STEVEN JAMES BARTLETT

VALIDITY

A Learning Game Approach to

Mathematical Logic

Steven James Bartlett

VALIDITY is an “autotelic” learning game: it is designed to give students a self-motivating experience in creative problem-solving. I designed the game because I was interested in improving the teaching of mathematical logic at the university level, and wished to supplement the standard approach to mathematical logic that asks students to prove pre-formulated problems. The latter type of challenge is indisputably worthwhile despite the fact that someone other than the student has devised these problems as pre-formulated exercises in proof construction. But a pure diet consisting exclusively of constructing proofs which have already been anticipated by someone else shortchanges the student who does not experience the creative task of formulating his or her own problems, for which constructing proofs then becomes the challenge.

The text-manual for VALIDITY consists of a general introduction that describes earlier studies made of autotelic learning games, paying particular attention to work done at the Law School of Yale University, called the ALL Project (Accelerated Learning of Logic). Following the introduction, the game of VALIDITY is described, first with reference to the propositional calculus, and then in connection with the first-order predicate calculus with identity. Sections in the text are devoted to discussions of the various rules of derivation employed in both calculi. Three appendices follow the main text; these provide a catalogue of sequents and theorems that have been proved for the propositional calculus and for the predicate calculus, and include suggestions for the classroom use of VALIDITY in university-level courses in mathematical logic.

I have used VALIDITY with great success in numerous classes in mathematical logic. Its success has been attested by my students, and by my own observations of the skills in constructing proofs that I have seen students develop when playing the game. Those specific skills include: improved ability and facility in constructing proofs—which are 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.

VALIDITY is not a parlor game. Although players become enthusiastic—indeed sometimes passionate—constructors of proofs, the game is serious and technically challenging.

The game, as I originally conceived and designed it, is intended to be used in conjunction with E. J. Lemmon’s text, Beginning Logic, but the game can be adapted to other texts. I recognize that a professor’s choice of logic text involves many factors, not least of which is the personal appeal of a certain approach to mathematical logic. There are no useful arguments, in my view, that can effectively persuade most mathematical logicians to adopt a system of logic to replace their own preferred system in courses they teach, and I shall not try to summon any. However, there is perhaps some value in sharing what it was and is that I find useful and attractive about Lemmon’s particular system of natural deduction.

First, in his system there is an optimum number of natural deduction rules, adequate for both the propositional and the predicate calculus with identity. Many other systems of logic err in favor of an excessive number of rules, in order to maximize convenience and make proofs maximally short; still other systems err in favor of mathematical elegance by admitting only a single rule, normally the rule of detachment, along with provisions for substitutivity.

I use the word ‘err’ advisedly in both contexts: If one is interested in encouraging students to develop and internalize logical skills that may spill over into other areas of their lives and studies, then a small set of rules, balancing convenience and elegance, and capable of being retained in the active memory of the average student, will turn out to be most desirable. Lemmon’s system, furthermore, recommends itself through the use of a method of assumption annotation, a notational device permitting students to check their proofs quickly for correctness, and exhibiting, for each line a proof, exactly what is presupposed. Such an assumption annotation system has clear-cut advantages both formally and in the context of philosophical argument. Some other systems of natural deduction provide similar methods to keep track of what each line in a proof depends upon.

For these reasons, and others which relate to economy of statement and aesthetics of structure, I adapted VALIDITY to serve as a companion to Lemmon’s book.

Any faculty member who is interested in incorporating a learning game approach within a rigorous course in mathematical logic has to roll up his or her sleeves, for the decreased formality that results when students play an academic game in class means giving up some of the control and structure that a standard class in mathematical logic provides. Too, the open-ended and inherently flexible nature of VALIDITY game playing will require not only the students, but the teacher also, to learn some new skills. It can sometimes be challenging in such a context for the teacher to stay ahead of the brightest students.

I definitely do not advocate using any learning game to the exclusion of work with a text and pre-formulated exercises. For my purposes, I used VALIDITY perhaps one-third of class time: In a class meeting three times a week, one class meeting each week might be devoted to the use of VALIDITY; the other two class meetings consisted of lecture combined with students assigned to construct proofs.

Students were graded on their performance when playing VALIDITY—an expression of its non-parlor-game purpose. Appendix III of the VALIDITY text-manual describes how this was done. However, it is certainly not written in stone that other faculty should do as I did if they wish to make use of the game.

Teachers who prefer an axiomatic approach to mathematical logic and who wish to think of ways to adapt a VALIDITY-like approach to complement their existing method of teaching will, I would expect, be somewhat challenged to adapt VALIDITY to their own needs. But I believe this can, with perseverance and intelligent thought, be done. Teachers who use a natural deduction approach but who do not wish to use Lemmon’s text will have an easier path in adapting VALIDITY to fit their interests.

But, clearly, whether it will be worthwhile to devote creative effort and time to adapt the basic approach of VALIDITY to your own teaching is a matter of your personal judgment.

VALIDITY has served hundreds of my students and their teacher very well. I hope the game as it exists or its underlying approach will be of value to others, and have decided to make VALIDITY freely available as an Open Access publication under the Creative Commons Attribution-NonCommercial-NoDerivs license.