The Paradoxes of Material Disjunction

In a previous post, I commented that truth-functional interpretations of conditionals are bothered by what are known as the "paradoxes of material implication". The problem arises because it is easy to form conditionals that, on the truth-functional interpretation, come out as true when, intuitively, they aren't true. What I want to point out now is that an exactly parallel problem afflicts truth-functional interpretations of disjunctions.

Recall that a disjunction is just an either-or proposition. Taking the "or" in the usual inclusive sense, a disjunction says, at a minimum, "Here's a set of options. At least one of these is true." For example, "Either the stoplight is red, green, or yellow" gives us a set of three options {red, green, yellow} and tells us that at least one of them obtains. It doesn't tell us which one obtains, nor does it tell us that only one obtains, but it does tell us that it's false that none of them obtains.

Now, on the standard truth-functional construal, a disjunction is true if and only if at least one of the options or "disjuncts" is in fact true. But it is not hard to form disjunctions for which this is counterintuitive. We'll call these the "paradoxes of material disjunction". Consider the following:

Either 2+2=4 or water is wet.
Either grass is green or the Eiffel Tower is in London.
Either the stoplight is red or it is green.

The problem with the first two is that the disjuncts are completely irrelevant to each other, yet each has at least one true disjunct. It would be very odd for anyone in normal discourse to utter either disjunction, unless they were being sarcastic or just plain silly. So when asked, "Is it true that either grass is green or the Eiffel Tower is in London?" most people would probably say, "Huh?" We might wonder, then, whether we should regard that disjunction as true. Perhaps we should conclude, instead, that it is meaningless and thus has no truth value. I'm not claiming to have shown that this is the proper response, but it seems at least arguably a plausible reaction.

The problem with the last one is that the disjuncts are obviously incomplete—the stoplight could be yellow (or off, for that matter). Now here I think that many people would respond by saying that "Either the stoplight is red or it is green" is just plain false. It's not hard to imagine a person saying, "Not necessarily. It could be yellow." What this suggests is that what makes a disjunction true is not simply that one or more of the disjunctions is in fact true, but also that there be no other (relevant) alternatives. In other words, "Either p or q" is true iff at least one of the alternatives must be true. We can accommodate this in parallel with C.I. Lewis's 'strict implication' by introducing a notion of 'strict disjunction'. On this construal, a disjunction is strictly true iff necessarily, at least one of the disjuncts is true.

Like strict implication, strict disjunction is not truth-functional. It does a good job of squaring with our intuitions on the latter two disjunctions above: both come out as false. It does not, however, explain our puzzlement with "Either 2+2=4 or water is wet." The latter comes out as true according to strict disjunction because 2+2=4 is a necessary truth, but it still sounds odd.