Formal Approaches to Modality

1. Modal logic Modal notions are pervasive in the meaning of a wide range of expressions from grammatical categories, such as tenses, to the lexical semantics of particular words, such as modal adverbials (probably, necessarily) and modal auxiliaries (must, may, can). The best known modalities are the alethic 1 modalities necessary and possible. Other modalities include temporal, de-ontic, epistemic, and doxastic ones, and modalities pertaining to disposition, ability, provability, mood, aspect, and so on. Temporal modalities deal with time. Deontic 2 modalities deal with obligation and permission. Epistemic 3 modalities deal with knowledge, and doxastic 4 modalities deal with belief. Although modals have been thoroughly studied since Aristotle, the formal theory of modality was revolutionized in the 1960's with the introduction of the possible world semantics by Hintikka and Kripke. In our formal approach to the semantic analysis of modal expressions, we will employ the model-theoretic apparatus of modal logic. In modal logic, modals correspond to sentential operators, whose semantic role is to qualify the truth of the sentences in their scope. For example, the sentence He is possibly right is represented by a formula that is roughly equivalent to It is possibly true that he is right. Likewise, the sentence Nature must obey necessity is represented as It is necessarily true that nature obeys necessity. Given a sentential symbol p, we use the square notation □p for the statement p is necessarily true, and the diamond notation ◊p for the statement p is possibly true. Thus, necessarily and possibly are formalized as modal operators that act on sentences. Most modal expressions, though by no means all, are treated similarly. The modal operators □ and ◊ are interdefinable. Even Aristotle in his De Interpretatione established that the negation of It is necessary that p is It is not possible that not-p. Similarly, It is possible that p is equivalent to It is