[CG] Information = Comprehension × Extension • Comment

Information = Comprehension × Extension • Comment 1 • https://inquiryintoinquiry.com/2024/10/11/information-comprehension-x-exten… Re: Information = Comprehension × Extension • Selection 1 • https://inquiryintoinquiry.com/2024/10/05/information-comprehension-x-exten… All, Selection 1 ends with Peirce drawing the following conclusion about the links between information, comprehension, inference, and symbolization. ❝Thus information measures the superfluous comprehension. And, hence, whenever we make a symbol to express any thing or any attribute we cannot make it so empty that it shall have no superfluous comprehension. ❝I am going, next, to show that inference is symbolization and that the puzzle of the validity of scientific inference lies merely in this superfluous comprehension and is therefore entirely removed by a consideration of the laws of information.❞ (Peirce 1866, p. 467) At this point in his inventory of scientific reasoning, Peirce is relating the nature of inference, information, and inquiry to the character of the signs mediating the process in question, a process he describes as “symbolization”. In the interest of clarity let's draw from Peirce's account a couple of quick sketches, designed to show how the examples he gives of conjunctive terms and disjunctive terms might look if they were cast within a lattice‑theoretic framework. Re: Information = Comprehension × Extension • Selection 5 • https://inquiryintoinquiry.com/2024/10/09/information-comprehension-x-exten… Looking back on Selection 5, let's first examine Peirce's example of a conjunctive term — “spherical, bright, fragrant, juicy, tropical fruit” — within a lattice framework. We have the following six terms. t₁ = spherical t₂ = bright t₃ = fragrant t₄ = juicy t₅ = tropical t₆ = fruit Suppose z is the logical conjunction of the above six terms. z = t₁ ∙ t₂ ∙ t₃ ∙ t₄ ∙ t₅ ∙ t₆ What on earth could Peirce mean by saying that such a term is “not a true symbol” or that it is “of no use whatever”? In particular, consider the following statement. ❝If it occurs in the predicate and something is said to be a spherical bright fragrant juicy tropical fruit, since there is nothing which is all this which is not an orange, we may say that this is an orange at once.❞ (Peirce 1866, p. 470). In other words, if something x is said to be z then we may guess fairly surely x is really an orange, in short, x has all the additional features otherwise summed up quite succinctly in the much more constrained term y, where y means “an orange”. Figure 1 shows the implication ordering of logical terms in the form of a “lattice diagram”. Figure 1. Conjunctive Term z, Taken as Predicate • https://inquiryintoinquiry.files.wordpress.com/2016/10/ice-figure-1.jpg What Peirce is saying about z not being a genuinely useful symbol can be explained in terms of the gap between the logical conjunction z, in lattice terms, the greatest lower bound of the conjoined terms, z = glb{t₁, t₂, t₃, t₄, t₅, t₆}, and what we might regard as the natural conjunction or natural glb of those terms, namely, y, “an orange”. In sum there is an extra measure of constraint which goes into forming the natural kinds lattice from the free lattice which logic and set theory would otherwise impose as a default background. The local manifestations of that global information are meted out over the structure of the natural lattice by just such abductive gaps as the one we observe between z and y. Reference — Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982. Resources — Inquiry Blog • Survey of Pragmatic Semiotic Information • https://inquiryintoinquiry.com/2024/03/01/survey-of-pragmatic-semiotic-info… OEIS Wiki • Information = Comprehension × Extension • https://oeis.org/wiki/Information_%3D_Comprehension_%C3%97_Extension C.S. Peirce • Upon Logical Comprehension and Extension • https://peirce.sitehost.iu.edu/writings/v2/w2/w2_06/v2_06.htm Regards, Jon cc: https://www.academia.edu/community/V91eDe