Plural Intensional Presuppositional predicate calculus (PIP)

One successful branch of natural language semantics uses first order predicate calculus and set theory, well-studied systems of mathematics in their own right, as meta-languages for natural-language meanings. Now this traditional approach suffers from certain well-known deficiencies in treating what we call improper scope anaphora, such as cross-sentential anaphora to indefinites, donkey pronouns (Geach 1962), and quantificational (Karttunen 1969) and modal (Roberts 1987) subordination. Dynamic plural logics, such as those proposed by van den Berg (1996) and Brasoveanu (2007), successfully solve the problems presented by improper scope phenomena, but at the cost of diverging sharply from the traditional predicate-calculus approach. This paper aims instead to present a treatment of improper scope that is accessible to the broad audience of linguists who are familiar chiefly with the traditional approach. Surprisingly, we require only two changes to standard logic with set theory: (i) a method to extend the scope of existentials, and (ii) a mechanism to store and retrieve subformulas. We implement these changes in a logic we call Plural Intensional Presuppositional predicate calculus (PIP). PIP also introduces (iii) a new operator for the sake of defining presuppositional felicity conditions (independent of truth conditions). We show that the resulting logic captures the full range of improper scope phenomena covered by state-of-the-art intensional dynamic plural logics. But PIP also provides a ready analysis for paycheck pronouns, which are difficult to capture in most dynamic logics, and presupposition projection, which has not received as much attention in dynamic plural logic.