Is Ross' paradox really a paradox?

Question

Consider the following disjunction introduction:

1. Your room is clean.
2. Therefore, your room is clean or your house is burnt down.

Ross' paradox allegedly arises when applying this inference to imperative logic:

1. Clean your room.
2. 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:

1. You should (clean your room or burn your house down).
2. (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

1. do(x)
2. Therefore, do(x) or do(y)

is valid but the inference

1. do(x)
2. 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

1. do(x)
2. x → y
3. 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:

1. not(x)
2. x → y
3. Therefore, not(y)

This would be a basic formal fallacy called denying the antecedent. It's not true for the modal necessity operator either:

1. Necessarily, 1 + 1 = 2.
2. 1 + 1 = 2 → Alexander invaded Persia.
3. Therefore, necessarily, Alexander invaded Persia.

EDIT 2:

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:

1. Mail the letter.
2. Therefore, mail the letter or burn it.

More formally,

1. OB (mail the letter)
2. 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)!

1. burn the letter → ~(mail the letter)
2. mail the letter → ~(burn the letter) [from 1 by contraposition]
3. OB(mail the letter) → OB(~(burn the letter)) [from 2 by axiom OB-RM]
4. OB(mail the letter)
5. OB(~(burn the letter)) [from 3 and 4 by modus ponens]
6. 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)

but

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.

Answer 1

The issue arises if we consider imperatives, i.e. statements in the form : "Clean your room".

In this case, the standard truth-functional accounts of connectives, that licenses inference patterns like disjunction introduction, seems at odds with natural language practice.

This conclusion is consistent with the standard assumption of mathematical logic that propositions stand for declarative sentences, excluding imperatives.

If we consider instead a sort of "modal" operator, like your "do(x)" (or "should"), things are different.

Consider  as the "modality" should; thus "You should (clean your room)" is : p.

From it we have truth-functionally : (p ∨ q), i.e. your 2. above.

But, according to your analysis, we have not : (p ∨ q), i.e. your 1. above.

Answer 2

I don't think your suggestion resolves the paradox. If you allow that do(x) entails do(x) or do(y), there remains the problem of explaining why you are not left with the option to do y. In ordinary English if Alice says to Bob, "Pay up or leave", we would understand this to mean that it is Bob's choice as to which he does. There does not seem to be any difference between representing this as Do(pay up OR leave) as opposed to Do(pay up) OR Do(leave). The key feature of the paradox is that an element of choice appears to have crept in where it does not belong. We need to find some way of explaining how the element of choice is specified or implied.

One option would be to accept that an imperative Do( A OR B ) always entails a choice. We might then attempt to develop a distinct imperative logic to account for the circumstances in which it is correct to infer Do( A OR B ). This logic would differ from classical logic, because it would have to exclude disjunction introduction in order to avoid the paradox.

An alternative approach might be to appeal to Grice's theory of implicature. We might hold that Do( A OR B ) only carries the conversational implicature of a choice. There is an analogy here with uttering a disjunction when answering a question. Suppose Alice asks Bob, "Where is Charlie?" and Bob knows that Charlie is in the library but responds, "Charlie is either in the library or the kitchen". Bob has not said anything untrue. Charlie's being in the library does indeed entail that he is either in the library or the kitchen. But Bob's answer is misleading. Bob is violating Grice's cooperative principle and being less informative than he could be in the circumstances. Alice is entitled to assume that Bob is obeying the cooperative principle and so it is reasonable for her to believe that Bob does not know whether Charlie is in the library or the kitchen. The answer "A or B" does not entail or mean "A or B and I don't know which", but in the circumstances it is a conversational implicature. One way to recognise the difference is that conversational implicatures are cancellable, but meanings are not. Suppose Bob had replied, "Charlie is in the library or the kitchen. I know which, but I'm choosing not to tell you." Here, Bob has expressly cancelled the implicature. Alice is no longer misled into believing that Bob does not know. Bob is still being less informative than he might, but at least he is being honest about it and not deceiving Alice.

Now, can we carry this over to imperatives? It depends on whether you think the following is plausible. Suppose Alice commands Bob, "Do A or B!" Ordinarily Bob is entitled to assume that Alice is giving him a choice, and it would be misleading, but not strictly incorrect, if Alice in fact wanted Bob to do A and not B. For the implicature to be cancellable, it would have to make sense for Alice to say something like, "Do A or B! I'm not giving you a choice: you must find out which is the right one to do and then do it." If you are willing to accept this, then we can say the same about your example. "Clean your room!" might be taken to entail "Clean your room or burn the house down!" but not "Clean your room or burn the house down and it is your choice which!"

Answer 3

Paul McNamara addresses Ross's paradox in deontic logic where there is an obligation operator, OB. His exposition may help explain why Ross's paradox appears paradoxical to some people even without that operator.

Consider the example provided by the OP:

1. Clean your room.
2. Therefore, clean your room or burn your house down.

McNamara used a similar example about mailing or burning a letter. As he put it

...it seems rather odd to say that an obligation to mail the letter entails an obligation that can be fulfilled by burning the letter (something presumably forbidden), and one that would appear to be violated by not burning it if I don't mail the letter.

Applying this to the example about cleaning one's room, it seems odd to say that the command to clean one's room can be fulfilled by burning down the house. And if one actually did burn the house down that would violate the command to clean the room.

The paradox follows from the OB-RM derived rule of standard deontic logic:

If ⊢ p → q, then ⊢ OBp → OBq

After substituting p for p and (p ∨ q) for q using OB-RM, one has ⊢ p → (p ∨ q), then ⊢ OBp → OB(p ∨ q). That is, if one is obligated to clean one's room then one is obligated to clean one's room or burn the house down. Furthermore, one could fulfill the obligation to clean one's room by substituting anything one wanted to do rather than burning the house down, such as taking a walk.

The OP suggests the paradox may be avoided by having a disjunction of two ought statements, OB p ∨ OB q, but that would not be derived from the OB-RM rule which is what is causing the problem. Besides, it would be odd for someone to make both of those commands rather than just the first one to clean one's room.

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Reference

McNamara, Paul, "Deontic Logic", The Stanford Encyclopedia of Philosophy (Fall 2018 Edition), Edward N. Zalta (ed.), URL = https://plato.stanford.edu/archives/fall2018/entries/logic-deontic/.