of Excluded MiddleThe fundamental law of excluded middle, which originated with Plato, states that for any proposition, that proposition is either true or its negation is true.
Such a statement is an essential component of Boolean algebra and can be written as the classic exclusive OR: A = (M ∨ ~M). A is equal to M or not M. The symbol ∨ is OR and the symbol ~ is the term “not.”
1.3 Building on the Past – From Those Who Laid the Foundation
As the centuries have rolled forward and technology has advanced, with thinking, reasoning, and learning yet at its roots, the works and ideas of the early philosophers laid the foundation for contemporary concepts and ideas. Yet, as we move forward, if we pause and reflect for a moment, we realize that learning is not limited to that proffered by the pendants of previous days but is a natural part of our daily existence. Each time that we encounter a fresh idea, make a new discovery, or solve one of life's many problems, we are revisiting our store of knowledge, we are reasoning, and, hopefully, we are learning and collecting new knowledge to add to that store. We are growing and enriching our perceived model of the world.
As we have seen, educators, scientists, and philosophers have debated, studied, and analyzed learning for centuries. Our deep understanding of this process remains embryonic, yet our learning continues. Following in the footsteps of early thinkers are Feigenbaum, Hegle, Newell, Minsky, Papert, Winston, McCulloch, Pitts, Rosenblatt, Hebb, Lukasiewicz, Hopfield, Knuth, and Zadeh. We will examine the roles that each has played and how they have contributed. The next few paragraphs present a small portion of their works.
1.4 A Learning and Reasoning Taxonomy
One can distinguish a number of different forms or methods of learning ranging from the most elementary rote learning to more complex processes of learning by analogy and discovery. Variations on this taxonomy have appeared in much of the recent literature. In the following paragraphs, the term “student” is frequently used. Consider that a student or learner could be a human person or, potentially, a machine.
In some of his earlier works, Edward Feigenbaum proposed a five‐phase learning process:
1 Request information.
2 Interpret the information.
3 Convert the information into a useable form.
4 Integrate the information into the existing knowledge store.
5 Apply the knowledge and evaluate the results.
The learning situation is composed of two parties, the learner and the teacher or environment, and a body of knowledge to be transferred from the environment to the learner. Based on the five criteria, a six‐level learning taxonomy was proposed. The taxonomy considers two extremes: no active learner participation and complete active learner participation. Examining the taxonomy, one can easily see the influence of Socrates.
1 Rote learning – A memorization process that requires little thought of meaning by the learner.
2 Learning with a teacher – Most of the information is provided by the teacher. Missing details must be inferred by the learner.
3 Learning by example – Specific conceptual instances are given; however, generalization must be achieved by the learner.
4 Learning by analogy or metaphor – Related conceptual instances are given. The learner must recognize the relation and apply it to the task at hand.
5 Learning by problem solving – Knowledge embedded in the problem may be gained by the learner through solving the problem.
6 Learning by discovery – Knowledge exists but must be hypothesized by the learner through theory formation and extracted by experiment.
1.4.1 Rote Learning
Habit or rote learning is the simplest learning process. The environment supplies all of the knowledge, and the learner merely accepts and stores it with no thought to meaning or content. Despite its elementary nature, the rote acquisition of knowledge is essential to all higher forms of learning. The learner must retain base information to be able to apply it to future problems.
B.F. Skinner took a somewhat different view. He suggested that no clear connection had been demonstrated in education between ends and means. He contended that the educational process should be reduced to defining goals or acts that the learner was able to perform. Based on the “present” state of knowledge, a sequence of acts could be created to move the learner from the present to the desired state. Often, the teacher would not be necessary.
One of the most familiar and perhaps best early instances of mechanized rote learning is Arthur Samuel's program designed to play the game of checkers. The program was initially equipped with a number of suggested procedures for playing the game correctly. The intent was to have the program learn by memorizing successful (deemed significant) board positions as it encountered them and then to use them properly and effectively in future games. Ultimately, the program progressed to the level of skilled novice.
1.4.2 Learning with a Teacher
Learning with a teacher is the first level of increased complexity in the learning hierarchy. Here, the learner is beginning to take an active role in several phases of the process, specifically she or he may request information from the teacher. In this situation, abstracted or general information is presented in an integrated manner by the teacher. The learner must accept the information and then complete the store of knowledge by inferring the missing details.
Many successful programs have been written using such a paradigm. In these, the program played the role of the student or learner. Several programs, including Mostow's FOO program for playing the card game “hearts” and Waterman's poker player, were oriented toward game playing. Davis's TEIRESIAS program presented an interesting variation on this scheme.
Rather than being autonomous, TEIRESIAS was designed to sit in front of the MYCIN program written earlier by E.H. Shortliffe. MYCIN was a large rule‐based system designed to assist physicians in the diagnosis and therapy of infectious diseases. The design and development process for any such large‐scale system is both iterative and refining. If the system makes a misdiagnosis or offers advice contrary to the physician's diagnosis, the knowledge base must be modified. Under such a condition, TEIRESIAS would interact with the user to correct the difficulty. Such a situation reduces to a two‐part task: first, explaining to the user the line of reasoning that led to the conclusion and then second, asking what additional or different information is needed to alter the result.
1.4.3 Learning by Example
Learning by example or induction increases the level of participation by the learner in the learning process. Unlike the previous example in which the teacher abstracted and then presented the material, here the student must assume the responsibility for the task. In such a context, specific conceptual instances are presented, and the student must recognize the significant or key features of the examples and then form the desired generalizations.
An early classic example of such an approach is Patrick Winston's work on “Learning Structural Descriptions from Examples.” The goal of Winston's program was to learn elementary geometrical constructs such as those one might build using toy blocks.
