Hyperintensional Semantics in A Higher-Order Logic with Definable Subtypes

Higher order grammar (HOG) is a linguistic formalism that aims to combine the advantages of existing constraint-based formalisms (such as HPSG) and proof-theoretic ones (such as categorial grammar) by using higher-order logic (HOL) as the description language; the underlying intuitionistic type system plays a role analogous to that of a categorial type logic, while the classical higher-order term logic serves to impose constraints (analogous to the role played by RSRL in HPSG). Here we focus on semantics, showing how the use of a HOL with definable substypes leads to a novel and surprisingly straightforward solution of the notorious granularity problem about natural-language (NL) meanings. We also call attention to a hitherto unnoticed problem in standard approaches to NL semantics having to do with nonprincipal ultrafilters and show why it does not arise under our proposal. The two main technical innovations that make the proposal work are (1) axiomatization of NL entailment as a preorder (as opposed to an order) on the set of (primitive) propositions, and (2) definition of the set of worlds as a certain subset of the powerset of the set of propositions. 1 Background on Higher Order Grammar Higher Order Grammar (HOG, [19], [18], [17]) is a framework for linguistic theory developed by the author together with Jirka Hana since early 2001. It belongs to a loose assemblage of approaches to natural language that might be called multistratal type-theoretic approaches, which also include Abstract Categorial Grammar (ACG, [3]), Lambda Grammar ([15]), and Grammatical Framework ([20], [13]), all of which emerged independently around the turn of the millenium. Shared characteristics of these approaches include: distinguishing between tectogrammar (abstract syntax) and phenogrammar (concrete syntax); separate type theories each with its own Curry-Howard proof term calculus, for each of (a) tectogrammar, (b) phenogrammar, and (c) meaning; and the interpretation of signs (tectostructures) into concrete linguistic forms (phenostructures) and meanings via structure-preserving translations of the term calculi. What distinguishes HOG from the other approaches in this family is the use of full intuitionistic propositional logic for the type logics rather than linear logic, more specifically a bivalent boolean version of Lambek and Scott’s higher-