Those who have taught a subject for many years have occasionally had the blessing of teaching a student who naturally has (or has developed) an intuitive insight into the subject. I was blessed this way by two of my logic students this morning!
I was teaching Intermediate Logic, Lesson 34, “Converting Truth Tables into Digital Logic Circuits.” We were going over the exercise, problem 3, for which the final answer in the text is the unsimplified proposition (~A • ~B • C) v (A • ~B • C). One of my students said, “I know that B has to be false and C has to be true, and it doesn’t matter about A. So that would simplify to ~B • C.” When I asked him if he has been reading ahead to future lessons, he said no, “I can just see it by looking at the proposition.” I was very impressed. He just saw future lessons on simplification techniques, without being taught them.
But the surprises for the morning were not yet over.
We went on to problem 4, for which the answer was (~A • B • ~C) v (A • ~B • ~C) v (A • B • ~C). I said, “This proposition can be simplified, significantly, but it’s a little more complicated now. Who can draw this circuit?” Another student said, “Can I just draw what I got?” I asked, “Did you get a different proposition than this?” He said, “It looks like the same thing, but I had it simplified.” When I asked him how he simplified it, he responded, “I pulled the ~C out, and then had A v B. I was left with (A v B) • ~C.” I looked over it, saw that he was right, and asked, “How did you come up with that?” He humbly replied, “I just figured, well, A or B had to be true and C had to be false in order for the proposition to be true.” Once again, this student just saw a way to simplify the proposition, without anyone teaching him how. I did not have anything like that insight into simplifying propositions until after studying them in detail in order to write the textbook. But this student just had the logical intuition to know what to do, and only so that he didn’t have to draw the ridiculously complicated circuit that the text asked for.
What a pleasure to teach students who understand logic better than I do!