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*To*: Ontology <ontology@ieee.org>*Subject*: ONT Re: Differential Analytic Turing Automata*From*: Jon Awbrey <jawbrey@att.net>*Date*: Mon, 08 Mar 2004 15:12:02 -0500*References*: <403F546C.D69B6918@att.net> <40476DB3.BC029F0@att.net> <404BB2D9.B81EF7A2@att.net> <404C0A1A.CFBAC3D@att.net> <404C8319.89C2BC35@att.net>*Sender*: owner-ontology@majordomo.ieee.org

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DATA. Note 15 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 'Tis a derivative from me to mine, | And only that I stand for. | | Winter's Tale, 3.2.43-44 We've talked about differentials long enough that I think it's past time we met with some. When the term is being used with its more exact sense, a "differential" is a locally linear approximation to a function, in the context of this logical discussion, then, a locally linear approximation to a proposition. I think that it would be best to just go ahead and exhibit the simplest form of a differential dF for the current example of a logical transformation F, after which the majority of the easiest questions will've been answered in visually intuitive terms. For F = <f, g> we have dF = <df, dg>, and so we can proceed componentwise, patching the pieces back together at the end. We have prepared the ground already by computing these terms: Ef = ((u + du)(v + dv)) Eg = ((u + du, v + dv)) Df = ((u)(v)) + ((u + du)(v + dv)) Dg = ((u, v)) + ((u + du, v + dv)) As a matter of fact, computing the symmetric differences Df = f + Ef and Dg = g + Eg has already taken care of the "localizing" part of the task by subtracting out the forms of f and g from the forms of Ef and Eg, respectively. Thus all we have left to do is to decide what linear propositions best approximate the difference maps Df and Dg, respectively. This raises the question: What is a linear proposition? The answer that makes the most sense in this context is this: A proposition is just a boolean-valued function, so a linear proposition is a linear function into the boolean space B. In particular, the linear functions that we want will be linear functions in the differential variables du and dv. As it turns out, there are just four linear propositions in the associated "differential universe" dU% = [du, dv], and these are the propositions that are commonly denoted: 0, du, dv, du + dv, in other words, (), du, dv, (du, dv). To be continued ... Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o inquiry e-lab: http://stderr.org/pipermail/inquiry/ o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

**Follow-Ups**:**ONT Re: Differential Analytic Turing Automata***From:*Jon Awbrey <jawbrey@att.net>

**References**:**ONT Differential Analytic Turing Automata***From:*Jon Awbrey <jawbrey@att.net>

**ONT Re: Differential Analytic Turing Automata***From:*Jon Awbrey <jawbrey@att.net>

**ONT Re: Differential Analytic Turing Automata***From:*Jon Awbrey <jawbrey@att.net>

**ONT Re: Differential Analytic Turing Automata***From:*Jon Awbrey <jawbrey@att.net>

**ONT Re: Differential Analytic Turing Automata***From:*Jon Awbrey <jawbrey@att.net>

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