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Jon Awbrey · @Inquiry

254 followers · 1811 posts · Server mathstodon.xyz

Inquiry Into Inquiry - Differential Logic

Differential Logic • The Logic of Change and Difference
• https://inquiryintoinquiry.com/2023/08/22/differential-logic-%ce%b1/

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
• https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Overview
• Part 1 ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1 )
• Part 2 ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_2 )
• Part 3 ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_3 )

Differential Propositional Calculus
• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Overview
• Part 1 ( https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_1 )
• Part 2 ( https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2 )

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overview
• Part 1 ( https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1 )
• Part 2 ( https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2 )
• Part 3 ( https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_3 )
• Part 4 ( https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_4 )
• Part 5 ( https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_5 )

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

#neuralnetworksystems #MinimalNegationOperators #differentialpropositions #QualitativeDynamics #booleandifferencecalculus #BooleanFunctions #leibniz #DiscreteDynamicalSystems #DifferentialLogic #LogicalGraphs #logic #Peirce

Last updated 1 year ago

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Jon Awbrey · @Inquiry

226 followers · 1602 posts · Server mathstodon.xyz

Survey of Differential Logic • 5
• https://inquiryintoinquiry.com/2023/04/25/survey-of-differential-logic-5/

This is a Survey of work in progress on Differential Logic, resources under development toward a more systematic treatment.

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.

Please follow the above link for the full set of resources.
Articles and blog series on the core ideas are linked below.

Differential Propositional Calculus
• https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Overview
1 https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_1
2 https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_2
• https://inquiryintoinquiry.com/2020/02/16/differential-propositional-calculus-overview/

Differential Logic
• https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Overview
1 https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1
2 https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_2
3 https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_3
• https://inquiryintoinquiry.com/2020/03/20/differential-logic-overview/

Differential Logic and Dynamic Systems
• https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overview
1 https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1
2 https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2
3 https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_3
4 https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_4
5 https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_5
• https://inquiryintoinquiry.com/2023/03/04/differential-logic-and-dynamic-systems-overview-2/

#Peirce #Logic #LogicalGraphs #DifferentialLogic #DynamicSystems
#BooleanFunctions #BooleanDifferenceCalculus #QualitativePhysics
#CactusCalculus #MinimalNegationOperators #NeuralNetworkSystems

#neuralnetworksystems #MinimalNegationOperators #CactusCalculus #qualitativephysics #booleandifferencecalculus #BooleanFunctions #dynamicsystems #DifferentialLogic #LogicalGraphs #logic #Peirce

Last updated 2 years ago

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Jon Awbrey · @Inquiry

182 followers · 1112 posts · Server mathstodon.xyz

@xameer

Cf. #DifferentialLogic • Discussion 3
• https://inquiryintoinquiry.com/2020/06/17/differential-logic-discussion-3/

#Physics once had a #FrameProblem (#Complexity of #DynamicUpdating) long before #AI did but physics learned to reduce complexity through the use of #DifferentialEquations and #GroupSymmetries (combined in #LieGroups). One of the promising features of #MinimalNegationOperators is their relationship to #DifferentialOperators. So I’ve been looking into that. Here’s a link, a bit in medias res, but what I’ve got for now.

#ai #dynamicupdating #complexity #physics #differentialoperators #MinimalNegationOperators #liegroups #groupsymmetries #differentialequations #frameproblem #DifferentialLogic

Last updated 2 years ago

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Jon Awbrey · @Inquiry

175 followers · 1069 posts · Server mathstodon.xyz

@bblfish @hochstenbach @josd

Here's the skinny on #MinimalNegationOperators —

• https://mathstodon.xyz/@Inquiry/109806663808536523

Minimal negation operators are a family of logical operators or #BooleanFunctions \(\nu(),\ \nu(x),\ \nu(x,y),\ \nu(x,y,z),\) etc.

In the so-called #ExistentialInterpretation of the brand of #LogicalGraphs I'll be using, \(\nu(x_1, \ldots, x_k)\) says exactly one of the \(x_i\) is equal to \(0\), that is, false.

#LogicalGraphs #ExistentialInterpretation #BooleanFunctions #MinimalNegationOperators

Last updated 2 years ago

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Jon Awbrey · @Inquiry

173 followers · 1043 posts · Server mathstodon.xyz

#MinimalNegationOperators
• https://oeis.org/wiki/Minimal_negation_operator

#Peirce #LogicalGraphs #MinimalNegations
#BooleanFunctions #PropositionalCalculus
#CactusGraphs #CactusLanguage #CactusSyntax

#cactussyntax #CactusLanguage #CactusGraphs #PropositionalCalculus #BooleanFunctions #minimalnegations #LogicalGraphs #Peirce #MinimalNegationOperators

Last updated 2 years ago

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Jon Awbrey · @Inquiry

170 followers · 1027 posts · Server mathstodon.xyz

@hochstenbach @josd

A program I worked on all through the 80s implemented a propositional modeler based on #Peirce's #LogicalGraphs, improving the efficiency of the Alpha level through the use of #MinimalNegationOperators. There's a collection of articles, blog posts, and group discussions about that linked on the following page.

#ThemeOneProgram • #SurveyPage
• https://inquiryintoinquiry.com/2022/06/12/survey-of-theme-one-program-4/

#surveypage #ThemeOneProgram #MinimalNegationOperators #LogicalGraphs #Peirce

Last updated 2 years ago

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Jon Awbrey · @Inquiry

157 followers · 858 posts · Server mathstodon.xyz

@bentnib

#IfThatDoesntWorkTry
#MinimalNegationOperators
• https://oeis.org/wiki/Minimal_negation_operator

#MinimalNegationOperators #ifthatdoesntworktry

Last updated 2 years ago

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Jon Awbrey · @Inquiry

157 followers · 858 posts · Server mathstodon.xyz

@xameer

#IfThatDoesntWorkTry
#MinimalNegationOperators
• https://oeis.org/wiki/Minimal_negation_operator

#MinimalNegationOperators #ifthatdoesntworktry

Last updated 2 years ago

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Jon Awbrey · @Inquiry

96 followers · 407 posts · Server mathstodon.xyz

@albertcardona @NicoleCRust @complexsystems @cogneurophys @PessoaBrain @SussilloDavid @carlosbrody @Neurograce @neuralreckoning @tyrell_turing @DrYohanJohn @cian @WiringtheBrain @tdverstynen @neuralengine

I'll just tag this for later when I've had a chance to collect my thoughts.

#WinnerTakeAll #Grossberg #CompetitionCooperation
#McClelland #Rumelhart #ParallelDistributedProcessing
#CactusLanguage #LogicalGraphs #MinimalNegationOperators

#MinimalNegationOperators #LogicalGraphs #CactusLanguage #paralleldistributedprocessing #Rumelhart #McClelland #competitioncooperation #grossberg #winnertakeall

Last updated 2 years ago

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Jon Awbrey · @Inquiry

82 followers · 269 posts · Server mathstodon.xyz

Open media

#ThemeOneProgram • #JetsAndSharks 1.4
• https://inquiryintoinquiry.com/2022/08/25/theme-one-program-jets-and-sharks-1/

Displayed below is the text expression of a #TraversalString which Theme One parses into a #CactusGraph #DataStructure in computer memory. The cactus graph represents a single #LogicalFormula in #PropositionalCalculus and this proposition embodies all the #LogicalConstraints defining the Jets and Sharks #DataBase.

• https://inquiryintoinquiry.files.wordpress.com/2022/08/theme-one-guide-e280a2-jets-and-sharks-e280a2-log-file.png

#Logic #LogicalGraphs #Peirce
#CactusSyntax #LogicalCacti
#MinimalNegationOperators

#MinimalNegationOperators #logicalcacti #cactussyntax #Peirce #LogicalGraphs #logic #database #logicalconstraints #PropositionalCalculus #logicalformula #datastructure #cactusgraph #traversalstring #JetsAndSharks #ThemeOneProgram

Last updated 2 years ago

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Jon Awbrey · @Inquiry

81 followers · 259 posts · Server mathstodon.xyz

Inquiry Into Inquiry - Theme One Program • Jets and Sharks 1

#ThemeOneProgram • #JetsAndSharks 1.3
• https://inquiryintoinquiry.com/2022/08/25/theme-one-program-jets-and-sharks-1/

The manner of representation may be illustrated by transcribing a well-known example from the #ParallelDistributedProcessing literature (#McClelland and #Rumelhart 1988) and working through a couple of the associated exercises as translated into #LogicalGraphs.

#Logic #Peirce #Semiotics #Semiosis
#Grossberg #CompetitionCooperation
#GraphTheory #ModelTheory #ProofTheory
#LogicalCacti #MinimalNegationOperators

#MinimalNegationOperators #logicalcacti #ProofTheory #modeltheory #graphtheory #competitioncooperation #grossberg #Semiosis #semiotics #Peirce #logic #LogicalGraphs #Rumelhart #McClelland #paralleldistributedprocessing #JetsAndSharks #ThemeOneProgram

Last updated 2 years ago

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Jon Awbrey · @Inquiry

81 followers · 258 posts · Server mathstodon.xyz

#ThemeOneProgram • #JetsAndSharks 1.2
• https://inquiryintoinquiry.com/2022/08/25/theme-one-program-jets-and-sharks-1/

One way to do this is to interpret the blank or #UnmarkedState as the #RestingState of a #NeuralPool, the bound or #MarkedState as its #ActivatedState, and to represent a mutually inhibitory pool of #Neurons \(a,b,c\) by the proposition \(\texttt{(}a\texttt{,}b\texttt{,}c\texttt{)}.\)

#Logic #LogicalGraphs #Peirce
#Grossberg #McClelland #Rumelhart
#GraphTheory #ModelTheory #ProofTheory
#LogicalCacti #MinimalNegationOperators

#MinimalNegationOperators #logicalcacti #ProofTheory #modeltheory #graphtheory #Rumelhart #McClelland #grossberg #Peirce #LogicalGraphs #logic #neurons #activatedstate #markedstate #neuralpool #restingstate #unmarkedstate #JetsAndSharks #ThemeOneProgram

Last updated 2 years ago

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Jon Awbrey · @Inquiry

80 followers · 256 posts · Server mathstodon.xyz

#ThemeOneProgram • #JetsAndSharks 1.1
• https://inquiryintoinquiry.com/2022/08/25/theme-one-program-jets-and-sharks-1/

Example 5. Jets and Sharks

The #PropositionalCalculus based on #MinimalNegationOperators can be interpreted in a way resembling the logic of #ActivationStates and #CompetitionConstraints in one class of #NeuralNetwork models.

#Logic #LogicalGraphs
#Peirce #Semiotics #Semiosis
#Grossberg #McClelland #Rumelhart
#ParallelDistributedProcessing #PDP
#GraphTheory #ModelTheory #ProofTheory

#ProofTheory #modeltheory #graphtheory #pdp #paralleldistributedprocessing #Rumelhart #McClelland #grossberg #Semiosis #semiotics #Peirce #LogicalGraphs #logic #neuralnetwork #competitionconstraints #activationstates #MinimalNegationOperators #PropositionalCalculus #JetsAndSharks #ThemeOneProgram

Last updated 2 years ago

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Jon Awbrey · @Inquiry

46 followers · 158 posts · Server mathstodon.xyz

Inquiry Into Inquiry - Differential Propositional Calculus • 5

#DifferentialPropositionalCalculus • 5.4
• https://inquiryintoinquiry.com/2020/02/29/differential-propositional-calculus-5/

The 2nd row of Figure 8 gives #VennDiagrams for the 3 #LinearPropositions of rank 2, expressed in terms of #MinimalNegationOperators by the following 3 forms, respectively:

\[\texttt{(}p\texttt{,}r\texttt{)}, \quad \texttt{(}q\texttt{,}r\texttt{)}, \quad \texttt{(}p\texttt{,}q\texttt{)}.\]

\(\text{Figure 8.2. Venn Diagram for}~\(\texttt{(}p\texttt{,}q\texttt{)}\)
• https://inquiryintoinquiry.files.wordpress.com/2020/02/venn-diagram-e280a2-p-q-r-e280a2-p-q.jpg

#MinimalNegationOperators #linearpropositions #venndiagrams #DifferentialPropositionalCalculus

Last updated 2 years ago

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Jon Awbrey · @Inquiry

38 followers · 134 posts · Server mathstodon.xyz

Survey of #DifferentialLogic
• https://inquiryintoinquiry.com/2022/11/20/survey-of-differential-logic-4/

This is a Survey of blog and wiki posts on Differential Logic, material I plan to develop toward a more compact and systematic account.

#Logic
#LogicalGraphs
#CactusLanguage
#QualitativeDynamics
#PropositionalCalculus
#LogicalTransformations
#MinimalNegationOperators
#DiscreteDynamicalSystems
#TransformationsOfDiscourse
#DifferentialPropositionalCalculus
#DifferentialAnalyticTuringAutomata

#DifferentialAnalyticTuringAutomata #DifferentialPropositionalCalculus #transformationsofdiscourse #DiscreteDynamicalSystems #MinimalNegationOperators #LogicalTransformations #PropositionalCalculus #QualitativeDynamics #CactusLanguage #LogicalGraphs #logic #DifferentialLogic

Last updated 2 years ago

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Jon Awbrey · @Inquiry

28 followers · 97 posts · Server mathstodon.xyz

Inquiry Into Inquiry - Differential Propositional Calculus • 3

#Differential Propositional Calculus • 3.4
• https://inquiryintoinquiry.com/2020/02/24/differential-propositional-calculus-3/

Table 7 summarizes the notations needed to describe ordinary propositional calculi in a systematic fashion.

Table 7. Propositional Calculus • Basic Notation
• https://inquiryintoinquiry.files.wordpress.com/2020/02/propositional-calculus-basic-notation.png

Applications —
#Cybernetics #SystemsTheory
#QualitativeDynamics
#DiscreteDynamics

#MinimalNegationOperators #CactusGraphs #BooleanFunctions #PropositionalCalculus #DifferentialLogic #LogicalGraphs #semiotics #logic #Peirce #DiscreteDynamics #QualitativeDynamics #systemstheory #cybernetics #differential

Last updated 2 years ago

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Jon Awbrey · @Inquiry

27 followers · 90 posts · Server mathstodon.xyz

#DifferentialPropositionalCalculus • 3
• https://inquiryintoinquiry.com/2020/02/24/differential-propositional-calculus-3/

Formal Development —

The preceding discussion outlined the ideas leading to the #DifferentialExtension of #PropositionalLogic. The next task is to lay out the concepts and terminology needed to describe various orders of differential propositional calculi.