To take a somewhat outrageous metaphor, the field of logic can be considered the child of two very dissimilar parents. These parents are the disciplines of philosophy and mathematics. They are perhaps the most influential subjects in the entire Western intellectual tradition, but they make an odd couple. One parent is concerned with such imprecise and all-embracing ideas as the existence of God, the nature of art, and the relationship between good and evil. This parent is thoughtful, eclectic, and word-centered, concerned with ethical principles and subjective consequences. The other parent cannot be bothered with the vagaries of emotion or aesthetics; it is cool, precise, and symbol-oriented, concerned with things that can be quantified and measured. Their progeny, logic, is a chip off both blocks: it makes symbolic constructions of philosophical principles. Like all brilliant children, it is a bit confusing at first.
The point that philosophy and mathematics share is that both devise and use abstract concepts to describe and understand the material world. It is precisely this point that often gets lost in elementary logic classes. To Jon Barwise, College Professor of Philosophy, Mathematics, and Computer Science at Indiana University Bloomington, the pedagogical difficulty of teaching beginning logic courses became the launching pad for an innovative use of computer software. In collaboration with John Etchemendy, a professor of philosophy at Stanford University, Barwise has developed three software packages that introduce students to the principles of logic. Each software package comes with a textbook and is designed for student use outside class. Barwise sees the courseware as "an intellectual sandbox or erector set where the students can go and explore the topics and work problems. The computer helps them do it efficiently and allows them to do things that you just couldn't do with a pencil and paper."
The first of these courseware packages was Turing's World, developed in 1986 to teach basic concepts of logic and computation. The software focuses on the Turing machine, a model for computation developed by the English logician Alan Turing in 1936. The Turing machine became the fundamental model for the computer and was extremely important in the computer revolution, but Turing had no such plans when he first designed the device. To the contrary, he was interested in solving abstract problems in theoretical mathematics. Barwise calls the development of the Turing machine "a classic case of pure research having enormous but unforeseen practical consequences."
Originally, the Turing machine was simply a mental construct. Turing envisioned it as a mechanical device that could read symbols written linearly on a long tape. The device could then respond to the symbols in a limited number of ways: it could write a symbol on the tape, move to the left, move to the right, and so on. The actions had to be simple, step-by-step procedures, as Barwise puts it, something "that a patient, obedient child, one with lots of time and memory but no originality or knowledge of the world" could do. The Turing machine never disobeys, never gets tired, and never needs to be told twice. However, because it lacks subtlety or independent thought, it must be given extremely precise instructions. Simple Turing machines became models for simple computers, such as those found in dishwashers; these computers perform only one function. Then Turing developed the idea of the universal Turing machine--a Turing machine that could do anything that any one Turing machine could do. A universal Turing machine is not designed to perform a single task, but can be given an infinite set of instructions. In other words, it is a programmable computer.
Creating a physical model of the Turing machine was not difficult; it was the conceptual apparatus that was revolutionary, though it seems obvious to us now. The Turing machine remains one of the fundamental principles of computer science and is the sine qua non of beginning computer science courses. It is difficult, however, to demonstrate a powerful Turing machine on the blackboard or with pencil and paper, for most Turing machines require hundreds of steps to complete a task of any complexity. It was obviously impractical to construct such a Turing machine during class or to assign such constructions as homework problems, so Barwise and Etchemendy had the novel idea of using the computer to teach about the computer. The computer's great advantage is its ability to perform rote tasks very quickly. With the computer software Turing's World, a student can design a Turing machine, check to see if it works, and find and correct any mistakes. The many laborious tasks that would take hours to represent with pencil and paper can be done at lightning speed by the computer.
The use of the computer to teach the principles of computer science was so successful that Barwise and Etchemendy went on to develop more software. Tarski's World teaches the language of elementary logic. For many years, introductory logic courses treated the subject in a highly formalized way, with many rules that had to be memorized and then applied to proofs. The unit of analysis was the sentence, and students were required to determine whether sentences were true or untrue by reference to the rules. Truth, the fundamental premise of the subject, was represented in such an abstract and symbolic way that the important relationship between sentences and the world they represented was often lost. Thus the proofs took on the appearance of a self-contained game, with little importance to the world at large. According to Barwise, this is precisely the wrong message. "Logic is not playing games with symbols," Barwise says. "Logic is the science of valid reasoning." As such, logic is of concern to every form of rational discourse, from science to law to ethics. How can a teacher convey the principles of logic in a way that is intellectually rigorous yet not so abstract that it loses all outside referents?
Barwise and Etchemendy began to question whether the sentence alone was sufficient to teach the principles of logic. Since logic is about the relationships between sentences and the world, they chose to depict the world in the form of diagrams. Tarski's World presents a variety of self-contained "worlds," represented by pictures of blocks on a grid. Sentences then have an obvious referent--the diagram of the world. Students can see for themselves if sentences such as "The small cube is behind the large tetrahedron" are true or not. Because the material is presented on a computer, the students can manipulate the worlds, change the worlds, and construct their own worlds. Tarski's World not only makes the material more immediate, more comprehensible, and more enjoyable, it enables the students to get far more practice (and thus develop their skills to a greater extent) than would be possible with more conventional methods. The computer does the tedious but conceptually simple steps quickly, checks the work for errors, and shows the student where mistakes have occurred.
Hyperproof, the third of Barwise and Etchemendy's courseware packages (developed with assistance from the Teaching and Learning Technologies Laboratory and the Center for Innovative Computer Applications, both at Indiana University Bloomington), is superficially similar to Tarski's World, though conceptually more sophisticated. Whereas Tarski's World merely teaches students the language of logic, Hyperproof teaches the principles of valid reasoning. In Hyperproof, students learn to construct proofs based on logical principles. The student is presented with a block world similar to the ones in Tarski's World, but a certain amount of information is left out; for example, a student may be told that a large block is in the back row of the grid, but not be told its shape. (This kind of block is represented by a cylinder, the idea being that the block is in the cylinder and therefore its shape is unknown.) Again, the computer's speed and patience allow the student to complete far more proofs than would be possible with pencil and paper. The computer checks the students' proofs and points out where errors have occurred, considerably lessening the anxiety and frustration that many students feel when first confronted with such abstract concepts.
With Tarski's World and Hyperproof, the abstract concepts of logic become visible and concrete--with the use of diagrams. Diagrams are not new to the world of mathematical pedagogy; Archimedes drew pictures in the sand to illustrate his lectures. Yet for many years, diagrams have been considered a sub-rosa form of representation, useful as heuristic tools, but neither as rigorous nor as respectable as words. After developing Hyperproof, Barwise realized that logic theory lacked diagrammatic representation. Though logicians taught the theories of logic by using diagrams, the theories themselves did not encompass or explain the world of graphical representation. "Reasoning does not depend on how you represent the information," Barwise says. In other words, diagrams should be treated with the same logical rigor as sentences or mathematical symbols. This innovative concept highlights the seamless relationship between research and teaching. In developing Hyperproof as a tool for teaching, Barwise and Etchemendy realized that the theories they taught were inadequate. This opened up a new area for research: developing theories of logic that would account for diagrammatic as well as word- and symbol-based information.
The introduction to Logical Reasoning and Diagrams (Oxford University Press, 1996), which Barwise edited with colleague Gerard Allwein, assistant director of the Visual Inference Laboratory and an adjunct professor of computer science at Indiana University Bloomington, conveys the sense of intellectual excitement that Barwise sees in this new endeavor. The introduction states, "The discovery of empirical data that does not fit within existing theory is (or should be) an exciting occurrence in any science." It is not surprising that the crossing of this intellectual frontier should have been occasioned by using computers. Barwise suggests that logicians' predilection for words and mathematical symbols may have been conditioned by centuries of considering the printed page the most important, eloquent, and intellectually rigorous form of communication. Such a preference may predispose scholars toward linear, word-based media. The computer, however, is a nonlinear device that can deal with graphic information as easily and completely as it can with words and symbols. As such, it is changing the boundaries of virtually every aspect of life.
Any new and powerful technology, from fire to nuclear fission, invites both wonder and caution. Barwise's view of computer technology, as might be expected from a logician, is temperate and reasoned. He is enthusiastic about its capabilities and fascinated by its workings; he is, after all, a professor of computer science. At the same time, he is concerned that continued use of the computer may undermine people's ability to read complicated texts--for millennia the backbone of intellectual endeavor. "Are people going to be able to read Ulysses in a hundred years?" he muses. Just as the Industrial Revolution created comfort and liberation for some and dislocation and misery for others, the bounty of the computer revolution may create unforeseen problems that future generations must solve.
We are all newcomers to the world of computers and, like all travelers in uncharted waters, we must create solutions that have no easy precedents. The principles of logic and reason are good places to start. Barwise's interest in logic comes both from his religious upbringing, which gave him a taste for philosophical questioning, and from high school mathematics teachers, who inspired in him an appreciation of the surety of mathematics. The unlikely partners, philosophy and mathematics, are essential to a world that requires both open-ended questions and precise answers. As scholars like Barwise remind us, logic remains an eloquent response to the shrill voices of ignorance, prejudice, and blind faith, which too often have their say.
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