p --> q
p
-------
therefore, q
(Law of Detachment)
p --> q
~q
-------
therefore, ~p
(Law of Contrapositive Inference)
I tried 2 or 3 or 1000 different ways of explaining it, and nothing stuck. I thought I hit gold with the following:
If the Dog is a Terrier, then the CHihuahua likes MENTos.
The Dog is a Terrier. (I thought this would help them connect it to detachment).
and
If the CON is a TRAitor, then I POSsibly like TV.
I hate TV. (for contrapositive inference).
So that helped some kids, but wasn't the raving success I thought it would be. Then just the other day, we were going over problems in tutoring, and I was circling things to make a point, and saw this:
Do you see it? The Detachment scribbles look like "D", and the Contrapositive scribbles look like "C". Well. They thought that was SO cool, and then they started accusing me, "why didn't you just show us this in the FIRST PLACE!". Like I was secretly storing away the "good stuff" and doing things the "hard way" until they finally wore me down to giving up the goods. Is this on ALL their minds: teachers always do things the hard way and hold back on us. Phwueey. I told them I just thought of it, and there was nothing to prevent them from figuring things out on their own and stop bothering me kid. .... They also passed their retests with flying colors.
Naming the rules of inference is one of the most useless things to do in a logic unit. I'd like students to be able to make correct inferences, but I could care less about whether they remember the completely arbitrary names associated with them. (You can reduce them all to just one rule: early theorem-proving programs just used "resolution".)
ReplyDeleteI think you are putting the effort into the wrong part of the unit.
Here's the reason I make them memorize the laws. They work up to doing flow proofs with logic statements and then flow proofs with geometry concepts, and I need them to know how the how proof goes together: you need statements and reasons/justifications of why you can make the claims you do in your proof. So I make them memorize the laws of inference to use as their proof reasons for multistepped proofs. Voila! entryway into other proofs.
ReplyDeleteReal problem here is that the names are not standard, so a lot of effort is going into remembering the special names your text uses for modus ponens and modus tollens.
ReplyDeleteHmmph. Even those names, old-fashioned, might not be universally recognized.
The most common texts (until recently) in New York State used "Law of Detachment" and "Modus Tollens". My hyper-traditional text and Discovering Geometry, at the other extreme, both use both Latin terms.
The methods course I took that covered some of this (10 or so years ago) used two different names, both English, neither "detachment"
Up to me? I'd explain what happens: "affirm the hypothesis" (or "affirm the antecedent") would be one form, and "deny the conclusion" would be the second. It is, after all, the forms that are immutable. The names range from whimsical to confusing to just strange.
(I know, I know, it's not really up to me. So I teach 'em my way first, and then I reveal the mystical names when the unit ends).
Jonathan
I agree with you that the names are random. I guess I'm more concerned with them learning to memorize something and attaching it to a meaningful implication in a proof, and knowing when to recognize and use which reason.
ReplyDelete