Is the "Law of Excluded Middle" Really a Law?

The so-called "Law of Excluded Middle" (LEM) is often taken to be one of the most fundamental laws of logic. It may be expressed as follows:

(LEM) For any proposition p, either p is true or p is not-true.

LEM is to be distinguished from the Principle of Bivalence (BV), which states:

(BV) For any proposition p, either p is true or p is false.

BV entails LEM, but the converse does not hold because "not-true" may not be equivalent to "false" (That they are equivalent is what BV claims. LEM makes no commitment on the matter.)

Anyway, it occurs to me that LEM works whenever things admit of sharp classification into either this or not-this, and fails insofar as things do not admit of sharp classification. In other words, LEM works insofar as things are precise and fails insofar as they are vague. To say that a concept is vague is to say that it has fuzzy boundaries, that there is a penumbra or "grey area" in which it is simply not clear whether a given item should fall inside or outside the boundary, or whether it should simply stay on the boundary.

For example, under normal conditions a given lightbulb must be either ON or OFF, with no in-between state. Consequently, in this case we have an instance where LEM applies. If p is "This lightbulb is on" then p is either true (the lightbulb is ON) or it is not true (the lightbulb is not-ON, i.e., OFF). Contrast that example with a man who has lost most, but not nearly all, of the hair on his head. Is he bald or not-bald? Well, it may not be so clear. Depending on how his remaining hair is distributed across his scalp we might lean more toward one answer or the other, but this case just isn't as sharply defined as with the lightbulb. Perhaps we should say that he's partly bald and partly non-bald or bald in some places and non-bald in others.

Anyway, I think my point is clear that vague cases create difficulties for LEM. The crucial question is whether all such cases merely pose difficulties in the application of LEM, or whether some of them reflect a genuine limitation on LEM itself.

To say that LEM ought to hold always and everywhere is tantamount to saying that reality is perfectly sharp and determinate all the way down and that vagueness is always merely epistemic, a function of the inexactness of our language and of our concepts. This is not obviously correct. In fact, there is evidence from physics that when we go down far enough things really do get "fuzzy" - think of wave-particle duality and such. Now this fuzziness may in the final analysis turn out to be merely epistemic, but it's going quite a ways beyond the available evidence at this point to suppose that this must be the case.

Continuity is another issue where LEM naturally comes into question. Our natural conception of space, time, and motion is that they are smoothly continuous and that points, lines, events, and such are interruptions of sorts in an already existing continuum. The arithmetization program in mathematics, however, proposes to analyze such continua into sets of discrete points. If we adopt this latter course as being the metaphysical truth of the matter, then we can apply LEM to such continua all the way down. But this is a large metaphysical assumption to make. It essentially amounts to saying that continua are not metaphysically basic and can, therefore, be reductively analyzed in terms of discrete non-continua (i.e., points). Conversely, one who takes our ordinary conception of continuity to be closer to the metaphysical truth of the matter - in other words, one who holds that continuity is metaphysically more basic than discontinuity - thereby commits himself to a denial of LEM at the continuum level.

One can't have it both ways. If LEM holds across the board, then continua cannot be metaphysically fundamental. If continua are metaphysically fundamental, then LEM can't hold across the board. Since I am partial to the latter view, I say that LEM is not really a law.