## Monday, June 13, 2011

### Disjunction introduction and conditionals with disjunctive antecedents

[Note: In the original version of this post, I made the embarrassing false claim that relevance logic denies disjunction introduction. This claim will explain Brandon's and my exchange in the comments. I have since edited the post.]

Consider this argument, a version of which I've already discussed:

1. I won't write a blog post today mainly on French cooking.
2. Therefore: I won't write a blog post today mainly on French cooking or tomorrow the world will come to an end (or both).
3. Therefore: If I write a blog post today mainly on French cooking, tomorrow the world will come to an end.
Premise (1) is true. Conclusion (3) sounds false. There are a couple of things that one can do about this odd argument. One can embrace the conclusion but insist that the conditional is only used materially, and is trivially true because the antecedent is false. One can—and I think this is going to be the most common reaction among philosophers—reject the inference of (3) from (2). But a lot of ordinary people will balk at (2)—the disjunction introduction step, where from p, we conclude p or q for any q.

Denying disjunction introduction neatly undercuts the above argument, as well as removing the oddity that everything can be proved from a contradiction.

But blocking disjunction introduction is a mistake, because we need disjunction introduction. Suppose that we say:

1. One has committed a violation of a school safety zone if one is (a) driving a motor vehicle in a school safety zone and (b) talking on a cellphone or driving at more than 20 miles per hour or both.
Now suppose:
1. Sam is driving a motor vehicle in a school safety zone.
2. Sam is talking on a cellphone.
We obviously want to conclude that Sam has committed a violation of a school safety zone. But to do that with modus ponens, we need to establish that the antecedent of the conditional in (4) is true for Sam:
1. Sam is (a) driving a motor vehicle in a school safety zone and (b) talking on a cellphone or driving at more than 20 miles per hour or both.
We get (7a) from (5). But the only information relevant to (7b) is (6), and to get to (7b) from (6), one needs disjunction-introduction. One can imagine the sleazy lawyer who contends:
We grant that my client was driving a motor vehicle in a school safety zone. The evidence adduced by the state, we concede, shows that he was talking on a cellphone, but no evidence was adduced by the state that he was talking on a cellphone or driving at more than 20 miles per hour or both.
This is obviously bad. We use conditionals with disjunctions in their antecedents quite regularly and so denying disjunction introduction is not very tenable.

One might try, instead, having additional inference rules for conditionals with special antecedents. For instance, one might allow this

1. From (i) if p or q, then r, and (ii) p, infer r.
2. From (i) if s and (p or q), then r, and (ii) s, and (iii) p, infer r.
Rule (9) would take care of the school safety zone case. But, first of all, lots of such rules would be needed to handle all cases. And, second, once we allowed such a rule we would be liable to let disjunction introduction in through the backdoor. For instance, if we allow (8), we can prove disjunction introduction from the plausible axiom: if A, then A.
1. p. (Premise)
2. if p or q, then p or q. (Axiom)
3. p or q. (Rule (8)).

Jon Kvanvig suggests to me that one might take care of this problem by replacing conditionals with disjunctive antecedents by conjunctions of conditionals. On this proposal, we would replace (4) with:

1. One has committed a violation of a school safety zone if one is driving a motor vehicle in a school safety zone and talking on a cellphone, and one has committed a violation of a school safety zone if one is driving a motor vehicle in a school safety zone and one is driving at more than 20 miles per hour.
But while we could, indeed, stop using locution (4) and use (13) instead, that is a pretty revisionary proposal. We do think Sam has violated a school safety zone given (4)-(6)—we don't need (13) to get that conclusion.

So, the upshot is this: in this case we have a pretty good argument that we would be mistaken to deny disjunction introduction.

Brandon said...

Very interesting. I know that at least some relevant logics are associated with the idea that disjunction is an equivocal notion, and so, for instance, I think there are versions that allow a form of disjunction (v) for which the rule of Addition is true (but Disjunctive Syllogism is not) and another (+) for which it is not (but Disjunctive Syllogism is). That is, there is a weak and a strong disjunction, and you can get weak disjunction by addition, but weak disjunction is not evenhanded eough, so to speak, to allow disjunctive syllogism. But I don't know much about relevant logic, either.

Alexander R Pruss said...

Thanks, Brandon.

I screwed up.

As I said, I really don't know almost anything about this stuff. I shouldn't have said that relevance logic doesn't allow disjunction intro. E, R and NR all do (for inclusive or). Embarrassing.

Alexander R Pruss said...

OK, I fixed the post. Less exciting, but truer.

Alexander R Pruss said...

Yay: I didn't write a blog post mainly on French cooking and the world didn't come to an end!

Brandon said...

That's a relief!

Eric S said...

Hmm, I'm not a logician, but I think Sam is guilty of the violation based on the evidence (or by how the statements are worded).

5. Sam is driving a motor vehicle in a school safety zone.
6. Sam is talking on a cellphone.

Both 5 and 6 describe what Sam "is" doing. If both are true, then he is doing them at the same time.

4'. One has committed a violation of a school safety zone if one is (a) driving a motor vehicle in a school safety zone and (b) "is" talking on a cellphone or "is" driving at more than 20 miles per hour or "is doing" both.

The lawyer's retort would be correct if the statements read,

5'. Sam was driving a motor vehicle in a school safety zone.
6'. Sam was talking on a cellphone.

Here, I wouldn't know when Sam "was" doing either activity.

Eric S said...

Taking another look...

A. I won't write a blog post today
B. The world will come to an end

1'. A
2'. Therefore (A or B)
3'. Therefore if ~A, then B

I'm not sure if this is correct, but I think the argument is self-refuting. The conclusion depends on the truth of A. So, if A is/was false, the truth of the conclusion is unknown.

a. If A is true, then it's not the case that I will write a blog post today
b. If I do write a blog post today, then A is false
c. If A is false, then (A or B) could be false
d. If (A or B) was false then (If ~A, then B) could be false
e. So, if I do write a blog post today, then (If ~A, then B) could be false