Jon, JA: I don't do off-list communication.
GR: Well, that is something new. In any event, Joe Ransdell established the practice of writing a list participant off-list before bringing the issue in question to the list. I follow this precedent. JA: If you have questions about the relevance of my posts to Peirce's lifelong work in logic, mathematics, and semiotics and to the development of pragmatic thought in general, please be so kind as to ask them on List. No. It is you who needs to provide the relevance of your posts to Peirce-L, not I nor anyone else. Best, Gary Richmond (writing to you about this matter for the last time on or off-list) “Let everything happen to you Beauty and terror Just keep going No feeling is final” ― Rainer Maria Rilke *Gary Richmond* *Philosophy and Critical Thinking* *Communication Studies* *LaGuardia College of the City University of New York* On Mon, Sep 27, 2021 at 4:08 PM Jon Awbrey <jawb...@att.net> wrote: > Dear Gary, > > I don't do off-list communication. > > If you have questions about the relevance of my posts > to Peirce's lifelong work in logic, mathematics, and > semiotics and to the development of pragmatic thought > in general, please be so kind as to ask them on List. > > Regards, > > Jon > > On 9/27/2021 3:46 PM, Gary Richmond wrote: > > off List > > > > Jon, > > > > Please remove Peirce-L from your list of sites if your messages aren't > > Peirce-L related in the sense in which the forum is conceived. If you > can't > > do that, given that I've repeatedly asked you to do so on and off List, > > then either leave the List or we'll do it for you if you prefer. > > > > I've Cc'd this note to Ben Udell. > > > > Gary Richmond (writing off-list as moderator) > > > > “Let everything happen to you > > Beauty and terror > > Just keep going > > No feeling is final” > > ― Rainer Maria Rilke > > > > *Gary Richmond* > > *Philosophy and Critical Thinking* > > *Communication Studies* > > *LaGuardia College of the City University of New York* > > > > > > > > > > > > > > > > On Mon, Sep 27, 2021 at 2:45 PM Jon Awbrey <jawb...@att.net> wrote: > > > >> Cf: Minimal Negation Operators • 4 > >> https://inquiryintoinquiry.com/2017/09/01/minimal-negation-operators-4/ > >> > >> All, > >> > >> I'm including a more detailed definition of minimal negation operators > >> in terms of conventional logical operations largely because readers of > >> particular tastes have asked for it in the past. But it can easily be > >> skipped until one has a felt need for it. Skimmed lightly, though, it > >> can serve to illustrate a major theme in logic and mathematics, namely, > >> the Relativity of Complexity or the Relativity of Primitivity to the > >> basis we have chosen for constructing our conceptual superstructures. > >> > >> ⁂ ⁂ ⁂ > >> > >> Defining minimal negation operators over a more conventional basis > >> is next in order of exposition, if not necessarily in order of every > >> reader’s reading. For what it’s worth and against the day when it may > >> be needed, here is a definition of minimal negations in terms of ∧, ∨, > >> and ¬. > >> > >> Formal Definition > >> ================= > >> > >> To express the general form of νₙ in terms of familiar operations, > >> it helps to introduce an intermediary concept. > >> > >> Definition. Let the function ¬ₘ : Bⁿ → B be defined for each > >> integer m in the interval [1, n] by the following equation. > >> > >> • ¬ₘ(x₁, …, xₘ, …, xₙ) = x₁ ∧ … ∧ xₘ₋₁ ∧ ¬xₘ ∧ xₘ₊₁ ∧ … ∧ xₙ. > >> > >> Then νₙ : Bⁿ → B is defined by the following equation. > >> > >> • νₙ(x₁, …, xₙ) = ¬₁(x₁, …, xₙ) ∨ … ∨ ¬ₘ(x₁, …, xₙ) ∨ … ∨ ¬ₙ(x₁, …, > xₙ). > >> > >> We may take the boolean product x₁ ∙ … ∙ xₙ or the logical conjunction > >> x₁ ∧ … ∧ xₙ to indicate the point x = (x₁, …, xₙ) in the space Bⁿ, in > >> which case the minimal negation νₙ(x₁, …, xₙ) indicates the set of > points > >> in > >> Bⁿ which differ from x in exactly one coordinate. This makes νₙ(x₁, …, > xₙ) > >> a discrete functional analogue of a point-omitted neighborhood in > ordinary > >> real analysis, more precisely, a point-omitted distance-one > neighborhood. > >> Viewed in that light the minimal negation operator can be recognized as > >> a differential construction, an observation opening a very wide field. > >> > >> The remainder of this discussion proceeds on the algebraic convention > >> making the plus sign (+) and the summation symbol (∑) both refer to > >> addition mod 2. Unless otherwise noted, the boolean domain B = {0, 1} > >> is interpreted for logic in such a way that 0 = false and 1 = true. > >> This has the following consequences. > >> > >> • The operation x + y is a function equivalent to the exclusive > >> disjunction of > >> x and y, while its fiber of 1 is the relation of inequality between > x > >> and y. > >> > >> • The operation ∑ₘ xₘ = x₁ + … + xₙ maps the bit sequence (x₁, …, xₙ) > >> to its parity. > >> > >> The following properties of the minimal negation operators > >> νₙ : Bⁿ → B may be noted. > >> > >> • The function ν₂(x, y) is the same as that associated with > >> the operation x + y and the relation x ≠ y. > >> > >> • In contrast, ν₃(x, y, z) is not identical to x + y + z. > >> > >> • More generally, the function νₙ(x₁, …, xₙ) for k > 2 > >> is not identical to the boolean sum ∑ₘ xₘ = x₁ + … + xₙ. > >> > >> • The inclusive disjunctions indicated for the νₙ of more than > >> one argument may be replaced with exclusive disjunctions without > >> affecting the meaning since the terms in disjunction are already > >> disjoint. > >> > >> Regards, > >> > >> Jon > >> > >
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