is still an imperative, without any ought judgment (i.e. not the same as "You ought to do A"). Following Castañeda's approach, the deontic operators (for ought) and (for all right, represented for permissible in some deontic logic notations) are applied to imperatives. This is opposed to many deontic logics which apply the deontic operators to indicatives. Doing so avoids a difficulty of many deontic logics to express conditional imperatives. An often given example is If you smoke, then you ought to use an ashtray. If the deontic operators and only attach to indicatives, then it is not clear that either of the following representations is adequate: However, by attaching the deontic operators to imperatives, we have unambiguously Belief logic symbols, when combined with imperative logic, allow beliefs and desires to be expressed. The notation is used for beliefs ("You believe A") and for desires ("You desire A"). In formal ethics, desire is taken in a strong sense when the agent of the belief is the same as the agent of the imperative. The following table shows the different interpretations for depending on the agent and the tense of the imperative: This strong interpretation of desires precludes statements such as "I want to get out of bed (right now), but I don't act to get out of bed". It does not, however, preclude "I want to get out of bed (right now), but I don't get out of bed". Perhaps I act to get out of bed (make my best effort), but can't for some reason (e.g. I am tied down, my legs are broken, etc.). Beliefs may be indicative, as above, or imperative (e.g. "Believe A", "Desire A"). They may also be combined with the deontic operators. For example, if means "God exists", then is "You ought to believe that God exists", and is "Everyone ought to believe that God exists". The modal operators and are used with their normal meanings in modal logic. In addition, to address the fact that logicians may disagree on what is logically necessary or possible, causal modal operators are separately defined to express that something is causally necessary or possible. The causal modal operators are represented and . In addition, an operator is used to mean "in every actual or hypothetic case". This is used, for example, when expressing deontic and prescriptive counterfactuals, and is weaker than . For example, means "In every actual