[CG] Re: Relations & Their Relatives

Relations & Their Relatives • 4 • https://inquiryintoinquiry.com/2024/08/03/relations-their-relatives-4-a/ From Dyadic to Triadic to Sign Relations — Peirce's notation for elementary relatives was illustrated earlier by a dyadic relation from number theory, namely, the relation written “i|j” for “i divides j”. Table 1 shows the first few ordered pairs of the relation on positive integers corresponding to the relative term, “divisor of”. Thus, the ordered pair i:j appears in the relation if and only if i divides j, for which the usual mathematical notation is “i|j”. Table 1. Elementary Relatives for the “Divisor Of” Relation • https://inquiryintoinquiry.files.wordpress.com/2015/02/elementary-relatives… Table 2 shows the same information in the form of a “logical matrix”. This has a coefficient of 1 in row i and column j when i|j, otherwise it has a coefficient of 0. (The zero entries have been omitted for ease of reading.) Table 2. Logical Matrix for the “Divisor Of” Relation • https://inquiryintoinquiry.files.wordpress.com/2015/02/logical-matrix-for-t… Just as matrices of real coefficients in linear algebra represent linear transformations, matrices of boolean coefficients represent logical transformations. The capacity of dyadic relations to generate transformations gives us part of what we need to know about the dynamics of semiosis inherent in sign relations. The “divisor of” relation x|y is a dyadic relation on the set of positive integers M and thus may be understood as a subset of the cartesian product M × M. It forms an example of a “partial order relation”, while the “less than or equal to” relation x ≤ y is an example of a “total order relation”. The mathematics of relations can be applied most felicitously to semiotics but there we must bump the “adicity” or “arity” up to three. We take any sign relation L to be subset of a cartesian product O × S × I, where O is the set of “objects” under consideration in a given discussion, S is the set of “signs”, and I is the set of “interpretant signs” involved in the same discussion. One thing we need to understand is the sign relation L ⊆ O × S × I relevant to a given level of discussion may be rather more abstract than what we would call a “sign process” proper, that is, a structure extended through a dimension of time. Indeed, many of the most powerful sign relations generate sign processes through iteration or recursion or similar operations. In that event, the most penetrating analysis of the sign process or semiosis in view is achieved by grasping the generative sign relation at its core. Resources — Relation Theory • https://oeis.org/wiki/Relation_theory Triadic Relations • https://oeis.org/wiki/Triadic_relation Sign Relations • https://oeis.org/wiki/Sign_relation Survey of Relation Theory • https://inquiryintoinquiry.com/2024/03/23/survey-of-relation-theory-8/ Peirce's 1870 Logic Of Relatives • https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Overview Regards, Jon cc: https://www.academia.edu/community/5wb323