Tuesday, September 15, 2009

The Monsters of Logic, Part Two

It's the time of the semester to think about this problem again: how to get students with only a few logic lectures under their belts to warm up to the very unintuitive truth-table for the material conditional. Here's my current favorite strategy:

Step 1: Point out the obviousness of holding P->Q true for true P and true Q, and holding P->Q false for true P and false Q.

Step 2: Point out the redundancy generated between conditionals and conjunctions if P->Q is held to be true only when P and Q are both true; and the redundancy generated between conditionals and equivalences when P->Q is held to be true whenever P and Q both have the same truth value. It's clear, at this point, that P->Q is going to need to be true when P is false and Q is true; the remaining question is what's best when P and Q are both false.

Step 3: Point out that if P->Q is held to be false when P and Q are both false, then we don't get to recover the intuitive equivalency between P<-->Q and [(P->Q)&(Q->P)]. Demonstrate with truth tables.

Step 4: Point out that if P->Q is held to be false when P and Q are both false, then we don't get to hold valid arguments as equivalent to tautological conditionals (which is a pretty sweet thing to be able to do). Demonstrate with truth tables for simple examples such as simplification [(P&Q)->P] and addition [P->(PvQ)].