## Revisions to enumerability and size of sets sections

As per issues 107 and 109, the material in sets-relations-functions/sizes-of-sets needs to be cleaned up. It was inconsistent in its assumption about whether functions are always total (they are, according to the definition in the preceding chapter), it gave an incorrect formal definition of enumerability (leaving out the empty set), and whenever it mentions $\mathbb{N}$ it inconsistently assumes that $\mathbb{N}$ begins at 1. Issue 109 deals with the problem that the section size-of-sets uses the cardinality notation $\left|X\right|$ which leads students to assume that they can manipulate cardinalities as they can in the finite case; the proposal is to replace $\left|X\right| \le \left|Y\right|$ with $X \preceq Y$ to avoid this.

Commit a6a70a4 fixes issue 109; commit 370cb02 fixes issue 107. The latter also reformulates the diagonal proofs to make them direct instead of reductio proofs.

If you teach these sections, these changes may affect you. Please comment on the issues in GitHub if you have concerns. They will be merged into the master branch in a week otherwise.

## Choosing an Open Introductory Logic Text

I will be teaching our Logic I course beginning in January 2017, and we have decided to modify an existing free and open logic text for use in that and (hopefully) future sections of Logic I here in Calgary. We have decided to take Tim Button’s Cambridge Version of PD Magnus’s forallx as a starting point. This is part of a larger project aimed at updating that introductory logic course. In this post I’ll say a bit about why I made chose that remix of forallx, point to some changes we intend to make to that text, and a bit about the other texts I looked at.

One of the first, and least content centred considerations was typesetting software. Picking a version of forallx as a starting point meant we could typeset the book in LaTeX. This is important for three reasons. First, Richard Zach and I are both used to working in LaTeX. Second, LaTeX allows for easy manipulation and restructuring of the text; swapping material between versions, or even inclusion of material in or from the Open Logic Project. Third, LaTeX is also free and open, which we think is important.

The result of sticking to books written in LaTeX was to move texts produced in LaTeX to the top of the pile, so to speak. So, for LaTeX-y and other considerations related to ease of use and modification, I choose to concentrate on the forallx family of open logic texts. As far as I’m aware, there are three versions of that text (not including the YYC remix that we’re developing) – the original by P.D. Magnus, the Cambridge version put together by Tim Button, and the Open Introduction to Logic aka the Lorain County Remix, put together by J. Robert Loftis. One major difference is worth noting at this point. Loftis’ version includes a great deal more material taken from Cathal Woods’ critical thinking text, that I would classify as belonging to an informal logic curriculum. The Magnus and Button versions, on the other hand, cover almost exactly the range of material that we cover in the Logic I course that I’ll be teaching.

Now we wanted to try to ensure that we wouldn’t be giving ourselves extra work when preparing forallx-YYC, so the goal was to pick a book that would need as little revision as possible. I suspect that changes will continue to be made to the text over the next couple of years at least, but having a text that meets our needs and will be ready for students to download in January is the first priority. This meant that simple things like using our preferred symbols for the connectives (¬, ∧, ∨, ↔, →), or using “first-order logic” instead of “predicate logic” put Button’s version ahead from the start. Most of us who’ve spent a large amount of time around symbolic logic have preferences when it comes to symbols and terminology, but in this case another major concern was continuity with the Open Logic Project. This was also a factor in making some of the other terminological and notational decisions, as it would be great to have it be possible for there to be continuity between the free and open texts for our first two logic courses, which are required for many students in philosophy and computer science. It turned out that Button’s book matched our preferences on many of these issues as well.

Although terminology and notational conventions are often important, especially for those students who go on to do more logic or philosophy of logic, more pedagogically and philosophically important are decisions related to how the proof theory and model theory (semantics) of first-order quantified predicate logic are set up.

I’ll start with proof theory. All three texts use Fitch-style natural deduction systems. So far so good – such systems are widely used, perspicuous, and useful. The first decision then was to decide whether to include a symbol for what Frege called the False, i.e. ‘⊥’ which I call “bottom”. Magnus and Loftis don’t, Button does. We went with ‘yes’ for a couple of reasons (Button gives some similar justifications). For one, it makes certain features of classical logic like the rule ex falso quod libet (explosion), that says that you can derive absolutely anything from a contradiction, more obvious. Relatedly, including bottom means that if students go on to study non-classical logics, and especially intuitionistic logic, the relationships between the logics are easy to see. Finally, it cleans up rules like those for negation by making the relationship between (classical) negation and truth clearer.

In this case, although Button includes bottom, he takes it to be defined by a canonical contradiction, whereas we will take it as primitive. Although this won’t be of great importance in this level course, it will hopefully make things less confusing, and raise fewer tough philosophical issues. (It also jives well with my ever more frequent Fregean tendencies.)

The other major issue in introducing the proof theory of first-order logic is how to deal with disjunction elimination. In introductory logic texts and courses the primitive disjunction elimination rule is very often disjunctive syllogism (¬A, A∨ B ⊢ B); it’s simple and there’s no doubt that DS plays an important role in classical deduction, but taking DS as primitive also inextricably ties disjunction to negation. Instead, I, like Button, and unlike Magnus and Loftis, prefer to use proof by cases (A∨ B, [A⊢C, B⊢C], ⊢C) for disjunction elimination. A further reason to do this is that proof by cases is a common proof strategy in mathematics, mathematical logic, and metalogic, and thus important in its own right.

In the case of model theory, I vacillated between wanting to include explicit set talk as per Magnus as Loftis, or instead eschew set talk in favour of plural locutions a la Button. Although there are benefits to including sets — that’s how model theory is usually done, students going on to take Logic II will need to learn about sets – we eventually decided that the difficulties in using sets, mostly related to added notational and conceptual complexity, outweighed the benefits. Additionally, I think there are good philosophical reasons for avoiding sets when doing model theory. Issues relating to intensionality, absolute generality, and indefinite extensibility come immediately to mind.

At the end of the day, the most significant goals in an introductory logic course are getting students used to doing the proof theory and model theory for formal languages, so given our preferences, Tim Button’s Cambridge Remix was the obvious choice as a starting point. We’ve sat down with printouts of all three versions and made a “Frankenbook,” slightly rearranging the material, adding some bits and pieces from Magnus’ original (mainly exercises) and the Woods/Loftis’ remix (exercises, a section on soundness and completeness, and the glossary), as well as a chapter on normal forms and expressive completeness from Button’s Metatheory book.  All of that’s now up on Richard’s GitHub (and the PDF here); we’ll continue to revise and add material, of course. (Magnus’ and Button’s source code is also available on GitHub through the OLP, and Loftis has his own repository).

Of course there are many other issues to consider when choosing or building an introductory logic text, but I hope to at least have given a good overview of how Richard and I are thinking about some of the relevant issues. I look forward to comments and suggestions from logic teachers current, past, and future. Keep your eye out for more updates from this project.

## For Ada Lovelace Day: Julia Bowman Robinson

Julia Bowman Robinson was an American mathematician. She is known mainly for her work on decision problems, and most famously for her contributions to the solution of Hilbert’s tenth problem. Robinson was born in St. Louis, Missouri on December 8, 1919. At a young age Robinson recalls being intrigued by numbers. At age nine she contracted scarlet fever and suffered from several recurrent bouts of rheumatic fever. This forced her to spend much of her time in bed, putting her behind in her education. Although she was able to catch up with the help of private tutors, the physical effects of her illness had a lasting impact on her life.

Despite her childhood struggles, Robinson graduated high school with several awards in mathematics and the sciences. She started her university career at San Diego State College, and transferred to the University of California, Berkeley as a senior. There she was highly influenced by the mathematician Raphael Robinson. They quickly became good friends, and married in 1941. As a spouse of a faculty member, Robinson was barred from teaching in the mathematics department at Berkeley. Although she continued to audit mathematics classes, she hoped to leave university and start a family. Not long after her wedding, however, Robinson contracted pneumonia. She was told that there was substantial scar tissue build up on her heart due to the rheumatic fever she suffered as a child. Due to the severity of the scar tissue, the doctor predicted that she would not live past forty and she was advised not to have children .

Robinson was depressed for a long time, but eventually decided to continue studying mathematics. She returned to Berkeley and completed her PhD in 1948 under the supervision of Alfred Tarski. The first-order theory of the real numbers had been shown to be decidable by Tarski, and from Gödel’s work it followed that the first-order theory of the natural numbers is undecidable. It was a major open problem whether the first-order theory of the rationals is decidable or not. In her thesis , Robinson proved that it was not.

Interested in decision problems, Robinson next attempted to find a solution Hilbert’s tenth problem. This problem was one of a famous list of 23 mathematical problems posed by David Hilbert in 1900. The tenth problem asks whether there is an algorithm that will answer, in a finite amount of time, whether or not a polynomial equation with integer coefficients, such as 3x2 − 2y + 3 = 0, has a solution in the integers. Such questions are known as Diophantine problems. After some initial successes, Robinson joined forces with Martin Davis and Hilary Putnam, who were also working on the problem. They succeeded in showing that exponential Diophantine problems (where the unknowns may also appear as exponents) are undecidable, and showed that a certain conjecture (later called “J.R.”) implies that Hilbert’s tenth problem is undecidable. Robinson continued to work on the problem for the next decade. In 1970, the young Russian mathematician Yuri Matijasevich finally proved the J.R. hypothesis. The combined result is now called the Matijasevich-Robinson-Davis-Putnam theorem, or MRDP theorem for short. Matijasevich and Robinson became friends and collaborated on several papers. In a letter to Matijasevich, Robinson once wrote that “actually I am very pleased that working together (thousands of miles apart) we are obviously making more progress than either one of us could alone” .

Robinson was the first female president of the American Mathematical Society, and the first woman to be elected to the National Academy of Science. She died on July 30, 1985 at the age of 65 after being diagnosed with leukemia.

(This short biography is part of the Open Logic Project; PDF here).

## How to Get (Printed) Open Textbooks to Your Students

One problem open textbooks (and instructors adopting open textbooks) face is how to make the texts available to their students. Of course, it’s easy to distribute electronic OERs. But if you want to provide your students a nice, printed version they can take to the coffee shop, you’re in a bind. First, you have to have it printed.  This is a bit of work, but with online print-on-demand services like lulu.com it’s possible.  But students would have to order the text themselves, and tax and shipping can almost double the (low) cost of a print-on-demand paperback, especially if you want it fast.

So big props to our campus bookstore, especially its manager Brent Beatty, who agreed to order 30 copies for my class and sell them at cost. Brent has been a member of UCalgary’s OER Working Group, so he’s attuned to the issues and challenges of open textbooks. All I had to do was send him the lulu.com order link; with volume discount, low volume shipping cost, and lulu.com’s frequent (constant?) promotional discount (25%) the shelf price is just a few cents above the list price on lulu (C$11). Now I just hope enough students buy it so they’re not making a loss! ## Line Art Portraits of Logicians You’ve probably seen some of the line art portraits of logicians we’ve commissioned. They were done by Calgary illustrator and graphic designer Matthew Leadbeater. We’re pleased to release them all now under a Creative Commons BY-NC license: anyone is free to use them in their own work, to create derivative works from them, and to share them, provided (a) credit to Matt Leadbeater is properly given and (see license terms!) (b) they are not used for any commercial purposes. They each come in two versions, one with a line below, and one with the portrait in a circle. You can download the original Adobe Illustrator files. For PNG and PDF formats, we have set up a GitHub repository. Commissioning these illustrations was made possible by a grant from the Alberta OER initiative. We gratefully acknowledge the support. [Bonus: an image file with all of them that tiles nicely, for your desktop background.] ## Student Satisfaction Survey Results In the Winter term 2016, I taught the University of Calgary’s second logic course from a textbook remixed from the Open Logic Project. Traditionally, Logic II has used Boolos, Burgess & Jeffrey’s Computability and Logic, and it was taught in Fall 2015 using that book as the required text by my colleague Ali Kazmi, and before that by him, Nicole, and me twice a year from that same book. One aim Nicole and I had specifically for the OLP was that it should provide a better text for Logic II, since neither we nor our students seemed to be very happy with “BBJ”. In order to ascertain that the OLP-derived text fares better with students, we did something radical: we asked them what they thought of it. Ali graciously gave permission to run the same textbook survey in his class, so we have something of a baseline. A direct comparison of the two books as textbooks for the course is not easily made, since Ali and I used the books differently: I stuck closer to my text than he did to BBJ; I assigned homework problems from the text; and we assessed students differently, so it’s difficult to control for or compare teaching outcomes. With small samples like ours the results are probably also not statistically significant. But the results are nevertheless interesting, I think, and also gratifying. We obtained clearance from the Conjoint Faculties Research Ethics Board for the study. All students in each section of Logic II in F15 and W16 were sent links to an electronic survey. As an incentive to participate, one respondent from each group was selected to receive a$100 gift certificate to the University of Calgary bookstore. The surveys were started in the last week of classes and remained open for 3 weeks each.  Response rates were comparable (23/43 in F15, 23/42 in W16). The survey was anonymous and developed with the help of the Taylor Institute for Teaching and Learning, who also administered the survey; results were not given to us until past the grade appeal deadline in W16.

We asked 23 questions.  The first three regarded how students accessed and used the textbooks. In the F15 section, the textbook was not made available electronically, but students were expected to buy their own copy (about $40). Most respondents did that, although almost a quarter apparently pirated electronic copies. In W16, the OLP-derived text was available for free in PDF and students had the option to buy a print copy at$10. Over half the respondents still opted to buy a copy.  We asked students how they used the texts in hardcopy and electronic form.

Those using the OLP-derived printed text underlined significantly less than those who used BBJ. I’m guessing the OLP text is better structured and so it’s not as necessary to provide structure & emphasis yourself by underlining. In fact, one student commented on BBJ as follows: “Very little in the way of highlighting, underlining, or separating the information. It was often just walls of text broken up by the occasional diagram.”

When using the electronic version (both PDF), students did not differ much in their habits between F15 and W16. More students took notes electronically in F15. I suspect it’s because the PDF provided in W16 was optimized for screen reading, with narrow margins, and so there was little space for PDF sticky notes as compared with a PDF of the print book in F15. Also notable: highlighting and bookmarking is not very common among users of the PDF.

The second set of questions concerned the frequency with which students consulted the textbook, generally and for specific purposes.  W16 students used the OLP-derived text significantly more often than F15 students did, and for all purposes.

The difference is especially striking for the questions about how often students consult the textbook for exams and homework assignments:

We next asked a series of questions about the quality of the texts. These questions were derived from the “Textbook Assessment and Usage Scale” by Regan Gurung and Ryan Martin. On all but one of these questions, the OLP-derived text scored positive (4 or 5 on a 5-point Likert scale) from over half the respondents. The discrepancy to students’ opinions of BBJ is starkest in the overall evaluations:

The one exception was the question “How well are examples used to explain the material?”:

This agrees with what we’ve heard in individual feedback: more, better examples!

Lastly, we were interested in how students think of the prices of textbooks for Logic II. We asked them how much they’d be willing to spend, how much the price influenced their decision to buy it. Interestingly, students seemed more willing to spend money on a textbook in the section (W16) in which they liked the textbook better. They also thought a free/cheap textbook was better value for money than the commercial textbook.

We also asked demographic data. Respondents from both sections were similar: almost all men in each (the course is mainly taken by Computer Science and Philosophy majors), evenly divided among 2nd, 3rd, 4th year students plus a couple of grad students in each (Logic II is required for the Philosophy PhD program). Student in W16 expected higher grades than those in F15, but that may well be just an effect of differences in assessment and grading style rather than better student performance.

If you care, there’s an interactive dashboard with all the graphs, and the raw data.

## Vote for your Favorite Logician

Suppose we got more illustrations of logicians. Who should we get?