Jon, My understanding is that the 'commutative triangle' is the simplest category.
The Peircean sign defined below (reproduced from http://www.iupui.edu/~arisbe/rsources/76DEFS/76defs.HTM) to me is a commutative triangle and a mathematical category: *"30 - 1905 - SS. pp. 192-193 - Letter to Lady Welby (Draft) presumably July 1905 .* A "sign" is anything, A, which, (1) in addition to other characters of its own, (2) stands in a dyadic relation Þ, to a purely active correlate, B, (3) and is also in a triadic relation to B for a purely passive correlate, C, this triadic relation being such as to determine C to be in a dyadic relation, µ, to B, the relation µ corresponding in a recognized way to the relation Þ." Which can be diagrammed as: Þ µ A --------> B ------- > C | ^ | | |___________________| h Figure 3. The Peircean sign as a mathematical triad. A = object; B = sign or representamen; C = interpretant, with the commutative condition satisfied, i.e., Þ x µ = h, where Þ = sign production; µ= sign interpretation, and h = information flow. If figure is right, information flow is possible only because the sign is a mathemtical category. Hence Figure 1 can be viewed as the definition of information, just as it is a definition of the sign. Let me know if you disagree. All the best. Sung On Wed, Feb 4, 2015 at 5:04 PM, Jon Awbrey <[email protected]> wrote: > Thread: > JA:http://permalink.gmane.org/gmane.science.philosophy.peirce/15622 > ET:http://permalink.gmane.org/gmane.science.philosophy.peirce/15623 > JA:http://permalink.gmane.org/gmane.science.philosophy.peirce/15625 > HR:http://permalink.gmane.org/gmane.science.philosophy.peirce/15626 > ET:http://permalink.gmane.org/gmane.science.philosophy.peirce/15627 > SJ:http://permalink.gmane.org/gmane.science.philosophy.peirce/15628 > > Sung, > > Your commutative triangle represents the decomposition of a function h > (which is a dyadic relation) as the functional composition of two other > functions f and g (which are also dyadic relations). This pictures the > decomposability, factoring, or reducibility of a dyadic relation and has > nothing to do with the irreducibility of any triadic relations. Nor does > it picture a category, it is only one commutative triangle in a category. > > Regards, > > Jon > > On 2/4/2015 4:14 PM, Sungchul Ji wrote: > >> Jon, >> >> An interesting post. I remember reading the quote about 5 years ago >> somewhere. >> >> The dative verb "give" nicely illustrates an irreducibly triadic >> relation >> (Figure 1) as compared to the sum of two dyadic relations (Figure 2): >> >> (1) Irreducibly triadic: A gives B to C (or A gives C B) which can be >> diagrammed as: >> >> f g >> A -------> B -------> C >> | ^ >> | | >> |_____________________| >> h >> >> Figure 1. Giving as a category, since f x g = h, where f = A let go of B; >> g = B becomes the possesion of D; h = A intended B to be possessed by C >> (and not by a stranger). >> >> (2) Sum of two independent dyadic relations: A let go of B which happens >> to be possessed by C which can be diagrammed as: >> >> f g >> A ------- > B -------> C >> >> Figure 2. Two independent dyadic relations connected only by f and g, >> where f = B leaving A, and g = B falling in the hands of C. >> >> >> With all the best. >> >> Sung >> >> >> >> >> On Wed, Feb 4, 2015 at 2:36 PM, Edwina Taborsky <[email protected]> >> wrote: >> >> I'm not sure, Jon, what your point is, in this post. Although there may >>> be >>> six 'mechanical' ways of relating three 'logical subjects', the reality >>> of >>> Mind as represented in the modal categories differentiates them by >>> function. >>> >>> Edwina >>> >>> ----- Original Message ----- From: "Jon Awbrey" <[email protected]> >>> To: "Peirce List" <[email protected]> >>> Sent: Wednesday, February 04, 2015 1:45 PM >>> Subject: [PEIRCE-L] Six Ways Of Looking At A Triadic Relation ⌬ 1 >>> >>> >>> Post : Six Ways Of Looking At A Triadic Relation ⌬ 1 >>> >>>> http://inquiryintoinquiry.com/2015/02/04/six-ways-of- >>>> looking-at-a-triadic-relation-%e2%8c%ac-1/ >>>> Posted : February 4, 2015 at 1:00 pm >>>> Author : Jon Awbrey >>>> >>>> Peircers, >>>> >>>> Here's a triadic factoid from the 1903 Harvard Lectures on Pragmatism >>>> that raises a number of important questions for me. I isolated Peirce's >>>> observation about the "ordinary logic of relations" from the context of >>>> his >>>> remarks that follow, partly in order to render the puzzle more striking. >>>> There's better copy on my blog, and I'll copy out more as I get time. >>>> >>>> --- >>>> >>>> A triadic relation has 3! = 6 ''converses'', six grammatically and >>>> rhetorically different ways of representing what is >>>> logically the same information. Peirce illustrates the situation as >>>> follows, with six variations on the theme of giving. >>>> >>>> <blockquote> >>>> >>>> So in a triadic fact, say, for example >>>> >>>> A gives B to C >>>> >>>> we make no distinction in the ordinary logic of relations between the >>>> ''subject nominative'', the ''direct object'', and >>>> the ''indirect object''. We say that the proposition has three >>>> ''logical >>>> subjects''. We regard it as a mere affair of >>>> English grammar that there are six ways of expressing this: >>>> >>>> A gives B to C >>>> A benefits C with B >>>> B enriches C at expense of A >>>> C receives B from A >>>> C thanks A for B >>>> B leaves A for C >>>> >>>> These six sentences express one and the same indivisible phenomenon. >>>> >>>> </blockquote> >>>> >>>> References >>>> >>>> * Peirce, C.S., “The Categories Defended”, Harvard Lectures on >>>> Pragmatism >>>> : Lecture 3 (MS 308, delivered on 9 April >>>> 1903). Published in Collected Papers (CP 5.66–81, 88–92, in part), >>>> Harvard Lectures (HL 167–188), Essential Peirce : >>>> Volume 2 (EP 2, 160–178). >>>> >>>> * Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1–6, >>>> Charles Hartshorne and Paul Weiss (eds.), vols. >>>> 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, >>>> 1931–1935, 1958. Volume 5 : Pragmatism and >>>> Pragmaticism, 1934. (Cited as CP). >>>> >>>> * Peirce, C.S., Pragmatism as a Principle and Method of Right Thinking : >>>> The 1903 Harvard Lectures on Pragmatism, >>>> Patricia Ann Turrisi (ed.), State University of New York Press, Albany, >>>> NY, 1997. (Cited as HL). >>>> >>>> Peirce, C.S., The Essential Peirce : Selected Philosophical Writings, >>>> Volume 2 (1893–1913), Peirce Edition Project >>>> (eds.), Indiana University Press, Bloomington and Indianapolis, IN, >>>> 1998. >>>> (Cited as EP 2). >>>> >>>> Add a comment to this post: >>>> http://inquiryintoinquiry.com/2015/02/04/six-ways-of- >>>> looking-at-a-triadic-relation-%e2%8c%ac-1/#respond >>>> >>>> > -- > > academia: http://independent.academia.edu/JonAwbrey > my word press blog: http://inquiryintoinquiry.com/ > inquiry list: http://stderr.org/pipermail/inquiry/ > isw: http://intersci.ss.uci.edu/wiki/index.php/JLA > oeiswiki: http://www.oeis.org/wiki/User:Jon_Awbrey > facebook page: https://www.facebook.com/JonnyCache > -- Sungchul Ji, Ph.D. Associate Professor of Pharmacology and Toxicology Department of Pharmacology and Toxicology Ernest Mario School of Pharmacy Rutgers University Piscataway, N.J. 08855 732-445-4701 www.conformon.net
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