List: I was hoping to post this on World Logic Day yesterday, but it took me until today to be satisfied with it. As I have noted before, citing the work (mostly in Spanish) of Arnold Oostra, Peirce came remarkably close to anticipating what is now known as *intuitionistic* logic. This name for it is unfortunate because of its association with the philosophical system of intuitionism, which Peirce almost certainly would *not* have endorsed. The common alternative of calling it *constructive* logic is perhaps less objectionable, but still does not really capture the Peircean motivations for exploring it.
Since the logic of true continuity is constructive/intuitionistic as reflected in the mathematical systems of synthetic differential geometry and smooth infinitesimal analysis, it seems appropriate instead to refer to it as *synechistic* logic. Accordingly, I am hereby trying out the new label of "synechistic existential graphs" (SEGs) for what I previously called IEGs. Preparing and further studying the ones that I attached to my previous post ( https://list.iupui.edu/sympa/arc/peirce-l/2021-01/msg00005.html) has prompted a few more observations that I think are worth mentioning. First, the fact that in SEGs "if X then Y" can be derived from "not-X or Y" but not vice-versa raises the question of whether inclusive disjunction (also called alternation) might be more fundamental than consequence. Reviewing Oostra's papers led me to realize that the scroll is a sign for *both* consequence and disjunction since having X in the outer close and two different loops, one with Y in its inner close and the other with Z in its inner close, signifies "if X then (Y or Z)." With the outer close empty instead, it signifies "if anything is true then (Y or Z)," which is equivalent to simply "Y or Z." Besides, it remains necessary for at least one primitive to be unsymmetrical like consequence rather than symmetrical like disjunction. Second, recall that the blank sheet for coexistence always has room for attaching another graph (tercoexistence) and a heavy line for identity always has room for attaching another branch (teridentity). In similar fashion, an evenly enclosed scroll always has room for attaching another loop and inner close by insertion. In other words, disjunction is a *continuous* logical relation like the other two (cf. R 499(s):33-35, 1906), which is presumably why it can be represented by the blank sheet in *entitative* graphs (CP 4.434, 1903). On the other hand, any loop and inner close of an oddly enclosed scroll may be erased, while any loop and inner close whatsoever may be iterated or deiterated within the outer close of the scroll to which it is attached. Third, when any scroll has multiple loops and one of them has a blackened inner close, the latter may be deleted since "X or falsity" is equivalent to simply "X." As a result, a loop with a blackened inner close only needs to remain in a scroll that has no *other* loops, thereby signifying negation rather than consequence and/or disjunction. Oostra *defines* a cut as equivalent to a negation scroll, but I still prefer the latter as more analytical in the sense that "the whole effort has been to dissect the operations of inference into as many distinct steps as possible" (CP 4.424, 1903). This is a feature that differentiates my SEGs from his IEGs. Fourth, any loop and inner close of an evenly enclosed scroll may become a nested and oddly enclosed negation scroll, which also causes the original scroll to become a negation scroll unless it has additional loops and inner closes. As Oostra shows, this "release" is not a distinct rule, it is obtained by carrying out a series of the usual permissions--insertion, iteration, deiteration, and erasure. Although he mentions the reverse transformation of "adherence," he does not spell out that it may only be implemented for a nested and evenly enclosed negation scroll, such that it becomes a loop and inner close of the next larger (oddly enclosed) scroll. Fifth, the practical advantage of shading is further magnified by these additional aspects of SEGs, since an evenly enclosed scroll is easily recognized as one having a shaded outer close. Only loops with unshaded inner closes may be inserted or released, and only loops with shaded inner closes may be erased or adhered. Following this convention is obviously much less tedious than counting enclosures, which is necessary when thin oval lines are used instead. It is also a more iconic way of distinguishing the surfaces that correspond to possibility (shaded) and actuality (unshaded). This is another feature that differentiates my SEGs from Oostra's IEGs. Finally, in classical EGs a scroll and nested cuts are logically equivalent, so release and adherence may be employed for any scroll accordingly. See attached (SEGs vs EGs.jpg) for how identity as "if X then X" and non-contradiction as "not-(X and not-X)" can be derived directly from the blank sheet in *both* EGs and SEGs, while double negation elimination as "if not-not-X then X" requires releasing a loop with a shaded inner close and excluded middle as "X or not-X" requires adhering a loop with an unshaded inner close, neither of which is allowed in SEGs. The upshot is that SEGs *become* classical EGs when either of these propositions--or equivalently, Peirce's Law (so-called) as "((P implies Q) implies P) implies P"--is treated as an additional axiom. Hence synechistic logic is to classical logic as non-Euclidean geometry is to Euclidean geometry, with excluded middle and its corollaries as the counterpart to the parallel postulate. Moreover, just as Peirce holds that "it seems for the present impossible to suppose the postulates of geometry precisely true" (CP 1.130, c. 1893), he also maintains that the "assumption" underlying excluded middle--namely, "that reality is so determinate as to verify or falsify every possible proposition"--is "utterly unwarranted" and not "strictly true" (NEM 3:758-760, 1893). Both these views are manifestations of his thoroughgoing synechism. As J. Jay Zeman puts it in his 1968 paper, "Peirce's Graphs--the Continuity Interpretation" (https://www.jstor.org/stable/40319551), "For Peirce ... mathematical or deductive reasoning is an integral part of reality itself. The development of a thorough understanding of mathematical reasoning is a first and absolutely essential step towards the development of a thorough understanding of reality" (p. 145). He later adds, "The Peirce of the existential graphs ... is as well the Peirce of synechism, the metaphysical doctrine asserting the reality of continua and the continuity of reality" (p. 148). Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt
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