Gary R, list, The very concept of teridentity is intriguing, and plays a central role in the interface between phaneroscopy and Existential Graphs (and thus logic) that I am trying to explore — which also makes it central to the “major shift in Peirce’s thinking” which I see occurring in the spring of 1906. So I guess it’s time I devoted an entire post to the subject of teridentity. To get the drift of it, I think we need to regard phaneroscopy, ontology, and the conventions of Existential Graphs as three parallel universes of discourse such that an assertion in one necessarily has implications in the other two, and they frequently draw terms from one another to make those assertions.
We begin with the Syllabus which accompanied the 1903 Lowell lectures: [[ Convention No. V. Every heavily marked point, whether isolated, the extremity of a heavy line, or at a furcation of a heavy line, shall denote a single individual, without in itself indicating what individual it is. A heavily marked line without any sort of interruption (though its extremity may coincide with a point otherwise marked) shall, under the name of a line of identity, be a graph, subject to all the conventions relating to graphs, and asserting precisely the identity of the individuals denoted by its extremities. Corollaries. It follows that no line of identity can cross a cut. Also, a point upon which three lines of identity abut is a graph expressing the relation of teridentity. ] CP 4.405-6 ] These definitions must be kept in mind as we proceed. The first “corollary” is important because Peirce will explicitly change his mind about it in the spring of 1906. Notice also that the extremity of a line of identity “denotes” an individual (in the strict logical sense of that word). But this is a somewhat peculiar use of “denote,” compared to its application to propositions, where the subject denotes the object of the propositional sign, and that subject is a symbol, typically a proper noun (a symbol crucially involving an index). In the EG system, the only means of denoting in that sense, i.e. ‘naming’ an individual, is a Selective, a letter placed at the extremity of a line of identity. As we know from the “Bedrock” manuscript, Peirce will by 1908 come to recognize this as a fundamental weakness of the whole EG system. Further on in the same document, we learn that a point of teridentity may also be a spot of teridentity: [[ 10. A spot is a graph any replica of which occupies a simple bounded portion of a surface, which portion has qualities distinguishing it from the replica of any other spot; and upon the boundary of the surface occupied by the spot are certain points, called the hooks of the spot, to each of which, if permitted, one extremity of one line of identity can be attached. Two lines of identity cannot be attached to the same hook; nor can both ends of the same line. 11. Any indefinitely small dot may be a spot replica called a spot of teridentity, and three lines of identity may be attached to such a spot. ] CP 4.416 ] In EGs, of course, a spot usually represents a logical rhema (or semiotic rheme) which has a valency, and is thus analogous to an element of the phaneron. The “boundary of the surface occupied by the spot” is purely imaginary and is not drawn on the graph-instance. It is visible in the graph only where a point on that boundary (a hook or peg) has a line of identity attached to it, and that point coincides with the extremity of the line, which also denotes an individual. The upshot of this, if I understand it, is that a spot of teridentity may be regarded either as a rhema, which is a general concept, or as a denoted individual, which (by definition) is not general, but is identical to the three extremities of the graph of teridentity. The question then is whether this ambiguity of the term represents a vagueness built into the system of EGs or an indefiniteness which is a constituent of the phaneron itself — perhaps even a real vagueness. I don’t know whether this investigation will arrive at an answer to (or an improved statement of) this question or not. The next stop on this guided tour is Peirce’s draft letter to Lady Welby dated March 9, 1906. I’ve placed this document on my website, http://www.gnusystems.ca/PeirceWelbyMarch1906.htm, and I would ask interested readers to view it with their browsers (rather than me trying to paste the many graphics into this email post) to see Peirce’s diagrams in their context, as many of my quotes from the letter will refer to one or more of those diagrams. You can of course keep your browser window open to facilitate moving back and forth between this post and the full version of the document. I might wish there were a better transcription of this document online, but I am not aware of its publication in any format other than the appendix of Semiotics and Significs, the source of my transcription. (A small part of it is given near the end of EP2, but that is not the part I will be focusing on.) Let me start with this remark that Peirce makes about the system of Existential Graphs: [[ If you take in all that I have said and what I am just about to say, you will begin to get an insight into the marvellous perfection and minute truth and profundity of this system. Logicians who analyze a proposition into “terms” and a “copula” are guilty of overlooking the truth I have just enunciated, that it is out of the nature of things for an object to be signified (and remember that the most solitary meditation is dialogue), otherwise than in relation to some actuality or existent in the commend. ] SS 197; I assume that “commend” is synonymous with commens and commind (a term he used earlier in the letter), which in EGs is represented by the Sheet of Assertion. ] If I may rephrase that: You can’t have a predicate without a subject of which it is predicated; that subject must be an actuality or existent in the commens, and is identical to the object of the proposition; that object is denoted by the subject of the proposition while the predicate is signified by that proposition. But that signified predicate can also be regarded as an “object,” because it can be an object of attention in phaneroscopy, namely a First, like the color of the sealing-wax in Peirce’s example that I posted earlier. The phaneroscopic point, I think, is that a color does not appear in the absence of something existent which actually has that color. More generally, we can recognize a First as an element of the phaneron only by prescinding it from the appearance of an existing thing, a Second which has that First as a quality. But also, recognition, or indeed any kind of cognition (or thought or sign), must also involve Thirdness. The semiotic corollary is that you can’t have a Sign without both an Object and an Interpretant as its correlates; you can’t have an Object (in the semiotic sense!) without a Sign which it determines and an Interpretant determined by the Object through the Sign; and you can’t have an Interpretant that doesn’t involve the Secondness between Sign and Object. In short, significance is a triadic relation — and this ‘feeds back’ into phaneroscopy because, as Peirce said in the quote I posted yesterday, it is “a necessary deduction from the fact that there are signs, that there must be an elementary triad.” I’m beginning to sense that this post is growing like Jack’s beanstalk, so I’ll end it with Peirce’s remarks about teridentity in the letter draft, and hope that readers can see how it is related to the above. [[ Introducing the graph of teridentity [diagram] … Quateridentity is obviously composed of two teridentities [diagram] but teridentity cannot be formed out of binidentity _________ where two lines merely make one longer line. Here let me point out what we mean in logic by equivalence and by composition. If one rheme, or verb, would be true in every conceivable case in which the other would be true, and conversely, then and only then those two verbs are logically equivalent. For logic has in view only the possible truth and falsity of signs. To say that a rheme is logically composed of two rhemes is to say that the first rheme is logically equivalent to the composite of the other two. It follows in the first place that every line of identity ought to be considered as bristling with microscopic points of teridentity; so that ___________ when magnified shall be seen to be ] SS 199 ] Now, what are the phaneroscopic and/or metaphysical implications of considering every line of identity as “bristling with microscopic points of teridentity”? If you’ve followed me this far, I’ll leave you with that question while I work on the next instalment. Gary f. From: Gary Richmond <[email protected]> Sent: 30-Mar-19 16:31 Gary F, List, I am intrigued by your commenting "the graph of teridentity (i.e. three-way identity), which occurs when a line of identity branches, cannot represent the basic triadic sign relation, because the Sign, Object and Interpretant are not identical to one another. Then adding: GF: Or are they, in some sense? Perhaps we should leave this question open, for now. I think we should keep it open for now. While I immediately tended to agree with your first statement, your question also immediately had we thinking about such things as the notion (developed by Peirce in places) that the Interpretant itself is a Sign; and some other half-baked notions came to mind. I'm not feeling up to addressing this question today, however. But I'd like to hear other's thoughts and your furthers thoughts about it, Gary F. Best, Gary R
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