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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4 Comments

  • I’ve never learned symbolic logic and am puzzled (in a good way) by the last line of the P -> Q truth table. Would a sample sentence for P -> Q be “If I go to the movies then I eat dinner”?

    The first line is clear enough… if I go to the movies and I eat dinner then I certainly haven’t disproved P -> Q
    The second line is clear too… if I go to the movies and I don’t each dinner I have definitely disproved P -> Q
    The third line is a little more confusing… if I don’t go to the movies and I eat dinner, why couldn’t P -> Q still be true? Does P -> Q mean that the only way Q can happen is if P happens? Why would that be P < -> Q?
    The fourth line is especially puzzling… if I neither go to the movies nor eat dinner, then how do we know anything about whether going to the movies relates to eating dinner? Couldn’t there be an “undetermined” in the truth table?

    It reminds me of a conversation I had as a geometry teacher. I told my students that if I won the lottery I would buy each of them a car. They said I was lying. I insisted I was telling the truth. Eventually one of them said, “he’s never going to play the lottery, so he can say that and it’s not a lie even though we’ll never get the cars.”

    Any help you or your students might be able to offer will be greatly appreciated!

  • David Wees wrote:

    The last two lines of the truth table are generally puzzling to my students as well, so I always frame it in the following way:

    If I say I’m going to do something, then I will do it.

    First line: If I say I’m going to do something, and then I do it, I’ve not lied so hence the statement in bold is true.

    Second line: If I say I’m going to do something, and then I do it, I’ve lied, so the statement in bold is false.

    Third line and fourth line: If I don’t say I’m going to do something, it doesn’t matter if I do it or not. If I do it, it’s a pleasant surprise, and if I don’t do it, no one can accuse me of making a false statement. So in both of these cases the statement in bold is still true.

     

  • David Wees wrote:

    It’s probably useful to point out in this discussion that I had an error in my truth table, and that I’ve fixed it. I have no idea why I put those two Fs in the table, it must just have been some sort of brain freeze.

  • It’s good to know that I trusted my own sense-making enough to ask good questions. I didn’t mean to find typos, just to figure out what was going on. Thanks.

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