Jon, There are several points that must be considered. The first is that all modern versions of modal logic after C. I. Lewis (including those based on post-1970 methods) are consistent with or variations of one or more of the versions specified by Lewis). That includes the versions of modal logic supported by the IKL logic of 2006. Further qualifiers such as wishing, hoping, fearing, specified in Holy Scriptures. . . may be ADDED to the specifications that determine possibility, actuality, or necessity.
Second, Lewis was inspired by Peirce's 1903 specifications, and no one knows how many other MSS Lewis may have read. But Lewis adopted the much more readable basic operators, represented by □ and ◇. For readability, they correspond to the words 'necessary' and 'possible' in English or their equivalents in other languages. Third, all of Peirce's 1903 combinations can be represented by combinations of those two symbols and negation. But the papers of Delta graphs can represent more information about each world, including the reasons why it happens to be possible, actual, necessary, or impossible. That is also true of the worlds specified by Hintikka, Dunn, IKL, and others. The specifications of those worlds can also add further information beyond just those two operators plus negation. Fourth, more issues of modality related to Peirce and modern variations were discussed at a workshop in Bogota hosted by invitation of Zalamea. Some of the presentations were published in the Journal of Applied Logics 5:5, 2018. http://www.collegepublications.co.uk/downloads/ifcolog00025.pdf . Others in the Journal Zalamea edited, Cuadernos de Sistemática Peirceana 8, 2016. https://ucaldas.academia.edu/CuadernosSistem%C3%A1ticaPeirceana . (Although this version is dated 2016, it was delayed by late submissions and editing until 2019.) Fifth, Risteen's background was significant. He was a former student of Peirce's at Johns Hopkins, and he was a paid assistant to Peirce for definitions in the Century Dictionary from S to Z. His most important contribution (at least for Delta graphs) was his note about Cayley's mathematical trees for the dictionary entry and in the discussions with Peirce in December 1911. It would have been wonderful to have a YouTube of their discussions on 3 Dec. 1911. The specifications about papers in L376 would allow a tree structure of papers. Risteen's knowledge of mathematical trees is a likely reason why Peirce had invited him to visit in December and why he was writing that letter to him shortly after the visit. And note the very strange coincidence that occurred shortly after Peirce began the letter L376: Juliette had washed and scrubbed the floor in December after a visitor had left. There were papers on the floor. Peirce slipped on the floor in an unusual fall that caused the kind of injury that occurs in a twisting motion. And the injury took six months to heal. Scientists, engineers, and crime investigators do not believe in strange coincidences that involve two or more unusual causes. They search for a hidden connection. John ---------------------------------------- From: "Jon Alan Schmidt" <jonalanschm...@gmail.com> List: As I continue contemplating my updated candidate for Delta EGs (see earlier posts below), I am finding that, in conjunction with the laws and facts semantics (LFS) developed by Dunn and Goble, it is very helpful for explicating the effects of adding various modal axioms to classical logic. For example, the distribution axiom K = □(p → q) → (□p → □q) that is included in all so-called "normal" modal logics is illustrated by the fact that if p → q is on every sheet for a possible state of things (PST) and p is also on every PST sheet, then q is likewise on every PST sheet or can be derived on any PST sheet where it is initially missing. As I have mentioned before, other axioms assign different properties of the binary alternativeness/accessibility relation (AR) between the actual state of things (AST) and any PSTs, as well as the latter and their higher-order PSTs when there are iterated modalities. - Serial, axiom D = □p → ◇p, or ◇⊤; every law-graph on the AST sheet is a fact-graph on at least one PST sheet, and any graph that can be derived from the blank on the AST sheet can also be derived from the blank on at least one PST sheet. - Reflexive, axiom T = □p → p, or p → ◇p; every law-graph on the AST sheet is also a fact-graph on the AST sheet, and every fact-graph on the AST sheet is a fact-graph on at least one PST sheet. - Symmetric, axiom B = ◇□p → p, or p → □◇p; every law-graph on any PST sheet is a fact-graph on the AST sheet, and every fact-graph on the AST sheet is a fact-graph on at least one second-order PST sheet for every first-order PST sheet. - Transitive, axiom 4 = □p → □□p, or ◇◇p → ◇p; every law-graph on the AST sheet is a law-graph on every PST sheet, and every fact-graph on a second-order PST sheet is a fact-graph on at least one first-order PST sheet. - Euclidean, axiom 5 = ◇□p → □p, or ◇p → □◇p; every law-graph on a PST sheet is a law-graph on the AST sheet, and every fact-graph on a PST sheet is a fact-graph on at least one second-order PST sheet for every first-order PST sheet. LFS effectively stipulates that the AR is serial because every law-graph on the AST sheet is a fact-graph on every PST sheet--its basic principle is that possibility is defined as consistency with the laws of the AST--and any classical tautology can be derived from the blank on every sheet. The AR properties and their corresponding axioms are then combined in different ways for different formal systems--serial for D (deontic logic), reflexive for T (or P with no iterated modalities), reflexive and symmetric for B, reflexive and transitive for S4, or reflexive and euclidean for S5. Any relation that is reflexive is also serial, while any relation that is reflexive and euclidean is also symmetric and transitive, and therefore an equivalence. As a result, in S5, every law-graph on the AST sheet is likewise a law-graph on every PST sheet, every second-order PST sheet, and vice-versa--i.e., every PST of any order has the very same laws as the AST. On the other hand, in S4, every law-graph on the AST sheet is likewise a law-graph on every PST sheet and every second-order PST sheet, but there might be additional law-graphs on those PST sheets--the set of relevant laws never shrinks when going to a higher-order PST, but it can grow. Applied to temporal logic, this is reminiscent of Peirce's hyperbolic cosmology in accordance with synechism. CSP: At present, the course of events is approximately determined by law. In the past that approximation was less perfect; in the future it will be more perfect. The tendency to obey laws has always been and always will be growing. We look back toward a point in the infinitely distant past when there was no law but mere indeterminacy; we look forward to a point in the infinitely distant future when there will be no indeterminacy or chance but a complete reign of law. But at any assignable date in the past, however early, there was already some tendency toward uniformity; and at any assignable date in the future there will be some slight aberrancy from law. (CP 1.409, EP 1:277, 1887-8) CSP: The state of things in the infinite past is chaos, tohu bohu, the nothingness of which consists in the total absence of regularity. The state of things in the infinite future is death, the nothingness of which consists in the complete triumph of law and absence of all spontaneity. Between these, we have on our side a state of things in which there is some absolute spontaneity counter to all law, and some degree of conformity to law, which is constantly on the increase owing to the growth of habit. (CP 8.317, 1891) In other words, the universe is constantly proceeding from a PST with only facts (no laws) toward a PST with only laws (no facts), while the AST always has both facts and laws. Regards, Jon
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