Dear Gary, Everything I have done in this area for the last 2021 – 1967 = 54 years is strongly related to Peirce and what his logic makes possible in ways no other approach does. That does not mean the natural development of this line of inquiry ends with Peirce’s last writings, anymore than any other live tradition ends with the passing of its pioneers.
If you have questions about the relation of my contributions to Peircean thought and pragmatism in general then I would be more than pleased to respond to them on List. Regards, Jon http://inquiryintoinquiry.com > On Jun 17, 2021, at 7:02 PM, Gary Richmond <[email protected]> wrote: > > off List, > > Jon, > > Please find a way to bring your very many posts around to Peirce, and when > there are those that can't be brought around, please omit Peirce-L from the > address. Again, Peirce-L is not a repository for everything and anything > pertaining to logic, to every bit of your work, for example, with cactus > graphs. And, in truth, I've seen so very much of this before. > > Thanks, > > Gary > > > “Let everything happen to you > Beauty and terror > Just keep going > No feeling is final” > ― Rainer Maria Rilke > > > Gary Richmond > Philosophy and Critical Thinking > Communication Studies > LaGuardia College of the City University of New York > > > > > > > >> On Thu, Jun 17, 2021 at 6:45 PM Jon Awbrey <[email protected]> wrote: >> Cf: Differential Logic • 4 >> https://inquiryintoinquiry.com/2020/03/26/differential-logic-4/ >> >> Differential Expansions of Propositions >> ======================================= >> https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Differential_Expansions_of_Propositions >> >> Bird’s Eye View >> =============== >> https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Bird.27s_Eye_View >> >> An efficient calculus for the realm of logic represented by boolean functions >> and elementary propositions makes it feasible to compute the finite >> differences >> and the differentials of those functions and propositions. >> >> For example, consider a proposition of the form “p and q” >> graphed as two letters attached to a root node, as shown below. >> >> Figure 1. Cactus Graph Existential p and q >> https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-existential-p-and-q.jpg >> >> Written as a string, this is just the concatenation “p q”. >> >> The proposition pq may be taken as a boolean function f(p, q) >> having the abstract type f : B × B → B, where B = {0, 1} is >> read in such a way that 0 means false and 1 means true. >> >> Imagine yourself standing in a fixed cell of the corresponding >> venn diagram, say, the cell where the proposition pq is true, >> as shown in the following Figure. >> >> Figure 2. Venn Diagram p and q >> https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagram-p-and-q.jpg >> >> Now ask yourself: What is the value of the proposition pq >> at a distance of dp and dq from the cell pq where you are >> standing? >> >> Don't think about it — just compute: >> >> Figure 3. Cactus Graph (p, dp)(q, dq) >> https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-pdpqdq-1.jpg >> >> The cactus formula (p, dp)(q, dq) and its corresponding graph arise >> by replacing p with p + dp and q with q + dq in the boolean product >> or logical conjunction pq and writing the result in the two dialects >> of cactus syntax. This follows because the boolean sum p + dp is >> equivalent to the logical operation of exclusive disjunction, which >> parses to a cactus graph of the following form. >> >> Figure 4. Cactus Graph (p, dp) >> https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-pdp-1.jpg >> >> Next question: What is the difference between the value of >> the proposition pq over there, at a distance of dp and dq from >> where you are standing, and the value of the proposition pq where >> you are, all expressed in the form of a general formula, of course? >> The answer takes the following form. >> >> Figure 5. Cactus Graph ((p, dp)(q, dq), pq) >> https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-pdpqdqpq-1.jpg >> >> There is one thing I ought to mention at this point: Computed over B, >> plus and minus are identical operations. This will make the relation >> between the differential and the integral parts of the appropriate >> calculus slightly stranger than usual, but we will get into that later. >> >> Last question, for now: What is the value of this expression from your >> current standpoint, that is, evaluated at the point where pq is true? >> Well, replacing p with 1 and q with 1 in the cactus graph amounts to >> erasing the labels p and q, as shown below. >> >> Figure 6. Cactus Graph (( , dp)( , dq), ) >> https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-dp-dq-1-1.jpg >> >> And this is equivalent to the following graph. >> >> Figure 7. Cactus Graph ((dp)(dq)) >> https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-dpdq-1.jpg >> >> We have just met with the fact >> that the differential of the AND >> is the OR of the differentials. >> >> • p and q ---Diff---> dp or dq >> >> Figure 8. Cactus Graph pq Diff ((dp)(dq)) >> https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-pq-diff-dpdq-1.jpg >> >> It will be necessary to develop a more refined analysis of >> that statement directly, but that is roughly the nub of it. >> >> If the form of the above statement reminds you of De Morgan's rule, >> it is no accident, as differentiation and negation turn out to be >> closely related operations. Indeed, one can find discussions of >> logical difference calculus in the Boole–De Morgan correspondence >> and Peirce also made use of differential operators in a logical >> context, but the exploration of these ideas has been hampered by >> a number of factors, not the least of which has been the lack of >> a syntax adequate to handle the complexity of expressions evolving >> in the process. >> >> Note. Due to the large number of Figures I won't attach them here, >> but see the blog post linked at top of the page for the Figures and >> also for the proper math formatting. >> >> Regards, >> >> Jon >> _ _ _ _ _ _ _ _ _ _ >> ► PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON >> PEIRCE-L to this message. PEIRCE-L posts should go to >> [email protected] . >> ► To UNSUBSCRIBE, send a message NOT to PEIRCE-L but to [email protected] >> with UNSUBSCRIBE PEIRCE-L in the SUBJECT LINE of the message and nothing in >> the body. More at https://list.iupui.edu/sympa/help/user-signoff.html . >> ► PEIRCE-L is owned by THE PEIRCE GROUP; moderated by Gary Richmond; and >> co-managed by him and Ben Udell.
_ _ _ _ _ _ _ _ _ _ ► PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON PEIRCE-L to this message. PEIRCE-L posts should go to [email protected] . ► To UNSUBSCRIBE, send a message NOT to PEIRCE-L but to [email protected] with UNSUBSCRIBE PEIRCE-L in the SUBJECT LINE of the message and nothing in the body. More at https://list.iupui.edu/sympa/help/user-signoff.html . ► PEIRCE-L is owned by THE PEIRCE GROUP; moderated by Gary Richmond; and co-managed by him and Ben Udell.
