Let's assume that S is some sentence (e.g. theorem). Are there efforts to quantify how well some agent (pupil, student, machine agent) understands this sentence S (or some sent of sentences)? E.g., if student is capable to derive many other sentences that are conclusions of the S+preexisting_knowlege_base_in_the_head_of_the_student, then, obviously, her understanding of S ir higher. Hence - maybe we can quantify the understanding by the cardinality (volume) of the set of conclusions that can be derived from compounded knowledge base (and use the volume of the initial base as the normalizing factor/divisor).
Am I alone in my efforts to formalize and quantify the understanding and comprehension of statements or are there other efforts? What are those efforts? What ar common terms, name, scientists in this field?
I guess, that such notions can have application in education sciences, e.g. to devise frameworks and tools for boosting the comprehension.