The law of the excluded middle, Part II

Those who would read the following should really have first read the first part, posted here on March 28th. I'm going to dive into the middle of material today without reviewing that. Ready? Let's go.

How have defenders of a middle choice (between true and false, EVEN for some well-formed propositions) defended that possibility against C.I. Lewis' critique?

Well, to begin, they note that any effort to prove something involves premises. To the extent that the law itself (we will call it LEM for short) is supposed -- as it is supposed by many -- to be a foundational premise in logic, it of course cannot be proven. If Lewis has actually proven LEM, it must have been by discovering other deeper premises, and turning LEM into a lemma, so to speak, a steppingstone on the path from some important truths to others.

In fact, the Lewis "explosion argument" takes it as a premise that every sentence entails the disjunction of itself and any other sentence. This is known as the principle of addition. It means that if we accept the proposition "the earth is spherical" we must accept the truth of the disjunction "the earth is spherical or unicorns exist" -- "the earth is spherical or the Packers went undefeated last season" and so forth. Since the earth is spherical, any sentence of that form is true.

This, the principle of addition, is what Lewis employs as a lever. If we accept any contradiction, then the principle of addition allows us to create such true sentences which in turn end up exploding the whole idea of truth, or of falsity.

But all this ensures is that anyone who really wants LEM as a firm part of a logical system might make the principle of addition a more fundamental premise and deduce LEM from it. It doesn't prove that we need either of them, or not in the direct form with which we've been working. LEM as I noted in our Part I is a useful heuristic and it might be well to keep it around in some modified form. Likewise with the principal of addition. If we modify one, we modify the other. But that is all that Lewis really proved here.

In circumstances in which we are tempted to say "it is both the case that A is true and that the negation of A is true" we ought to be careful about taking either half of that conjunction and joining it to a random disjunction of the form "A is true and B". With that understood, the creation of non-binary notions of truth and falsehood may proceed undaunted by the feared explosion.

At this moment, I should show some appreciation to the philosopher Jc Beale, whose communications have helped me clarify my thoughts on this matter. Thank you, sir.

One more brief point. There are arguments for "truth gluts" on the one hand, and there are arguments for "truth gaps" on the other. A truth gap is a situation in which some well-formed A is neither true nor false. There is no truth about A. A certain quantity of chin hair is both a beard and a non-beard. A truth glut is a situation in which some well-formed sentence is both true and false. There is a glut of truth about A. The hair is a beard and it is a non-beard. Some logicians who propound a non-binary logic are interested in the truth gluts, some in the truth gaps. The LEM as stated denies the possibility of either.

Professor Beale advocates for a logic with both gaps and gluts.

Examples? Aside from hairs that may and/or may not be a beard? Consider Zeno's paradoxes such as the movement of an arrow. It may well be both true and false to say that the arrow is exactly at point X at this moment. Indeed, accepting that truth glut may be a defining characteristic of motion in time.

Our Part III will bring political philosophy into the discussion.

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