Who's Counting: Alternative Voting Methods and Mitt Romney's Mathematical/Political Gaffes

The large number of candidates running for president in both parties splinters voter support. Two unfortunate consequences of this are that good second-tier candidates often quickly fall by the wayside and that not so impressive first-tier candidates are anointed early by the prevailing poobahs and pundits.

Alternative Voting Methods

A partial solution to the first problem of losing good second-tier candidates prematurely is to use a method different than the standard plurality way of determining winners in the various primaries and caucuses. There are many.

Voters might, for example, rank their favorite candidates, giving, say, three points to their first choice, two to their second, and one to their third, and the one with the highest point total would be the winner. In this way voters could give support to both Obama and Clinton, say, or indulge their secret liking for Ron Paul.

Alternatively, voters might vote for as many of the candidates as they wish and the one with the highest approval percentage would be the winner. The principle of "one person, one vote" might be replaced with "one candidate, one vote." Scenarios in which, for example, two liberal candidates split the liberal vote, say 32 percent to 28 percent, and allow a conservative candidate to win with 40 percent of the vote would not develop. This method might favor consensus candidates and work against polarizing ones.

Yet another method would have the voters rank the candidates -- their first, second, third, fourth choices, etc. -- and if none of the candidates received a majority of the voters' first-place votes, the candidate with the fewest first-place votes would be eliminated and the votes adjusted upward according to the original rankings. This would be repeated until a candidate obtains a majority of the adjusted first-place votes. The method provides, in effect, an instant runoff election and is used in various municipalities from Minneapolis to San Francisco.

All methods have their flaws (in fact, a mathematical result known as Arrow's theorem says so), but some are usually better than others, especially in multicandidate races.

The Null Set, Non Sequiturs and Mitt Romney

A partial solution to the second problem of the early anointing of front-runners is radical. It is to focus on the actual content of the candidates' pronouncements and not on their hair, posture, jaw lines or rankings in the polls.

Let me step back and first note the definition of two elementary terms from formal logic.

The null set is a notion common in mathematics and is generally taken to be a collection having no members, an empty set. The set of humans who are over 12 feet in height is a null set as is the set of square circles as is the set of even prime numbers bigger than 2. Occasionally, it is used to indicate a set that is sparse in some other way. But, in any case, the empty set is not a difficult notion and is employed widely in mathematics and formal logic.

A non sequitur is an argument in which the conclusion does not follow from the assumptions. A simple example is 1. If I'm in Bangkok, then I'm in Thailand. 2. I'm in Thailand. 3. Therefore, I'm in Bangkok. Conclusion 3 does not follow from assumptions 1 and 2. Occasionally a statement lacking any relation at all to statements preceding it is deemed a non sequitur, but, as with the null set, it is not a difficult notion and is widely used in mathematics, formal logic and informal discourse.

I bring up these bits of elementary logic here because of the answer Republican presidential candidate Mitt Romney gave to a question posed by Wolf Blitzer during the Republican candidates' debate last month.

The question was: "Knowing everything you know right now, was it a mistake for us to invade Iraq?"

Romney's answer was, "Well, the question is, kind of, a non sequitur, if you will. What I mean by that -- or a null set -- that is that if you're saying let's turn back the clock and Saddam Hussein had opened up his country to IAEA inspectors and they'd come in and they'd found that there were no weapons of mass destruction, had Saddam Hussein therefore not violated United Nations resolutions, we wouldn't be in the conflict we're in. But he didn't do those things, and we knew what we knew at the point. We made the decision to get in."

Not satisfied, Blitzer reposes the question, to which Romney replied, "Well, I answered the question by saying it's a non sequitur. It's a non -- null set kind of question, because you can go back and say, 'If we knew then what we know now,' by virtue of inspectors having been let in and giving us that information, by virtue of if Saddam Hussein had followed the U.N. resolutions, we wouldn't be having this discussion. So it's a hypothetical that I think is an unreasonable hypothetical."

Pretentious or Obfuscatory, but Wrong

Romney was either being pretentious by using terms he didn't understand in order to impress his audience or he was being intentionally obfuscatory because he didn't want to answer the question. Either way, his response, despite his being a first-tier candidate, does not portend a candidacy distinguished by candor or eloquence.

But the deeper problem with his answer was that it was clearly wrong. Saddam Hussein did allow U.N. inspectors into Iraq, they did not find any WMDs, and they were ordered out by President Bush while they were still looking. Bush's unwillingness to give the inspectors a few extra months of time (presumably Romney would have agreed) contrasts starkly with his willingness to give the war effort years and years of extra time.

Romney's gaffe brings to mind former President Ford's ill-informed remark during his debate with Jimmy Carter in 1976 that Poland was a free country not dominated by the Soviet Union.

In any case, more focus on the content of the candidates' words and a primary voting method more suitable to races with many candidates might improve our presidential electoral process.

Sticky rice and mango taste great. That's a real non sequitur.

John Allen Paulos, a professor of mathematics at Temple University, has written such bestsellers as "Innumeracy" and "A Mathematician Plays the Stock Market." His "Who's Counting?" column on ABCNEWS.com appears the first weekend of every month.