What is the semantic type of the noun phrase 'every student' in formal semantics?
Ae — it denotes the individual that is every student
B⟨e, t⟩ — it denotes a predicate that applies to individuals
C⟨⟨e, t⟩, t⟩ — it takes a predicate and returns a truth value
Dt — it directly expresses a truth value
'Every student' is a generalized quantifier, not an individual. There is no single entity it names, so type e is impossible. Instead, it combines with a predicate like 'passed' (type ⟨e, t⟩) and returns a truth value — true if every student is in the extension of 'passed.' This gives it type ⟨⟨e, t⟩, t⟩. The common misconception is treating quantifier phrases as referring expressions (type e) by analogy with names like 'John.' The type-lifting from apparent noun-phrase simplicity to quantifier complexity is one of Montague semantics' central contributions.
Question 2 Multiple Choice
The transitive verb 'admires' has type ⟨e, ⟨e, t⟩⟩. When it combines with its object 'Bach' (type e), what type does the resulting expression have?
At — the combination produces a complete sentence
Be — Bach remains an individual in the resulting expression
C⟨e, t⟩ — the result is a predicate waiting for a subject
D⟨⟨e, t⟩, t⟩ — the result must be a generalized quantifier
Semantic composition here is function application: applying 'admires' (type ⟨e, ⟨e, t⟩⟩) to 'Bach' (type e) consumes the first argument and returns ⟨e, t⟩ — a predicate meaning 'is someone who admires Bach.' This predicate then waits for the subject (e.g., 'John', type e) to yield a full truth value (type t). This illustrates currying: functions take one argument at a time, and the intermediate type ⟨e, t⟩ is not yet a complete sentence but a well-formed semantic object awaiting further composition.
Question 3 True / False
According to type theory, the expression 'The ham sandwich wants the check' is semantically very difficult to compose because the types are incompatible.
TTrue
FFalse
Answer: False
'The ham sandwich' has type e (it denotes an individual, at least syntactically), and 'wants' has a type that can take type-e arguments, so the types actually do compose. The anomaly is not a type-level failure — it is a pragmatic one, resolved through coercion: in restaurant contexts, 'the ham sandwich' is reinterpreted as referring to the person who ordered it. Type theory identifies where and why composition can go wrong, but pragmatic coercion can override or repair apparent mismatches without blocking derivation entirely.
Question 4 True / False
In a well-typed semantic derivation, a complete declarative sentence like 'John runs' denotes a value of type t.
TTrue
FFalse
Answer: True
'John' has type e (an individual). 'Runs' has type ⟨e, t⟩ (a function from individuals to truth values). Applying 'runs' to 'John' yields type t — a truth value representing whether it is true that John runs. This is the foundational compositionality claim: sentences denote truth conditions, and the type system ensures that grammatical sentences reduce to type t when fully composed. Non-sentential expressions like 'John' or 'runs' have sub-propositional types that only reach t through complete composition.
Question 5 Short Answer
Why can't noun phrases like 'every student' or 'some professor' have semantic type e, and what type must they have instead?
Think about your answer, then reveal below.
Model answer: Type e is reserved for expressions that denote specific individuals — names like 'John' or 'Mary.' Quantifier phrases like 'every student' do not pick out a single individual; they express a relationship between two sets (e.g., the set of students and the set of things that passed). They must have type ⟨⟨e, t⟩, t⟩: they take a predicate (type ⟨e, t⟩) and return a truth value indicating whether the quantificational condition is satisfied.
This type-lifting is not merely formal machinery — it reflects a deep semantic difference between reference (picking out an entity) and quantification (asserting something about the relationship between sets). Treating 'every student' as type e would mean it names some particular individual, which fails for universal and existential quantifiers that range over sets. Montague's insight was that natural language noun phrases uniformly have the higher type ⟨⟨e, t⟩, t⟩, with proper names 'lifted' to that type for uniformity, rather than a heterogeneous system mixing e and quantifier types.