STEVEN JAMES BARTLETT

MATHEMATICAL LOGIC AND

PROBLEM-SOLVING

Steven James Bartlett

MATHEMATICAL LOGIC

Beginning with my first teaching position at the University of Florida, I became interested in the possibility of improving the logical thinking skills of students by means of so-called autotelic learning games. ‘Autotelic’ means setting one’s own ends or goals, that is, to be self-motivating. The first of such games that I became aware of was Layman E. Allen’s WFF ’N PROOF: The Game of Modern Logic, which I used in conjunction with a standard text in mathematical logic. WFF ’N PROOF, however, is limited to the propositional calculus, and is further limited for an American audience by the fact that it uses Polish notation.

I spent many hours devising a very different game of mathematical logic, VALIDITY: A Learning Game Approach to Mathematical Logic. Unlike WFF ’N PROOF, VALIDITY can be used to teach skills in both propositional and first-order predicate calculus; VALIDITY is also a good deal more challenging and more appropriate to university-level classes; and VALIDITY employs the more familiar and widely used system of logic notation found in Anglo-American countries.

I had truly excellent results using VALIDITY in classes that I taught in mathematical logic; the game, I need to emphasize, was used as an adjunct learning experience, supplementing a standard text in mathematical logic. When I say “excellent results,” this is what I mean: First, students were highly motivated to learn to play the game well. Since the game stresses and builds skills in formulating proofs, those were the kinds of skills that the game helped students to develop and hone. Second, the game was used in a serious/non-entertainment context: Students were divided into groups of three (with the instructor sitting in as a player when any group had less than three student participants), and their skills in constructing proofs were evaluated and graded by recording each individual student’s frequency of wins and losses. More about the grading procedure will be found on the web page describing VALIDITY.

Short of a statistical experimental vs. control group series of trials which is was not possible to make part of my teaching, it is of course difficult to assess the value of using a supplemental learning approach like VALIDITY. However, most teachers are able to weigh the pros and cons of a teaching methodology based on close observations of students. This unquestionably involves subjective judgment, but not all such judgments are useless. Having used VALIDITY to teach hundreds of students skills in constructing proofs in mathematical logic, it has been clear to me that specific skills are developed to a higher degree in the average student when using VALIDITY than can be expected when using a standard text in mathematical logic alone.

Those specific skills include: improved ability and facility in constructing proofs—which are of course the main goals of VALIDITY; improved mental efficiency—that is, the ability quickly to see directly through to an effective proof strategy; improved mental anticipation and retention— that is, increased ability to hold the whole anticipated proof in mind; and improved cognitive flexibility—that is, the ability to “re-group” and to re-formulate a proof strategy when the moves of other players change the framework within which a proof needs to be developed.

All of these skills are clearly basic to efficient, effective, and intelligent problem-solving. Certainly, any teaching approach that claims to succeed in improving the degree to which students are able to learn such skills should be tried.

I have not—perhaps I should be ashamed to say—been motivated to submit VALIDITY for publication by a large publisher. Instead, I made use of the limited print runs of the first two editions, which were sufficient to supply the needs of my own classes. However, I have come to feel that my general aversion to the tedium of the submission/publishing process have stood in the way of VALIDITY’s ability to gain a foothold in university-level courses in mathematical logic.

Potentially to remedy this, I’ve decided to make VALIDITY available as a free open access publication. University faculty who may be interested in using VALIDITY in the context of their classes are invited to consider its use by clicking here.

PROBLEM-SOLVING

While a professor of philosophy at Saint Louis University, I received grants from the National Science Foundation and the Lilly Endowment, which led to the development of a campus-wide course in problem-solving. Part of the content of this course was adapted from UCLA engineering professor Moshe Rubinstein’s approach to classroom teaching and his excellent text, Patterns of Problem-Solving. Still another part of the content reflected my concomitant teaching of computer programming for the Department of Mathematics. And still another part expressed my interest in studying whether cognitive intelligence could, to any meaningful, testable extent, be taught.

Thanks to the support I received and freedom I was given by the Department of Philosophy and by the University administration, I offered Patterns of Problem-solving every semester for most of a decade. The course was granted the unusual status of a campus-wide class that satisfied a course requirement for nearly every major offered by the University. The class was always over-subscribed by students.

My experience in offering this class led to two main convictions:

First, I became convinced that university-level instruction in a wide variety of largely quantitatively based problem-solving skills and techniques, emphasizing underlying psychologically-based positive attitudes toward problem-solving, can significantly profit students of all majors. There are unfortunately very few university courses that specifically seek to improve the generalizable ability of students to think well, as this can be measured objectively when quantitatively based problem-solving is the focus. Certainly some courses, for example, in mathematical logic, are believed to do this, and perhaps also to some extent classes in mathematics and computer programming.

Professors of philosophy frequently believe that training in philosophical thought increases the ability of students to think critically and logically, and I am sure that this is also true to an important degree. However, there is no substitute for class content that permits all class participants to decide objectively whether any instance of reasoning is or is not valid, and here it is my experience that training in general, quantitatively based problem-solving and in mathematical logic offer especially good ways to bring about clear-headed, systematic, careful, and disciplined skills and habits of thought in students. In my conception of an ideal world, basic but rigorous, mentally challenging classes in symbolic logic and general problem-solving would be required in all universities of all students.

Second, as a result of pre- and post-testing of IQ of my students (excepting those who did not wish to participate in this, of whom there were very few), I found that, on average, students meaningfully improved their IQ scores (and they very much enjoyed the process). By “meaningfully,” I mean statistically significant improvements. For data that show such improvements, see my report Patterns of Problem Solving: A Final Report of Work Undertaken with the Support of the Lilly Endowment, which summarizes the initial pre- and post-test IQ study.* The results achieved were replicated in subsequently offered classes. Whether such IQ improvements would remain with students in their future lives is an open question, since it was not possible to follow them by means of longitudinal IQ testing.

Related to my work in problem-solving, here are several papers that can be downloaded:

“A Metatheoretical Basis for Interpretations of Problem-Solving Behavior”

“Protocol Analysis in Creative Problem Solving”

“The Use of Protocol Analysis in Philosophy”

Patterns of Problem Solving: A Final Report of Work Undertaken with the Support of the Lilly Endowment

LOWER BOUNDS OF AMBIGUITY AND REDUNDANCY

The elimination of ambiguity and redundancy are unquestioned goals in the exact sciences, and yet, as this paper shows, there are inescapable lower bounds that constrain our wish to eliminate them. The author discusses contributions by Richard Hamming (inventor of the Hamming code) and Satosi Watanabe (originator of the Theorems of the Ugly Duckling). Utilizing certain of their results, the author leads readers to recognize the unavoidable, central roles in effective communication, of redundancy, and of ambiguity of meaning, reference, and identification. Originally published in Poznań Studies in the Philosophy of Science, Vol. 4, Nos. 1-4, 1978, 37-48.

To download a copy, click here. Also available from PhilPapers.

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* See also Moshe F. Rubinstein, “A Decade of Experience in Teaching an Interdisciplinary Problem-solving Course,” in Tuma, D. T. and Reif, F. (Eds.), Problem Solving and Education: Issues in Teaching and Research, pp. 25-38, esp. pp. 34-36. Hillsdale, NJ: Lawrence Erlbaum, 1980.