Semantics Week 5

What is the purpose of typed lambda calculus in semantics?

It provides a formal language to represent the meanings of natural language expressions compositionally. Types lambda handles functions as meanings, enabling precise modelling on how sentence meanings are built from word meanings.

What are the basic types used in lambda?

e: type of entities (individuals)

t: type of truth values (propositions)

⟨a, b⟩: type of functions from type a to type b

Example: ⟨e, t⟩ is a function from entities to truth values (e.g., dog → true/false)

What does it mean for a predicate to be of type ⟨e, t⟩?

It means the predicate denotes a function from entities to truth values.

Example: dog(x) is true iff x is a dog.

What type is a transitive verb like ‘loves’?

Type ⟨e, ⟨e, t⟩

It first takes an object of type e, and returns a predicate of type ⟨e, t⟩ that expects a subject.

What is beta-reduction in lambda calculus?

Beta-reduction is the process of applying a function to an argument by substituting the argument for the variable in the function’s body.

Example:

(λx.dog(x))(fido) → dog(fido)

What is the purpose of curried functions in lambda?

Curried functions allow multi-argument functions to be modelled as nested single-argument functions, facilitating functional application and composition in semantic trees.