Consider the following disjunction introduction:
- Your room is clean.
- Therefore, your room is clean or your house is burnt down.
- Clean your room.
- Therefore, clean your room or burn your house down.
Intuitively, 2 doesn't appear to follow from 1. However, it seems to me that this is just due to a syntactic ambiguity in the conclusion. That is, the conclusion could be interpreted in one of two ways:
- You should (clean your room or burn your house down).
- (You should clean your room) or (you should burn your house down).
The conclusion that follows from "clean your room" is really 2, not 1. Thus I don't see where the paradox arises. More generally, consider a do operator, so that do(x) denotes "x is obligatory" or "I command you to do x". Then the inference
- Therefore, do(x) or do(y)
is valid but the inference
- Therefore, do(x or y)
is not. Am I missing something about this purported paradox?
EDIT: Is the underlying problem here the implicit assumption that
- x → y
- Therefore, do(y)
is a valid inference? If so, that doesn't seem right. To see why it's not true in general, consider replacing the do operator with the negation operator:
- x → y
- Therefore, not(y)
- Necessarily, 1 + 1 = 2.
- 1 + 1 = 2 → Alexander invaded Persia.
- Therefore, necessarily, Alexander invaded Persia.
On second thought, I've been thinking about Ross' paradox the wrong way. First, define the following deontic operators:
- it is obligatory that (OB)
- it is impermissible that (IM)
- it is omissible that (OM)
- it is permissible that (PE)
- it is optional that (OP)
All of these operators can be expressed in terms of OB alone:
- IM p ↔ OB ~ p
- OM p ↔ ~ OB p
- PE p ↔ ~ OB ~ p
- OP p ↔ (~ OB p) ∧ (~ OB ~ p)
Consider the simple example:
- Mail the letter.
- Therefore, mail the letter or burn it.
- OB (mail the letter)
- OB (mail the letter ∨ burn the letter)
I don't see anything wrong with 2 anymore, because it's not really presenting a choice! It's not saying that burning the letter is permissible! Just because an obligation entails a second obligation that would be satisfied by an action, it doesn't mean that the first obligation would be satisfied by that action. If you burn the letter, you would have fulfilled a different obligation entailed by the original obligation, but not the original obligation itself. In fact, doing so would make it impossible to fulfill the original obligation of mailing the letter. Thus burning the letter is impermissible, even though mailing the letter or burning it is permissible (and in fact obligatory)!
- burn the letter → ~(mail the letter)
- mail the letter → ~(burn the letter) [from 1 by contraposition]
- OB(mail the letter) → OB(~(burn the letter)) [from 2 by axiom OB-RM]
- OB(mail the letter)
- OB(~(burn the letter)) [from 3 and 4 by modus ponens]
- IM(burn the letter) [from 5 and the definition of IM]
The key point here is that OB (p ∨ q) is perfectly consistent with IM q, because it's not actually expressing a real choice. But how do you express a real choice then, like "pay up or leave" in the context of a restaurant? The proper way to express a real choice between p and q is not
OB (p ∨ q)
OB (p ∨ q) ∧ OP p ∧ OP q
In particular, you must specify that p and q are each optional, lest it turns out that one of them was actually obligatory (or impermissible) individually.