Jon, Your recent note reminded me of one I had started in reply to one of yours on April 9th. But I got distracted by miscellaneous details, such as income tax. So I finished the earlier note and include it here.
I have much more to say about modal logic, but I'll save that for another note. For a quick overview, see the 6-page article on "Five Questions on Epistemic Logic", http://jfsowa.com/pubs/5qelogic.pdf . For more, see "Worlds, Models, and Descriptions" http://jfsowa.com/pubs/worlds.pdf . Both articles relate the discussions of 21st century methods to pioneering work by Peirce. John ---------------------- JAS: According to Peirce, classical logic as implemented using the Alpha (propositional) and Beta (first-order predicate) parts of his Existential Graphs (EG) is applicable only to a universe of discourse that is definite, individual, and real. Important point: Peirce's writings are precise. A single word or phrase, if omitted or ignored, may distort or even reverse the meaning of the whole. In the following quotation, the words creator and fictive are critical. CSP: The sheet on which the graphs are written (called the sheet of assertion), as well as each portion of it, is a graph asserting that a recognized universe is definite (so that no assertion can be both true and false of it), individual (so that any assertion is either true or false of it), and real (so that what is true or false of it is independent of any judgment of man or men, unless it be that of the creator of the universe, in case this is fictive); any graph written upon this sheet is thereby asserted of that universe; and any multitude of graphs written disconnectedly upon the sheet are all asserted of the universe. (R 491:29, 1903) In the same year, Peirce defined fictive: "For the fictive is that whose characters depend upon what characters somebody attributes to it" (EP2:209). That is true of every theory of pure mathematics. Its creator is the mathematician who specifies the axioms. When mathematics is applied to physical reality, the fictive universe of discourse (Uod) may be more definite than the physical UoD. But there is a third UoD of experimental observations. Because of errors in measurement, mappings among all three UoDs are, at best, approximations. In the book Photometric Researches (1878), Peirce described his methods for using logic and mathematics to analyze, relate, and reason about the three UoDs: mathematical, physical, and experimental. Although he had not yet developed his terminology of UoDs, the methods he describes in that book are good illustrations of his later theories. See http://jfsowa.com/peirce/PRexcerpts.pdf JAS: However, [Peirce] also maintains that reality itself is general rather than strictly individual, such that some assertions are legitimately indeterminate. That comment is not implied by the quotation you selected: CSP: To speak of the actual state of things implies a great assumption, namely that there is a perfectly definite body of propositions which, if we could only find them out, are the truth, and that everything is really either true or in positive conflict with the truth. This assumption, called the principle of excluded middle, I consider utterly unwarranted, and do not believe it. Still, I hold that there is reason for thinking it to be very nearly true. (NEM 3:758, 1893) You quoted everything up to, but not including the sentence in italics. It explains why Newtonian mechanics, a definite mathematical theory, can be "very nearly true" for anything we observe without special instruments. JAS: In these passages, Peirce refers to the "great assumption" that every proposition is either true or false as "the principle of excluded middle." However, in today's standard logical terminology, this is instead designated as the semantic principle of bivalence and distinguished from the (so-called) law of excluded middle, which is that either a proposition or its negation must be true-or, equivalently, that a proposition and its negation cannot both be false. Accordingly, there are at least two basic approaches for deviating from classical logic to facilitate treating some propositions as neither true nor false. That paragraph requires clarification and qualification. I'll comment on the issues as they arise. JAS: Intuitionistic logic rejects the law of excluded middle by denying that the negation of a false proposition must be true. That is not what Brouwer said. He claimed that without a constructive proof, nobody can know whether a theory is true or false. Therefore, intuitionism prohibits theories that assume entities without any method for constructing them. But in any theory that can be proved by constructive methods, the negation of a false statement is indeed true. For more about intuitionism, search for "intuition" in the Handbook on Mathematical Logic, edited by Jon Barwise: http://jfsowa.com/temp/Barwise77.pdf JAS: Arnold Oostra ... implemented in the Alpha, Beta, and Gamma (modal) parts of EG by using scrolls for implication instead of cuts for negation, consistent with Peirce's explicit derivation of the latter from the former. Two points: (1) For the EG syntax from 1897 to 1913, a scroll may be replaced by a nest of two ovals or vice-versa without the slightest change of meaning. In the early days, Peirce mistakenly thought that a "sign of illation" was important. But in 1911, he recognized that the rules of inference are fundamental, and he demoted the scroll to an optional synonym for a nest of two ovals. The derivation of negation from a scroll is an identity; it can be used in either direction. Deriving a scroll from two negations is the direction Peirce preferred after May 1911. He repeated that preference in 8 November 1913. 2.That is Oostra's theory. He is entitled to use any notation he prefers and define it any way he chooses. But his theory is unrelated to anything Peirce wrote. For intuitionism, Brouwer's insistence on constructions prohibits theories that do not have constructive proofs: Cantor's infinities, Peirce's theory of continuity and infinitesimals, and the theory of infinitesimals by Abraham Robinson (1966). It's unlikely that Peirce left "hints" about a theory that contradicts one of his most important innovations. JAS: Multi-valued logics reject the principle of bivalence by introducing at least one intermediate truth value (ITV) besides true and false. The word reject is too strong. See below for the last line of page 344r. JAS: Peirce's Logic Notebook includes a late entry that states the following before going on to provide the very first known truth tables for a three-valued logic. CSP: Triadic Logic is that logic which, though not rejecting entirely the Principle of Excluded Middle, nevertheless recognizes that every proposition, S is P, is either true, or false, or else S has a lower mode of being such that it can neither be determinately P, nor determinately not-P, but is at the limit between P and not P. (R 339:515 [344r], 1909) The ending of of page 344r: "Triadic Logic is universally true. But Dyadic Logic is not absolutely false - it is only L." [L is that lower mode of being.] For many applications, L may be considered unknown or unknowable. That is an admission of incomplete knowledge. In quantum mechanics, L is the result of Heisenberg's uncertainty principle. But scientists and engineers don't use triadic logic. They use precise, but continuous mathematics. JAS: Jan Lukasiewicz eventually did much more extensive work on three-valued logics, associating the ITV with possibility. Yes. Peirce deserves credit for anticipating Lukasiewicz. But the truth-table format is too restrictive to state all the significant axioms of modality. Clarence Irving Lewis and Arthur Prior developed versions of modal logic that were strongly influenced by Peirce. But they expressed their axioms in conventional algebraic notations, not in truth tables. Today, logicians do not use truth tables to represent and reason about any version of modal logic. JAS: Two-valued intuitionistic logic, three-valued triadic logic, and four-valued L-modal are thus all candidate formal systems for reasoning about a universe of discourse that is definite, general, and real. That sentence lumps together three unrelated systems: (1) Intuitionistic logic is a metatheory about pure mathematics. It says nothing about the physical world, and it is inconsistent with important mathematical theories by Peirce and others. (2) Although Peirce made some comments about N-valued logics, he never showed any practical applications. Section 17 of NEM (3:729 to 763) has some fragmentary notes about them, but with no examples of how they might be used for any practical or theoretical studies. Today, scientists in every branch of the physical, psychic, and practical sciences use theories of probability. (3) For two-valued Boolean logic, truth-tables are convenient for summarizing axioms and implications. For modal logic, tables are too restrictive to express the many variations of axioms and their implications. In the Logic Notebook, the entry on Triadic Modal Logic (p. 645, 344r) has question marks on five statements. The next page (646) has the title "Note on the Tinctures." It begins with one short paragraph, followed by a crossed-out copy of the word Let. The rest of the page is empty. There are no further remarks on modal logic in the LNB. That is not a convincing endorsement of truth tables for modal logic. On a related note, fuzzy set theory has had many applications for reasoning about borderline cases. I was invited to contribute an article to the Festschrift for Lotfi Zadeh's 90th birthday. Since I wanted to be kind in a Festschrift, I did not make harsh criticisms. Instead, I emphasized the need for further research and discussed related issues in Peirce's writings. See "What is the source of fuzziness?" http://jfsowa.com/pubs/fuzzy.pdf This ends my comments on Jon's remarks of April 9th. While writing this commentary, I did some further study of the 1909 entries in Peirce's Logic Notebook, NEM volume 3 on probability, and some 20th and 21st century writings on related logics. Short summary: By August 1909, Peirce chose to develop probability as his primary method for reasoning about that lower limit L. I'll write more about this issue later,
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