List: This submission addresses the potential connections between category theory, formal chemical logic and critical propositions of CSP's philosophy of pragmaticism, such as notions of the trichotomy and existential graphs. At issue is the connection, if any, between the vast plethora of modern logics, modern chemical logic and existential graphs.
In particular, it addresses the dictionary of terms deployed in drawing conclusions from semiosis of singular (unique) objects termed “sin-signs”. The first row of the trichotomy proposes that the representation of pragmatic objects as signs - qualisigns, sin-signs and legisigns. Presumably, these three terms are pragmatically coherent, that is, all refer to the same object. Presumably, all three terms could refer to a multitude of scientific objects and sematic terms for any particular “sin-sign” that are relevant to pragmatic inquiry. The second row of the trichotomy proposes three constraints on the meaning of signs - icons, indices, and symbols. Again, presumably, each of these semantic terms could refer to a vast multitude of scientific terms that represent the unique object of the existent sin-sign. Of particular importance to the logic of chemistry is the term “index.” This term has implications for measurements, counting, listing, illations, relations, categorizing, organizing, and critically the artificial name assigned to the singularity. Again, presumably, each of the semantic terms associated with the indices of a sin-sign are relevant to the sinsign and can be symbolized (and also vaguely iconized). The third row of the trichotomy proposes that a proposition can be categorically composed from the relevant terms of the second row that describe the interpretations of the semiosis of the interpreter. Again, presumably each of the three terms could be composed from the potential logical implications of the plethora of terms emanating from the first and second rows of the trichotomy, CSP’s personal weltanschaung. >From these three antecedents, a plethora of many to one and one to many >semantic mappings emerge as possible propositional sentences. (In special >cases,matheamtical mappings are also necessary) The logical terms of each >possible propositional sentence is constrained to the relevant indices of the >sin-sign . That is, the attributes of sin-sign are necessary terms in the >argument if the sin-sign is be logically coherent with the relevant qualities >and quantities and the name (legisign) of the emanating sign-generator. One very very special case of trichotomistic reasoning is the formal logic of chemistry which is based on atomic and molecular sentences (Russell/Whitehead). Each symbol for a chemical element represents a triad, an informed logical constant, an atomic number, an informed integer. That is, a symbol, an index and an icon. The set of atomic number follow Peano’s successor relation, each is commensurable with numerous qualities-signs and each is incommensurable with all other atomic numbers. The perplexity of table of chemical elements was well-known to CSP; in his earlier writings, be often gives exact chemical descriptions to his examples of logical reasoning. Later, after the discovery of unique parts of atoms (electrons, JJ Tompson, 1897), the chemical examples faded from his writings. The mystical nature of the plethora of electrical forms does not rest comfortably in this semantic milieu. The formal inquiries of chemistry are referenced to and indexed by the chemical table of elements (which CSP acknowledged as the bedrock of his reasoning.) A formal logic of the chemical sciences is based on five indices. These five indices are the molecular weight, the molecular formula, the molecular number, the molecular structure, and the handedness. The name of a chemical molecule (inclusive of proteins and genetic sequences) is commensurable with exactly one permutation group that matches these quali-signs. The molecular permutation group matches the index with the sin-sign. The indices of the permutation group also match the existential bipolar graphs representing atomic and molecular sentences as well as metabolic graphs. The formal logic of chemistry matches the meanings of these terms with the signs emanating from the sin-sign. The one to many and many to one mappings unconcealed (Wittgenstein, Heidegger, Whitehead) by the quali-signs reveal the composition of the identity of the object. The proof of the validity of the matchings of numerical terms is valided by the adjunctive synthesis of the object designated by the sin-sign. The antecedent and consequent are related by analysis and synthesis. Metaphorically, one can propose a “cybernetic” illation between the sin-sign and the legisign. Hence, closure is approached. This post is merely a short introduction to certain aspects of the formal logic of chemistry that are relevant to the semiotic aspect of existential graphs as essential components of compounding atomic sentences. (As an aside, this implies that these matchings of logical terms imply that chemical logic is exclusively semiotic.) While the obligatory formal logic of chemistry was developed pragmatically (see Coulomb and Lavosier works in 18th Century for the quantitative roots,) only rudimentary fragments of the concepts of logical constants were known to CSP. Apparently, he was able to compose the convoluted semantics of quantitative logical constants (the trichotomy of terms) from these scanty fragments. A remarkable logical feat! For readers not familiar with the roots of relevant logic, see the works of the medieval logician Swinehead as interpreted and modern interpretations of relevant logic by S. Read, and Topics in Conditional Logic by Nute (1980), The Philosophical Status of Diagrams,Greaves (2002) and Epistemic Friction by G. Sher (2016). In summary, I conclude that the perplex bedrock of the 19th Century trichotomy is grounded on the abstract concept of chemical elements as logical constants that are mathematically composable by addition of atomic numbers. In closing, I would like to acknowledge to list members. Ben U. planted an tetradic seed decades ago. Over the years, Edwinia T. has contributed substantially. Cheers Jerry Research Professor Krasnow Institute for Advanced Study George Mason University > On Jun 3, 2021, at 10:13 PM, John F. Sowa <s...@bestweb.net> wrote: > > Jon AS. List, > > Your comments helped me state the following four points more clearly. > I believe they have been established beyond any reasonable doubt. If > you're not convinced, I suggest that you forward this note to any Peirce > scholar(s) anywhere in the world and ask whether they have any doubts or > concerns that should be addressed. > > 1. Peirce's insight of June 1911: All EG permissions (rules of > inference) depend only on negations. In R670 (June 7), he chose > negation as primitive and defined the conditional as a nest of two > negations. In L231 (June 22), he dropped the adjective 'illative' for > all three permissions. In that MS, he specified EGs with greater > clarity. precision, and generality than ever before. > > 2. For formal logic, the choice of primitives is irrelevant. But the > rules and definitions are simpler with negation as primitive. For > determining the raw data of any empirical science (starting with > phanerosccopy), the only logical operators that can be derived without > methodeutic are existence, conjunction, and negation. > > 3. Peirce's early views (based on reading Whately's Elements of Logic in > 1851) led him to chose the scroll as an illative sign. But in every MS > from 1896 to the end, the scroll is logically equivalent to a nest of > negations, and every rule of inference is based on negations. Replacing > a scroll with a nest of negations has no effect on the meaning of any EG. > > 4. In the century after Peirce, some authors developed variations of > EGs. Anybody can claim that their notation was inpired by Peirce. But > nobody can claim Peirce's authority for any graphs unless they conform > to the syntax, semantics (endoporeutic), and rules of inference > (permissions) of some text (MS or publication) by Peirce. > > JAS> I just realized another undeniable fact that I find intriguing (and > somewhat surprising) ... [Peirce] does not discuss or show Alpha EGs at > all in the "clean version" of RL 231 (June 1911), nor in RL 378 > (September 1911)... > > L231 allows EGs to have any mixture of medads, monads, and polyads > (relations with two or more pegs). For examples of Alpha graphs (only > medads), see L231 Fig. 3 for "If it thunders, it lightens" and Fig. 4 > for "It thunders and it does not lighten". For convenience, I posted a > copy of the L231 EGs at http://jfsowa.com/peirce/eg1911.pdf > > JFS> I am not attributing any opinions to CSP. I am just saying that an > illative sign has no more significance than the letters QED. It has no > effect on any logical or philosophical idea. > > JAS> Thereby disagreeing with Peirce, which is fine; all I am requesting > is that this be acknowledged. > > About EGs, I agree with Peirce's version of June 1911, and I disagree > with some statements about EGs that he made in R669 and earlier. I have > said that repeatedly. > > JAS> As Francesco Bellucci put it back in March "A simpler hypothesis is > available: [Peirce] was offering a simpler presentation of the logical > graphs, one which neither reflects nor refutes one of the most stable > ideas of Peirce's philosophy of logic." > > That is the only significant objection to point #1 that anyone has ever > mentioned. Since Francesco has collaborated with Ahti in developing > excellent resources about EGs, their opinions must be taken seriously. > > Prior to June 1911, Peirce had defined negation in terms of a scroll > (material implication) and a pseudograph (a symbol for falsehood). The > June 11 version of EGs demoted that definition to an equivalence. Points > #2 and #3 explain why the difference between a definition (which must be > true) and an equivalence (which happens to be true) is irrelevant for > formal logic. Since all true statements in the earlier EGs (definitions, > axioms, theorems, and proofs) are equivalent to true statements in the > 1911 EGs, every true logical idea remains true. > > As an example, the article by Ma and Pietarinen (Peirce's calculi for > classical propositional logic) is about issues in formal logic. > Therefore, every logical idea in that article remains true when the > symbol := for definition is replaced by the symbol ≡ for equivalence. But > some of the commentary may have to be changed. See the attached file > mm_avp.png, which includes my changes with the red text and the red > symbol for equivalence. Below that is the equivalent statement in the > 1911 EG notation. > > JFS> Only three logical operators can be observed directly: any > sensation is evidence that something A exists; an observation of a > difference in sensations is evidence for the existence of at least two > things (A,B). That requires three logical operators: (Exist A,B), (A > AND B), (A is NOT B). Those operators are the basis for the 1911 EGs. > > JAS> This indicates another disagreement with Peirce. As I have pointed > out before, he plainly maintains that such "Familiar universal > elementary relations of logic" cannot be observed directly. > > That opinion is not by Peirce, and it ignores my description of direct > observation as a multi-step process. For existence, "an observation of > [1] any sensation [2] is evidence that [3] something A exists." For > conjunction and negation, "observation [1] of a difference in sensations > [2] is evidence for [3] the existence of [4] at least two things (A,B). > [5] That requires three logical operators: [6] (Exist A,B) [7] (A AND > B) [8] (A is NOT B)." > > That summary shows that recognizing the 1911 operators takes several > semeiotic steps. But recognizing if-then requires more complex > methodeutic. To determine what Peirce "plainly maintains" in EP 2:485, > see the definitions in EP 2:484: > > CSP> Objects may be presented in three ways, thus: First, as mere > Ideas, or what might be if things were not as they are... Second, > as Brutely compelling attention. Third, as Rationally recommending > themselves, or as Habitudes to which one is already reconciled... > Adopting this enumeration as a basis of a division of Signs, I obtain: > A. Descriptives... B. Designatives... which in this case cannot be > given by independent reasoning. C. Copulants, which neither describe > nor denote their Objects, but merely express universally the logical > sequence of these latter upon something otherwise referred to. Such, > among linguistic signs, as "If __ then __", "___is___", "___causes___", > "___would be___", "___is relative to___for__", "Whatever", etc. (EP > 2:484, letter to Lady Welby, 25 Dec 1908) > > The phrase "Brutely compelling attention" aptly describes the steps > for recognizing existence, conjunction, and negation. > > CSP> But if we attempt to analyze "possesses the character" in like > manner, we get "A possesses the character of the possession of the > character of Redness"; and so on ad infinitum. So it is, with "A > implies B," "A implies its implication of B," etc. So with "It rains > and hails," "It rains concurrently with hailing," "It rains concurrently > with the concurrence of hailing," and so forth. I call all such signs > Continuants. They are all Copulants and are the only pure Copulants. > These signs cannot be explicated: they must convey Familiar universal > elementary relations of logic. We do not derive these notions from > observation, nor by any sense of being opposed, but from our own reason. > (EP 2:485, 1908) > > In this paragraph, Peirce wrote that both 'AND' and 'IF-THEN' are > copulants derived "from our own reason." But the steps for deriving > 'AND' are much simpler than the methodeutic for deriving 'IF-THEN'. > > JFS> Every version of Alpha and Beta EGs is purely classical. > > JAS> Every version of Alpha and Beta EGs developed by Peirce himself is > purely classical. Others have developed non-classical versions of Alpha > and Beta EGs, most notably the intuitionistic versions by Oostra for > both (as well as Gamma) and by Ma and Pietarinen for Alpha only. > > On the first sentence, we agree. Re: Ma & Pietarinen, see above. > For any EGs that differ from Peirce's, see point #4 at the top. > > Finally, the 1911 EGs are an excellent foundation for mainstream > developments in logic, linguistics, and computer science today. > They were the basis for solving a research problem stated in 1988. > See slide 65 of "Peirce, Polya, and Euclid: Integrating logic, > heuristics, and geometry", http://jfsowa.com/talks/ppe.pdf > > That talk, which was presented at the April 2015 conference of the > American Philosophical Association, was the basis for a 76-page article > in the Journal of Applied Logics. See the references in those slides. > > For relating 1911 EGs to linguistics, see "From existential graphs to > conceptual graphs", http://jfsowa.com/pubs/eg2cg.pdf > > For relating Peirce's writings to issues of modal logic, see "Five > questions on epistemic logic", http://jfsowa.com/pubs/5qelogic.pdf , and > "Worlds, models, and descriptions", http://jfsowa.com/pubs/worlds.pdf > > For relating 1911 EGs to research issues in computer science, see the > slides for a keynote speech at the European Semantic Web Conference in > June 2020: http://jfsowa.com/talks/eswc.pdf (This conference, which had > been scheduled for Crete, was Zoomified.) > > For these purposes, the 1911 EGs are superior to the earlier versions. > The scroll is an optional notation that has no special significance. > > John > <mm_avp.png>_ _ _ _ _ _ _ _ _ _ > ► PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON > PEIRCE-L to this message. 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