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Added a recent reference on Peirce’s Gamma graphs for modal logic. This describes his first approach via broken cuts rather than the later tinctured sheet approach. I keep meaning to see if there’s anything in the latter close to LSR 2-category of modes approach.
According to the broken-cut method, possibility is broken cut surrounding solid cut, while necessity is solid cut surrounding broken cut. Since solid cut is negation, broken cut signifies not-necessarily. Easy to see $\Box \neg = \neg \lozenge$ as the same pattern of three cuts, etc.
In the Alpha case, we’re to think of negated propositions as though written elsewhere on another sheet (or the back of the sheet). There seems to be a three-dimensionality to the graphs, e.g., the conditional as like a tube from one sheet to another, Wikipedia. I gather his later ideas on tinctured graphs had this idea of being inscribed on different sheets.
Added three references
Nathan Haydon and Paweł Sobociński, Compositional diagrammatic first-order logic, in A.-V. Pietarinen, P. Chapman, L. Bosveld-de Smet, V. Giardino, J. Corter, and S. Linker, editors, Diagrammatic Representation and Inference, pages 402–418, Cham, 2020. Springer International Publishing, [pdf]
Filippo Bonchi, Alessandro Di Giorgio, Nathan Haydon and Paweł Sobociński. Diagrammatic algebra of first order logic To appear at LICS, 2024, [arXiv:2401.07055]
Filippo Bonchi, Alessandro Di Giorgio, Davide Trotta, When Lawvere meets Peirce: an equational presentation of boolean hyperdoctrines [arXiv:2404.18795]
- Filippo Bonchi, Alessandro Di Giorgio, Davide Trotta, When Lawvere meets Peirce: an equational presentation of boolean hyperdoctrines [arXiv:2404.18795]
Remember to check your links. removed spurious colon.
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