Inquiry Calculus and the Issue of Negative Higher Order Informations

Abstract

Abstract:In this paper, we will give the derivation of an inquiry calculus, or, equivalently, a Bayesian information theory. From simple ordering follow lattices, or, equivalently, algebras. Lattices admit a quantification, or, equivalently, algebras may be extended to calculi. The general rules of quantification are the sum and chain rules. Probability theory follows from a quantification on the specific lattice of statements that has an upper context. Inquiry calculus follows from a quantification on the specific lattice of questions that has a lower context. There will be given here a relevance measure and a product rule for relevances, which, taken together with the sum rule of relevances, will allow us to perform inquiry analyses in an algorithmic manner.

Lattices and Their Consistent Quantification

“…That is, it is not generally true that for elements x ≤ y we will have q (x) ≤ q (y). An example of this is the co-information lattice in information theory [39] and relevance among questions [40] for which some quantities are negative.…”

“…Another application involves quantifying the degree to which one question answers another. [16,17,29,30,40,[54][55][56][57] By defining a question in terms of the set of all possible logical statements that answer it, [54] one can construct the lattice of questions [56] as a free distributive lattice. [31,32,58] For example, consider a problem in which I have collected one piece of fruit that could be an apple, a banana, a cantaloupe, or a date.…”

“…Due to the exclusion term that is subtracted, the relevance of a question that does not partition the answers, such as this one, can be negative. [16,39,40] However, restricting oneself to partition questions, which partition the top answers, such as (AB ∨ CD), (A ∨ BCD), and I = (A ∨ B ∨ C ∨ D), one is left with a bi-valuation that has non-negative relevance values.…”

“…Another application involves quantifying the degree to which one question answers another [10,13,32,34,36,35,37,60,59]. By defining a question in terms of the set of all possible logical statements that answer it [10], one can construct the lattice of questions [32] as a free distributive lattice [6,19,12].…”

“…Our earlier results, obtained for lattices in general indicate that we will have a sum rule. The question (AB ∨ BCD) can be written as the join of two questions Due to the exclusion term that is subtracted, the relevance of a question that does not partition the answers, such as this one, can be negative [3,35,59]. However, restricting oneself to partition questions, which partition the top answers, such as (AB∨CD), (A∨BCD), and I = (A∨B∨C∨D), one is left with a bi-valuation that has non-negative relevance values.…”