RanaldClouston · @RanaldClouston
278 followers · 1304 posts · Server fediscience.orgA first #CallForPapers has gone out for #FSCD #FSCD2024 (Formal Structures for Computation and Deduction) on at least one mailing list, although the website https://cs.ioc.ee/fscd24/ is not yet updated with it. The conference "covers all aspects of formal
structures for computation and deduction from theoretical foundations to
applications" and will be held in Talinn in July next year. #TypeTheory #ProofTheory
#ProofTheory #typetheory #fscd2024 #fscd #callforpapers
hxameer :qed: · @xameer
197 followers · 5457 posts · Server mathstodon.xyz#prooftheory folks any pointers?
The relationship between Herbrand’s theorem and Gentzen’s
“versch ̈after Haupsatz” is difficult to pinpoint, also for proof-theory specialists: to Gentzen,
Herbrand’s is a particular case of his own, with empty antecedent and only one prenex
formula in the consequent; but Gentzen’s Hauptsatz holds only for prenex formulae, though
it is extendable to intuitionistic logic; both give cut elimination; Herbrand’s theorem is
perhaps more informative on the Mittelsequenz
http://www.ailalogica.it/archive/preprint/preprint18_lolli.pdf
Yoriyuki Yamagata · @yoriyuki
3 followers · 3 posts · Server mathstodon.xyzI'm watching "Proof Complexity and Meta-Mathematicshttps" bit by bit
https://www.youtube.com/playlist?list=PLgKuh-lKre10CktjVrpIX9VEvX6Rog36N
#ProofTheory #simonsinstitute #mathematicallogic
Jon Awbrey · @Inquiry
165 followers · 970 posts · Server mathstodon.xyzPropositions 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. Moreover, 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 information.
• https://oeis.org/wiki/Propositions_As_Types_Analogy
#CurryHowardIsomorphism #LambdaCalculus #ProofTheory #TypeTheory
#typetheory #ProofTheory #lambdacalculus #CurryHowardIsomorphism #propositionsastypes #propositionsastypesanalogy
nilesh · @nilesh
1 followers · 8 posts · Server mathstodon.xyz
I stumbled upon one guy proposing a new foundation for computer mathematics called "Abstraction logic". As a newbie, I found it far more more comprehensible than usual any material on intuitionistic type theory versions.
But there's no social proof of it.
Any #logic #typetheory #lambdacalculus #foundation of #mathematics #automatedreasoning #prooftheory people who are willing to have a look to weight on it?
Links:
https://arxiv.org/pdf/2207.05610.pdf
https://youtu.be/LbFKSaPhBSA
https://obua.com/publications/philosophy-of-abstraction-logic/2/
#ProofTheory #automatedreasoning #mathematics #foundation #lambdacalculus #typetheory #logic
Jon Awbrey · @Inquiry
129 followers · 624 posts · Server mathstodon.xyz
#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
#ProofTheory #modeltheory #graphtheory #BooleanFunctions #PropositionalCalculus #lawsofform #spencerbrown #Peirce #logic #initialequation #dualgraphs #alphagraphs #duality #LogicalGraphs
Jon Awbrey · @Inquiry
109 followers · 499 posts · Server mathstodon.xyz
#LogicalGraphs • 16
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no#Duality
#Duality • Logical and Topological
Turning now to the #InitialEquation or #LogicalAxiom whose text expression is \(\texttt{(}~\texttt{)(}~\texttt{)}=\texttt{(}~\texttt{)}\), Figure 8 shows the planar maps and their #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
#ProofTheory #modeltheory #graphtheory #BooleanFunctions #PropositionalCalculus #lawsofform #spencerbrown #Peirce #logic #dualgraphs #logicalaxiom #initialequation #duality #LogicalGraphs
Richard Zach · @rrrichardzach
776 followers · 357 posts · Server mathstodon.xyz
Get the English original from OUP https://global.oup.com/academic/product/an-introduction-to-proof-theory-9780192895943 or electronically at https://academic.oup.com/book/42130 #logic #ProofTheory
Richard Zach · @rrrichardzach
776 followers · 357 posts · Server mathstodon.xyzGet the English original from OUP https://global.oup.com/academic/product/an-introduction-to-proof-theory-9780192895943 or electronically at https://academic.oup.com/book/42130 #logic #ProofTheory
Richard Zach · @rrrichardzach
776 followers · 354 posts · Server mathstodon.xyz
Mon livre sur la théorie de la démonstration avec Paolo Mancosu et Sergio Galvan est disponible en français à partir d'aujourd'hui https://www.vrin.fr/livre/9782711630912/introduction-a-la-theorie-de-la-demonstration #Logic #Logique #ProofTheory
Jon Awbrey · @Inquiry
97 followers · 425 posts · Server mathstodon.xyz#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
#ProofTheory #modeltheory #graphtheory #BooleanFunctions #PropositionalCalculus #lawsofform #spencerbrown #Peirce #logic #traversalstring #duality #LogicalGraphs
Jon Awbrey · @Inquiry
96 followers · 414 posts · Server mathstodon.xyz#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
#ProofTheory #modeltheory #graphtheory #BooleanFunctions #PropositionalCalculus #lawsofform #spencerbrown #logic #rootedtrees #dualgraph #alphagraphs #Peirce #duality #LogicalGraphs
Jon Awbrey · @Inquiry
96 followers · 408 posts · Server mathstodon.xyz
#LogicalGraphs • 11
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no#Duality
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
#ProofTheory #modeltheory #graphtheory #BooleanFunctions #PropositionalCalculus #lawsofform #spencerbrown #Peirce #logic #initialequation #dualgraphs #alphagraphs #LogicalGraphs
Jon Awbrey · @Inquiry
94 followers · 376 posts · Server mathstodon.xyz
#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
#ProofTheory #modeltheory #graphtheory #BooleanFunctions #PropositionalCalculus #lawsofform #spencerbrown #Peirce #logic #formalequations #LogicalGraphs
Jon Awbrey · @Inquiry
94 followers · 375 posts · Server mathstodon.xyz#LogicalGraphs • 5
• https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no#Abstract_POV
#AbstractPointOfView (cont.)
In particular, though we may note in passing such historical details as the circumstance that Charles Sanders #Peirce used a #StreamerCross symbol where George #SpencerBrown used a #CarpentersSquare marker, the theme of principal interest at the abstract level of form is neutral with regard to variations of that order.
#Logic #PropositionalCalculus #BooleanFunctions
#LawsOfForm #GraphTheory #ModelTheory #ProofTheory
#ProofTheory #modeltheory #graphtheory #lawsofform #BooleanFunctions #PropositionalCalculus #logic #carpenterssquare #spencerbrown #streamercross #Peirce #abstractpointofview #LogicalGraphs
Jon Awbrey · @Inquiry
90 followers · 359 posts · Server mathstodon.xyz#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.
#ProofTheory #modeltheory #graphtheory #logic #lawsofform #spencerbrown #alphagraphs #Peirce #formalsystems #LogicalGraphs
Jon Awbrey · @Inquiry
89 followers · 338 posts · Server mathstodon.xyz#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
#ProofTheory #modeltheory #graphtheory #BooleanFunctions #PropositionalCalculus #ExistentialGraphs #EntitativeGraphs #QualitativeLogic #logic #Peirce #logicalgraph #LogicalGraphs
Jon Awbrey · @Inquiry
81 followers · 259 posts · Server mathstodon.xyz
#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
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
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