Robert, List: I have one technical quibble.
RM: Gamma graphs, isomorphic to normal modal logic What the relevant Wikipedia article ( https://en.wikipedia.org/wiki/Existential_graph) actually says is that Gamma graphs are "(nearly) isomorphic to normal modal logic," and even this is not quite right. As another Wikipedia article explains ( https://en.wikipedia.org/wiki/Normal_modal_logic), "normal modal logic" encompasses any formal system that (among other things) includes the "necessitation rule" by which every theorem is *necessarily *true. As I discuss at length in another thread ( https://list.iupui.edu/sympa/arc/peirce-l/2021-04/msg00092.html), Gamma graphs only satisfy this criterion when the sole axiom is changed from the blank sheet to an empty broken cut within an empty solid cut, and this further requires restricting iteration/deiteration across broken cuts to only certain kinds of modal graphs. When the blank sheet is retained as the only axiom as in Alpha and Beta, and iteration/deiteration is allowed for any graph across any cuts, Gamma graphs are instead isomorphic to the four-valued Ł-modal system of Łukasiewicz, which *is not* a "normal modal logic" because it *does not* include the "necessitation rule." For all the details and some examples, see Zeman's dissertation ( https://isidore.co/calibre/get/pdf/4481, 1964, pp. 160-175). His Gamma-4/4.2/5 versions are isomorphic to the normal modal logics S4/S4.2/S5, while his Gamma-MR version is isomorphic to Ł-modal. More recently, Ma and Pietarinen have identified a total of 15 different normal modal logics that can be implemented with Gamma graphs, although they treat the blank sheet as the lone axiom and the "necessitation rule" as an additional permission ( https://www.researchgate.net/publication/315949046_Gamma_graph_calculi_for_modal_logics, 2018). They do not mention Gamma-MR at all. Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt On Thu, May 13, 2021 at 3:46 AM robert marty <[email protected]> wrote: > List, > I agree with John. His statement could be called "the proper use of > citations." I would add, however, a condition that is not necessary but > desirable: citations should produce mainly to show the adequacy of a new > theoretical thought, or even a single formalized argument, with Peirce's > thought. This is the "citation as illustration." Examples dedicated by > Wikipedia : > > - Alpha graphs, isomorphic to sentential logic and to the two-element > Boolean algebra > - Beta graphs, isomorphic to first-order logic with identity, with all > closed formulas > - Gamma graphs, isomorphic to normal modal logic > > Another example that concerns me is the relations between the classes of > signs within their lattice and their adequacy with Peirce's affinities (no > dedicated ...) > This is a way to help constructive criticism countered by arguing with > other quotes and different arguments. This is why the elaboration > (individually but preferably collectively) of complete and validated > Thesauri on essential terms is indispensable. Thus, for example, I found it > necessary to do so to define "sign." Indeed, it appeared to me that the > individual choices on which the researchers relied could fuel endless > debates that the pooling of such a set could clarify, even settle. Indeed, > a Thesaurus always produces an effect of meaning through the > categorizations it allows and the historical perspective it introduces. > This is why, whatever the dispute one may have with him, Jon Alan is to be > congratulated for the 61 maxims of pragmatism that he has assembled and > commented on with objectivity, for he has produced an indisputable common > good. > > Best regards, > Robert Marty > Honorary Professor; Ph.D. Mathematics; Ph.D. Philosophy > fr.wikipedia.org/wiki/Robert_Marty > *https://martyrobert.academia.edu/ <https://martyrobert.academia.edu/>* >
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