoinquiry.com/2026/04/09/reflection-on-recursion-2/Turning to the form of a simple recursive function f(n) = m(n, f(p(n))), the clause we used to define it earns the title of “syntactic recursion” due to the way the function name “f” occurring in the defined phrase “f(n)” re‑occurs in the defining phrase “m(n, f(p(n)))”.
Simple Recursion
• https://inquiryintoinquiry.com/wp-content/uploads/2026/03/simple-recursion-fn-mn-fpn.pngIt needs to be clear there is no circle in the definition — each instance of the type f is defined in terms of an instance one step simpler until the base case is reached and fixed by fiat. Instead of a circle then we have two gyres, the gyre down via the precedent function p and the gyre up via the modifier function m.
cc: https://www.academia.edu/community/L24rvm
cc: https://www.academia.edu/community/LE2mrr
cc: https://www.researchgate.net/post/Reflection_On_Recursion#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations -
Reflection On Recursion • 2
• https://inquiryintoinquiry.com/2026/04/09/reflection-on-recursion-2/Turning to the form of a simple recursive function f(n) = m(n, f(p(n))), the clause we used to define it earns the title of “syntactic recursion” due to the way the function name “f” occurring in the defined phrase “f(n)” re‑occurs in the defining phrase “m(n, f(p(n)))”.
Simple Recursion
• https://inquiryintoinquiry.com/wp-content/uploads/2026/03/simple-recursion-fn-mn-fpn.pngIt needs to be clear there is no circle in the definition — each instance of the type f is defined in terms of an instance one step simpler until the base case is reached and fixed by fiat. Instead of a circle then we have two gyres, the gyre down via the precedent function p and the gyre up via the modifier function m.
cc: https://www.academia.edu/community/L24rvm
cc: https://www.academia.edu/community/LE2mrr
cc: https://www.researchgate.net/post/Reflection_On_Recursion#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations -
Reflection On Recursion • 2
• https://inquiryintoinquiry.com/2026/04/09/reflection-on-recursion-2/Turning to the form of a simple recursive function f(n) = m(n, f(p(n))), the clause we used to define it earns the title of “syntactic recursion” due to the way the function name “f” occurring in the defined phrase “f(n)” re‑occurs in the defining phrase “m(n, f(p(n)))”.
Simple Recursion
• https://inquiryintoinquiry.com/wp-content/uploads/2026/03/simple-recursion-fn-mn-fpn.pngIt needs to be clear there is no circle in the definition — each instance of the type f is defined in terms of an instance one step simpler until the base case is reached and fixed by fiat. Instead of a circle then we have two gyres, the gyre down via the precedent function p and the gyre up via the modifier function m.
cc: https://www.academia.edu/community/L24rvm
cc: https://www.academia.edu/community/LE2mrr
cc: https://www.researchgate.net/post/Reflection_On_Recursion#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations -
Reflection On Recursion • 2
• https://inquiryintoinquiry.com/2026/04/09/reflection-on-recursion-2/Turning to the form of a simple recursive function f(n) = m(n, f(p(n))), the clause we used to define it earns the title of “syntactic recursion” due to the way the function name “f” occurring in the defined phrase “f(n)” re‑occurs in the defining phrase “m(n, f(p(n)))”.
Simple Recursion
• https://inquiryintoinquiry.com/wp-content/uploads/2026/03/simple-recursion-fn-mn-fpn.pngIt needs to be clear there is no circle in the definition — each instance of the type f is defined in terms of an instance one step simpler until the base case is reached and fixed by fiat. Instead of a circle then we have two gyres, the gyre down via the precedent function p and the gyre up via the modifier function m.
cc: https://www.academia.edu/community/L24rvm
cc: https://www.academia.edu/community/LE2mrr
cc: https://www.researchgate.net/post/Reflection_On_Recursion#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations -
Reflection On Recursion • 1.3
• https://inquiryintoinquiry.com/2026/04/06/reflection-on-recursion-1/Comment 5 —
Recursion is rife in mathematics and computation, typically sporting its recursive character on its sleeve in the fashion of syntax sketched above.
But mathematics and computation are overlearned subjects and practices, enjoying long histories of being gone over with an eye to articulating every last detail of any way they might be conceived and conducted.
So it's fair to ask whether all that artifice truly tutors nature or only creates a rationalized reconstruction of it. Then again, even if that's all it does, is there anything of use to be learned from it?
Comment 6 —
The prevalence of recursion in mathematics arises from the architecture of mathematical systems.
Mathematical systems grow from a fourfold root.
• “Primitives” are taken as initial terms.
• “Definitions” expound ever more complex terms in relation to the primitives.
• “Axioms” are taken as initial truths.
• “Theorems” follow from the axioms by way of inference rules.
Recursive definitions of mathematical objects and inductive proofs of the corresponding theorems follow closely parallel patterns. And again, in computation, recursive programs follow the same patterns in action.
#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations -
Reflection On Recursion • 1.3
• https://inquiryintoinquiry.com/2026/04/06/reflection-on-recursion-1/Comment 5 —
Recursion is rife in mathematics and computation, typically sporting its recursive character on its sleeve in the fashion of syntax sketched above.
But mathematics and computation are overlearned subjects and practices, enjoying long histories of being gone over with an eye to articulating every last detail of any way they might be conceived and conducted.
So it's fair to ask whether all that artifice truly tutors nature or only creates a rationalized reconstruction of it. Then again, even if that's all it does, is there anything of use to be learned from it?
Comment 6 —
The prevalence of recursion in mathematics arises from the architecture of mathematical systems.
Mathematical systems grow from a fourfold root.
• “Primitives” are taken as initial terms.
• “Definitions” expound ever more complex terms in relation to the primitives.
• “Axioms” are taken as initial truths.
• “Theorems” follow from the axioms by way of inference rules.
Recursive definitions of mathematical objects and inductive proofs of the corresponding theorems follow closely parallel patterns. And again, in computation, recursive programs follow the same patterns in action.
#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations -
Reflection On Recursion • 1.3
• https://inquiryintoinquiry.com/2026/04/06/reflection-on-recursion-1/Comment 5 —
Recursion is rife in mathematics and computation, typically sporting its recursive character on its sleeve in the fashion of syntax sketched above.
But mathematics and computation are overlearned subjects and practices, enjoying long histories of being gone over with an eye to articulating every last detail of any way they might be conceived and conducted.
So it's fair to ask whether all that artifice truly tutors nature or only creates a rationalized reconstruction of it. Then again, even if that's all it does, is there anything of use to be learned from it?
Comment 6 —
The prevalence of recursion in mathematics arises from the architecture of mathematical systems.
Mathematical systems grow from a fourfold root.
• “Primitives” are taken as initial terms.
• “Definitions” expound ever more complex terms in relation to the primitives.
• “Axioms” are taken as initial truths.
• “Theorems” follow from the axioms by way of inference rules.
Recursive definitions of mathematical objects and inductive proofs of the corresponding theorems follow closely parallel patterns. And again, in computation, recursive programs follow the same patterns in action.
#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations -
Reflection On Recursion • 1.3
• https://inquiryintoinquiry.com/2026/04/06/reflection-on-recursion-1/Comment 5 —
Recursion is rife in mathematics and computation, typically sporting its recursive character on its sleeve in the fashion of syntax sketched above.
But mathematics and computation are overlearned subjects and practices, enjoying long histories of being gone over with an eye to articulating every last detail of any way they might be conceived and conducted.
So it's fair to ask whether all that artifice truly tutors nature or only creates a rationalized reconstruction of it. Then again, even if that's all it does, is there anything of use to be learned from it?
Comment 6 —
The prevalence of recursion in mathematics arises from the architecture of mathematical systems.
Mathematical systems grow from a fourfold root.
• “Primitives” are taken as initial terms.
• “Definitions” expound ever more complex terms in relation to the primitives.
• “Axioms” are taken as initial truths.
• “Theorems” follow from the axioms by way of inference rules.
Recursive definitions of mathematical objects and inductive proofs of the corresponding theorems follow closely parallel patterns. And again, in computation, recursive programs follow the same patterns in action.
#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations -
Reflection On Recursion • 1.3
• https://inquiryintoinquiry.com/2026/04/06/reflection-on-recursion-1/Comment 5 —
Recursion is rife in mathematics and computation, typically sporting its recursive character on its sleeve in the fashion of syntax sketched above.
But mathematics and computation are overlearned subjects and practices, enjoying long histories of being gone over with an eye to articulating every last detail of any way they might be conceived and conducted.
So it's fair to ask whether all that artifice truly tutors nature or only creates a rationalized reconstruction of it. Then again, even if that's all it does, is there anything of use to be learned from it?
Comment 6 —
The prevalence of recursion in mathematics arises from the architecture of mathematical systems.
Mathematical systems grow from a fourfold root.
• “Primitives” are taken as initial terms.
• “Definitions” expound ever more complex terms in relation to the primitives.
• “Axioms” are taken as initial truths.
• “Theorems” follow from the axioms by way of inference rules.
Recursive definitions of mathematical objects and inductive proofs of the corresponding theorems follow closely parallel patterns. And again, in computation, recursive programs follow the same patterns in action.
#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations -
Reflection On Recursion • 1.2
• https://inquiryintoinquiry.com/2026/04/06/reflection-on-recursion-1/Comment 3 —
If we discard from the idea of recursion what is not of its essence, we find recursion occurs when our understanding of one situation has recourse to our understanding of other situations.
Very typically, the object situation presents itself as complex, difficult, or unfamiliar while the resource situations are regarded as being better understood.
It must be appreciated, however, that any ranking of situations by level of understanding is contingent on the circumstances in view and may vary radically in alternate settings.
Comment 4 —
Recursion occurs more markedly in “syntactic recursion”, where the recursive process shows its character as such in the symbols of its syntactic expression.
A sense of the difference can be gained by looking at a case of “ostensible syntactic recursion”. (How much substance backs the ostentation is a subject we'll take up, maybe at length, but later …)
Consider the following diagram for the computation of a simple recursive function.
Simple Recursion
• https://inquiryintoinquiry.com/wp-content/uploads/2026/03/simple-recursion-fn-mn-fpn.pngFor example, the factorial function f(n) = n! has a definition in terms of the predecessor function p(n) = n-1 and the multiplier function m(j, k) = j∙k.
#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations -
Reflection On Recursion • 1.2
• https://inquiryintoinquiry.com/2026/04/06/reflection-on-recursion-1/Comment 3 —
If we discard from the idea of recursion what is not of its essence, we find recursion occurs when our understanding of one situation has recourse to our understanding of other situations.
Very typically, the object situation presents itself as complex, difficult, or unfamiliar while the resource situations are regarded as being better understood.
It must be appreciated, however, that any ranking of situations by level of understanding is contingent on the circumstances in view and may vary radically in alternate settings.
Comment 4 —
Recursion occurs more markedly in “syntactic recursion”, where the recursive process shows its character as such in the symbols of its syntactic expression.
A sense of the difference can be gained by looking at a case of “ostensible syntactic recursion”. (How much substance backs the ostentation is a subject we'll take up, maybe at length, but later …)
Consider the following diagram for the computation of a simple recursive function.
Simple Recursion
• https://inquiryintoinquiry.com/wp-content/uploads/2026/03/simple-recursion-fn-mn-fpn.pngFor example, the factorial function f(n) = n! has a definition in terms of the predecessor function p(n) = n-1 and the multiplier function m(j, k) = j∙k.
#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations -
Reflection On Recursion • 1.2
• https://inquiryintoinquiry.com/2026/04/06/reflection-on-recursion-1/Comment 3 —
If we discard from the idea of recursion what is not of its essence, we find recursion occurs when our understanding of one situation has recourse to our understanding of other situations.
Very typically, the object situation presents itself as complex, difficult, or unfamiliar while the resource situations are regarded as being better understood.
It must be appreciated, however, that any ranking of situations by level of understanding is contingent on the circumstances in view and may vary radically in alternate settings.
Comment 4 —
Recursion occurs more markedly in “syntactic recursion”, where the recursive process shows its character as such in the symbols of its syntactic expression.
A sense of the difference can be gained by looking at a case of “ostensible syntactic recursion”. (How much substance backs the ostentation is a subject we'll take up, maybe at length, but later …)
Consider the following diagram for the computation of a simple recursive function.
Simple Recursion
• https://inquiryintoinquiry.com/wp-content/uploads/2026/03/simple-recursion-fn-mn-fpn.pngFor example, the factorial function f(n) = n! has a definition in terms of the predecessor function p(n) = n-1 and the multiplier function m(j, k) = j∙k.
#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations -
Reflection On Recursion • 1.2
• https://inquiryintoinquiry.com/2026/04/06/reflection-on-recursion-1/Comment 3 —
If we discard from the idea of recursion what is not of its essence, we find recursion occurs when our understanding of one situation has recourse to our understanding of other situations.
Very typically, the object situation presents itself as complex, difficult, or unfamiliar while the resource situations are regarded as being better understood.
It must be appreciated, however, that any ranking of situations by level of understanding is contingent on the circumstances in view and may vary radically in alternate settings.
Comment 4 —
Recursion occurs more markedly in “syntactic recursion”, where the recursive process shows its character as such in the symbols of its syntactic expression.
A sense of the difference can be gained by looking at a case of “ostensible syntactic recursion”. (How much substance backs the ostentation is a subject we'll take up, maybe at length, but later …)
Consider the following diagram for the computation of a simple recursive function.
Simple Recursion
• https://inquiryintoinquiry.com/wp-content/uploads/2026/03/simple-recursion-fn-mn-fpn.pngFor example, the factorial function f(n) = n! has a definition in terms of the predecessor function p(n) = n-1 and the multiplier function m(j, k) = j∙k.
#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations -
Reflection On Recursion • 1.2
• https://inquiryintoinquiry.com/2026/04/06/reflection-on-recursion-1/Comment 3 —
If we discard from the idea of recursion what is not of its essence, we find recursion occurs when our understanding of one situation has recourse to our understanding of other situations.
Very typically, the object situation presents itself as complex, difficult, or unfamiliar while the resource situations are regarded as being better understood.
It must be appreciated, however, that any ranking of situations by level of understanding is contingent on the circumstances in view and may vary radically in alternate settings.
Comment 4 —
Recursion occurs more markedly in “syntactic recursion”, where the recursive process shows its character as such in the symbols of its syntactic expression.
A sense of the difference can be gained by looking at a case of “ostensible syntactic recursion”. (How much substance backs the ostentation is a subject we'll take up, maybe at length, but later …)
Consider the following diagram for the computation of a simple recursive function.
Simple Recursion
• https://inquiryintoinquiry.com/wp-content/uploads/2026/03/simple-recursion-fn-mn-fpn.pngFor example, the factorial function f(n) = n! has a definition in terms of the predecessor function p(n) = n-1 and the multiplier function m(j, k) = j∙k.
#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations -
Reflection On Recursion • 1.1
• https://inquiryintoinquiry.com/2026/04/06/reflection-on-recursion-1/Ongoing conversations with Dan Everett on Facebook have me backtracking to recurring questions about the relationship between formal language theory (as I once learned it) and the properties of natural languages as they are found occurring in the field.
A point of particular interest is the role of recursion in formal and natural languages, along with collateral questions about its role in the cognitive sciences at large.
It has taken me quite a while to bring my reflections up to the threshold of minimal coherence — and the inquiry remains ongoing — but it may catalyze the thinking process if I simply share what I've thought so far …
Comment 1 —
Recursion is where you find it — so, myself not being a natural language researcher, when someone who is says they don't find it in a given corpus I just take them at their word …
Comment 2 —
The question to which I keep returning has to do with the relationship between two ways we find recursion occurring.
One way I'd call “pragmatic recursion” — if I wanted to be precise and cover its full scope — since so many of its operations occur without conscious direction, but for now I'll defer to more familiar language, calling it “cognitive” or “conceptual” recursion.
Resources —
Inquiry Driven Systems • Inquiry Into Inquiry
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_OverviewReflective Interpretive Frameworks
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_10#RIF_1The Phenomenology of Reflection
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_11#The_Phenomenology_of_ReflectionHigher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_Relations#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations -
Reflection On Recursion • 1.1
• https://inquiryintoinquiry.com/2026/04/06/reflection-on-recursion-1/Ongoing conversations with Dan Everett on Facebook have me backtracking to recurring questions about the relationship between formal language theory (as I once learned it) and the properties of natural languages as they are found occurring in the field.
A point of particular interest is the role of recursion in formal and natural languages, along with collateral questions about its role in the cognitive sciences at large.
It has taken me quite a while to bring my reflections up to the threshold of minimal coherence — and the inquiry remains ongoing — but it may catalyze the thinking process if I simply share what I've thought so far …
Comment 1 —
Recursion is where you find it — so, myself not being a natural language researcher, when someone who is says they don't find it in a given corpus I just take them at their word …
Comment 2 —
The question to which I keep returning has to do with the relationship between two ways we find recursion occurring.
One way I'd call “pragmatic recursion” — if I wanted to be precise and cover its full scope — since so many of its operations occur without conscious direction, but for now I'll defer to more familiar language, calling it “cognitive” or “conceptual” recursion.
Resources —
Inquiry Driven Systems • Inquiry Into Inquiry
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_OverviewReflective Interpretive Frameworks
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_10#RIF_1The Phenomenology of Reflection
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_11#The_Phenomenology_of_ReflectionHigher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_Relations#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations -
Reflection On Recursion • 1.1
• https://inquiryintoinquiry.com/2026/04/06/reflection-on-recursion-1/Ongoing conversations with Dan Everett on Facebook have me backtracking to recurring questions about the relationship between formal language theory (as I once learned it) and the properties of natural languages as they are found occurring in the field.
A point of particular interest is the role of recursion in formal and natural languages, along with collateral questions about its role in the cognitive sciences at large.
It has taken me quite a while to bring my reflections up to the threshold of minimal coherence — and the inquiry remains ongoing — but it may catalyze the thinking process if I simply share what I've thought so far …
Comment 1 —
Recursion is where you find it — so, myself not being a natural language researcher, when someone who is says they don't find it in a given corpus I just take them at their word …
Comment 2 —
The question to which I keep returning has to do with the relationship between two ways we find recursion occurring.
One way I'd call “pragmatic recursion” — if I wanted to be precise and cover its full scope — since so many of its operations occur without conscious direction, but for now I'll defer to more familiar language, calling it “cognitive” or “conceptual” recursion.
Resources —
Inquiry Driven Systems • Inquiry Into Inquiry
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_OverviewReflective Interpretive Frameworks
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_10#RIF_1The Phenomenology of Reflection
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_11#The_Phenomenology_of_ReflectionHigher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_Relations#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations -
Reflection On Recursion • 1.1
• https://inquiryintoinquiry.com/2026/04/06/reflection-on-recursion-1/Ongoing conversations with Dan Everett on Facebook have me backtracking to recurring questions about the relationship between formal language theory (as I once learned it) and the properties of natural languages as they are found occurring in the field.
A point of particular interest is the role of recursion in formal and natural languages, along with collateral questions about its role in the cognitive sciences at large.
It has taken me quite a while to bring my reflections up to the threshold of minimal coherence — and the inquiry remains ongoing — but it may catalyze the thinking process if I simply share what I've thought so far …
Comment 1 —
Recursion is where you find it — so, myself not being a natural language researcher, when someone who is says they don't find it in a given corpus I just take them at their word …
Comment 2 —
The question to which I keep returning has to do with the relationship between two ways we find recursion occurring.
One way I'd call “pragmatic recursion” — if I wanted to be precise and cover its full scope — since so many of its operations occur without conscious direction, but for now I'll defer to more familiar language, calling it “cognitive” or “conceptual” recursion.
Resources —
Inquiry Driven Systems • Inquiry Into Inquiry
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_OverviewReflective Interpretive Frameworks
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_10#RIF_1The Phenomenology of Reflection
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_11#The_Phenomenology_of_ReflectionHigher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_Relations#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations -
Reflection On Recursion • 1.1
• https://inquiryintoinquiry.com/2026/04/06/reflection-on-recursion-1/Ongoing conversations with Dan Everett on Facebook have me backtracking to recurring questions about the relationship between formal language theory (as I once learned it) and the properties of natural languages as they are found occurring in the field.
A point of particular interest is the role of recursion in formal and natural languages, along with collateral questions about its role in the cognitive sciences at large.
It has taken me quite a while to bring my reflections up to the threshold of minimal coherence — and the inquiry remains ongoing — but it may catalyze the thinking process if I simply share what I've thought so far …
Comment 1 —
Recursion is where you find it — so, myself not being a natural language researcher, when someone who is says they don't find it in a given corpus I just take them at their word …
Comment 2 —
The question to which I keep returning has to do with the relationship between two ways we find recursion occurring.
One way I'd call “pragmatic recursion” — if I wanted to be precise and cover its full scope — since so many of its operations occur without conscious direction, but for now I'll defer to more familiar language, calling it “cognitive” or “conceptual” recursion.
Resources —
Inquiry Driven Systems • Inquiry Into Inquiry
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_OverviewReflective Interpretive Frameworks
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_10#RIF_1The Phenomenology of Reflection
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_11#The_Phenomenology_of_ReflectionHigher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_Relations#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations -
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_OverviewReflective Interpretive Frameworks
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_10#Reflective_Interpretive_FrameworksThe Phenomenology of Reflection
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_11#The_Phenomenology_of_ReflectionHigher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_RelationsNotes 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 -
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_OverviewReflective Interpretive Frameworks
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_10#Reflective_Interpretive_FrameworksThe Phenomenology of Reflection
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_11#The_Phenomenology_of_ReflectionHigher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_RelationsNotes 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 -
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_OverviewReflective Interpretive Frameworks
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_10#Reflective_Interpretive_FrameworksThe Phenomenology of Reflection
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_11#The_Phenomenology_of_ReflectionHigher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_RelationsNotes 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 -
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_OverviewReflective Interpretive Frameworks
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_10#Reflective_Interpretive_FrameworksThe Phenomenology of Reflection
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_11#The_Phenomenology_of_ReflectionHigher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_RelationsNotes 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 -
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_OverviewReflective Interpretive Frameworks
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_10#Reflective_Interpretive_FrameworksThe Phenomenology of Reflection
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_11#The_Phenomenology_of_ReflectionHigher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_RelationsNotes 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 -
Differential Logic • The Logic of Change and Difference
• https://inquiryintoinquiry.com/2026/03/14/differential-logic-the-logic-of-change-and-difference-a/“Differential logic is the logic of variation — the logic of change and difference.”
Differential logic is the component of logic whose object is the description of variation — the aspects of change, difference, distribution, and diversity — in universes of discourse subject to logical description. A definition as broad as that naturally incorporates any study of variation by way of mathematical models, but differential logic is especially charged with the qualitative aspects of variation pervading or preceding quantitative models.
To the extent a logical inquiry makes use of a formal system, its differential component treats the use of a “differential logical calculus” — a formal system with the expressive capacity to describe change and diversity in logical universes of discourse.
A simple case of a differential logical calculus is furnished by a “differential propositional calculus”, a formalism which augments ordinary propositional calculus in the same way the differential calculus of Leibniz and Newton augments the analytic geometry of Descartes.
See —
Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/Survey of Differential Logic
• https://inquiryintoinquiry.com/2025/05/03/survey-of-differential-logic-8/Differential Logic
• https://oeis.org/wiki/Differential_Logic_%E2%80%A2_OverviewDifferential Propositional Calculus
• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_OverviewDifferential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overviewcc: https://www.academia.edu/community/VXoNQ9
cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_and_Difference2#Peirce #Logic #Mathematics #LogicalGraphs #DifferentialLogic #DynamicSystems
#Inquiry #PropositionalCalculus #BooleanFunctions #BooleanDifferenceCalculus
#EquationalInference #MinimalNegationOperators #CalculusOfLogicalDifferences -
Differential Logic • The Logic of Change and Difference
• https://inquiryintoinquiry.com/2026/03/14/differential-logic-the-logic-of-change-and-difference-a/“Differential logic is the logic of variation — the logic of change and difference.”
Differential logic is the component of logic whose object is the description of variation — the aspects of change, difference, distribution, and diversity — in universes of discourse subject to logical description. A definition as broad as that naturally incorporates any study of variation by way of mathematical models, but differential logic is especially charged with the qualitative aspects of variation pervading or preceding quantitative models.
To the extent a logical inquiry makes use of a formal system, its differential component treats the use of a “differential logical calculus” — a formal system with the expressive capacity to describe change and diversity in logical universes of discourse.
A simple case of a differential logical calculus is furnished by a “differential propositional calculus”, a formalism which augments ordinary propositional calculus in the same way the differential calculus of Leibniz and Newton augments the analytic geometry of Descartes.
See —
Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/Survey of Differential Logic
• https://inquiryintoinquiry.com/2025/05/03/survey-of-differential-logic-8/Differential Logic
• https://oeis.org/wiki/Differential_Logic_%E2%80%A2_OverviewDifferential Propositional Calculus
• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_OverviewDifferential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overviewcc: https://www.academia.edu/community/VXoNQ9
cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_and_Difference2#Peirce #Logic #Mathematics #LogicalGraphs #DifferentialLogic #DynamicSystems
#Inquiry #PropositionalCalculus #BooleanFunctions #BooleanDifferenceCalculus
#EquationalInference #MinimalNegationOperators #CalculusOfLogicalDifferences -
Differential Logic • The Logic of Change and Difference
• https://inquiryintoinquiry.com/2026/03/14/differential-logic-the-logic-of-change-and-difference-a/“Differential logic is the logic of variation — the logic of change and difference.”
Differential logic is the component of logic whose object is the description of variation — the aspects of change, difference, distribution, and diversity — in universes of discourse subject to logical description. A definition as broad as that naturally incorporates any study of variation by way of mathematical models, but differential logic is especially charged with the qualitative aspects of variation pervading or preceding quantitative models.
To the extent a logical inquiry makes use of a formal system, its differential component treats the use of a “differential logical calculus” — a formal system with the expressive capacity to describe change and diversity in logical universes of discourse.
A simple case of a differential logical calculus is furnished by a “differential propositional calculus”, a formalism which augments ordinary propositional calculus in the same way the differential calculus of Leibniz and Newton augments the analytic geometry of Descartes.
See —
Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/Survey of Differential Logic
• https://inquiryintoinquiry.com/2025/05/03/survey-of-differential-logic-8/Differential Logic
• https://oeis.org/wiki/Differential_Logic_%E2%80%A2_OverviewDifferential Propositional Calculus
• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_OverviewDifferential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overviewcc: https://www.academia.edu/community/VXoNQ9
cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_and_Difference2#Peirce #Logic #Mathematics #LogicalGraphs #DifferentialLogic #DynamicSystems
#Inquiry #PropositionalCalculus #BooleanFunctions #BooleanDifferenceCalculus
#EquationalInference #MinimalNegationOperators #CalculusOfLogicalDifferences -
Differential Logic • The Logic of Change and Difference
• https://inquiryintoinquiry.com/2026/03/14/differential-logic-the-logic-of-change-and-difference-a/“Differential logic is the logic of variation — the logic of change and difference.”
Differential logic is the component of logic whose object is the description of variation — the aspects of change, difference, distribution, and diversity — in universes of discourse subject to logical description. A definition as broad as that naturally incorporates any study of variation by way of mathematical models, but differential logic is especially charged with the qualitative aspects of variation pervading or preceding quantitative models.
To the extent a logical inquiry makes use of a formal system, its differential component treats the use of a “differential logical calculus” — a formal system with the expressive capacity to describe change and diversity in logical universes of discourse.
A simple case of a differential logical calculus is furnished by a “differential propositional calculus”, a formalism which augments ordinary propositional calculus in the same way the differential calculus of Leibniz and Newton augments the analytic geometry of Descartes.
See —
Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/Survey of Differential Logic
• https://inquiryintoinquiry.com/2025/05/03/survey-of-differential-logic-8/Differential Logic
• https://oeis.org/wiki/Differential_Logic_%E2%80%A2_OverviewDifferential Propositional Calculus
• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_OverviewDifferential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overviewcc: https://www.academia.edu/community/VXoNQ9
cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_and_Difference2#Peirce #Logic #Mathematics #LogicalGraphs #DifferentialLogic #DynamicSystems
#Inquiry #PropositionalCalculus #BooleanFunctions #BooleanDifferenceCalculus
#EquationalInference #MinimalNegationOperators #CalculusOfLogicalDifferences -
Differential Logic • The Logic of Change and Difference
• https://inquiryintoinquiry.com/2026/03/14/differential-logic-the-logic-of-change-and-difference-a/“Differential logic is the logic of variation — the logic of change and difference.”
Differential logic is the component of logic whose object is the description of variation — the aspects of change, difference, distribution, and diversity — in universes of discourse subject to logical description. A definition as broad as that naturally incorporates any study of variation by way of mathematical models, but differential logic is especially charged with the qualitative aspects of variation pervading or preceding quantitative models.
To the extent a logical inquiry makes use of a formal system, its differential component treats the use of a “differential logical calculus” — a formal system with the expressive capacity to describe change and diversity in logical universes of discourse.
A simple case of a differential logical calculus is furnished by a “differential propositional calculus”, a formalism which augments ordinary propositional calculus in the same way the differential calculus of Leibniz and Newton augments the analytic geometry of Descartes.
See —
Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/Survey of Differential Logic
• https://inquiryintoinquiry.com/2025/05/03/survey-of-differential-logic-8/Differential Logic
• https://oeis.org/wiki/Differential_Logic_%E2%80%A2_OverviewDifferential Propositional Calculus
• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_OverviewDifferential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overviewcc: https://www.academia.edu/community/VXoNQ9
cc: https://www.researchgate.net/post/Differential_Logic_The_Logic_of_Change_and_Difference2#Peirce #Logic #Mathematics #LogicalGraphs #DifferentialLogic #DynamicSystems
#Inquiry #PropositionalCalculus #BooleanFunctions #BooleanDifferenceCalculus
#EquationalInference #MinimalNegationOperators #CalculusOfLogicalDifferences -
Differential Logic • 2.2
• https://inquiryintoinquiry.com/2026/02/06/differential-logic-2-b/Cactus Language for Propositional Logic (cont.)
The second kind of connective is a concatenated sequence of propositional expressions, written e₁ e₂ … eₖ₋₁ eₖ to mean all the propositions e₁, e₂, …, eₖ₋₁, eₖ are true, in short, their “logical conjunction” is true. An expression of that form is associated with a cactus structure called a “node” and is “painted” with the colors e₁, e₂, …, eₖ₋₁, eₖ as shown below.
Node Connective
• https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-ej-node-connective.jpgAll other propositional connectives can be obtained through combinations of the above two forms. As it happens, the parenthesized form is sufficient to define the concatenated form, making the latter formally dispensable, but it's convenient to maintain it as a concise way of expressing more complicated combinations of parenthesized forms. While working with expressions solely in propositional calculus, it's easiest to use plain parentheses for logical connectives. In contexts where ordinary parentheses are needed for other purposes an alternate typeface (…) may be used for the logical operators.
Resources —
Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/Minimal Negation Operator
• https://oeis.org/wiki/Minimal_negation_operatorSurvey of Differential Logic
• https://inquiryintoinquiry.com/2025/05/03/survey-of-differential-logic-8/Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/#Peirce #Logic #Mathematics #LogicalGraphs #DifferentialLogic #DynamicSystems
#Inquiry #PropositionalCalculus #BooleanFunctions #BooleanDifferenceCalculus
#EquationalInference #MinimalNegationOperators #CalculusOfLogicalDifferences -
Differential Logic • 2.2
• https://inquiryintoinquiry.com/2026/02/06/differential-logic-2-b/Cactus Language for Propositional Logic (cont.)
The second kind of connective is a concatenated sequence of propositional expressions, written e₁ e₂ … eₖ₋₁ eₖ to mean all the propositions e₁, e₂, …, eₖ₋₁, eₖ are true, in short, their “logical conjunction” is true. An expression of that form is associated with a cactus structure called a “node” and is “painted” with the colors e₁, e₂, …, eₖ₋₁, eₖ as shown below.
Node Connective
• https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-ej-node-connective.jpgAll other propositional connectives can be obtained through combinations of the above two forms. As it happens, the parenthesized form is sufficient to define the concatenated form, making the latter formally dispensable, but it's convenient to maintain it as a concise way of expressing more complicated combinations of parenthesized forms. While working with expressions solely in propositional calculus, it's easiest to use plain parentheses for logical connectives. In contexts where ordinary parentheses are needed for other purposes an alternate typeface (…) may be used for the logical operators.
Resources —
Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/Minimal Negation Operator
• https://oeis.org/wiki/Minimal_negation_operatorSurvey of Differential Logic
• https://inquiryintoinquiry.com/2025/05/03/survey-of-differential-logic-8/Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/#Peirce #Logic #Mathematics #LogicalGraphs #DifferentialLogic #DynamicSystems
#Inquiry #PropositionalCalculus #BooleanFunctions #BooleanDifferenceCalculus
#EquationalInference #MinimalNegationOperators #CalculusOfLogicalDifferences -
Differential Logic • 2.2
• https://inquiryintoinquiry.com/2026/02/06/differential-logic-2-b/Cactus Language for Propositional Logic (cont.)
The second kind of connective is a concatenated sequence of propositional expressions, written e₁ e₂ … eₖ₋₁ eₖ to mean all the propositions e₁, e₂, …, eₖ₋₁, eₖ are true, in short, their “logical conjunction” is true. An expression of that form is associated with a cactus structure called a “node” and is “painted” with the colors e₁, e₂, …, eₖ₋₁, eₖ as shown below.
Node Connective
• https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-ej-node-connective.jpgAll other propositional connectives can be obtained through combinations of the above two forms. As it happens, the parenthesized form is sufficient to define the concatenated form, making the latter formally dispensable, but it's convenient to maintain it as a concise way of expressing more complicated combinations of parenthesized forms. While working with expressions solely in propositional calculus, it's easiest to use plain parentheses for logical connectives. In contexts where ordinary parentheses are needed for other purposes an alternate typeface (…) may be used for the logical operators.
Resources —
Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/Minimal Negation Operator
• https://oeis.org/wiki/Minimal_negation_operatorSurvey of Differential Logic
• https://inquiryintoinquiry.com/2025/05/03/survey-of-differential-logic-8/Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/#Peirce #Logic #Mathematics #LogicalGraphs #DifferentialLogic #DynamicSystems
#Inquiry #PropositionalCalculus #BooleanFunctions #BooleanDifferenceCalculus
#EquationalInference #MinimalNegationOperators #CalculusOfLogicalDifferences -
Differential Logic • 2.2
• https://inquiryintoinquiry.com/2026/02/06/differential-logic-2-b/Cactus Language for Propositional Logic (cont.)
The second kind of connective is a concatenated sequence of propositional expressions, written e₁ e₂ … eₖ₋₁ eₖ to mean all the propositions e₁, e₂, …, eₖ₋₁, eₖ are true, in short, their “logical conjunction” is true. An expression of that form is associated with a cactus structure called a “node” and is “painted” with the colors e₁, e₂, …, eₖ₋₁, eₖ as shown below.
Node Connective
• https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-ej-node-connective.jpgAll other propositional connectives can be obtained through combinations of the above two forms. As it happens, the parenthesized form is sufficient to define the concatenated form, making the latter formally dispensable, but it's convenient to maintain it as a concise way of expressing more complicated combinations of parenthesized forms. While working with expressions solely in propositional calculus, it's easiest to use plain parentheses for logical connectives. In contexts where ordinary parentheses are needed for other purposes an alternate typeface (…) may be used for the logical operators.
Resources —
Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/Minimal Negation Operator
• https://oeis.org/wiki/Minimal_negation_operatorSurvey of Differential Logic
• https://inquiryintoinquiry.com/2025/05/03/survey-of-differential-logic-8/Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/#Peirce #Logic #Mathematics #LogicalGraphs #DifferentialLogic #DynamicSystems
#Inquiry #PropositionalCalculus #BooleanFunctions #BooleanDifferenceCalculus
#EquationalInference #MinimalNegationOperators #CalculusOfLogicalDifferences -
Differential Logic • 2.2
• https://inquiryintoinquiry.com/2026/02/06/differential-logic-2-b/Cactus Language for Propositional Logic (cont.)
The second kind of connective is a concatenated sequence of propositional expressions, written e₁ e₂ … eₖ₋₁ eₖ to mean all the propositions e₁, e₂, …, eₖ₋₁, eₖ are true, in short, their “logical conjunction” is true. An expression of that form is associated with a cactus structure called a “node” and is “painted” with the colors e₁, e₂, …, eₖ₋₁, eₖ as shown below.
Node Connective
• https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-ej-node-connective.jpgAll other propositional connectives can be obtained through combinations of the above two forms. As it happens, the parenthesized form is sufficient to define the concatenated form, making the latter formally dispensable, but it's convenient to maintain it as a concise way of expressing more complicated combinations of parenthesized forms. While working with expressions solely in propositional calculus, it's easiest to use plain parentheses for logical connectives. In contexts where ordinary parentheses are needed for other purposes an alternate typeface (…) may be used for the logical operators.
Resources —
Logic Syllabus
• https://inquiryintoinquiry.com/logic-syllabus/Minimal Negation Operator
• https://oeis.org/wiki/Minimal_negation_operatorSurvey of Differential Logic
• https://inquiryintoinquiry.com/2025/05/03/survey-of-differential-logic-8/Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2025/05/02/survey-of-animated-logical-graphs-8/#Peirce #Logic #Mathematics #LogicalGraphs #DifferentialLogic #DynamicSystems
#Inquiry #PropositionalCalculus #BooleanFunctions #BooleanDifferenceCalculus
#EquationalInference #MinimalNegationOperators #CalculusOfLogicalDifferences