A 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
#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
I'm watching "Proof Complexity and Meta-Mathematicshttps" bit by bit
https://www.youtube.com/playlist?list=PLgKuh-lKre10CktjVrpIX9VEvX6Rog36N
#ProofTheory #simonsinstitute #mathematicallogic
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. 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
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
#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
#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
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
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
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
#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
#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
#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
#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
#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
#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
#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
#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
#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
#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