## Teaching Logical Methods

#### November 14, 2019

It’s been a big year. At the start of 2019, Shawn Standefer and I decided to throw all our cards in the air and upend the curriculum for the Level 2 logic unit in the philosophy program here at Melbourne. We wrote 200 pages of a draft textbook (while I really should have been finishing my other book). Shawn designed and implemented a whole raft of multiple choice practice questions, and we worked on a range of class activities to help our class of 60 students grapple with the material. I recorded hours of video lectures covering the content. We stuffed all of this into the LMS. And we spent hours in the classroom teaching 60 students the ins and outs of proof theory and model theory for propositional logic, modal logic (including two-dimensional modal logic), and first-order predicate logic. Like I said, it was a big year putting all of this together. Now we’ve wrapped up our first semester teaching the new unit, so we can sit back, breathe and reflect on how things went.

While I was taking that breath and thinking about how things went, Liam Kofi Bright posted a thoughtful reflection on what he hopes to achieve when he teaches formal methods to his students at the LSE. Our aims for our logic class were similar to Liam’s, but we come at things from a slightly different angle. Since I found reading Liam’s reflections helpful, it makes sense to put my thoughts down in public, in case others might benefit in some way.

Our Logical Methods unit is unashamedly a logic class. (We don’t try to teach the wide range of formal methods. There’s no probability calculus, decision theory or anything beyond logic.) Our subject is designed for philosophy students, though at least a third of the enrolment were students coming from other majors, and even other degree programs. Still, our aim was to give philosophy students the skills and the vocabulary from logic that they will find useful in the rest of their engagement with philosophy, but at the same time, get a sense of logic as a field of philosophical reflection all of its own. Yes, we wanted students to come away able to both construct proofs and make models in formal systems for propositional, modal and predicate logic, and to get a real feel for what we can do with proofs and with models, and how we can use tools from logic—from proof theory and from model theory—to understand and analyse arguments, and to explain and explore the connections between the claims we make. One of our aims was for students to acquire (or strengthen) some logical reflexes. To become familiar with basic inference principles, and to become comfortable with combining them. To get a sense of how to build a model and to construct a counterexample to an argument. To understand what kinds of moves are appropriate when reasoning about possibility or necessity, or with the quantifiers, and what kinds of mistakes you can make if you fail to pay attention to scope distinctions and ambiguity. These reflexes are useful when it comes to philosophical reflection, and it’s good to have a place to practice with them in their own right, before wielding them out in the Real World of other philosophy tutorials.

But in addition to those useful skills, we wanted to give the class a sense of some of the active debates in contemporary work in logic. Since Shawn and I are pluralists and not partisans by temperament, it was natural to us to introduce both intutitionistic logic (when we started venturing into propositional logic with Prawitz-style natural deduction as our account of proofs) and classical logic (when we introduced boolean valuations and truth tables as our account of models). This makes the mismatch between validity-as-given-by-proofs and validity-as-given-by-absence-of-counterexample a problem to be investigated. Do we close the gap by adding more proofs, or by adding more models, or do we live with this gap? That is a question that is worth asking, since working out an answer helps you articulate what sorts of things proofs and models are for. So, we tried to set up the curriculum so that students could understand proofs and models well enough to be able to both use those techniques whenever they’re doing reasoning, and then also to understand the kinds of questions that occupy those of us who work in these areas now.

In his reflections, Liam described the tension between the two distinct aims of helping students overcome ‘math anxiety’ on the one hand, and helping induce wonder and humility before the world’s complexities, on the other. There is something curious about the seeming opposition between these two goals, but as I thought about Liam’s piece, it struck me that I have never experienced these two goals to be in any way opposed in practice. If you think of logic as some finite collection of simple tools and techniques to be practiced and honed and mastered (like, say, learning the alphabet, or the addition and multiplication tables for the numbers from 1 to 12—or the truth tables for the boolean connectives), then once you’ve dealt with your math anxiety, there would be no humility before any awesome complexity, because the field would be all rather humdrum. But that’s not how logic has developed, and to stay at the level of multiplication tables (or boolean truth tables) is to miss out on what logic has become, and to be blind to the enormous range of the kinds of questions we are able to ask, and which we’re fortunate enough to occasionally be able to answer.

Liam’s reflections pointed me to a nice paper by Sheila Tobias on “math anxiety”. According to Tobias’ research, the significant variables associated with students’ inability to do college level mathematics are (1) fear, (2) the belief that mathematics is a white male domain, and (3) the belief that you are either good at mathematics or good at languages and never both. These concerns ring true to my experience of dealing with student concerns about logic teaching. To those worries, I will add another concern that turns discourages many from doing logic: (4) the belief that to do well in philosophy or mathematics you have to be brilliant. Worries like these have been on my mind for some time.

We’ve tried to design a subject that addresses concerns like these. We are nowhere near done with them–especially addressing the gender and race issues–but the results of our initial steps have been encouraging. Here is a little of what we’ve done so far.

• On the brilliance effect: At every stage of learning a new technique or method or result, we try to operationalise what you need to learn, and to break these things down into practice tasks. When we teach the basics of natural deduction, there are separate tasks involving (a) reading proofs, (b) checking proofs to spot mistakes, (c) filling in the gaps in proofs, (d) making your own proofs. We don’t throw students into the deep end, hoping that they’ll “get it.” Whenever there are separable components to a skill, we break things down, bit by bit, and slowly build them up. We show what you need to learn in able to do something well, and then we give them tasks to practice. To master this stuff, you don’t need to just “get it”. There are skills to learn and practice, and you can get there if you put in the time and the work. Thanks to Shawn’s work on the Learning Management System, we have a range of practice exercises that students can hack away at, getting quick feedback on their work. Regular checkpoints show when a student has mastered reading proofs, or writing their own, checking truth in a model, or identifying facts about logical consequence.

This is one place where logic has it relatively easy, compared to other areas in philosophy. It’s a disciplinary norm to define things precisely and work things out piece by piece. It’s our job as educators to introduce those concepts in a manageable way. If we do that well, we show as well as say that these are tools and techniques that can be mastered by practice.

• On the perceived difference between linguistic and mathematical expertise: Here we try to use a range of different metaphors and explanatory strategies to explain what we’re doing when we’re doing formal logic. Thankfully, it is relatively easy to make clear that we’re not doing mathematics in the sense of doing a lot of calculations or heavy duty algebra. We’re up front that we teach formal logic, but we’re careful to explain that patterns are everywhere, and not just in mathematics classes. Here, the example of our colleagues in linguistics is helpful to us. We explain that we’re looking at human activities of speaking, thinking and representing, and identifying patterns and structures that recur in what we can say and mean, in order to activate the imagination and expertise of those who are confident with language.

Given that we have an interdisciplinary class (some from the Sciences, but most are Humanities students), we try to walk both sides of the street when it comes to explanatory metaphors. Instead of always talking about algorithms or programs, we describe things as recipes or routines. When we talk about model building, sometimes we emphasise the routine and the systematic, but we also describe specifying a model as an act of imagination, engaging our creativity. Given that the entire subject is built around the duality between proof theory and model theory, we take the perspective-switching nature of looking at a phenomenon from more than one side seriously, so it comes naturally to attempt to describe things in more than one way.

• On gender and race: Here we have much more to do than what we’ve done so far. We own the fact that we have taught the subject as two white males. It is a challenge for us to expand our citations beyond the usual small circle of white male experts, but we’ve expanded it just a little. We point to women logicians along the way, and our students’ final project involved the logic and metaphysics of putting together modal logic and first-order predicate logic, it focussed on Ruth Barcan Marcus and the Barcan formulas. So, we put her work front and centre as the culmination of the semester. There is much more to be done–especially on decolonising our curriculum–but the results of these tiny steps have been encouraging so far. With all the changes we made, our retention rate for non-male students was significantly higher than it has been in previous years. The class is still male dominated, but significantly less so than in previous occasions.

• On fear: We have not measured students’ anxiety or fears in any way, but we’ve attempted to address that fear in a number of ways. We are fortunate, at Melbourne, to have very good undergraduate students. They are smart, and they are engaged and they are willing to put in the work. So we explain that the subject will involve effort, but that the effort will pay off if they put in the time. We structured the assessments to be predictable and manageable. Not only were the practice tasks broken down into bite-size repeatable skills to master. We assessed them on the same basis. At four points through the semester, there was a multiple choice test, worth 15% of their final score, on just these skills. They weren’t all easy (they added up to some quite challenging tasks) but they were manageable, they were predictable and most importantly, they weren’t an exam. 60% of the assessment task could be done by grinding out effort. It is a way to manage the fear of not “getting it”.

The remaining 40% of the assessment was two projects. The second project was on quantified modal logic and Ruth Barcan Marcus, as I mentioned, and the first was on truth tables and proofs for three-valued logics, and the ways these could be used to model different sorts of phenomena. Both of these projects were clearly related to the work they’d done in class, but they allowed students to reach beyond what they’d practiced, and learn to apply their skills in different environments. The results have been much better than we had expected they’d be. The students loved being stretched, they took things seriously, and wrote quality work. Passing turned out to be relatively easy, if they put in the effort to grind. Then with the fear of passing dealt with, they could make the effort to excel if they wanted to. From the looks of it, almost all of them really wanted to excel, and many of them have.

So, that’s been our experience of teaching logical methods in 2019. It’s been a wild ride, and it’s been such a pleasure to be on that ride with Shawn and 60 willing students. Thanks to friends and colleagues, like Dave Ripley, François Schroeter, Allen Hazen, and my current and former graduate students, Kai, Timo, Lian, John, Sakinah and Adam who have helped us sort out some of our thinking about these issues.

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