#prooftheory — Public Fediverse posts

My *next* talk in this spring/summer of research combines some longstanding interests of mine (Graham Priest’s Logic of Paradox) and more recent interests (natural deduction and the sequent calculus). I bet you didn’t think that you could creatively apply Gentzen’s thoroughly standard rules of natural deduction to give you a sound and complete calculus for Priest’s LP, but it turns out that you can.

https://consequently.org/presentation/2026/lp-subst-arche/

#prooftheory #NaturalDeduction #paradox #philosophy

My *next* talk in this spring/summer of research combines some longstanding interests of mine (Graham Priest’s Logic of Paradox) and more recent interests (natural deduction and the sequent calculus). I bet you didn’t think that you could creatively apply Gentzen’s thoroughly standard rules of natural deduction to give you a sound and complete calculus for Priest’s LP, but it turns out that you can.

https://consequently.org/presentation/2026/lp-subst-arche/

#prooftheory #NaturalDeduction #paradox #philosophy

My *next* talk in this spring/summer of research combines some longstanding interests of mine (Graham Priest’s Logic of Paradox) and more recent interests (natural deduction and the sequent calculus). I bet you didn’t think that you could creatively apply Gentzen’s thoroughly standard rules of natural deduction to give you a sound and complete calculus for Priest’s LP, but it turns out that you can.

https://consequently.org/presentation/2026/lp-subst-arche/

#prooftheory #NaturalDeduction #paradox #philosophy

My *next* talk in this spring/summer of research combines some longstanding interests of mine (Graham Priest’s Logic of Paradox) and more recent interests (natural deduction and the sequent calculus). I bet you didn’t think that you could creatively apply Gentzen’s thoroughly standard rules of natural deduction to give you a sound and complete calculus for Priest’s LP, but it turns out that you can.

https://consequently.org/presentation/2026/lp-subst-arche/

#prooftheory #NaturalDeduction #paradox #philosophy

My *next* talk in this spring/summer of research combines some longstanding interests of mine (Graham Priest’s Logic of Paradox) and more recent interests (natural deduction and the sequent calculus). I bet you didn’t think that you could creatively apply Gentzen’s thoroughly standard rules of natural deduction to give you a sound and complete calculus for Priest’s LP, but it turns out that you can.

https://consequently.org/presentation/2026/lp-subst-arche/

#prooftheory #NaturalDeduction #paradox #philosophy

It’s neat to see that an old (fiddly, complicated) decidability argument I wrote up in the 1990s is getting some attention. Here, Raj Goré and Anthony Peigné formalise (and generalise) my decidability argument for display formulations of some substructural logics. This is interesting work, worth looking into.

https://link.springer.com/article/10.1007/s11225-026-10239-8

#logic #prooftheory #rocqprover

It’s neat to see that an old (fiddly, complicated) decidability argument I wrote up in the 1990s is getting some attention. Here, Raj Goré and Anthony Peigné formalise (and generalise) my decidability argument for display formulations of some substructural logics. This is interesting work, worth looking into.

https://link.springer.com/article/10.1007/s11225-026-10239-8

#logic #prooftheory #rocqprover

It’s neat to see that an old (fiddly, complicated) decidability argument I wrote up in the 1990s is getting some attention. Here, Raj Goré and Anthony Peigné formalise (and generalise) my decidability argument for display formulations of some substructural logics. This is interesting work, worth looking into.

https://link.springer.com/article/10.1007/s11225-026-10239-8

#logic #prooftheory #rocqprover

It’s neat to see that an old (fiddly, complicated) decidability argument I wrote up in the 1990s is getting some attention. Here, Raj Goré and Anthony Peigné formalise (and generalise) my decidability argument for display formulations of some substructural logics. This is interesting work, worth looking into.

https://link.springer.com/article/10.1007/s11225-026-10239-8

#logic #prooftheory #rocqprover

It’s neat to see that an old (fiddly, complicated) decidability argument I wrote up in the 1990s is getting some attention. Here, Raj Goré and Anthony Peigné formalise (and generalise) my decidability argument for display formulations of some substructural logics. This is interesting work, worth looking into.

https://link.springer.com/article/10.1007/s11225-026-10239-8

#logic #prooftheory #rocqprover

I’m looking forward to spending time today with @ohad, @modaltype and other folks at the LFCS at Edinburgh, and getting to talk about some weird substructural modal logic.

https://consequently.org/presentation/2026/tlmh-edi/

#logic #prooftheory

Propositions As Types Analogy • 1
• https://inquiryintoinquiry.com/2013/01/29/propositions-as-types-analogy-1/

One of my favorite mathematical tricks — it almost seems too tricky to be true — is the Propositions As Types Analogy. And I see hints the 2‑part analogy can be extended to a 3‑part analogy, as follows.

Proof Hint ∶ Proof ∶ Proposition
∷
Untyped Term ∶ Typed Term ∶ Type

or

Proof Hint ∶ Untyped Term
∷
Proof ∶ Typed Term
∷
Proposition ∶ Type

See my working notes on the Propositions As Types Analogy —
• https://oeis.org/wiki/Propositions_As_Types_Analogy

#Mathematics #CategoryTheory #ProofTheory #TypeTheory
#Logic #Analogy #Isomorphism #PropositionalCalculus
#CombinatorCalculus #CombinatoryLogic #LambdaCalculus
#Peirce #LogicalGraphs #GraphTheory #RelationTheory

Propositions As Types Analogy • 1
• https://inquiryintoinquiry.com/2013/01/29/propositions-as-types-analogy-1/

One of my favorite mathematical tricks — it almost seems too tricky to be true — is the Propositions As Types Analogy. And I see hints the 2‑part analogy can be extended to a 3‑part analogy, as follows.

Proof Hint ∶ Proof ∶ Proposition
∷
Untyped Term ∶ Typed Term ∶ Type

or

Proof Hint ∶ Untyped Term
∷
Proof ∶ Typed Term
∷
Proposition ∶ Type

See my working notes on the Propositions As Types Analogy —
• https://oeis.org/wiki/Propositions_As_Types_Analogy

#Mathematics #CategoryTheory #ProofTheory #TypeTheory
#Logic #Analogy #Isomorphism #PropositionalCalculus
#CombinatorCalculus #CombinatoryLogic #LambdaCalculus
#Peirce #LogicalGraphs #GraphTheory #RelationTheory

Propositions As Types Analogy • 1
• https://inquiryintoinquiry.com/2013/01/29/propositions-as-types-analogy-1/

One of my favorite mathematical tricks — it almost seems too tricky to be true — is the Propositions As Types Analogy. And I see hints the 2‑part analogy can be extended to a 3‑part analogy, as follows.

Proof Hint ∶ Proof ∶ Proposition
∷
Untyped Term ∶ Typed Term ∶ Type

or

Proof Hint ∶ Untyped Term
∷
Proof ∶ Typed Term
∷
Proposition ∶ Type

See my working notes on the Propositions As Types Analogy —
• https://oeis.org/wiki/Propositions_As_Types_Analogy

#Mathematics #CategoryTheory #ProofTheory #TypeTheory
#Logic #Analogy #Isomorphism #PropositionalCalculus
#CombinatorCalculus #CombinatoryLogic #LambdaCalculus
#Peirce #LogicalGraphs #GraphTheory #RelationTheory

Propositions As Types Analogy • 1
• https://inquiryintoinquiry.com/2013/01/29/propositions-as-types-analogy-1/

One of my favorite mathematical tricks — it almost seems too tricky to be true — is the Propositions As Types Analogy. And I see hints the 2‑part analogy can be extended to a 3‑part analogy, as follows.

Proof Hint ∶ Proof ∶ Proposition
∷
Untyped Term ∶ Typed Term ∶ Type

or

Proof Hint ∶ Untyped Term
∷
Proof ∶ Typed Term
∷
Proposition ∶ Type

See my working notes on the Propositions As Types Analogy —
• https://oeis.org/wiki/Propositions_As_Types_Analogy

#Mathematics #CategoryTheory #ProofTheory #TypeTheory
#Logic #Analogy #Isomorphism #PropositionalCalculus
#CombinatorCalculus #CombinatoryLogic #LambdaCalculus
#Peirce #LogicalGraphs #GraphTheory #RelationTheory

Propositions As Types Analogy • 1
• https://inquiryintoinquiry.com/2013/01/29/propositions-as-types-analogy-1/

One of my favorite mathematical tricks — it almost seems too tricky to be true — is the Propositions As Types Analogy. And I see hints the 2‑part analogy can be extended to a 3‑part analogy, as follows.

Proof Hint ∶ Proof ∶ Proposition
∷
Untyped Term ∶ Typed Term ∶ Type

or

Proof Hint ∶ Untyped Term
∷
Proof ∶ Typed Term
∷
Proposition ∶ Type

See my working notes on the Propositions As Types Analogy —
• https://oeis.org/wiki/Propositions_As_Types_Analogy

#Mathematics #CategoryTheory #ProofTheory #TypeTheory
#Logic #Analogy #Isomorphism #PropositionalCalculus
#CombinatorCalculus #CombinatoryLogic #LambdaCalculus
#Peirce #LogicalGraphs #GraphTheory #RelationTheory

Readings shared July 12, 2025. https://jaalonso.github.io/vestigium/posts/2025/07/13-readings_shared_07-12-25 #ASP #ATP #CLP #CategoryTheory #CoqProver #Emacs #FunctionalProgramming #Haskell #ITP #IsabelleHOL #LLMs #LeanProver #Logic #LogicProgramming #Math #Prolog #ProofTheory #Prover9

"A major function [of deductive #logic is in] assessing exactly what is involved in asserting some set of propositions. […] By omitting some premiss without which the deduction of some conclusion is not valid, it misrepresents the premiss from which this conclusion is obtained, and hence responsibility for the conclusion. To agree to accept partial responsibility as good enough here is like agreeing to say that somebody was responsible for the dinner when he peeled potatoes and the cook did the rest. The first statement cannot be accepted as an elliptical, but allowable, way of making the second statement. And similarly suppression [of some premiss] enables us to obtain as causally responsible a partially sufficient rather than a fully sufficient causal condition."

Valerie Plumwood in Australasian Journal of Logic, 2023: https://ojs.victoria.ac.nz/ajl/issue/view/894 v @rrrichardzach

#Plumwood #causality #correlations #economics #reason #ProofTheory #PhilSci #truth #science #ethics #ecofeminism #freedom

This afternoon I had the pleasure of sneaking in to a session of Scottish Programming Languages and Verification Summer School, organised by @edwinb and colleagues in Computer Science at St Andrews.

@dorchard gave a neat talk on graded modalities. It’s neat to see substructural logics applied in the wild, and there was some logical insight, too, on the different behaviour of box-type and diamond-type modalities in a constructive setting.

#logic #modality #prooftheory

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

#Propositions As #Types • 1
• https://inquiryintoinquiry.com/2013/01/29/propositions-as-types-1/

One of my favorite mathematical tricks — it almost seems too tricky to be true — is the #PropositionsAsTypesAnalogy. And I see hints the 2-part analogy can be extended to a 3-part analogy, as follows.

\(\text{proof hint : proof : proposition :: untyped term : typed term : type}\)

See my notes on #PropositionsAsTypes for more.
• https://oeis.org/wiki/Propositions_As_Types_Analogy

#Logic #Combinators #ProofTheory #TypeTheory
#CurryHowardIsomorphism #LambdaCalculus

#Propositions As #Types • 1
• https://inquiryintoinquiry.com/2013/01/29/propositions-as-types-1/

One of my favorite mathematical tricks — it almost seems too tricky to be true — is the #PropositionsAsTypesAnalogy. And I see hints the 2-part analogy can be extended to a 3-part analogy, as follows.

\(\text{proof hint : proof : proposition :: untyped term : typed term : type}\)

See my notes on #PropositionsAsTypes for more.
• https://oeis.org/wiki/Propositions_As_Types_Analogy

#Logic #Combinators #ProofTheory #TypeTheory
#CurryHowardIsomorphism #LambdaCalculus

#Propositions As #Types • 1
• https://inquiryintoinquiry.com/2013/01/29/propositions-as-types-1/

One of my favorite mathematical tricks — it almost seems too tricky to be true — is the #PropositionsAsTypesAnalogy. And I see hints the 2-part analogy can be extended to a 3-part analogy, as follows.

\(\text{proof hint : proof : proposition :: untyped term : typed term : type}\)

See my notes on #PropositionsAsTypes for more.
• https://oeis.org/wiki/Propositions_As_Types_Analogy

#Logic #Combinators #ProofTheory #TypeTheory
#CurryHowardIsomorphism #LambdaCalculus

#Propositions As #Types • 1
• https://inquiryintoinquiry.com/2013/01/29/propositions-as-types-1/

One of my favorite mathematical tricks — it almost seems too tricky to be true — is the #PropositionsAsTypesAnalogy. And I see hints the 2-part analogy can be extended to a 3-part analogy, as follows.

\(\text{proof hint : proof : proposition :: untyped term : typed term : type}\)

See my notes on #PropositionsAsTypes for more.
• https://oeis.org/wiki/Propositions_As_Types_Analogy

#Logic #Combinators #ProofTheory #TypeTheory
#CurryHowardIsomorphism #LambdaCalculus

#Propositions As #Types • 1
• https://inquiryintoinquiry.com/2013/01/29/propositions-as-types-1/

One of my favorite mathematical tricks — it almost seems too tricky to be true — is the #PropositionsAsTypesAnalogy. And I see hints the 2-part analogy can be extended to a 3-part analogy, as follows.

\(\text{proof hint : proof : proposition :: untyped term : typed term : type}\)

See my notes on #PropositionsAsTypes for more.
• https://oeis.org/wiki/Propositions_As_Types_Analogy

#Logic #Combinators #ProofTheory #TypeTheory
#CurryHowardIsomorphism #LambdaCalculus

#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 • 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

#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 • 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 • 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

#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

#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 • 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

#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 1.1
• 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

#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

#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