All, One of the more disconcerting developments, I might even say "devolutions", I've observed over the last 20 years has been the general slippage back to absolutist and dyadic ways of thinking, all of it due to the stubborn pull of unchecked reductionism, a failure to comprehend the relational paradigm, especially triadic relations, their irreducibility, and its consequences.
With all that in mind, I'll return to a point in our earlier discussions, add a bit more on the concept of closure, and continue from there to its bearing on the pragmatic maxim. > Cf: The Difference That Makes A Difference That Peirce Makes : 23 > At: https://inquiryintoinquiry.com/2019/09/20/the-difference-that-makes-a-difference-that-peirce-makes-23/ > > A fundamental question in applications of mathematical logic > is the threshold of complexity between dyadic (binary) and > triadic (ternary) relations, in particular, whether 2-place > relations are universally adequate or whether 3-place relations > are irreducible, minimally adequate, and even sufficient as > a basis for all higher dimensions. > > One of Peirce's earliest arguments for the sufficiency > of triadic relative terms occurs at the top of his > 1870 "Logic of Relatives". > > Cf: https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Overview > Cf: https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1 > Cf: https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#Selection_1 > > <QUOTE> > > The conjugative term involves the conception of "third", > the relative that of second or "other", the absolute term > simply considers "an" object. No fourth class of terms exists > involving the conception of "fourth", because when that of "third" > is introduced, since it involves the conception of bringing objects > into relation, all higher numbers are given at once, inasmuch as the > conception of bringing objects into relation is independent of the > number of members of the relationship. Whether this "reason" for the > fact that there is no fourth class of terms fundamentally different > from the third is satisfactory or not, the fact itself is made > perfectly evident by the study of the logic of relatives. > (Peirce, CP 3.63). > > </QUOTE> > > Peirce's argument invokes what is known as a "closure principle", > as I remarked in the following comment: > > What strikes me about the initial installment this time around is > its use of a certain pattern of argument I can recognize as invoking > a "closure principle", and this is a figure of reasoning Peirce uses > in three other places: his discussion of "continuous predicates", his > definition of "sign relations", and in the "pragmatic maxim" itself. > > * https://oeis.org/wiki/Continuous_predicate > * https://oeis.org/wiki/Sign_relation > * https://inquiryintoinquiry.com/2008/08/07/pragmatic-maxim/ > > In mathematics, a "closure operator" is one whose repeated application > yields the same result as its first application. > > If we take an arbitrary operator A, the result of applying A > to an operand x is Ax, the result of applying A again is AAx, > the result of applying A again is AAAx, and so on. In general, > it is perfectly possible each application yields a novel result, > distinct from all previous results. > > But a closure operator C is defined by the property CC = C, > so nothing new results beyond the first application. > Cf: The Difference That Makes A Difference That Peirce Makes : 24 At: https://inquiryintoinquiry.com/2019/09/22/the-difference-that-makes-a-difference-that-peirce-makes-24/ The concepts of "closure" and "idempotence" are closely related. We usually speak of a "closure operator" in contexts where the objects acted on are the primary interest, as in topology, where the objects of interest are open sets, boundaries, closed sets, etc. In contexts where we abstract away from the operand space, as in algebra, we tend to say "idempotence" for the detached application CC = C. (If I recall right, it was actually Charles Peirce's father Benjamin who coined the term "idempotence".) At any rate, I'll have to mutate the principle a bit to cover the uses Peirce makes of it. Regards, Jon
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