I tend to take violations of logic personally, almost viscerally. It's a reflex, a mental gag. So I was less than ecstatic as I was grading my students' attempts at basic proofs in voting theory.
Voting theory (a.k.a. social choice) provides a great opportunity to demonstrate key features of mathematical reasoning to students who otherwise flinch at the very mention of math. It works especially well in an election year.
It demonstrates mathematical modeling. For instance, the preference model specifies that each voter can rank all candidates according to her preference, from most preferred to least. That sounds reasonable, although I dare anyone to even name all the presidential candidates, let alone rank them. A more complicated model allows the voters to distinguish how much they would prefer one candidate to another.
Voting theory demonstrates that the choice of procedure matters. As Stalin said, “It's not who vote that counts, but who counts the votes.” With preference ballots, where voters list their preferences from most to least, there are many reasonable procedures to determine the winner, and in contested elections different procedures announce different winners. Such examples allow for discussions about the very nature of democracy. Can reasonable people ultimately disagree on their preferences? Do some voting procedures capture the will of the electorate better than others? By what standards?
Voting theory demonstrates the value of precisely defined modeling. One can state precisely what one would like of a voting procedure. Some common criteria of an ideal voting procedure:
- If a candidate has more than a half of first-place votes, that candidate should win. (Majority)
- An ideal voting procedure shouldn't allow spoilers---candidates who themselves are not winners, but whose presence or absence changes the outcome. (Independence of Irrelevant Alternatives)
- If there is a candidate who beats every other candidate in a head-to-head comparison, that candidate should win. (Condorcet)
- If everyone would rather have candidate X than candidate Y, Y shouldn't win. (Pareto)
Precisely defined model allows for a logical exploration and clear statements which can be logically demonstrated, like:
- The only voting procedure based on ranking which guarantees to satisfy both the Pareto and the Independence of Irrelevant Alternatives criteria is a dictatorship. (Arrow's impossibility theorem)
I guess that says something about democracy.
I assigned less dramatic statements as homework: “Does such-and-such method guarantee to satisfy such-and-such criterion?” We did quite a few of those in class. The requisite proofs are direct and simple (although “simple” is not the same as “easy”), and rest primarily on the following basic tenet of logic:
- One example does not prove the rule, but one counter-example is sufficient to disprove it.
- To prove that a criterion is guaranteed, where the criterion is of the “if (premise) then (conclusion)” form, one assumes the premise and demonstrates that the conclusion follows.
Do I really need to recount the specifics of the painful experience of reading my students' work? Turning each page, not knowing whether a fresh violation of logic awaits?
I've seen it all. Proof by example: “Runoff procedure guarantees Majority criterion, because here's an example where it does.”
Failing to assume the premise: “Head-to-head comparison procedure doesn't guarantee the Condorcet criterion, because here's an example where there isn't even such a candidate.”
That, and failing to read the criterion: “Runoff procedure does not guarantee the Pareto criterion, because here's an election where no candidate has the majority of first place votes.”
Misusing Arrow's theorem: “Only dictatorship satisfies both Pareto and Independence of Irrelevant Alternatives, so Borda can't satisfy Pareto, because it's not a dictatorship.”
Failing to even read Arrow's theorem: “Borda method does not guarantee Pareto, because Arrow's theorem says that only dictatorship guarantees Pareto.”
Not to mention, vague statements like “Runoff method doesn't guarantee Condorcet, because I guess it could happen...”, which is more of logic harassment than rape.
Now I wonder, how much of this is because the context is outside of their common experience, and how much of it is really a problem with logic? When I give a similar logical violation in the more familiar context, none have a difficulty with it. Yet that is one of the hall-marks of a rational person, that she does not abandon logic no matter the context.
Then again, how much of it comes from pure neglect, not caring enough about one's work to make sure it's sound? Not so much a rape of logic, more like abuse through negligence?
Either way, it takes me a few moments to remember not to take such horrors to heart. I gave the students an option to improve their fledgling attempts at proofs, which most of them will take since they tend to be very grade-conscious around here.
And they will not pass my course without demonstrating some basic respect towards logic.
