Imperative logic

Imperative logic is the field of logic concerned with imperatives. In contrast to declaratives, it is not clear whether imperatives denote propositions or more generally what role truth and falsity play in their semantics. Thus, there is almost no consensus on any aspect of imperative logic.
Jørgensen's dilemma.
One of a logic's principal concerns is logical validity. It seems that arguments with imperatives can be valid. Consider:
However, an argument is valid if the conclusion follows from the premises. This means the premises give us reason to believe the conclusion, or, alternatively, the truth of the premises determines truth of the conclusion. Since imperatives are neither true nor false and since they are not proper objects of belief, none of the standard accounts of logical validity apply to arguments containing imperatives.
Here is the dilemma. Either arguments containing imperatives can be valid or not. On the one hand, if such arguments can be valid, we need a new or expanded account of logical validity and the concomitant details. Providing such an account has proved challenging. On the other hand, if such arguments cannot be valid (either because such arguments are all invalid or because validity is not a notion that applies to imperatives), then our logical intuitions regarding the above argument (and others similar to it) are mistaken. Since either answer seems problematic, this has come to be known as Jørgensen's dilemma, named after Jørgen Jørgensen ().
While this problem was first noted in a footnote by Frege, it received a more developed formulation by Jørgensen.
Ross's paradox.
Alf Ross observed that there is a potential problem for any account of imperative inference. Classical logic validates the following inference:
This inference is called disjunction introduction. However, a similar inference does not seem to be valid for imperatives. Consider:
Ross's paradox highlights the challenge faced by anyone who wants to modify or add to the standard account of validity. The challenge is what we mean by a valid imperative inference. For valid declarative inference, the premises give you a reason to believe the conclusion. One might think that for imperative inference, the premises give you a reason to do as the conclusion says. While Ross's paradox seems to suggest otherwise, its severity has been subject of much debate.
Mixed inferences.
The following is an example of a pure imperative inference:
In this case, all the sentences making up the argument are imperatives. Not all imperative inferences are of this kind. Consider again:
Notice that this argument is composed of both imperatives and declaratives and has an imperative conclusion.
Mixed inferences are of special interest to logicians. For instance, Henri Poincaré held that no imperative conclusion can be validly drawn from a set of premises which does not contain at least one imperative. While R.M. Hare held that no declarative conclusion can be validly drawn from a set of premises which cannot validly be drawn from the declaratives among them alone. There is no consensus among logicians about the truth or falsity of these (or similar) claims and mixed imperative and declarative inference remains vexed.
Applications.
Aside from intrinsic interest, imperative logic has other applications. The use of imperatives in moral theory should make imperative inference an important subject for ethics and metaethics.