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A constructivist approach to symbolic logic

First, some background. Symbolic logic is a way of taking ordinary sentences, and turning them into mathematical statements, and then examining the truth value of the sentences by working in the symbolic form. An example of a complex argument which was completed using almost all symbolic logic, see Gödel’s theorem.

Normally when I teach symbolic logic, I stand up at the front of the room and carefully go through each logical connective covered in the course, and the students take notes, then work on some problems to solidify their understanding. Today I decided to try something different.

I went through one example logical connective, specifically the If P then Q connective, normally shown as P → Q. The truth table for this logical connective is shown below.

P Q P Q
T T T
T F F
F T T
F F T

Update: I’ve fixed the entries in the truth table above (shown in red) as for some reason I put in the wrong values. Thanks to @suburbanlion for pointing out my goof.

We talked about the different possible situations for this one connective, and then I wrote down the other connectives we need to cover for our course, asked the students to break into small groups, and decide in their groups what the truth value should be for these connectives.

Logical connective Sample sentence Symbolic sentence
 ∧  I go to the movies and I eat dinner.  P ∧ Q
 ∨  I go to the movies or I eat dinner.  P ∨ Q
 ↔  If and only if I go to the movies, then I eat dinner.  P ↔ Q
   Either I go to the movies or I eat dinner.  P ⊻ Q
 ¬  I do not go to the movies.  ¬P

 (Symbols may not display correctly in all browsers)

What was interesting was that all but one group came up with exactly what mathematicians have agreed upon for the different logical symbols, which is by far a much higher success rate than I’ve had in presenting this topic. The one group that had a disagreement, only had a problem with one line of one of their logical connectives, so even they were mostly correct. Once the groups were done, I asked them to write down an explanation of why they felt like they were correct, and to check with other groups for agreement. There were some great discussions that happened during class, with students arguing about why there particular version was correct.

One of the reasons that I think this approach works is because I am making more use of the prior knowledge of students. Teenagers spend a lot of time arguing semantics and these types of sentences are frequently embedded in the language they use. So rather than ignoring their prior experience, I make critical use of it in this discussion approach. Further, the number of discussions and opportunities for students to draw connections between what they know already, and these new ideas is increased by the fact that they have more than just me to argue with over the logic. I’ve essentially taken an activity which has students talking in series (each of them taking turns to ask me questions, and to bring up their counter-arguments) to one that has them working in parallel (they talk to each other, and at one time, there are multiple students talking and discussing the point).

Here’s a quote from this class that I think sums up the experience nicely for me as well: "This is fun! I like trying to figure this out."

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