Jon, John, List: The attempts to interpret the on going discussions leads to simple questions about meaning of symbols and logics.
Given a graphic object, how does one decipher the logical content of it? What types of semantics can be associated with what types of visual distinctions? How many distinctions are to be associated with a graphic object? And how are these distinctions associated with the forms embodied in the objects with logical premises OF ANY ORDER? I pose these questions because as the discussion unfolds into the vast richness of modal logics within the modern forms of symbolic logics, the roles of individual minds in expressing semes appears to become dominant. In other words, the boundaries between symbols and icons seems to disappearing... Cheers Jerry > On Mar 8, 2024, at 9:45 PM, Jon Alan Schmidt <jonalanschm...@gmail.com> wrote: > > Jeff, List: > > Indeed, as Don Roberts summarizes, "The Gamma part of EG corresponds, > roughly, to second (and higher) order functional calculi, and to modal logic. > ... By means of this new section of EG Peirce wanted to take account of > abstractions, including qualities and relations and graphs themselves as > subjects to be reasoned about" > (https://www.felsemiotica.com/descargas/Roberts-Don-D.-The-Existential-Graphs-of-Charles-S.-Peirce.pdf, > 1973, p. 64). Likewise, according to Ahti-Veikko Pietarinen, "In the Gamma > part Peirce proposes a bouquet of logics beyond the extensional, > propositional and first-order systems. Those concern systems of modal logics, > second-order (higher-order) logics, abstractions, and logic of multitudes and > collections, among others" (LF 2/1:28). Jay Zeman says a bit more about Gamma > EGs for second-order logic in his dissertation. > > JZ: There is also another suggestion, in 4.470, which is interesting but to > which Peirce devotes very little time. Here he shows us a different kind of > line of identity, one which expresses the identity of spots rather than of > individuals. This is an intriguing move, since it strongly suggests at least > the second order predicate calculus, with spots now acquiring > quantifications. Peirce did very little with this idea, so far as I am able > to determine, but it seems to me that there would not be too much of a > problem in working it into a graphical system which would stand to the higher > order calculi as beta stands to the first-order calculus. The continuity > interpretation of the "spot line of identity" is fairly clear; it maps the > continuity of a property or a relation. The redness of an apple is the same, > in a sense, as the redness of my face if I am wrong; the continuity of the > special line of identity introduced in 4.470 represents graphically this > sameness. This sameness or continuity is not the same as the identity of > individuals; although its representation is scribed upon the beta sheet of > assertion, its "second intentional" nature seems to cause Peirce to classify > it with the gamma signs. (https://isidore.co/calibre/get/pdf/4481, 1964, pp. > 31-32) > > The CP reference here is to the paragraph right before the one where Peirce > suggests the notation of a dotted oval and dotted line to assert a > proposition about a proposition (CP 4.471, 1903), similar to the first EG on > RLT 151 (1898), as John and I discussed recently > (https://list.iupui.edu/sympa/arc/peirce-l/2024-02/msg00141.html). Here is > what Peirce says (and scribes) in that text; the image is from LF 2/1:165, > with Peirce's handiwork on the right and Pietarinen's reproduction on the > left. > > CSP: Convention No. 13. The letters ρ0, ρ1, ρ2, ρ3, etc. each with a number > of hooks greater by one than the subscript number, may be taken as rhemata > signifying that the individuals joined to the hooks, other than the one > vertically above the ρ taken in their order clockwise are capable of being > asserted of the rhema indicated by the line of identity joined vertically to > the ρ. > Thus, Fig. 57 expresses that there is a relation in which every man stands to > some woman to whom no other man stands in the same relation; that is, there > is a woman corresponding to every man or, in other words, there are at least > as many women as men. The dotted lines between which, in Fig. 57, the line of > identity denoting the ens rationis is placed, are by no means necessary.  > > On the other hand, as I keep pointing out, Peirce's only stated purpose for > needing to add a new Delta part was "in order to deal with modals" (R L376, > 1911 Dec 6), so I doubt that it would have had anything to do with > higher-order logics. John Sowa seems to be convinced that Peirce had in mind > a more generalized situation/context logic using metalanguage, but so far, I > see no evidence for this in the extant 19 pages of that letter to Risteen. > Pietarinen speculates, "Perhaps he planned the Delta part on quantificational > multi-modal logics as can be discerned in his theory of tinctured graphs that > was fledgling since 1905" (LF 1:21), but that also seems unlikely to me since > Peirce ultimately describes the tinctures as "nonsensical" (R 477, 1913 Nov > 8). > > As far as I know, the only new notation that Peirce ever proposes for > representing modal propositions with EGs after abandoning broken cuts (1903) > and tinctures (1906) is the one in his Logic Notebook that I have been > advocating (R 339:[340r], 1909 Jan 7). Echoing Zeman's remark in the > quotation above, the sameness or continuity of a possible state of things > (PST) as represented by a heavy line of compossibility (LoC) in my candidate > for Delta EGs is not the same as the identity of individuals as represented > by a heavy line of identity in Beta EGs. > > Regards, > > Jon Alan Schmidt - Olathe, Kansas, USA > Structural Engineer, Synechist Philosopher, Lutheran Christian > www.LinkedIn.com/in/JonAlanSchmidt > <http://www.linkedin.com/in/JonAlanSchmidt> / twitter.com/JonAlanSchmidt > <http://twitter.com/JonAlanSchmidt> > On Fri, Mar 8, 2024 at 5:11 PM Jeffrey Brian Downard <peirce-l@list.iupui.edu > <mailto:peirce-l@list.iupui.edu>> wrote: >> Hello John, Jon, List, >> >> >> >> Peirce examines both first and second intentional logics. The distinction >> appears to be similar, in some respects, to the contemporary distinction >> between first and second order logics. Here, for instance, is an SEP entry >> on higher order logics: >> https://seop.illc.uva.nl/entries/logic-higher-order/#HighOrdeLogiVisVisTypeTheo >> >> >> >> Does Peirce’s explorations in the Gamma system of the EG, and his >> contemplation of a possible Delta system, bear some similarities to >> contemporary discussions of higher order logics, such as third order, or >> fourth order, etc.? >> >> >> >> --Jeff D >> > _ _ _ _ _ _ _ _ _ _ > ARISBE: THE PEIRCE GATEWAY is now at > https://cspeirce.com and, just as well, at > https://www.cspeirce.com . It'll take a while to repair / update all the > links! > ► PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON > PEIRCE-L to this message. PEIRCE-L posts should go to peirce-L@list.iupui.edu > . > ► To UNSUBSCRIBE, send a message NOT to PEIRCE-L but to l...@list.iupui.edu > with UNSUBSCRIBE PEIRCE-L in the SUBJECT LINE of the message and nothing in > the body. 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