#modeltheory — Public Fediverse posts

In the mountains, I love exploring nature and climbing any little hidden corner.

My academic interests include mathematics (#mathematics) and mathematical logic ( #mathematicallogic #logic), particularly model theory ( #modeltheory), category theory ( #categorytheory), higher-order logic and metalogic ( #metalogic), as well as the philosophy of mathematics and logic ( #philosophyofmathematics #philosophyoflogic), epistemology and ontology of them.

#introduction

@LeoTsai14 While they do not actually call it so, set theoreticians do a lot of work in a category in which the objects are the models of set theory and the arrows are the elementary embeddings (https://en.wikipedia.org/wiki/Elementary_equivalence#Elementary_embeddings) between them.
Models of (ZFC-like) set theories have the interesting property that the maps between them are to some amount determined by the mappings between their classes of ordinals: If this map is an isomorphism, the whole map is one (https://en.wikipedia.org/wiki/Critical_point_(set_theory)).
You may also have a look at inner model theory (https://en.wikipedia.org/wiki/Inner_model_theory), I think.

#SetTheory #ModelTheory #MathematicalLogic #Categories

A new Ph.D thesis prize is being established in honor of Zoé Chatzidakis, by Fondation Sciences Mathématiques de Paris.

This yearly award for an outstanding thesis in model theory will honor her scientific legacy and her commitment to the younger generation.

https://www.helloasso.com/associations/fondation-sciences-mathematiques-de-paris/formulaires/7

#Math #ModelTheory #Logic

Very excited about this new preprint, with Kyle Gannon and Krzysztof Krupinski!

"Definable convolution and idempotent Keisler measures III. Generic stability, generic transitivity, and revised Newelski's conjecture"
https://arxiv.org/abs/2406.00912

Classical work by Wendel, Rudin, Cohen (before inventing forcing) and others classifies idempotent Borel measures on locally compact abelian groups, showing that they are precisely the Haar measures of compact subgroups.
We are interested in a counterpart of this phenomenon in the definable category. In the same way as e.g. algebraic or Lie groups are important in algebraic or differential geometry, the understanding of groups definable in a given first-order structure (or in certain classes of first-order structures) is important for model theory and its applications. The class of stable groups is at the core of model theory, and the corresponding theory was developed in the 1970s-1980s borrowing many ideas from the study of algebraic groups over algebraically closed fields. More recently, many of the ideas of stable group theory were extended to the class of NIP groups, which contains both stable groups and groups definable in o-minimal structures or over the p-adics. This led to multiple applications, e.g. a resolution of Pillay’s conjecture for compact o-minimal groups, or Hrushovski’s work on approximate subgroups. This brought to light the importance of the study of invariant measures on definable subsets of the group, as well as the methods of topological dynamics. In particular, deep connections with tame dynamical systems as studied by Glasner, Megrelishvili and others have emerged.

#Mathematics #ModelTheory #TopologicalDynamics

After a short summer break exploring Scotland, it’s time to slowly get back into the saddle, giving a few talks, and preparing for the new academic year’s teaching.

First up, a short visit to Bochum for a PhD exam, and an impromptu talk on non-classical models for the identity predicate.

https://consequently.org/presentation/2023/exploring-three-valued-models-for-identity/

#logic #ModelTheory #ProofTheory #PhilosophicalLogic

#DifferentialPropositionalCalculus • 7
• https://inquiryintoinquiry.com/2020/03/05/differential-propositional-calculus-7/

In our #Model of #Propositions as #Mappings of a #UniverseOfDiscourse to a set of 2 values, in other words, #IndicatorFunctions of the form \(f:X\to\mathbb{B},\) #SingularPropositions are those singling out the #MinimalDistinctRegions of the universe, represented by single cells of the corresponding #VennDiagram.

#Logic #LogicalGraphs #DifferentialLogic
#PropositionalCalculus #BooleanFunctions
#ModelTheory #ProofTheory #Semiotics

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2023/03/28/survey-of-animated-logical-graphs-5/

This is a Survey of blog and wiki posts on Logical Graphs, encompassing several families of graph-theoretic structures originally developed by Charles S. Peirce as graphical formal languages or visual styles of syntax amenable to interpretation for logical applications.

#Peirce #Logic #LogicalGraphs #EntitativeGraphs #ExistentialGraphs
#Boole #BooleanAlgebra #BooleanFunctions #ModelTheory #ProofTheory
#SpencerBrown #LawsOfForm #PropositionalCalculus #LogicAsSemiotics

#LogicalGraphs • 14
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no#Duality

#Duality • Logical and Topological

The procedure just described is called “traversing” the tree and the string read off is called the “#TraversalString” of the tree. The reverse operation of going from the string to the tree is called “parsing” the string and the tree constructed is called the “#ParseGraph” of the string.

#Logic #Peirce #SpencerBrown #LawsOfForm
#PropositionalCalculus #BooleanFunctions
#GraphTheory #ModelTheory #ProofTheory

The video of my yesterday's talk at the #IAS on some uses of #ModelTheory in #Erdős #Geometry is now online. (Please ignore the nonsense that I've said instead of the definition of modularity!)

https://youtu.be/w5ITepQZL8U

#LogicalGraphs • 12
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no#Duality

#Duality • Logical and Topological

Once we make the connection between one of #Peirce's #AlphaGraphs and its character string expression it's not too big a leap to see how the character string codes up the structure of the topological #DualGraph in the space of #RootedTrees.

#Logic #Peirce #SpencerBrown #LawsOfForm
#PropositionalCalculus #BooleanFunctions
#GraphTheory #ModelTheory #ProofTheory

#LogicalGraphs • 11
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no#Duality

#Duality • Logical and Topological

Editing the composite picture of #AlphaGraphs and #DualGraphs in Figure 4 to bring out the dual graphs by themselves affords a view of the second #InitialEquation shown in Figure 5.

Figure 5
• https://oeis.org/w/images/4/46/Logical_Graph_Figure_5_Visible_Frame.jpg

#Logic #Peirce #SpencerBrown #LawsOfForm
#PropositionalCalculus #BooleanFunctions
#GraphTheory #ModelTheory #ProofTheory

#LogicalGraphs • 3
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no

We begin on a low but expansive plateau of #FormalSystems #Peirce mapped out in his system of #AlphaGraphs \((\alpha),\) a platform so abstract in its mathematical forms as to support at least two interpretations for use in the conduct of logical reasoning. Along the way, we incorporate the later contributions of George #SpencerBrown, who revived and augmented Peirce's system in his book #LawsOfForm.

#Logic #GraphTheory #ModelTheory #ProofTheory

#DifferentialPropositionalCalculus • 7
• https://inquiryintoinquiry.com/2020/03/05/differential-propositional-calculus-7/

In our #Model of #Propositions as #Mappings of a #UniverseOfDiscourse to a set of 2 values, in other words, #IndicatorFunctions of the form \(f:X\to\mathbb{B},\) #SingularPropositions are those singling out the #MinimalDistinctRegions of the universe, represented by single cells of the corresponding #VennDiagram.

#Logic #LogicalGraphs #DifferentialLogic
#PropositionalCalculus #BooleanFunctions
#ModelTheory #ProofTheory #Semiotics

#DifferentialPropositionalCalculus • 7
• https://inquiryintoinquiry.com/2020/03/05/differential-propositional-calculus-7/

In our #Model of #Propositions as #Mappings of a #UniverseOfDiscourse to a set of 2 values, in other words, #IndicatorFunctions of the form \(f:X\to\mathbb{B},\) #SingularPropositions are those singling out the #MinimalDistinctRegions of the universe, represented by single cells of the corresponding #VennDiagram.

#Logic #LogicalGraphs #DifferentialLogic
#PropositionalCalculus #BooleanFunctions
#ModelTheory #ProofTheory #Semiotics

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

A program I worked on all through the 80s aimed to develop #Algorithms and #DataStructures for integrating #EmpiricalLearning and #LogicalReasoning. I had earlier developed programs for #Learning #TwoLevelFormalLanguages and for #PropositionalLogic #ConstraintSatisfaction based on C.S. #Peirce's #LogicalGraphs. Getting those 2 faculties to work together meant integrating the ways of #Empiricism and #Rationalism.

#ThemeOneProgram #Learning #Reasoning
#Logic #LogicalGraphs #FormalLanguages
#Algorithm #DataStructure #GraphTheory
#Peirce #PragmaticSemioticInformation
#Empiricism #Rationalism #Pragmatism
#ModelTheory #ProofTheory #Semiotics
#ParallelDistributedProcessing #PDP
#Grossberg #McClelland #Rumelhart

#LogicalGraphs • 1
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no

A #LogicalGraph is a graph-theoretic structure in one of the systems of graphical syntax Charles Sanders #Peirce developed for #Logic.

In his papers on #QualitativeLogic, #EntitativeGraphs, and #ExistentialGraphs, Peirce developed several versions of a graphical formalism, or a graph-theoretic formal language, designed to be interpreted for logic.

#PropositionalCalculus #BooleanFunctions
#GraphTheory #ModelTheory #ProofTheory

#LogicalGraphs • 17
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no#Duality

#Duality • Logical and Topological

Editing the composite picture of #AlphaGraphs and #DualGraphs in Figure 8 to bring out the dual graphs by themselves affords a view of the first #InitialEquation shown in Figure 9.

Figure 9
• https://oeis.org/w/images/8/85/Logical_Graph_Figure_9_Visible_Frame.jpg

#Logic #Peirce #SpencerBrown #LawsOfForm
#PropositionalCalculus #BooleanFunctions
#GraphTheory #ModelTheory #ProofTheory

#LogicalGraphs • 17
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no#Duality

#Duality • Logical and Topological

Editing the composite picture of #AlphaGraphs and #DualGraphs in Figure 8 to bring out the dual graphs by themselves affords a view of the first #InitialEquation shown in Figure 9.

Figure 9
• https://oeis.org/w/images/8/85/Logical_Graph_Figure_9_Visible_Frame.jpg

#Logic #Peirce #SpencerBrown #LawsOfForm
#PropositionalCalculus #BooleanFunctions
#GraphTheory #ModelTheory #ProofTheory

#LogicalGraphs • 17
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no#Duality

#Duality • Logical and Topological

Editing the composite picture of #AlphaGraphs and #DualGraphs in Figure 8 to bring out the dual graphs by themselves affords a view of the first #InitialEquation shown in Figure 9.

Figure 9
• https://oeis.org/w/images/8/85/Logical_Graph_Figure_9_Visible_Frame.jpg

#Logic #Peirce #SpencerBrown #LawsOfForm
#PropositionalCalculus #BooleanFunctions
#GraphTheory #ModelTheory #ProofTheory

#LogicalGraphs • 16
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no#Duality

#Duality • Logical and Topological

Turning to the #InitialEquation or #LogicalAxiom whose text expression is \(``\texttt{(}~\texttt{)(}~\texttt{)}=\texttt{(}~\texttt{)}",\) Figure 8 shows the planar maps and their corresponding #DualGraphs superimposed.

Figure 8
• https://oeis.org/w/images/0/09/Logical_Graph_Figure_8_Visible_Frame.jpg

#Logic #Peirce #SpencerBrown #LawsOfForm
#PropositionalCalculus #BooleanFunctions
#GraphTheory #ModelTheory #ProofTheory

#LogicalGraphs • 16
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no#Duality

#Duality • Logical and Topological

Turning to the #InitialEquation or #LogicalAxiom whose text expression is \(``\texttt{(}~\texttt{)(}~\texttt{)}=\texttt{(}~\texttt{)}",\) Figure 8 shows the planar maps and their corresponding #DualGraphs superimposed.

Figure 8
• https://oeis.org/w/images/0/09/Logical_Graph_Figure_8_Visible_Frame.jpg

#Logic #Peirce #SpencerBrown #LawsOfForm
#PropositionalCalculus #BooleanFunctions
#GraphTheory #ModelTheory #ProofTheory

#LogicalGraphs • 16
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no#Duality

#Duality • Logical and Topological

Turning to the #InitialEquation or #LogicalAxiom whose text expression is \(``\texttt{(}~\texttt{)(}~\texttt{)}=\texttt{(}~\texttt{)}",\) Figure 8 shows the planar maps and their corresponding #DualGraphs superimposed.

Figure 8
• https://oeis.org/w/images/0/09/Logical_Graph_Figure_8_Visible_Frame.jpg

#Logic #Peirce #SpencerBrown #LawsOfForm
#PropositionalCalculus #BooleanFunctions
#GraphTheory #ModelTheory #ProofTheory

#LogicalGraphs • 11
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no#Duality

#Duality • Logical and Topological

Editing the composite picture of #AlphaGraphs and #DualGraphs in Figure 4 to bring out the dual graphs by themselves affords a view of the second #InitialEquation shown in Figure 5.

Figure 5
• https://oeis.org/w/images/4/46/Logical_Graph_Figure_5_Visible_Frame.jpg

#Logic #Peirce #SpencerBrown #LawsOfForm
#PropositionalCalculus #BooleanFunctions
#GraphTheory #ModelTheory #ProofTheory

#LogicalGraphs • 11
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no#Duality

#Duality • Logical and Topological

Editing the composite picture of #AlphaGraphs and #DualGraphs in Figure 4 to bring out the dual graphs by themselves affords a view of the second #InitialEquation shown in Figure 5.

Figure 5
• https://oeis.org/w/images/4/46/Logical_Graph_Figure_5_Visible_Frame.jpg

#Logic #Peirce #SpencerBrown #LawsOfForm
#PropositionalCalculus #BooleanFunctions
#GraphTheory #ModelTheory #ProofTheory

#LogicalGraphs • 11
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no#Duality

#Duality • Logical and Topological

Editing the composite picture of #AlphaGraphs and #DualGraphs in Figure 4 to bring out the dual graphs by themselves affords a view of the second #InitialEquation shown in Figure 5.

Figure 5
• https://oeis.org/w/images/4/46/Logical_Graph_Figure_5_Visible_Frame.jpg

#Logic #Peirce #SpencerBrown #LawsOfForm
#PropositionalCalculus #BooleanFunctions
#GraphTheory #ModelTheory #ProofTheory

#LogicalGraphs • 3
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no

We begin on a low but expansive plateau of #FormalSystems #Peirce mapped out in his system of #AlphaGraphs \((\alpha),\) a platform so abstract in its mathematical forms as to support at least two interpretations for use in the conduct of logical reasoning. Along the way, we incorporate the later contributions of George #SpencerBrown, who revived and augmented Peirce's system in his book #LawsOfForm.

#Logic #GraphTheory #ModelTheory #ProofTheory

#LogicalGraphs • 3
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no

We begin on a low but expansive plateau of #FormalSystems #Peirce mapped out in his system of #AlphaGraphs \((\alpha),\) a platform so abstract in its mathematical forms as to support at least two interpretations for use in the conduct of logical reasoning. Along the way, we incorporate the later contributions of George #SpencerBrown, who revived and augmented Peirce's system in his book #LawsOfForm.

#Logic #GraphTheory #ModelTheory #ProofTheory

#LogicalGraphs • 3
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no

We begin on a low but expansive plateau of #FormalSystems #Peirce mapped out in his system of #AlphaGraphs \((\alpha),\) a platform so abstract in its mathematical forms as to support at least two interpretations for use in the conduct of logical reasoning. Along the way, we incorporate the later contributions of George #SpencerBrown, who revived and augmented Peirce's system in his book #LawsOfForm.

#Logic #GraphTheory #ModelTheory #ProofTheory

#LogicalGraphs • 3
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no

We begin on a low but expansive plateau of #FormalSystems #Peirce mapped out in his system of #AlphaGraphs \((\alpha),\) a platform so abstract in its mathematical forms as to support at least two interpretations for use in the conduct of logical reasoning. Along the way, we incorporate the later contributions of George #SpencerBrown, who revived and augmented Peirce's system in his book #LawsOfForm.

#Logic #GraphTheory #ModelTheory #ProofTheory

#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

#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

#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

#LogicalGraphs • 5
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no#Abstract_POV

#AbstractPointOfView —

We may note in passing historical details like the fact Charles Sanders #Peirce used a #StreamerCross symbol where George #SpencerBrown used a #CarpentersSquare marker but the themes of primary interest at the abstract level of form are indifferent to variations of that order.

#Logic #PropositionalCalculus #BooleanFunctions
#Form #Idea #Isomorphism #MathematicalPerspective
#LawsOfForm #GraphTheory #ModelTheory #ProofTheory

Hi! I am a philosopher and logician, and if you would like to understand the kind of work I do, the easiest way to get up to speed is to start with my little slip of a book “Proofs and Models in Philosophical Logic”. https://consequently.org/writing/pmpl-elements/

#introduction #philosophy #logic #prooftheory #modeltheory

A workshop and a conference on #ModelTheory in Wroclaw, Poland this Fall!

Model Theory Workshop: 15 - 18 September 2023
https://math.uni.wroc.pl/~pkowa/registrationw23.html

Model Theory Conference: 19 - 23 September 2023
https://math.uni.wroc.pl/~pkowa/registrationc23.html

#ModelTheory Conference in Seoul, in commemoration of Byunghan Kim’s 60th birthday.

Aug. 28-30, 2023

The registration deadline is June 15, 2023.
This conference is ASL-sponsored, so students (who are ASL members) are eligible for ASL travel support. Partial travel expenses can also be provided to those who present a poster.

https://sites.google.com/yonsei.ac.kr/modeltheoryseoul2023/home?authuser=0

A model of a theory. First volume is out!
#ModelTheory

Slides from my talk at the #ModelTheory Conference in celebration of Ludomir Newelski's 60th birthday in Będlewo, Poland. It's about recent work with Kyle Gannon on convolution semigroups of measures in NIP groups, which turn out to be particularly structured!

https://www.math.ucla.edu/~chernikov/slides/Newelski2022.pdf

Now that my author copies of Logical Methods (written with @standefer) have arrived, I no longer have to worry about last-minute Christmas gift buying.

If you order *your* copies from MIT Press (or wherever else you get your books), you won’t get them in time for Christmas. It’s released to the whole world on January 3.

https://mitpress.mit.edu/9780262544849/

#logic #PhilLogic #ProofTheory #ModelTheory #Textbook

#LogicalGraphs • 6
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no#Progenesis

In Lieu of a Beginning —

Consider the #FormalEquations indicated in Figures 1 and 2.

Figure 1
• https://oeis.org/w/images/8/81/Logical_Graph_Figure_1_Visible_Frame.jpg

Figure 2
• https://oeis.org/w/images/6/66/Logical_Graph_Figure_2_Visible_Frame.jpg

For the time being these two forms of transformation may be referred to as axioms or initial equations.

#Logic #Peirce #SpencerBrown #LawsOfForm
#PropositionalCalculus #BooleanFunctions
#GraphTheory #ModelTheory #ProofTheory

All in another day’s work, talking about proofs and models in philosophical logic, this time focusing on boolean valuations, supervaluations, and tri-valuations.

#PhilLogic #ModelTheory

Intro to Mathematical Logic: from the basic first-order #logic, through #Gödel's Completeness, to the fundamentals of #ModelTheory! All in 28 videos from a course I've taught at #UCLA:

https://youtube.com/playlist?list=PL54Pt_mZzBqibWHgesgEICeQHnwHom8xz

#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

#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

#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

#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 pool of #MutuallyInhibitoryNeurons \(a,b,c\) by the proposition \(\texttt{(}a\texttt{,}b\texttt{,}c\texttt{)}.\)

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

#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

#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

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

A program I worked on all through the 80s aimed to develop #Algorithms and #DataStructures for integrating #EmpiricalLearning and #LogicalReasoning. I had earlier developed programs for #Learning #TwoLevelFormalLanguages and for #PropositionalLogic #ConstraintSatisfaction based on C.S. #Peirce's #LogicalGraphs. Getting those 2 faculties to work together meant integrating the ways of #Empiricism and #Rationalism.

#ThemeOneProgram #Learning #Reasoning
#Logic #LogicalGraphs #FormalLanguages
#Algorithm #DataStructure #GraphTheory
#Peirce #PragmaticSemioticInformation
#Empiricism #Rationalism #Pragmatism
#ModelTheory #ProofTheory #Semiotics
#ParallelDistributedProcessing #PDP
#Grossberg #McClelland #Rumelhart

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

A program I worked on all through the 80s aimed to develop #Algorithms and #DataStructures for integrating #EmpiricalLearning and #LogicalReasoning. I had earlier developed programs for #Learning #TwoLevelFormalLanguages and for #PropositionalLogic #ConstraintSatisfaction based on C.S. #Peirce's #LogicalGraphs. Getting those 2 faculties to work together meant integrating the ways of #Empiricism and #Rationalism.

#ThemeOneProgram #Learning #Reasoning
#Logic #LogicalGraphs #FormalLanguages
#Algorithm #DataStructure #GraphTheory
#Peirce #PragmaticSemioticInformation
#Empiricism #Rationalism #Pragmatism
#ModelTheory #ProofTheory #Semiotics
#ParallelDistributedProcessing #PDP
#Grossberg #McClelland #Rumelhart