Mary,
The subject line got truncated in your post so I made up a new and shorter one to continue the thread. I can only speak for myself - I read your post carefully more than once, but left it to others to reply to it (which Gary R had already done, actually) because I had no answers to the questions you raised in it. I couldn't make a connection between your suggestion of “would-be hypothetical situations, such as the mobius strip” and Peirce's idea of using the verso of the sheet of assertion as the area inside a cut. In fact I still don't see a connection. A mobius strip, being a bounded surface with only one side, doesn't have a verso, and I don't see how it relates to Peirce's “discovery” that the verso of the sheet represents “a kind of possibility” and not just the negation of the graph within the cut. I also couldn't get a handle on your question “Would boundedness exist in a mobius strip?” or its relevance to the issue we’ve been discussing yesterday and today. Maybe it’s just my obtuseness, but you’ll need to explain what you were driving at before I can see its relevance to Jeff’s post that I did reply to. (I assume you want to be given credit for more than just mentioning the “verso” in your post, but I don’t yet see what else in it anticipates Jeff’s post). gary f. -----Original Message----- From: Libertin, Mary [mailto:mli...@ship.edu] Sent: 20-Jan-15 8:36 AM To: biosemiot...@lists.ut.ee Subject: [biosemiotics:7975] Re: Contradictories, contraries, etc. WAS Jeffrey, Gary R, Lists, I brought up significance of the verso side of the existential graphs in my most recent post last week, but have not been acknowledged as the initiator of this thread. Gary R. responded to my comment on the distributed and undistributed significance of the immediate and direct objects of the dicisign. I wrote: "I do think we should go on. Stjernfelt places his discussion of the dicisign in as large a Universe of Discourse as is practical for his audience. We need to be more tolerant of interdisciplinary analogies. I also think we need some instruction when we find it necessary, which means we should ask. Here are some of the questions that came to mind after the third time reading NP: how is the sheet of assertion, recto and verso sides, to be understood in various ³would¹be² hypothetical situations, such as the mobius strip. Would boundedness exist in a mobius strip? The concepts of in/out, the whole or the part of the universe of discourse are in chapter 8, along with many other important thoughts, juxtapositions, questions, and musings. . . ” I have been researching this area and find it surprising that my initial discussion has been overlooked. If this is the first time the issue has been discussed I wish to be given credit or acknowledged in the discussion. Mary Libertin Mary Libertin, PhD Professor of English Shippensburg University of PA Shippensburg PA 17257 <mailto:mli...@ship.edu> mli...@ship.edu On 1/19/15, 11:16 PM, "Jeffrey Brian Downard" < <mailto:Jeffrey.Downard@nau. edu> jeffrey.down...@nau.edu> wrote: >Gary F., Lists, > >You’ve provided a sketch of some of the developments you see in >Peirce’s account of how we should interpret the two sides of the sheet >of assertion. One amendment I’d like to add to your sketch is that, as >early as the Lowell Lectures of 1903, Peirce described a book of >multiple sheets that are tacked together at the corners. As such, he >was already thinking of multiple related pages where things are being >asserted and denied as related actualities, necessities and possibilities. > >The idea of using both sides of a single sheet is important for the >following reasons. As we know, the development of the existential >graphs is largely motivated by the goal of drawing on topological ideas >as a way of gaining a more graphical system of logic than is available >using a more symbolic and algebraic approach. Let me ask: what is the >topological import of having a system that uses two separate sides of a >sheet? My hunch is that the two sides are being treated as separate >(literally, disconnected in some respects) because they are not being >conceived as part of a single non-orientable surface. That is, the >topology of the sheet does not have one or more cross-caps. As such, >it has the topological characteristic of a sphere or a torus (perhaps >with more than one hole in the donut) that is orientable with respect >to the two sides of the surface. > >The reason I point out that Peirce was already describing a book with >many sheets in 1903 is that, from the get-go with the gamma graphs, he >was consistently moving back and forth between a simple system with one >page and a more complex version with multiple pages. We shouldn’t be >surprised to find Peirce doing this. Like any good mathematician, he >is moving from a more complex version of a problem that is stated in a >higher number of dimensions (say 3, 4 or more) to a simpler version of >the problem stated in only 2 dimensions―and then back up again to a >system of higher dimensions once we’ve cleared matters up by working >with the simpler case. > >Let’s separate two sets of questions: one set stems from a specific >set of points Peirce makes in the 1906 essay “An Improvement on the >Gamma Graphs,” and another set stems from trying to understand how this >essay fits into the larger trajectory of the development of his views >from 1903 up to 1906 and through to the end of his productive career as a logician. > For starters, I will focus on the first set of questions about the >1906 essay itself. In that essay, as he draws out the second group of >consequences that follow from his new discovery, he adds the following >remarks: > >First: “As soon, however, as I discovered that the verso of the sheet >represents a universe of possibility, I saw clearly that such a graph >was not only interpretable, but that it fills the great lacuna in all >my previous developments of the logic of relatives. For although I have >always recognized that a possibility may be real, that it is sheer >insanity to deny the reality of the possibility of my raising my arm, >even if, when the time comes, I do not raise it;” > >Second: “and although, in all my attempts to classify relations, I have >invariably recognized, as one great class of relations, the class of >references, as I have called them, where one correlate is an existent, >and another is a mere possibility; yet whenever I have undertaken to >develop the logic of relations, I have always left these references out >of account, notwithstanding their manifest importance, simply because >the algebras or other forms of diagrammatization which I employed did >not seem to afford me any means of representing them. I need hardly say >that the moment I discovered in the verso of the sheet of Existential >Graphs a representation of a universe of possibility, I perceived that >a reference would be represented by a graph which should cross a cut, >thus subduing a vast field of thought to the governance and control of >exact logic.” (CP, >4.579) > >So, the initial point I’d like to make is that it is clear Peirce is >keen to think about the two sides of the sheet of assertion because he >sees that it might supply him with a way of bringing some clarity to >reference. Let’s see what he has to say about reference in 1903. > >In “Nomenclature and Division of Dyadic Relations,” Peirce >characterizes reference as a dyadic relation between two subjects of >different categories of being (e.g., one subject is an actual existent, >and another subject is a possible). This is what we have when >something like a quality, as reference to a ground, is brought into a >dyadic relation with a token diagram (thereby yielding a double >reference to ground and to object). A dyadic relation proper is what >obtains when both subjects belong to the same category of being. One >species of a dyadic relation is a referential relation. This kind of >relation is what we have when the two subjects of the dyadic relation >belong to different universes of discourse. If both are from the same >universe of discourse, then it is a rerelation. >With that bit of terminology in hand, let’s see what Peirce seems to be >saying about the relation of reference―and how it might compare to a >referential relation as a species of a dyadic relation proper. If we >think of the two sides of a sheet of assertion as representing, on the >front, existential assertions, and, on the back, a related set of >possibilities, then the connection between the two sides of the page is >a reference relation. On the other hand, if we think about the >relationship between two such double-sided pages as so many pages in a >larger book, then each of the pages might be related to one another as >representing somewhat different universes of discourse. That is, the >pages that are farther along in the book represent what might possibly >obtain if something were different from the way it is represented in >the universe of discourse of the first page in the book. So, we have a >related set of universes of discourse that stand, one to another, as so >many maps > > of a photograph (see “Gamma Part of the Existential Graphs,” CP, 4.513). > >Given the hour and the length of this email, let me hold off until >tomorrow to talk with you about the development of these ideas from >1906 on to the end of his career. The development I’m particular >interested in thinking about is his treatment of a book of pages as >connected by a pencil. This is another idea he is taking from topology >and projective geometry and using to develop and refine his graphical >logic. The reason I find this idea so interesting is that the pages >are now connected to one another as a set of continuous rotations >around a line. This rotation around a line is a movement in thought >that starts with a surface and then―by rotation--generates what is called a moduli space. > >So, before turning to the idea of a book of pages connected by a >pencil, let me see if you have questions about reference and the two >sides of the sheet of assertion. You seem to be puzzled, for instance, >about Peirce’s apparent movement from thinking about the two sides as >representative of positive assertion on the front and denial on the >back, to an system where he is treating the front as positive >existential assertion and the back as possibilities. I’ve been >assuming that the two ideas are intimately related. The possibilities >on the back are what obtain when some part of what is stated on the >front is denied (and vice versa). As such, we would say that if some >part of what is denied on the back were not the case (e.g., if the >denial were in error), then there would be something on the front of >the sheet that represents what would be possible under such a >condition. The real developments in Peirce’s thinking come in the way >he is diagramming how these positive exist > > ential assertions, denials, and possibilities under different sets of >conditions are related to one another via lines, branches, indexed >branches of a system of connections, areas with fuzzy borders, etc. > >Hope that helps, > >Jeff >
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