Friday, February 10, 2006

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.


At 2/12/2006 10:32 AM, Blogger Macuquinas d' Oro said...

Alan (if I may),

If you teach Intro Logic, think about doing some testing of your student’s inferential intuitions at the beginning of the course. ( Perhaps you already do )

When I would offer
“The butler did it. Therefore, either the butler did it or the maid did it”, with the possible evaluations 1. correct or 2 . incorrect,
I would get virtually complete rejection of the inference.
The only exceptions were people who had previously had some formal sentential logic. (That began to raise for me the question of whether studying truth functional Sentential Logic had really improved those people’s understanding of inference in natural languages.)

The same result obtained with “The butler did it. Therefore, if the butler didn’t, the maid did.” I think in the ten years that I did some kind of entry testing, I found two students who approved that inference.

When I would offer, on the supposition that it is known to be true that the butler did it, the statement “either the Butler or the maid it”, with the possible evaluations 1. true 2. false 3. nonsense/meaningless
I would get a distribution of replies favoring false, then true, then meaningless. More than 4 out of five thought it had truth value (t or f).

When I offered on the same supposition “Either the Butler did it or 2+2=4”, the distribution skewed sharply in favor of nonsense, with a few favoring true. Those who called it true seem to think it a jokingly emphatic way of saying the butler did it. As if it meant the same as “either the butler did it or 2+2 don’t= 4!”
The majority, though, thought the sentence a completely nonsensical attempt to state a pair of alternatives.

None of this, I suspect, is surprising to you. Does it also begin to hint the strict disj (and Lewis modal systems) are not going to be any more useful than truth functional disj in explicating disjunctive inference in ordinary language?

At 2/12/2006 7:18 PM, Blogger Alan Rhoda said...

Hi Oudeis. (It's Phil, right?)

I do teach an intro to critical thinking course. Mostly informal logic. I do talk about conditionals and disjunctions, but I don't even attempt to persuade them to view those truth-functionally. I prefer an approach that stays closer to natural language when possible.

Your examples are good ones for illustrative the ways in which truth-functional sentential logic departs from our commonsense intuitions.

I do think that modal logic gets things closer, but (as I'm sure you know) just as there are the so-called paradoxes of material implication, there are also paradoxes of strict implication--cases where the antecedent is either a necessary falsehood or the consequent is a necessary truth--that yield intuitively wrong results.

Relevance logics strengthen the conditional even further, but the difficulty they face is that relevance is terribly hard (I suspect impossible) to explicate in purely formal terms. That's because relevance is an inherently intensional notion. Since it's inception with Aristotle, logic has been primarily extensional--applied set theory, if you will. Truth-functional logic works fine as long as you're operating in purely extensional contexts, as mathematicians typically do. But natural language, it seems, cannot be captured in the language of sets and classes. Modal logic partially accommodates by introducing higher-order sets (i.e., possible worlds), but ultimately keeps everything purely extensional--and, as D. Lewis has shown, it's possible to translate modal logic into truth-functional sentential logic by extending our domain of quantification to cover possible worlds.

The way I sum it up is this: If you want a logic that stays as close as possible to our commonsense intuitions, go with relevance logic. If, however, you're working in a purely extensional context or you merely want a logic that will give you an effective (no false negatives) test for invalidity, then truth-functional sentential or modal logic is the way to go.


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