How to express modalities in lambda calculus - are some extensions required?

Question

Lambda calculus can be used for encoding semantics of natural language, e.g. http://yoavartzi.com/tutorial/ contains full details about semantic parsing of natural language: converting natural language texts into lambda expressions - Cornell Semantic Parsing Framework is used https://github.com/cornell-lic/spf.

My question is - how to express modalities in lambda calculus - are some extensions required? E.g. usual classical modal logics can express beliefs/duties-permissions/knowledge and other kind of modalities. Relational (Kripke) semantics is used for modal logics. So - how can we extend of combine lambda calculus to express modalities?

There is idea to express modalities as just another kind of predicates https://www.amazon.co.uk/Toward-Predicate-Approaches-Modality-Trends-ebook/dp/B016XM48V0 but are there some alternatives, more "natrual" ways to introduce modalities in lambda calculus? Full handling of natural language certainly requires this.