Monday, March 29, 2010

Disjunction introduction

The following argument is valid by disjunction introduction:

  1. I won't play the lottery.
  2. I won't play the lottery or I'll win the lottery.
But the conclusion sounds wrong.

Now, here is a hypothesis. The wrong-sound of the or-sentences corresponds precisely to the appearance of falsity in the conditional:

  1. If I play the lottery, I'll win the lottery.
Now if the indicative conditional is the material conditional, then (3) is true assuming I won't play the lottery. This is one of the standard objections to taking the indicative conditional to be a material conditional. But I think this objection is weakened if (2) also sounds wrong, since (2) is the material conditional analysis of (3). If (3) sounds wrong, and (2) is the analysis of (3), then if one of them sounds wrong, so should the other. And that's what we observe.

The following would be an interesting test. Consider this argument:

  1. I won't play the lottery.
  2. So: I won't play the lottery or I'll win the lottery.
  3. So: If I do play the lottery, I'll win the lottery.
Where would ordinary folks situate the apparent fallacy? Would it be at step (5) or at step (6)? I don't know. But if only at (5), then the material conditional analysis of indicative conditionals is not challenged by the apparent wrongness of (3).

On the other hand, (3) sounds false to ordinary folks (unless the lottery is rigged!), but I don't know if (2) sounds false to ordinary folks.


Alexander R Pruss said...

Here's a variant--and more compelling--argument.

I am about to lose a chess game to a grandmaster. I decide to quit. (Note: Below, I distinguish losing from conceding.)

1. (Premise) I will quit.
2. I will quit or I will win. (Disjunction introduction)
3. If I don't quit, I will win. (Material conditional introduction)

Most of the students in my upper level metaphysics class were suspicious of step 2, but were OK with step 3. As a colleague pointed out to me, we can expect ordinary speakers to object to 2 by saying: "That's not right! You will quit or you will lose." He, though, objected to step 3.

Douglas said...

Perhaps your upper level metaphysics students fault step 2 because their use of disjunction-introduction is constrained by some rule like this: Whenever you have A, introduce 'A or B' only if B would have held if not-A had held.

Alexander R Pruss said...

Good suggestion. I suspect that people sometimes hear "p or q" as "if not p, q".

Perhaps when we teach logic, we should paraphrase "p v q" not as "p or q" but as: "At least one of the following is the case: p, q"? A colleague mentioned that he routinely has logic students who question the propriety of disjunction introduction.

With this paraphrase:
1. (Premise) I will quit.
2. At least one of the following is the case: I will quit, I will win.
3. If I don't quit, I will win.

I conjecture that people will accept the inference of 2 from 1. The inference of 3 from 2 is harder, because the "At least one of the following" phrase is unusual, but I conjecture they'll accept the inference after a bit of thought. And then I conjecture that they'll be really puzzled.

Douglas said...

Disjunction-introduction always meets resistance from some of my logic students too. Your paraphrase of 'p v q' would, I think, decrease that resistance. The inference of 2 from 1 is easy to see now. I agree that students will appreciate the inference of 3 from 2 given some time.

The benefit of your paraphrase is that it reveals the truth-functionality of 'p v q'. Perhaps we should teach both your paraphrase and the usual textbook one (to motivate the project of formalization).

khomus said...

I've been talking about this in another forum, concerning the ability of contradictions to prove anything. I say they don't, because I distrust DI. Consider:

Either 2+2=4 or 3+5=8

We know both of these to be true.

Either 2+2=4 or 3+5=17

This is true because 2+2=4, thus satisfying a condition of DI.

Either 2+2=4 or 3+5=17

I surely haven't proven that 3+5=17. But this is precisely what supposedly let's us prove anything from a contradiction, thus:

2+2=4 and 2+2=5
Either 2+2=4 or 3+5=17 (true because 2+2=4)
Either 2+2=4 or 3+5=17 (2+2 is false because 2+2=5, because P and !P are true)
Thus, 3+5=17 is true, via DI

But that seems obviously wrong, because it was wrong when we had no contradiction whatsoever. One can multiply examples indefinitely, the Wikipedia example is: Socrates is a man. Either Socrates is a man or pigs are flying in formation over England.

Just because I say this, one of these two options is true, I don't see how it follows that if we suddenly discover Socrates to be a god, let's say, that it follows that pigs are in fact flying in formation over England. I guess I don't see any necessary relation between the two terms of the disjunction. It seems obvious that one or both CAN be true, but it does not seem at all obvious that one or both MUST be true.

khomus said...

Sorry, I was cut off. I realize this isn't precisely what you're discussing, but if you could enlighten me, I'd greatly appreciate it. I'm just a bumbling idiot with an interest, so I've surely gotten something wrong, but beats me what it actually is.