Building the Knowledge Structure in Practice

Abstract

In Chapter 5, we established the equivalence of two seemingly quite different concepts: on the one hand the knowledge spaces, and on the other hand the entailments for Q. Recall that the latter are the relations {IE275-1} that satisfy the following two conditions: for all q ∈ Q and A, B ∈ 2^Q\{∅},

• (1) if q ∈ A, then {IE274-2};

• (2) if {IE274-3} and {IE274-4} holds whenever b ∈ B, then {IE274-5} (see Theorem 5.5). The unique entailment {IE274-6} derived from some particular space {IE274-7}, is defined by the formula {IE274-7} where A ∈ 2^Q \ {∅} and q ∈ Q. An empirical interpretation of an entailment is suggested by this formula, in terms of the responses to a class of questions or queries that an expert may be asked. In the field of education, these queries may take the form:

[Q1] Suppose that a student under examination has just provided wrong responses to all the questions in some set A. Is it practically certain that this student will also fail item q? Assume that the conditions are ideal in the sense that errors and lucky guesses are excluded.