08/27/2012 03:42 pm ET | Updated Oct 27, 2012

The Curious Use of Language in the Lance Armstrong Decision

Did Lance Armstrong dope or use blood transfusions during his professional cycling career? I have no idea. Nor, it appears, does anyone else except for Lance and perhaps a few members of his team. But as a mathematician with expertise in the use of language in reasoning, I find the much-touted central pillar of the United States Anti-Doping Agency's case against him does not stand up to even a cursory examination.

Apart from hearsay evidence from two disgraced former cycling teammates of Armstrong, the USADA bases its case (at least according to what they have said) on the blood and urine samples taken from the cyclist in 2009 and 2010, when he made a brief comeback to the sport after four years in retirement. In a June letter to Armstrong, subsequently made public, the USADA said those samples were "fully consistent with blood manipulation including EPO use and/or blood transfusions."

Though a recreational cyclist, my interest in this case is fairly minimal. It is that term "fully consistent with" that piqued my mathematician's interest. It is a very odd phrase to use in a situation like this, not least because it has absolutely no evidentiary force. It says nothing of any significance.

[Certainly, after two years deliberation, including testimony from former team-mates obtained under oath through a grand jury, the U.S. federal criminal investigation of the allegations made against him finally dropped the case early this year, saying there was no real evidence against him.]

Though the layperson typically thinks of mathematicians as being focused on numbers, that is actually not the case. That false view is a consequence of the mathematics taught in high school. Only at university are you likely to encounter the mathematics done by the professionals. High among our real areas of expertise are logical reasoning, rigorous proof, and the precise use of language.

Incidentally, I am not referring here to using language and reasoning precisely in esoteric discussions of arcane mathematical topics. Yes, we do that too. But we also apply our expertise in everyday, practical domains. (Homeland Security, to name one domain I myself have worked on.)

There are a number of terms we use to describe evidence. The strongest is "proof" (or "conclusive proof", but any mathematician will tell you the adjective is superfluous.) We might say that, "Evidence X proves that Y happened."

An alternative that might seem weaker, but in actuality is not, is that "Evidence X implies that Y happened."

Definitely weaker, is "Evidence X suggests (or indicates) that Y happened."

All of these have evidentiary power of differing degrees. And there are others.

At the other end of the spectrum, we can say, "Evidence X contradicts Y having happened." X proves Y did not occur.

Evidence collected to uncover wrong-doing, such as doping controls in sport, by virtue of their design, rarely (if at all) provide proof of innocence. At best, when a doping test does not come up positive, the most you can say is it did not yield proof. It does not rule out (i.e., does not contradict) doping, just as a negative result from a cancer screening does not mean you are cancer free, merely that the test did not detect any cancer.

So what does that USADA term "fully consistent with" mean? Well, first of all, let's drop the "fully"; it's superfluous. Consistency is a definitive term. Something is either consistent or not; no half measures. It's also a term mathematicians like myself are very familiar with -- again for real world uses as much if not more than within theoretical mathematics. It means "does not contradict". Nothing more, nothing less.

Given the availability of terms such as "proves," "indicates," "suggests," or more evocative terms such as "raises the distinct possibility that," why did the USADA decide to use the curious term "consistent with"? Since they surely spent a lot of time, and consulted with a number of lawyers, in drafting their letter, their choice of wording was clearly deliberate. Why choose a term that means "does not contradict"?

After all, I can say "Drinking milk as a child is (fully) consistent with using crack cocaine as an adult." Should we take that as evidence that milk producers are to blame for adult drug use? Of course not. But this example has exactly the same logical heart, and the same evidentiary force, as the USADA letter's "fully consistent with blood manipulation including EPO use and/or blood transfusions."

Why not say "suggest" or "indicates"? They fall well short of "proof", but they do carry some weight.

"Does not contradict" is, then, it appears, a key part of their case against Armstrong. In which case, I find it troubling. The USA should have far higher standards of proof than that.