Twenty-three centuries ago, the Greek philosopher Aristotle asserted "everything must either be or not be, whether in the present or in the future." This assertion became the central proposition of Aristotelian logic, which in turn became the dominant mode of reasoning in Western civilization.
Aristotle's logic is thus based on the law of the excluded middle, which holds that all "well-formed" statements are either true or false. It is difficult to overstate the impact this way of thinking about the nature of truth has had on Western culture. From the software of our computers to the structure of our legal systems to the design of our voting booths, our culture operates on the assumption that a thing either is or isn't the case: Every number in the software is either a one or a zero; the lawyer's argument either is or isn't the law; the voter either did or didn't vote for president of the United States.
This assumption is so deeply rooted in our thinking that even the act of treating it as an assumption seems superfluous or perverse: For us, the law of the excluded middle isn't anything so humble as a useful device to aid reasoning or a pragmatic fiction that helps reduce the otherwise unbearable complexity of the world. Rather, it is a fundamental (what philosophers call a "metaphysical") assertion about the way things really are.
The law of the excluded middle has been an enormously powerful tool in the development of — to name just three of its most obvious applications — science, law and politics. What reasonable objection could be raised against a logical law that has enjoyed such remarkable success in so many human endeavors? Only one: It isn't true.
This is not to say that the law of the excluded middle is false. (Such an assertion would plunge its asserter into a labyrinth of logical paradox from which there is no non-contradictory escape.) Rather, it is to say that, like everything else in this world, it is true only to some extent.
Another Greek philosopher, Zeno, proposed the following thought experiment. Begin with a heap of sand. Now remove one grain. Is it still a heap? Yes. Remove another grain. Is it still a heap? Yes. If you keep removing single grains then, eventually, the heap will no longer be a heap — but at what point? We can't say, because the border between what we recognize as a heap and not-a-heap is too fuzzy to be defined by the removal of any particular grain of sand.
This is called a "sorites paradox." Some variation on this paradox potentially troubles or infects all either-or propositions, which is to say all invocations of the law of the excluded middle. Consider voting for president. Is a completely punched-out ballot a valid vote? Yes. Is an untouched chad a valid vote? No. At the logical extremes, the law of the excluded middle works perfectly, which is another way of saying that the midpoint of any definitional system presents no problems for that system — as long as one excludes it from consideration. Any serious consideration of the pregnant chad must give birth to nothing less than a crisis of Western metaphysics, let alone of presidential politics. Whether dimpled, or two-cornered or swinging, the chad that both has and has not been punched by a citizen who has both voted and failed to vote for president confounds our logic as thoroughly as it confuses our legal system.
As the courts prepare to lose themselves in a labyrinth of logical, legal and political contradictions, it would do well to remember the poet's apocalyptic query: And what plump chad, its hour come round at last, Slouches toward Fort Lauderdale to be born?
Paul Campos is a professor of law at the University of Colorado. He can be contacted at paul.campos@colorado.edu.