Dynamic Logic in Natural Language

Standard first-order logic defines truth in a three-part scheme: a language, structures D of objects with relations and operations, and maps from language to structures that drive semantic evaluation. In particular, “interpretation functions” I map predicate letters to real predicates, while variable assignments s map individual variables to objects. Logicians often lump D and I together into a “model” M, and then interpret formulas:

formula M is true in model M under assignment s (M, s |= M) with a recursive definition matching syntactic construction steps with semantic operations for connectives and quantifiers. This pattern has been applied to natural language since Montague 1974, stating under which conditions a sentence is true. Compositional interpretation in tandem with syntactic construction works even beyond logical and natural languages: it is also a well-known design principle for programs (van Leeuwen 1990). And the paradigm finds an elegant mathematical expression in universal algebra and category theory.