A completeness proof in full DDL

Dynamic doxastic logicians — not a large community — have trodden gingerly within the area of full DDL or, as with the present author, have not trodden at all. However, the latter, after having written up the final version of [6], realised that the proof given in that paper for two varieties of basic DDL can be extended to cover full DDL; in fact, the «full» proof is simpler than the «basic» one. The extended proof is outlined in Section 1–3. In Section 4, the relationship to AGM is considered. Section 5 puts the importance of the proof into perspective.This note — an extended abstract rather than a full-fledged paper — should be read as an appendix to [6]. Although some definitions are repeated here, many are not. Readers who require more detail are referred to [6], a copy of which they should have on hand

A festival of facts

Consider yourself invited to a filmfestival — a festival of shorts. Regrettably it is a festival with a limitation: you will not be able actually to see the films. However, this limitation is offset by an unusual feature: the films to be shown are not only short but ultrashort — with one exception, each film consists of only a few frames. Instead of showing the films we shall be able to give a complete description of them

Modal logic and philosophy

Modal logic is one of philosophy’s many children. As a mature adult it has moved out of the parental home and is nowadays straying far from its parent. But the ties are still there: philosophy is important to modal logic, modal logic is important for philosophy. Or, at least, this is a thesis we try to defend in this chapter. Limitations of space have ruled out any attempt at writing a survey of all the work going on in our field—a book would be needed for that. Instead, we have tried to select material that is of interest in its own right or exemplifies noteworthy features in interesting ways. Here are some themes that have guided us throughout the writing: • The back-and-forth between philosophy and modal logic. There has been a good deal of give-and-take in the past. Carnap tried to use his modal logic to throw light on old philosophical questions, thereby inspiring others to continue his work and still others to criticise it. He certainly provoked Quine, who in his turn provided—and continues to provide—a healthy challenge to modal logicians. And Kripke’s and David Lewis’s philosophies are connected, in interesting ways, with their modal logic. Analytic philosophy would have been a lot different without modal logic! • The interpretation problem. The problem of providing a certain modal logic with an intuitive interpretation should not be conflated with the problem of providing a formal system with a model-theoretic semantics. An intuitively appealing model-theoretic semantics may be an important step towards solving the interpretation problem, but only a step. One may compare this situation with that in probability theory, where definitions of concepts like ‘outcome space’ and ‘random variable’ are orthogonal to questions about “interpretations” of the concept of probability. • The value of formalisation. Modal logic sets standards of precision, which are a challenge to—and sometimes a model for—philosophy. Classical philosophical questions can be sharpened and seen from a new perspective when formulated in a framework of modal logic. On the other hand, representing old questions in a formal garb has its dangers, such as simplification and distortion. • Why modal logic rather than classical (first or higher order) logic? The idioms of modal logic—today there are many!—seem better to correspond to human ways of thinking than ordinary extensional logic. (Cf. Chomsky’s conjecture that the NP + VP pattern is wired into the human brain.) In his An Essay in Modal Logic (1951) von Wright distinguished between four kinds of modalities: alethic (modes of truth: necessity, possibility and impossibility), epistemic (modes of being known: known to be true, known to be false, undecided), deontic (modes of obligation: obligatory, permitted, forbidden) and existential (modes of existence: universality, existence, emptiness). The existential modalities are not usually counted as modalities, but the other three categories are exemplified in three sections into which this chapter is divided. Section 1 is devoted to alethic modal logic and reviews some main themes at the heart of philosophical modal logic. Sections 2 and 3 deal with topics in epistemic logic and deontic logic, respectively, and are meant to illustrate two different uses that modal logic or indeed any logic can have: it may be applied to already existing (non-logical) theory, or it can be used to develop new theory

Blueprint for a dynamic deontic logic

AbstractExtending the idiom of dynamic logic we outline a deontic logic in which deontic operators operate on terms rather than on formulæ. In a second step we distinguish between what we call real and deontic actions