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: P1. Take all the books off the table! P2. Foundations of Arithmetic is on the table. C1. Therefore, take Foundations of Arithmetic off the table! 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 (da). While this problem was first noted in a footnote by Frege, it received a more developed formulation by Jørgensen. Deontic logic takes the approach of adding a modal operator to an argument with imperatives such that a truth-value can be assigned to the proposition. For example, it may be hard to assign a truth-value to the argument "Take all the books off the table!", but ("take all the books off the table"), which means "It is obligatory to take all the books off the table", can be assigned a truth-value, because it is in the indicative mood. Ross's paradox Alf Ross observed that applying the classical rule of disjunction introduction under the scope of an imperative operator leads to unintuitive (or apparently absurd) results. When applied to simple declaratives, the result appears to be valid deduction. P1. The