Formal semantics[edit]
Linguistic modality has been one of the central concerns in formal semantics and philosophical logic. Research in these fields has led to a variety of accounts of the propositional content and conventional discourse effects of modal expressions. The predominant approaches in these fields are based on modal logic. In these approaches, modal expressions such as must and can are analyzed as quantifiers over a set of possible worlds. In classical modal logic, this set is identified as the set of worlds accessible from the world of evaluation. Since the seminal work of Angelika Kratzer, formal semanticists have adopted a more finely grained notion of this set as determined by two conversational background functions called the modal base and ordering source respectively.[3]: 79–90
For an epistemic modal like English must or might, this set is understood to contain exactly those worlds compatible with the knowledge that the speaker has in the actual world. Assume for example that the speaker of sentence (4) above knows that John just bought a new luxury car and has rented a huge apartment. The speaker also knows that John is an honest person with a humble family background and doesn't play the lottery. The set of accessible worlds is then the set of worlds in which all these propositions which the speaker knows about John are true. The notions of necessity and possibility are then defined along the following lines: A proposition P follows necessarily from the set of accessible worlds, if all accessible worlds are part of P (that is, if p is true in all of these worlds).[3]: 80 Applied to the example in (4) this would mean that in all the worlds which are defined by the speaker's knowledge about John, it is the case that John earns a lot of money (assuming there is no other explanation for John's wealth). In a similar way a proposition p is possible according to the set of accessible worlds (i.e. the modal base), if some of these worlds are part of P.
Recent work has departed from this picture in a variety of ways. In dynamic semantics, modals are analyzed as tests which check whether their prejacent is compatible with (or follows from) the information in the conversational common ground. Probabilistic approaches motivated by gradable modal expressions provide a semantics which appeals to speaker credence in the prejacent. Illocutionary approaches assume a sparser view of modals' propositional content and look to conventional discourse effects to explain some of the nuances of modals' use.