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#minimalnegationoperators — Public Fediverse posts

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  1. Reflective Interpretive Frameworks • Incident 2
    inquiryintoinquiry.com/2026/08

    Re: Terence Tao • Modular Arithmetic Challenge
    terrytao.wordpress.com/2026/06
    competition.sair.foundation/co

    The Modular Arithmetic Challenge asks a simple question:

    • Can a neural network learn to do modular multiplication efficiently?

    Incidental Reflection 1 —

    There are alternative models of neural networks which do not depend on threshold neurons and endlessly fiddling with weights.

    Incidental Reflection 2 —

    The series of three blog posts linked below present a case study comparing two ways of handling a classic example from the Parallel Distributed Processing paradigm, namely, the “Jets and Sharks” database problem, first taking up the original treatment by McClelland and Rumelhart and then proceeding according to a program I developed for propositional logic modeling. The latter method makes use of ideas from Grossberg's competition‑cooperation and winner‑take‑all dynamics, but is purely propositional‑logic based, involving no extraneous weights.

    Theme One Program • Jets and Sharks
    (1) inquiryintoinquiry.com/2024/06
    (2) inquiryintoinquiry.com/2024/06
    (3) inquiryintoinquiry.com/2024/06

    Resources —

    Survey of Theme One Program
    inquiryintoinquiry.com/2025/05

    Differential Analytic Turing Automata
    oeis.org/wiki/Differential_Ana

    #Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
    #Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations
    #PropositionalModels #MinimalNegationOperators #DifferentialAnalyticTuringAutomata
    #CactusGraphs #CactusLanguage #DifferentialLogic #NeuralNetworks #SequenceLearning

  2. Reflective Interpretive Frameworks • Incident 2
    inquiryintoinquiry.com/2026/08

    Re: Terence Tao • Modular Arithmetic Challenge
    terrytao.wordpress.com/2026/06
    competition.sair.foundation/co

    The Modular Arithmetic Challenge asks a simple question:

    • Can a neural network learn to do modular multiplication efficiently?

    Incidental Reflection 1 —

    There are alternative models of neural networks which do not depend on threshold neurons and endlessly fiddling with weights.

    Incidental Reflection 2 —

    The series of three blog posts linked below present a case study comparing two ways of handling a classic example from the Parallel Distributed Processing paradigm, namely, the “Jets and Sharks” database problem, first taking up the original treatment by McClelland and Rumelhart and then proceeding according to a program I developed for propositional logic modeling. The latter method makes use of ideas from Grossberg's competition‑cooperation and winner‑take‑all dynamics, but is purely propositional‑logic based, involving no extraneous weights.

    Theme One Program • Jets and Sharks
    (1) inquiryintoinquiry.com/2024/06
    (2) inquiryintoinquiry.com/2024/06
    (3) inquiryintoinquiry.com/2024/06

    Resources —

    Survey of Theme One Program
    inquiryintoinquiry.com/2025/05

    Differential Analytic Turing Automata
    oeis.org/wiki/Differential_Ana

    #Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
    #Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations
    #PropositionalModels #MinimalNegationOperators #DifferentialAnalyticTuringAutomata
    #CactusGraphs #CactusLanguage #DifferentialLogic #NeuralNetworks #SequenceLearning

  3. Reflective Interpretive Frameworks • Incident 2
    inquiryintoinquiry.com/2026/08

    Re: Terence Tao • Modular Arithmetic Challenge
    terrytao.wordpress.com/2026/06
    competition.sair.foundation/co

    The Modular Arithmetic Challenge asks a simple question:

    • Can a neural network learn to do modular multiplication efficiently?

    Incidental Reflection 1 —

    There are alternative models of neural networks which do not depend on threshold neurons and endlessly fiddling with weights.

    Incidental Reflection 2 —

    The series of three blog posts linked below present a case study comparing two ways of handling a classic example from the Parallel Distributed Processing paradigm, namely, the “Jets and Sharks” database problem, first taking up the original treatment by McClelland and Rumelhart and then proceeding according to a program I developed for propositional logic modeling. The latter method makes use of ideas from Grossberg's competition‑cooperation and winner‑take‑all dynamics, but is purely propositional‑logic based, involving no extraneous weights.

    Theme One Program • Jets and Sharks
    (1) inquiryintoinquiry.com/2024/06
    (2) inquiryintoinquiry.com/2024/06
    (3) inquiryintoinquiry.com/2024/06

    Resources —

    Survey of Theme One Program
    inquiryintoinquiry.com/2025/05

    Differential Analytic Turing Automata
    oeis.org/wiki/Differential_Ana

    #Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
    #Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations
    #PropositionalModels #MinimalNegationOperators #DifferentialAnalyticTuringAutomata
    #CactusGraphs #CactusLanguage #DifferentialLogic #NeuralNetworks #SequenceLearning

  4. Reflective Interpretive Frameworks • Incident 2
    inquiryintoinquiry.com/2026/08

    Re: Terence Tao • Modular Arithmetic Challenge
    terrytao.wordpress.com/2026/06
    competition.sair.foundation/co

    The Modular Arithmetic Challenge asks a simple question:

    • Can a neural network learn to do modular multiplication efficiently?

    Incidental Reflection 1 —

    There are alternative models of neural networks which do not depend on threshold neurons and endlessly fiddling with weights.

    Incidental Reflection 2 —

    The series of three blog posts linked below present a case study comparing two ways of handling a classic example from the Parallel Distributed Processing paradigm, namely, the “Jets and Sharks” database problem, first taking up the original treatment by McClelland and Rumelhart and then proceeding according to a program I developed for propositional logic modeling. The latter method makes use of ideas from Grossberg's competition‑cooperation and winner‑take‑all dynamics, but is purely propositional‑logic based, involving no extraneous weights.

    Theme One Program • Jets and Sharks
    (1) inquiryintoinquiry.com/2024/06
    (2) inquiryintoinquiry.com/2024/06
    (3) inquiryintoinquiry.com/2024/06

    Resources —

    Survey of Theme One Program
    inquiryintoinquiry.com/2025/05

    Differential Analytic Turing Automata
    oeis.org/wiki/Differential_Ana

    #Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
    #Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations
    #PropositionalModels #MinimalNegationOperators #DifferentialAnalyticTuringAutomata
    #CactusGraphs #CactusLanguage #DifferentialLogic #NeuralNetworks #SequenceLearning

  5. Reflective Interpretive Frameworks • Incident 2
    inquiryintoinquiry.com/2026/08

    Re: Terence Tao • Modular Arithmetic Challenge
    terrytao.wordpress.com/2026/06
    competition.sair.foundation/co

    The Modular Arithmetic Challenge asks a simple question:

    • Can a neural network learn to do modular multiplication efficiently?

    Incidental Reflection 1 —

    There are alternative models of neural networks which do not depend on threshold neurons and endlessly fiddling with weights.

    Incidental Reflection 2 —

    The series of three blog posts linked below present a case study comparing two ways of handling a classic example from the Parallel Distributed Processing paradigm, namely, the “Jets and Sharks” database problem, first taking up the original treatment by McClelland and Rumelhart and then proceeding according to a program I developed for propositional logic modeling. The latter method makes use of ideas from Grossberg's competition‑cooperation and winner‑take‑all dynamics, but is purely propositional‑logic based, involving no extraneous weights.

    Theme One Program • Jets and Sharks
    (1) inquiryintoinquiry.com/2024/06
    (2) inquiryintoinquiry.com/2024/06
    (3) inquiryintoinquiry.com/2024/06

    Resources —

    Survey of Theme One Program
    inquiryintoinquiry.com/2025/05

    Differential Analytic Turing Automata
    oeis.org/wiki/Differential_Ana

    #Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
    #Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations
    #PropositionalModels #MinimalNegationOperators #DifferentialAnalyticTuringAutomata
    #CactusGraphs #CactusLanguage #DifferentialLogic #NeuralNetworks #SequenceLearning

  6. Transformations of Logical Graphs • Discussion 1
    inquiryintoinquiry.com/2024/05

    Re: Laws of Form
    groups.io/g/lawsofform/topic/t

    Mauro Bertani
    groups.io/g/lawsofform/message

    Dear Mauro,

    The couple of pages linked below give the clearest and quickest introduction I've been able to manage so far when it comes to the elements of logical graphs, at least, in the way I've come to understand them. The first page gives a lot of detail by way of motivation and computational implementation, so you could easily put that off till you feel a need for it. The second page lays out the precise axioms or initials I use — the first algebraic axiom varies a bit from Spencer Brown for a better fit with C.S. Peirce — and also shows the parallels between the dual interpretations.

    Logical Graphs • First Impressions
    inquiryintoinquiry.com/2023/08

    Logical Graphs • Formal Development
    inquiryintoinquiry.com/2023/09

    Additional Resources —

    Logic Syllabus
    inquiryintoinquiry.com/logic-s

    Survey of Animated Logical Graphs
    inquiryintoinquiry.com/2024/03

    Survey of Semiotics, Semiosis, Sign Relations
    inquiryintoinquiry.com/2024/01

    #Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
    #SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
    #CactusSyntax #MinimalNegationOperators #MathematicalDuality #Form

  7. Mathematical Duality in Logical Graphs • 1.2
    inquiryintoinquiry.com/2024/05

    It was in this context that Peirce's systems of logical graphs developed, issuing in dual interpretations of the same formal axioms which Peirce referred to as “entitative graphs” and “existential graphs”, respectively. He developed only the existential interpretation to any great extent, since the extension from propositional to relational calculus appeared more natural in that case, but whether there is any logical or mathematical reason for the symmetry to break at that point is a good question for further research.

    Resources —

    Duality Indicating Unity
    inquiryintoinquiry.com/2013/01

    C.S. Peirce • Logic of Number
    inquiryintoinquiry.com/2012/09

    C.S. Peirce • Syllabus • Selection 1
    inquiryintoinquiry.com/2014/08

    References —

    • Peirce, C.S., [Logic of Number — Le Fevre] (MS 229), in Carolyn Eisele (ed., 1976), The New Elements of Mathematics by Charles S. Peirce, vol. 2, 592–595.

    • Spencer Brown, G. (1969), Laws of Form, George Allen and Unwin, London, UK.

    #Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
    #SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
    #CactusSyntax #MinimalNegationOperators #MathematicalDuality #Form

  8. Mathematical Duality in Logical Graphs • 1.1
    inquiryintoinquiry.com/2024/05

    “All other sciences without exception depend upon the principles of mathematics; and mathematics borrows nothing from them but hints.”

    — C.S. Peirce • “Logic of Number”

    “A principal intention of this essay is to separate what are known as algebras of logic from the subject of logic, and to re‑align them with mathematics.”

    — G. Spencer Brown • “Laws of Form”

    The duality between entitative and existential interpretations of logical graphs tells us something important about the relation between logic and mathematics. It tells us the mathematical forms giving structure to reasoning are deeper and more abstract at once than their logical interpretations.

    A formal duality points to a more encompassing unity, founding a calculus of forms whose expressions can be read in alternate ways by switching the meanings assigned to a pair of primitive terms. Spencer Brown's mathematical approach to “Laws of Form” and the whole of Peirce's work on the mathematics of logic shows both thinkers were deeply aware of this principle.

    Peirce explored a variety of dualities in logic which he treated on analogy with the dualities in projective geometry. This gave rise to formal systems where the initial constants, and thus their geometric and graph‑theoretic representations, had no uniquely fixed meanings but could be given dual interpretations in logic.

    #Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
    #SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
    #CactusSyntax #MinimalNegationOperators #MathematicalDuality #Form

  9. Operator Variables in Logical Graphs • Discussion 1
    inquiryintoinquiry.com/2024/04

    Re: Operator Variables in Logical Graphs • 1
    inquiryintoinquiry.com/2024/04

    Re: Academia.edu • Stephen Duplantier
    academia.edu/community/Lxn1Ww?

    SD:
    ❝The best way for me to read Peirce is as if he was writing poetry. So if his algebra is poetry — I imagine him approving of the approach since he taught me abduction in the first place — there is room to wander. With this, I venture the idea that his “wide field” is a local algebraic geography far from the tended garden. There, where weeds and wild things grow and hybridize are the non‑dichotomic mathematics.❞

    Stephen,

    “Abdeuces Are Wild”, as they say, maybe not today, maybe not tomorrow, but soon …

    As far as my own guess, and a lot of my wandering in pursuit of it goes, I'd venture Peirce's field of vision opens up not so much from dichotomic to trichotomic domains of value as from dyadic to triadic relations, and all that with particular significance into the medium of reflection afforded by triadic sign relations.

    Resources —

    Logic Syllabus
    inquiryintoinquiry.com/logic-s

    Semeiotic
    oeis.org/wiki/Semeiotic

    Sign Relations
    oeis.org/wiki/Sign_relation

    Triadic Relations
    oeis.org/wiki/Triadic_relation

    #Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
    #SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
    #CactusSyntax #MinimalNegationOperators #LogicalOperatorVariables

  10. Operator Variables in Logical Graphs • 1.2
    inquiryintoinquiry.com/2024/04

    Consider De Morgan's rules:

    • ¬(A ∧ B) = ¬A ∨ ¬B

    • ¬(A ∨ B) = ¬A ∧ ¬B

    The common form exhibited by the two rules could be captured in a single formula by taking “o₁” and “o₂” as variable names ranging over a family of logical operators, then asking what substitutions for o₁ and o₂ would satisfy the following equation.

    • ¬(A o₁ B) = ¬A o₂ ¬B

    We already know two solutions to this “operator equation”, namely, (o₁, o₂) = (∧, ∨) and (o₁, o₂) = (∨, ∧). Wouldn't it be just like Peirce to ask if there are others?

    Having broached the subject of “logical operator variables”, I will leave it for now in the same way Peirce himself did:

    ❝I shall not further enlarge upon this matter at this point, although the conception mentioned opens a wide field; because it cannot be set in its proper light without overstepping the limits of dichotomic mathematics.❞ (Peirce, CP 4.306).

    Further exploration of operator variables and operator invariants treads on grounds traditionally known as second intentional logic and “opens a wide field”, as Peirce says. For now, however, I will tend to that corner of the field where our garden variety logical graphs grow, observing the ways in which operative variations and operative themes naturally develop on those grounds.

    #Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
    #SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
    #CactusSyntax #MinimalNegationOperators #LogicalOperatorVariables

  11. Operator Variables in Logical Graphs • 1.1
    inquiryintoinquiry.com/2024/04

    In lieu of a field study requirement for my bachelor's degree I spent two years in various state and university libraries reading everything I could find by and about Peirce, poring most memorably through reels of microfilmed Peirce manuscripts Michigan State had at the time, all in trying to track down some hint of a clue to a puzzling passage in Peirce's “Simplest Mathematics”, most acutely coming to a head with that bizarre line of type at CP 4.306, which the editors of Peirce's “Collected Papers”, no doubt compromised by the typographer's reluctance to cut new symbols, transmogrified into a script more cryptic than even the manuscript's original hieroglyphic.

    I found one key to the mystery in Peirce's use of “operator variables”, which he and his students Christine Ladd‑Franklin and O.H. Mitchell explored in depth. I will shortly discuss that theme as it affects logical graphs but it may be useful to give a shorter and sweeter explanation of how the basic idea typically arises in common logical practice.

    #Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
    #SpencerBrown #LawsOfForm #BooleanFunctions #PropositionalCalculus
    #CactusSyntax #MinimalNegationOperators #LogicalOperatorVariables

  12. Differential Logic • The Logic of Change and Difference
    inquiryintoinquiry.com/2023/08

    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, for example, the aspects of change, difference, distribution, and diversity, in universes of discourse subject to qualitative logical description. In its formalization, differential logic treats the principles governing the use of a “differential logical calculus”, in other words, 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. This augments ordinary propositional calculus in the same way the differential calculus of Leibniz and Newton augments the analytic geometry of Descartes.

    Resources —

    Differential Logic
    oeis.org/wiki/Differential_Log
    • Part 1 ( oeis.org/wiki/Differential_Log )
    • Part 2 ( oeis.org/wiki/Differential_Log )
    • Part 3 ( oeis.org/wiki/Differential_Log )

    Differential Propositional Calculus
    oeis.org/wiki/Differential_Pro
    • Part 1 ( oeis.org/wiki/Differential_Pro )
    • Part 2 ( oeis.org/wiki/Differential_Pro )

    Differential Logic and Dynamic Systems
    oeis.org/wiki/Differential_Log
    • Part 1 ( oeis.org/wiki/Differential_Log )
    • Part 2 ( oeis.org/wiki/Differential_Log )
    • Part 3 ( oeis.org/wiki/Differential_Log )
    • Part 4 ( oeis.org/wiki/Differential_Log )
    • Part 5 ( oeis.org/wiki/Differential_Log )

    #Peirce #Logic #LogicalGraphs #DifferentialLogic #DiscreteDynamicalSystems
    #Leibniz #BooleanFunctions #BooleanDifferenceCalculus #QualitativeDynamics
    #DifferentialPropositions #MinimalNegationOperators #NeuralNetworkSystems