Jon,Jerry,list. Not to interfere here. But it can get more complicated since both non-relatives and relatives are terms of singular reference. So, the term "A" and the term "A:B" are both individuals with singular reference. Yet Peirce says that "every term of singular reference is general." How to resolve this! How to get generality out of both? 1) The non-relative term "A" as well as "A:B" could both be treated as Boolean classes; universal classes. Thus, singular reference is to a class. 2) As a "system of objects," however, the non-relative might be the logical sum of the term with itself. This would give a singleton A. ( or maybe A:A with respect to the next n+1 arity. "A:B" is easier in that the logical sum is four. But the degenerate case "A:A" is part of the system of objects. How to get generality? Maybe try Induction along with the assumption that the universe is greater than one. Jim W > Date: Fri, 13 Feb 2015 00:48:44 -0500 > From: [email protected] > To: [email protected]; [email protected] > Subject: Re: [PEIRCE-L] Peirce’s 1880 “Algebra Of Logic” Chapter 3 • > Selection 4 > > Inquiry Blog > http://inquiryintoinquiry.com/2015/01/30/peirces-1880-algebra-of-logic-chapter-3-%E2%80%A2-preliminaries/ > http://inquiryintoinquiry.com/2015/02/01/peirces-1880-algebra-of-logic-chapter-3-%E2%80%A2-selection-1/ > http://inquiryintoinquiry.com/2015/02/03/peirces-1880-algebra-of-logic-chapter-3-%E2%80%A2-selection-2/ > http://inquiryintoinquiry.com/2015/02/11/peirces-1880-algebra-of-logic-chapter-3-%E2%80%A2-selection-3/ > http://inquiryintoinquiry.com/2015/02/12/peirces-1880-algebra-of-logic-chapter-3-%E2%80%A2-selection-4/ > > Peirce List > JA:http://permalink.gmane.org/gmane.science.philosophy.peirce/15566 > JA:http://permalink.gmane.org/gmane.science.philosophy.peirce/15607 > JA:http://permalink.gmane.org/gmane.science.philosophy.peirce/15608 > JA:http://permalink.gmane.org/gmane.science.philosophy.peirce/15616 > JW:http://permalink.gmane.org/gmane.science.philosophy.peirce/15624 > JA:http://permalink.gmane.org/gmane.science.philosophy.peirce/15634 > JW:http://permalink.gmane.org/gmane.science.philosophy.peirce/15636 > JW:http://permalink.gmane.org/gmane.science.philosophy.peirce/15639 > JW:http://permalink.gmane.org/gmane.science.philosophy.peirce/15641 > JA:http://permalink.gmane.org/gmane.science.philosophy.peirce/15646 > JW:http://permalink.gmane.org/gmane.science.philosophy.peirce/15657 > JA:http://permalink.gmane.org/gmane.science.philosophy.peirce/15659 > JA:http://permalink.gmane.org/gmane.science.philosophy.peirce/15660 > JW:http://permalink.gmane.org/gmane.science.philosophy.peirce/15664 > JLRC:http://permalink.gmane.org/gmane.science.philosophy.peirce/15665 > > Jerry, List, > > When in doubt, please refer to my blog versions where > I can use the appropriate math formatting and typefaces. > > Peirce uses roman (non-italic) typeface for constants or > individuals, that is, proper names of individual entities. > So I use LaTeX \mathrm{A}, \mathrm{B}, \mathrm{C}, etc. > for these. Think "Ann", "Bob", "Cal", etc. for these. > > Math style sheets dictate italics for variables, > so this is the default in LaTeX math settings. > The only variable used in CP 3.220 is ''x''. > > A general term is one that denotes any and all the elements of a set. > Peirce uses terms like "aggregate" or "logical sum" to refer to sets. > > General terms and variables are not exactly the same thing, > although there is a relation between them in some contexts > since a variable ranges over the members of a set. > > So if Bob and Cal and Don each love only Ann, > but Ann loves only Don, then the dual relative > ℓ = lover of = B:A + C:A + D:A + A:D. > > I think the problem is that "member" has two meanings. > It can mean "member of a set" or "member of a tuple". > > Regards, > > Jon > > On 2/12/2015 11:55 PM, Jerry LR Chandler wrote: > > List, Jon: > > > > If read from a technical perspective, this passage (CP 3.220) can be > > interpreted as nearly self-contradictory or utterly ambiguous. It appears > > that CSP is unable to distinguish between nouns as Proper Nouns and nouns > > as generals. But this can be a perplex gloss. > > > > Contrast: > >> Every relative, ..., is general; > > (seemingly inferring that a relative is a general, that is, a mathematical > > variable.) > > > >> An individual relative refers > >> to a system all the members of which are individual. > > > > (seemingly inferring that and INDIVIDUAL relative is not a general (!) > > and further, requiring that the concept of "an individual relative" also > > makes necessary a system of relations such that only individuals (that is, > > perhaps non-generals? ) are the only members allowed.) > > > >> The expressions > >> > >> (A : B) > >> > >> (A : B : C) > >> > >> may denote individual relatives. > > > > are expressions of pairings of symbols that are not to be interpreted as > > variables, at least as I read it. > > (The notation is NOT the usual notation for variables for multiplication or > > addition. Nor is the subsequent table of extensions...) > > > > Nevertheless, I find the meaning of CP3.220 to clear and straight forward. > > It is analogous to the meaning of natural relations among atomic relatives. > > > > "Individual relative" may refer to the Proper Name of an individual > > chemical element. > > > > It is also consistent with his later paper on the logic of relatives in > > relation to graph theory: > > C. S. Peirce (1897 Jan.), "The Logic of Relatives", The Monist, v. VII, n. > > 2 pp. 161-217. > > > > In short, in CSP's terminology, the nature of addition and of > > multiplication, differ for general relatives and individual relatives, > > vaguely similar to Boolean notions. > > > > Cheers > > > > Jerry > > > > > > > > On Feb 12, 2015, at 3:28 PM, Jon Awbrey wrote: > > > >> Post : Peirce's 1880 “Algebra Of Logic” Chapter 3 • Selection 4 > >> http://inquiryintoinquiry.com/2015/02/12/peirces-1880-algebra-of-logic-chapter-3-%e2%80%a2-selection-4/ > >> Posted : February 12, 2015 at 4:00 pm > >> > >> <blockquote> > >> > >> Chapter 3. The Logic of Relatives (cont.) > >> > >> §2. Relatives (cont.) > >> > >> 220. Every relative, like every term of singular reference, is general; > >> its definition describes a system in general terms; and, as general, it > >> may be conceived either as a logical sum of individual relatives, or as > >> a logical product of simple relatives. An individual relative refers > >> to a system all the members of which are individual. The expressions > >> > >> (A : B) > >> > >> (A : B : C) > >> > >> may denote individual relatives. Taking dual individual relatives, > >> for instance, we may arrange them all in an infinite block, thus, > >> > >> A:A A:B A:C A:D A:E etc. > >> B:A B:B B:C B:D B:E etc. > >> C:A C:B C:C C:D C:E etc. > >> D:A D:B D:C D:D D:E etc. > >> E:A E:B E:C E:D E:E etc. > >> etc. etc. etc. etc. etc. etc. > >> > >> In the same way, triple individual relatives may be arranged > >> in a cube, and so forth. The logical sum of all the relatives > >> in this infinite block will be the relative universe, ∞, where > >> > >> x ─< ∞, > >> > >> whatever dual relative x may be. It is needless to distinguish > >> the dual universe, the triple universe, etc., because, by adding > >> a perfectly indefinite additional member to the system, a dual > >> relative may be converted into a triple relative, etc. Thus, for > >> ''lover of a woman'', we may write ''lover of a woman coexisting > >> with anything''. In the same way, a term of single reference is > >> equivalent to a relative with an indefinite correlate; thus, > >> ''woman'' is equivalent to ''woman coexisting with anything''. > >> Thus, we shall have > >> > >> A = A:A + A:B + A:C + A:D + A:E + etc. > >> > >> A:B = A:B:A + A:B:B + A:B:C + A:B:D + etc. > >> > >> </blockquote> > >> > > -- > > 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
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