List: Having discussed the Alpha and Beta parts of Peirce's system of Existential Graphs (EG) in previous threads, including their relevance for what is now known as intuitionistic logic ( https://list.iupui.edu/sympa/arc/peirce-l/2021-02/msg00022.html) and Peirce's long-sought proof of his pragmatism ( https://list.iupui.edu/sympa/arc/peirce-l/2021-03/msg00015.html), I am now moving on to the Gamma part.
In his 1903 Lowell Lectures and accompanying Syllabus, Peirce incorporates certain aspects of modal logic into EG by introducing a new sign, the broken cut (CP 4.515-516). While a proposition is asserted as *actually true* where unenclosed (Fig. 181) or within a double solid cut (Fig. 180), and *actually false* within a single solid cut (Fig. 185), it is asserted as *possibly false* within a single broken cut (Fig. 186). With double negation elimination, the remaining modalities correspond to different combinations of cuts--*necessarily true* (not possibly false) for a broken cut within a solid cut (Fig. 179), *possibly true* (possibly not false) for a solid cut within a broken cut (Fig. 182), and *necessarily false* (not possibly true) for a solid cut within a broken cut within another solid cut (Fig. 183). All six of these graphs are shown in the attachment (gamma.jpg). The permission for erasure and insertion is the same with broken cuts as with solid cuts. Moreover, an evenly enclosed solid cut may be partially "erased" to become a broken cut, and an oddly enclosed broken cut may have additional cuts "inserted" in its gaps to become a solid cut. Along with the double (solid) cut rule, this properly ensures that the graph of a *necessarily* true or false proposition can be directly transformed into one that is *actually* true or false, and that the graph of an *actually* true or false proposition can be directly transformed into one that is *possibly* true or false. On the other hand, according to Peirce, "The Rule of Iteration and Deiteration does not apply to the broken cut" (R 478:127[158]). Don Roberts initially interprets this to mean "that Peirce did not allow a graph to be iterated or deiterated across a broken cut" ( https://www.felsemiotica.com/descargas/Roberts-Don-D.-The-Existential-Graphs-of-Charles-S.-Peirce.pdf, 1973, p. 82). However, he later acknowledges that "[Jay] Zeman developed a Gamma version of [David] Lewis' modal systems S4, S4.2, and S5 by tinkering, in a clever way, with the rules of iteration and deiteration" ( https://core.ac.uk/download/pdf/82124291.pdf, 1992, p. 662). For the details, see Zeman's dissertation (https://isidore.co/calibre/get/pdf/4481, 1964, pp. 160-175). Rather than the blank sheet, signifying that every true proposition is *actually* true, the sole axiom in these versions is an empty broken cut within a solid cut, signifying that every true proposition is *necessarily* true. The blank sheet is derived by filling in the oddly enclosed broken cut and then moving the empty double cut out of sight. The problem is that the usual iteration/deiteration of any graph across any cuts would improperly enable transforming every *actually* true proposition into one that is *necessarily* true, and every *possibly* false proposition into one that is *actually* false. To prevent this, each system has certain restrictions on iteration/deiteration across broken cuts. - Gamma-4 only allows it for necessarily true propositions (broken cut within solid cut). - Gamma-4.2 also allows it for possibly necessarily true propositions (double broken cut). - Gamma-5 allows it for all modal propositions (at least one broken cut). Arnold Oostra discusses these three specific versions in his paper about adapting Gamma EG for intuitionistic modal logic ( https://www.academia.edu/36792039/Cuadernos_de_Sistem%C3%A1tica_Peirceana_4, 2012, pp. 27-50). As with propositional logic (Alpha) and first-order predicate logic (Beta), the key adjustment is distinguishing the continuous scroll for implication (one inner close) and inclusive disjunction (multiple inner closes) from nested cuts, rather than treating them as equivalent. However, a scroll can now have broken outer and/or inner loops, resulting in the following translations with propositions A and B in the outer and inner closes, respectively. - Solid outer and inner loops – If A is actually true, then B is actually true. - Solid outer loop, broken inner loop – If A is actually true, then B is necessarily true. - Broken outer loop, solid inner loop – It is possibly true that if A is actually true, then B is actually true. - Broken outer and inner loops – It is possibly true that if A is actually true, then B is necessarily true. Solid and broken negation scrolls are derived as the actual and possible implication of falsity, each with a solid inner loop enclosing the pseudograph--either a blackened inner close or an empty solid cut. Any inner loop of an evenly enclosed implication or disjunction scroll may be "detached" to become an oddly enclosed negation scroll, and any evenly enclosed negation scroll within another negation scroll may be "adhered" to form an oddly enclosed implication scroll. However, a negation scroll may not be "adhered" to a surrounding implication or disjunction scroll, i.e., one that already has at least one inner loop containing a graph. Note that since double negation elimination is not valid in intuitionistic logic, possibility and necessity must now be independently defined, rather than designating one as a primitive and deriving the other from it. Zeman also proposes a fourth version of Gamma EG that neither Roberts nor Oostra mentions, which is a subject for another post. Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt
_ _ _ _ _ _ _ _ _ _ ► 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.
