Jon, I don't have time for details right now. I'll comment on your note later, either today or tomarrow.
Just one point. I admit that my example about a triangle was too hasty. It's a definition, not a postulate. But I'll clarify the distinctions about axioms and postulates, as they developed from Aristotle's day through Peirce's day to the present In Euclid, axioms are supposed to be universal principles that are so obvious that no justification is needed. And postulates are supposed to be true of geometry. Peirce accepted that terminology, but he recognized that there were non-Euclidean geometries, and therefore the term 'postulate' had to be limited to a specific theory -- because different theories of geometry can have different postulates. In most 20th C mathematics, the fact that (1) many or even most axioms are not self-evident and (2) different theories may have different definitions for the same terms means that there is no need for two different words. For modern mathematics, 'axiom' is the only word that is used. Furthermore, in every branch of mathematics from Euclid to the present, both axioms and postulates are stated as if-then or universally quantified sentences. For understanding Peirce, however, we have to recognize his distinctions. Since he knew the work on non-Euclidean geometry and he was talking about different possible worlds in many MSS (certainly in 2006), we can assume that in 1911, he recognized that different possible worlds could have different postulates. Again, I urge you to read the IKL manuscripts in https://jfsowa.com/ikl . As I said, you don't have to believe anything I said, but you should read the consensus of 9 very competent logicians. A subset of IKL is equivalent to Peirce's Delta graphs. If you just add the option of quantifying over relations, Delta graphs would be almost exactly equivalent to IKL. There is just one rarely used option in IKL that goes further. There is more to say about Risteen, who had a very strong understanding of what Peirce was doing. That is the main reason why Peirce could describe Delta graphs in a note to him in a minimum number of words John ---------------------------------------- From: "Jon Alan Schmidt" <jonalanschm...@gmail.com> Sent: 3/1/24 2:12 PM To: Peirce-L <peirce-l@list.iupui.edu> Subject: [PEIRCE-L] Definitions, Axioms, and Postulates (was Delta Existential Graphs) John, List: I changed the subject line since your reply did not actually have anything to do with my updated candidate for Delta EGs. As always, I would welcome any feedback on that in the other thread. JFS: For any theory of any kind with any logic of any kind, axioms are always stated in an if-then form. The if-part (shaded) states the condition, and the then part states the conclusion. Even definitions are stated as if-then statements in EGs. This reflects just how fundamental if-then is as a logical operation, which is why in his expositions of EGs until June 1911, Peirce consistently recognizes that the scroll is a primitive--along with the blank sheet and (in Beta) the line of identity--and then derives the cut for negation from it as the implication of absurdity or falsity. However, as you have often pointed out, Peirce changes his strategy for explaining EGs in June 1911, between writing R 669 and R 670. He realizes that shading an oddly enclosed area is a more iconic way to convey that it is a different surface from an evenly enclosed area than drawing a thin line as its boundary, and that it would be easier for audiences unfamiliar with EGs to understand and use the notation if he instead treated shading as a primitive for negation. The two approaches are philosophically different, but fully equivalent for implementing classical logic. "Out of the conceptions of non-relative deductive logic, such as consequence, coexistence or composition, aggregation, incompossibility, negation, etc., it is only necessary to select two, and almost any two at that, to have the material needed for defining the others. What ones are to be selected is a question the decision of which transcends the function of this branch of logic" (CP 2.379, 1902). Peirce initially selects coexistence (sheet) and consequence (scroll) for theoretical reasons, but eventually switches from the latter to negation (shading) for practical purposes. JFS: For example: "If x=y and y=z, then x=z." "Every triangle has three sides" is equivalent to "If x is a triangle, x has three sides." In that example, the proposition (pheme) about equality is an axiom, since it must be true of every possible world. But the pheme about triangles is a postulate that is true in geometry, but it might not be a postulate in some other possible world. On the contrary, there is no possible world in which a triangle does not have three sides, unless we are allowing for a world in which the word "triangle" has a different definition. In that case, there would also be some possible worlds in which x=y and y=z but it is not the case that x=z, because the symbol "=" is defined as denoting a non-transitive relation (such as incompossibility or negation) instead of a transitive one (such as equality, identity, coexistence, or implication). Of course, "Every triangle has three sides" could also be stated as an if-then statement--"If something is a triangle, then it has three sides." JFS: The distinction between axioms and postulates is one that Peirce adopted from Euclid ... Peirce himself describes the distinction between definitions, postulates, and axioms as follows. CSP: A definition is the logical analysis of a predicate in general terms. It has two branches, the one asserting that the definitum is applicable to whatever there may be to which the definition is applicable; the other (which ordinarily has several clauses), that the definition is applicable to whatever there may be to which the definitum is applicable. A definition does not assert that anything exists. A postulate is an initial hypothesis in general terms. It may be arbitrarily assumed provided that (the definitions being accepted) it does not conflict with any principle of substantive possibility or with any already adopted postulate. By a principle of substantive possibility, I mean, for example, that it would not be admissible to postulate that there was no relation whatever between two points, or to lay down the proposition that nothing whatever shall be true without exception. For though what this means involves no contradiction, it is in contradiction with the fact that it is itself asserted. An axiom is a self-evident truth, the statement of which is superfluous to the conclusiveness of the reasoning, and which only serves to show a principle involved in the reasoning. It is generally a truth of observation; such as the assertion that something is true. (EP 2:302, c. 1901) A definition "does not assert that anything exists," only that if something exists to which the definition is applicable, then the definitum is likewise applicable to that thing, and vice-versa--i.e., logical equivalence, represented in EGs as two scrolls (nested cuts or ring-shaped shaded areas) with the antecedent and consequent trading places between them. A postulate is "an initial hypothesis" that "may be arbitrarily assumed," so in accordance with R 514, it is scribed in the (shaded) margin of the sheet for a possible state of things and thereby "merely asserted to be possible." An axiom is "a self-evident truth," "generally a truth of observation," so it is scribed on the (unshaded) sheet for the actual state of things and thereby asserted to be true, often as an if-then proposition represented by a scroll. Notice how the if-then operation is integral to all three of these. JFS: After re-reading Don Roberts' chapter on Gamma graphs (which I hadn't read for years), I realize that there is no conflict between that chapter and his writings about Delta graphs in L376. ... Furthermore, what Peirce wrote about Delta graphs in L376 is consistent with his 1903 version of modal logic in every possible world. Again, what Peirce writes about Delta graphs in R L376 is only the single statement, "I shall now have to add a Delta part in order to deal with modals." Everything else in the 19 extant pages of that letter to Risteen is applicable to every part of EG--there is nothing dealing with modals or otherwise unique to Delta. As Roberts summarizes it, "Peirce gives a sketch of the history of EG, reaffirms his opinion that all reasoning is dialogical, and points out that the purpose of EG was not to serve as a calculus, but 'to facilitate the anatomy, and thereby the physiology of deductive reasonings.' This manuscript contains the only reference I have found to a proposed Delta part of EG which would deal with modal logic" (p. 135). JFS: But the "papers" of L376 allow the "postulates" in the margins to state additional information about the nested graphs. For example that the nested graphs, may be wished, hoped, feared, imagined, or occurring at some time in the past, present, future in the real word or in heaven, hell, Wonderland, or the Looking Glass. Again, that is not how postulates work. They are not metalanguage about other propositions, they are hypothetical premisses from which other propositions would follow necessarily as conclusions; or in an if-then proposition, they constitute the antecedent (in the shaded margin) from which other propositions follow necessarily as the consequent (in the remaining unshaded area). That is how all the theorems of Euclidean geometry are derived from its five postulates. Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt / twitter.com/JonAlanSchmidt On Thu, Feb 29, 2024 at 6:15 PM John F Sowa <s...@bestweb.net> wrote: John, Some observations: For any theory of any kind with any logic of any kind, axioms are always stated in an if-then form. The if-part (shaded) states the condition, and the then part states the conclusion. Even definitions are stated as if-then statements in EGs. For example: "If x=y and y=z, then x=z." "Every triangle has three sides" is equivalent to "If x is a triangle, x has three sides." In that example, the proposition (pheme) about equality is an axiom, since it must be true of every possible world. But the pheme about triangles is a postulate that is true in geometry, but it might not be a postulate in some other possible world. The distinction between axioms and postulates is one that Peirce adopted from Euclid, but modern logicians use the word 'axiom' for the starting assumptions of any theory. They rarely use the word 'postulate. After re-reading Don Roberts' chapter on Gamma graphs (which I hadn't read for years), I realize that there is no conflict between that chapter and his writings about Delta graphs in L376. And L376 is completely consistent with the IKL logic of 2006. But IKL has some features that go beyond L376. Anything stated in Delta graphs may be mapped to IKL, but some IKL statements cannot be mapped to Delta graphs. Furthermore, what Peirce wrote about Delta graphs in L376 is consistent with his 1903 version of modal logic in every possible world. But the "papers" of L376 allow the "postulates" in the margins to state additional information about the nested graphs. For example that the nested graphs, may be wished, hoped, feared, imagined, or occurring at some time in the past, present, future in the real word or in heaven, hell, Wonderland, or the Looking Glass. Wonderland, for example, would be a possible world that could not be actualized -- as Peirce said in CP 8.192, stated below. John
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