Abstract
The issue of rewarding partially correct answers has been addressed by many authors (Guzman, E. & Conejo, R., 2004, Gardner-Medwin, A.R. 1995, Huffman, D, Goldberg, F., & Michlin, M. 2003). Intelligent systems have been designed to assign scores related to the importance of missing or incorrect part of an answer. Such systems are meant to facilitate the process of knowledge assessment. While trying to be efficient in evaluating students’ responses these systems operate with the answers to a single question addressing learning a new term, understanding a new concept or mastering a new skill. However, experimental practice shows that asking several questions about the same item results in inconsistent and/or incomplete feedback, i.e. some of the answers are correct while others are partially correct or even incorrect. A large number of computer based systems and thus automated assessment systems lack the ability to reason with inconsistent information. Such a situation occurs when, f. ex. a student answers to two questions about one item and one of the answers is correct and the other one is incorrect or missing. Reasoning by applying classical logic cannot solve the problem because the presence of contradiction leads to trivialization, i. e. anything follows from ‘correct and incorrect’ and thus all inconsistencies are treated as equally bad (Priest, 2001). In this paper we discuss how to assess students’ understanding of new terms and concepts, shortly after they have been introduced in a subject. Application of many-valued logic allows the system to give meaningful responses in the presence of inconsistencies. Decision making rules, an intelligent agent is applying for assessing students’ understanding of new terms and concepts are presented. Such rules distinguish between students’ hesitation in the process of giving an answer and lack of knowledge. We propose use of the generalized Lukasiewicz’s logic in a Web-based assessment system as a way of resolving problems with inconsistent and/or incomplete input.
TopBackground
A brief overview of a six-valued logic, which is a generalized Kleene’s logic (Kleene, S., 1952), has been first presented by Moussavi, M. & Garcia, N., 1989. Fitting, 1991 developed further this logic by assigning probability estimates to formulas instead of non-classical truth values.
The six-valued logic distinguishes two types of unknown knowledge values - permanently or eternally unknown value and a value representing current lack of knowledge about a state (Garcia, O.N. & Moussavi, M., 1990).
Two kinds of negation, weak and strong negation are discussed in Wagner, G., 1994. Weak negation or negation-as-failure refers to cases when it cannot be proved that a sentence is true. Strong negation or constructable falsity is used when the falsity of a sentence is directly established.
The semantic characterization of a four-valued logic for expressing practical deductive processes is presented by Belnap N.J., 1977. In Gurfinkel, A. & Chechik, M. 2005, it is shown that additional reasoning power can be obtained without sacrificing performance, by building a prototype software model-checker using Belnap’s logic.
Bi-dimensional systems representing and reasoning with temporal and uncertainty information have appeared also in Felix, P., Fraga, S., Marin, R., & Barro, S., 1999, and Mulsliner, D.J., Durfee, E.H., Shin, K.G., 1993.
Key Terms in this Chapter
Belnap’s Logic: It has four truth values ‘T, F, Both, None’. The meaning of these values can be described as follows: an atomic sentence is stated to be true only (T), an atomic sentence is stated to be false only (F), an atomic sentence is stated to be both true and false, for instance, by different sources, or in different points of time (Both), and an atomic sentences status is unknown. That is, neither true, nor false (None).
XML-RPC: It is remote procedure calling using HTTP as the transport and XML as the encoding.
Six-Valued Logic: The six-valued logic obtained as an extension of the Kleene’s logic has six truth values - true, false, unknown, unknownt - intermediate level of truth between unknown and true, unknownf - intermediate level of truth between unknown and false, contradiction.
Lukasiewicz’s Generalized Logic: It is done by inserting evenly spaced division points in the interval between 0 and 1.
Kleene’s Logic: Kleene’s logic has three truth values, truth, unknown and false, where unknown indicates a state of partial vagueness. These truth values represent the states of a world that does not change.
LAMP Web Server: It is a combination of free software tools of an Apache Web server, a database server and a scripting programming platform on a Linux operating environment. Lukasiewicz’s Three-Valued Logic: Lukasiewicz’s three-valued valued logic has a third value, 1/2, attached to propositions referring to future contingencies. The third truth value can be construed as ‘intermediate’ or ‘neutral’ or ‘indeterminate’.