# What Should Philosophers Teach in Quantitative Reasoning Courses? (guest post by Landon D.C. Elkind)

The following is a guest post* by Landon D.C. Elkind (University of Iowa) about the content of philosophy courses that satisfy general education requirements in quantitative or formal reasoning. It originally appeared on his blog.

### What Should Philosophers Teach in Quantitative Reasoning Courses?

by Landon D.C. Elkind

Most philosophy departments offer some logic course that satisfies a mathematical, quantitative, or formal reasoning general education requirement. My university describes their quantitative and formal reasoning requirement as follows:

*To help you develop important analytical skills, these courses focus on the presentation and evaluation of evidence and argument, the understanding of the use and misuse of data, and the organization of information in quantitative or other formal symbolic systems.*

Given that these are quantitative and general education requirements, what should we, as philosophers, teach in these courses?

Almost all of these courses choose topics from the following categories:

- categorical logic (square of opposition, categorical inferences)
- categorical logic (diagrams)
- informal fallacies and reasoning
- propositional logic (proofs)
- propositional logic (truth-tables)
- propositional logic (tableaux)
- inductive and causal reasoning, including causation vs correlation vs explanation
- probabilistic reasoning, including the probability calculus

Most of them, in my experience, do not proceed to quantification theory. Perhaps the thought is that this much content is too difficult for this level (but categorical logic is an easier, albeit more cumbersome, way to teach universal and existential reasoning).

So: given the quantitative reasoning aims of “organization of information in quantitative or other formal symbolic systems”, which of these should we select?

I think that categorical logic should be struck from this list. Categorical reasoning is too unwieldy to be justifiable, even if the diagrams are fun. Formulas need not have the categorical form, and they should not be forced into it, even for pedagogical purposes.

Propositional logic, on the other hand, is woefully inexpressive. But it is a useful tool for teaching *transferable* analytical skills using syntactic and semantic methods. I cannot sing the virtues of proofs, tables, and tableaux too highly for imparting the analytical habit of mind.

What about informal fallacies? I say that they should be struck. They do not, first, teach any analytical skills. They do help one *identify* mistakes in reasoning, which may then be leveraged to evaluate arguments and assess the use (or misuse) of data. But they do not in themselves demand much development of analytical skills. They are just labels that bring out faulty reasoning more clearly.

Also, they can be *introduced and taught* in a week. Familiarity with them is the only real virtue of teaching them. Informal fallacies do not seem to be worth *assessing* using multiple choice tests or projects outside of class. That seems to be more about testing our ability to remember the names of fallacies—which is not a useful thing to teach—as opposed to sharpening our wits so that we can catch mistakes in reasoning.

If we are concerned to teach one to reason well—to cultivate the analytical habit of mind—then it seems to me that we are better off teaching inductive and causal reasoning, plus the probability calculus. These do cultivate *active *analytical thinking in a way that passively identifying fallacies does not.

So I will be minimizing or leaving out informal fallacies altogether in my critical thinking class! I am quite excited about what the results will be.

Art: Koos Verhoeff, “Gordian Knot”

Before I alter syllabi, I’d like to see some evidence that categorical logic and informal fallacies don’t “cultivate the analytical habit of mind” that go beyond how things seem to one logic instructor. Also, some kind of operationalization of “analytical habits of mind” might be useful too.

I agree that the author should clarify their notions of analytical skills or habits of mind. They write:

“What about informal fallacies? I say that they should be struck. They do not, first, teach any analytical skills. They do help one identify mistakes in reasoning, which may then be leveraged to evaluate arguments and assess the use (or misuse) of data. But they do not in themselves demand much development of analytical skills. They are just labels that bring out faulty reasoning more clearly.”

I’m having a hard time seeing why identifying mistakes in reasoning doesn’t count as an analytical skill. And supposing the author is right that it doesn’t, why isn’t the ability to do so a worthwhile pedagogical goal anyway?

I’m not at all committed to teaching informal fallacies. But without some elaboration, the argument here, as stated, doesn’t increase my doubts about it.

“What about informal fallacies? I say that they should be struck. They do not, first, teach any analytical skills. They do help one identify mistakes in reasoning, which may then be leveraged to evaluate arguments and assess the use (or misuse) of data. But they do not in themselves demand much development of analytical skills. They are just labels that bring out faulty reasoning more clearly.”

This may be the case if you treat them as a list of mistakes not to make and leave it at that, rather than, as is currently the practice in informal logic and argumentation, as argument schemes that may or may not be appropriate.

For a window into the current view in informal logic by two of its leading experts, I recommend people read this:

https://cdn.ymaws.com/www.apaonline.org/resource/collection/808CBF9D-D8E6-44A7-AE13-41A70645A525/v09n1Teaching.pdf

A longer version here:

https://ojs.uwindsor.ca/index.php/informal_logic/article/view/454

I’d also recommend a close read of the “Critical Thinking” entry in the SEP, again by a leading scholar in argumentation studies.

https://plato.stanford.edu/entries/critical-thinking/

These are very useful resources. Their view of informal fallacies as, as you put it, “argument schemes that may or may not be appropriate” is a far better use of that subject matter in a critical thinking course.

You are right that I was dismissing the “taxonomy of informal fallacies” use of that subject matter, which is how I was taught informal fallacies. And I think that we agree that the taxonomy approach is not a good one. But I did not address this use of informal fallacies as part of the broader topic of informal argument schemes that is described in these pieces.

I will review the literature that is referenced in these linked pieces in reconsidering the virtues of teaching informal fallacies as part of this broader topic. I encourage others to do the same. So: thank you for posting these articles.

Wait, your experience is that people *don’t* cover predicate logic in their logic courses? I’m *very* surprised by this; I’ve had exactly the opposite experience.

For context: I’ve taught dozens of intro logic courses and some half dozen different institutions, and in each of them I’ve either covered quantification or nonclassical logics. I’ve also had no difficulty getting students through the material. Students do often find full predicate logic difficult (as they should!) but by and large they learn it. Having also taught in math departments it’s worth comparing where doing this leaves logic as compared to other courses students take to fulfill their quantitative reasoning requirement. It seems to me that if you cover quantification this lands the course (for the students) at a point that makes it somewhat harder than college algebra (for most of them) but somewhat easier than an intro stats course. I’d say *not* covering quantification would leave the course as massively less difficult than even college algebra.

If I were to press for one thing here, it would be *removing* propositional logic in favor of predicate logic. I’ve become more and more convinced that stopping at predicate logic is really just an unnecessary detour.

One final thing: you lump together all of natural deduction under the heading `proofs’. But there’s some reason to think there are differences among the available methods: the consensus seems to be that Hilbert systems are a very poor choice, Prawitz-style natural deduction is ok, a full sequent calculus would be better, but is probably too hard, and that Fitch- or Lemmon-style natural deduction systems give a mostly-happy medium. There was a recent discussion about this on logic twitter. A part of that discussion can be found in this thread: https://twitter.com/consequently/status/1036398046509064192

Thanks for linking to that discussion thread. I followed that thread in real time, and it reinforced my decision to choose a text that covered propositional logic trees.

At my institution, there is a separate course (Symbolic Logic) that covers quantification theory (including higher-order logic), properties of relations, descriptions, and some meta-logic results. In that institutional context, the quantitative reasoning course that I discuss in this piece covers other areas to avoid duplication. (I say that without offering a judgment as to whether that is the most appropriate breakdown of course content in quantitative reasoning courses, and with awareness that the courses will have different content based on the institutional setting.)

You are right that I did not break down different methods of deduction. I teach and prefer Lemmon-style myself, but I did not weigh in on that issue here.

I *think* that Adam Elga (Pton) did a diagramming-arguments course that looked to me to be about the perfect way to go for Elkind’s gen-ed requirements. If I were taking that route (and I’d like to) I’d also do appropriate pointing to fun (and live) issues in logic and philosophy of logic — the sort of stuff that Shay Allen was on about above (viz., Oct 23 2018 @ 1.17pm). But for getting people to look for arguments, and in turn be in position to assess them, I reckon that Elga is on the right track — if indeed it was Elga. (I think it was.) I also think that folks should be encouraged to go on from the diagramming-arguments stage to explore logic (and its philosophy).

I completely agree that it’s a good idea to remove categorical logic and deemphasize informal fallacies. That’s what I’ve done when I’ve taught critical thinking. I also think it’s a good idea to cover cognitive biases and important statistical fallacies and facts (e.g. the base rate fallacy, Simpson’s paradox, the fact that correlation isn’t transitive, etc.), and perhaps simple decision theory.

About the informal fallacies. I’m not impressed by students who have successfuly learned to identify them (I don’t get too many of them, since these Critical Thinking courses seem to be a largely American phenomenon). Typically, when these students are confronted with an argument, and they can identify a fallacy, they name it and feel they are done with the text (“it’s fallacious!” they proudly proclaim, like they scored in some kind of game). The questions whether the argument is sound when the fallacious bit is struck, or whether the argument can be salvaged etc. don’t seem to occur to them. It’s fallacious after all.

So I’m guessing — for what my little evidence may be worth — that there is some truth in the claim that learning these fallacies doesn’t sharpen analytic skills. It seems to encourage rather shallow engagements with a text.

Just one quick comment: if you are teaching a course that has been approved for a particular general education requirement, then you should make sure that any changes you make are consistent with the course as approved by the campus gen ed committee. Otherwise, you endanger the course’s approval for the future.