Francesco, List: Welcome to the List, Francesco! Your posts are refreshingly original.
Is it possible that the following assertion is a consequence of modern notions of set theory and symbolic logic rather than the state of logical thought in the latter part of the 19 th Century? > the proper treatment of multiply quantified sentences is only possible once a > proper notation for variables and quantifiers is adopted. And this notation > requires that individuals may be denoted by variables that range over a > domain, and a variable is an index. Peirce's reference to "the logic of > triadic and higher relations failed" is a clear reference to his General > Algebra of Logic (1885), where an apparatus of quantification was > systematically presented I question this because of a simple notational counter-example. A multiply quantified sentence is necessary to represent a chemical structure with multiple atoms and the forms of relations among the different atoms. The molecular formula can represent multiple atoms of the same name / atomic weight. The molecular structure can represent multiple relations among either pairs of the same atom or multiple relations between two different atoms. These chemical facts were known to CSP. The “proper treatment” of these chemical facts is through diagrammatic logic where two different symbols are used to represent two different classes of abstract signs, one class of symbols for atoms representing names and another class of symbols for relations representing the uniting of the atomic signs into a singular molecular object. Note that these chemical symbols are used differently than the concepts of variables ranging over a domain. Note that the names of the atoms are indexed within the atomic table of elements. Associated with each chemical atom is a unique set of quali-signs. Icons were associated with the names of metals since Greek times. In summary, it is my belief that the epistemology of the matter is consistent with a notation for representing multiply quantified sentences and that this representation differs from the set theoretical logic of variables related by functions. The form of the breadth and depth of the logical quantifiers are representations of observation - physical measurements. Cheers Jerry > On Sep 9, 2018, at 2:16 PM, Francesco Bellucci > <bellucci.france...@googlemail.com> wrote: > > On Sun, Sep 9, 2018 at 11:20 AM, Francesco Bellucci > <bellucci.france...@googlemail.com > <mailto:bellucci.france...@googlemail.com>> wrote: > Jeffrey, Gary F., Jon, List > > CSP "Accordingly, we find that indices are absolutely indispensable in > mathematics; and until this truth was comprehended, all efforts to reduce to > rule the logic of triadic and higher relations failed; while as soon as it > was once grasped the problem was solved" > > I take this to mean: until the fact that indices are indispensable in > mathematics was comprehended, it was impossible to give a satisfactory > treatment of the logic of relations. A triadic relative, like "Gxyz", may > occur in a multiply quantified sentence. Now (as e.g. Dummett has > magisterially explained in his book on Frege, but the explanation holds > mutatis mutandis for Peirce), the proper treatment of multiply quantified > sentences is only possible once a proper notation for variables and > quantifiers is adopted. And this notation requires that individuals may be > denoted by variables that range over a domain, and a variable is an index. > Peirce's reference to "the logic of triadic and higher relations failed" is a > clear reference to his General Algebra of Logic (1885), where an apparatus of > quantification was systematically presented which was capable of expressing > not only dyadic relations, as his previous system of Algebra of Dyadic > Relations (1883), but also triadic and polyadic relations. >
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