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)].
An complementary strategy: Use it as an occasion to discuss the arbitrariness and artificiality of logical systems. Then show why taking the material conditional as true when P is false is *useful for certain purposes*, e.g., those evident in steps 2-4. As an extra credit assignment, invite students to formulate some consequences of an alternative logic that treats the material conditional as false when P is false.
ReplyDeleteOh, of course. That's totally the way to go.
ReplyDeleteOne thing that's worked for me is appealing to rule-following intuitions. Take "if you drink you must be 21." The rule can't be broken unless you drink and you aren't 21. Otherwise, the rule has been followed.
ReplyDeleteHi Dan,
ReplyDeleteIt's good to hear from you. Hopefully, I see you at Brendan's and/or CUNY in the near future.
Any way, I've found that sort of thing to be quite helpful, too. I've been inspired based on the psychological results regarding the Wason cards and the differential effects of social and non-social content in the conditionals. I even discuss that a bit with the students.