The law of the excluded middle, Part I

This is the first in a projected three-part series about the law of the excluded middle, a (disputed) logical principle according to which, for any well-defined proposition, either the proposition is true or it is false. Today I will explain the law in question and offer a classical argument for its validity. On another day (not tomorrow, I assure you): I will explain why I do not believe that argument is a strong one and why we might want to allow for violations of this LEM. When I get around to Part III, I will discuss how the dispute feeds into political philosophy and certain questions about the legitimate regulation of markets.

So ... let us proceed.

When it seems that a proposition IS both true and false, or is neither, and in either case is in violation of this law, various remedies are available: the proposition in question might just be nonsense (as for example the claim that the Jabberwocky has a frumious Bandersnatch), it might embed a false premise (as for example the premise that France has a contemporary King who might either be bald or not), or it may be ambiguous (as for example the claim that days are 24 hours long).

Sticking with that last example for a moment: the word "day" in English can refer either to the whole of a planetary rotation, as from one midnight to the next, or it can refer to the period of light from sunrise to sunset. In the one case, our above proposition is true, in the other it is false. Thus the old scholastic saying, sometimes quoted by William James, "if you think you have found a contradiction you must make a distinction."

Allowing for all of that within the adjective "well-defined," none of those examples directly challenges the law of the excluded middle.

There is no doubt that the law -- or rather the presumption -- of the excluded middle can be a useful heuristic pressing us to review premises and define terms -- still, IS it a law? An ironclad law? Why should we decree it so?

C.I. Lewis, a great mid-20th-century philosopher whose Ph.D. supervisor was Josiah Royce, offered an argument for excluding any middles. His argument has come to be known as the principle of explosion. The idea is that if we accept even one seemingly innocuous contradiction into our worldview, everything explodes, because literally ANYTHING can be true.

The proof -- as applied to the seemingly harmless contradiction "all lemons are yellow" and "not all lemons are yellow" goes roughly like this:

1. Not all lemons are yellow -- by assumption.

2. All lemons are yellow -- also by assumption.

3. The two-part disjunction "all lemons are yellow or unicorns exist" must be true, since the word "or" means that it doesn't depend on unicorns and we have presumed lemons are yellow.

4. Since we also presumed that not all lemons are yellow we have to infer that the first part of the disjunction is false, so in order for the whole disjunction to be true, unicorns must exist.

5. In case you are fine with the inference that unicorns exist, consider that procedure can be repeated for any second statement, i.e. "all lemons are yellow or the Green Bay Packers went undefeated in the 2024-25 season." Thereby proving the truth of anything we wish to put into that second spot of the disjunction. I have it on good authority the Packers lost six times that season.

6. This explosion of absurd "truths" is intolerable, so we must reject the proposition that both (1) and (2) above can be true. Which is what was to be proved.

But perhaps we can save our double assumption about lemons without either re-writing the Packers season or discovering unicorns. Maybe? More later.

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