expected of good structural proof theories, such as purity (the proof theory does not introduce extra-logical notions such as labels) and analyticity (the logical rules support a clean notion of analytic proof). More complex calculi have been applied to modal logic to achieve generality. Decision methods Analytic tableaux provide the most popular decision method for modal logics. Modal logics in philosophy Alethic logic Modalities of necessity and possibility are called alethic modalities. They are also sometimes called special modalities, from the Latin species. Modal logic was first developed to deal with these concepts, and only afterward was extended to others. For this reason, or perhaps for their familiarity and simplicity, necessity and possibility are often casually treated as the subject matter of modal logic. Moreover, it is easier to make sense of relativizing necessity, e.g. to legal, physical, nomological, epistemic, and so on, than it is to make sense of relativizing other notions. In classical modal logic, a proposition is said to be possible if it is not necessarily false (regardless of whether it is actually true or actually false); necessary if it is not possibly false (i.e. true and necessarily true); contingent if it is not necessarily false and not necessarily true (i.e. possible but not necessarily true); impossible if it is not possibly true (i.e. false and necessarily false). In classical modal logic, therefore, the notion of either possibility or necessity may be taken to be basic, where these other notions are defined in terms of it in the manner of De Morgan duality. Intuitionistic modal logic treats possibility and necessity as not perfectly symmetric. For example, suppose that while walking to the convenience store we pass Friedrich's house, and observe that the lights are off. On the way back, we observe that they have been turned on. "Somebody or something turned the lights on" is necessary. "Friedrich turned the lights on", "Friedrich's roommate Max turned the lights on" and "A burglar named Adolf broke into Friedrich's house and turned the lights on" are contingent. All of the above statements are possible. It is impossible that Socrates (who has been dead for over two thousand years) turned the lights on. (Of course, this analogy does not apply alethic modality in a truly rigorous fashion; for it to do so, it would have to axiomatically make such statements as "human beings cannot rise from the dead", "Socrates was a