Questions: Quantifier Scope and Binding: Formal Treatment

Question 1

Consider 'Every professor assigned some paper.' A student says this means 'there is one specific paper that every professor assigned — they all assigned the same one.' Which formal representation corresponds to this reading?

• A. 'Some paper' takes narrow scope under 'every professor': ∀x[professor(x) → ∃y[paper(y) ∧ assigned(x,y)]]
• B. 'Some paper' takes wide scope over 'every professor': ∃y[paper(y) ∧ ∀x[professor(x) → assigned(x,y)]]
• C. The sentence is unambiguous because quantifiers always take surface-syntactic scope order
• D. Both readings have identical truth conditions, so scope does not matter for this sentence

Answer: (B)

The reading 'there is one specific paper all professors assigned' is the wide-scope reading for 'some paper' — the existential quantifier takes scope over the universal: ∃y[paper(y) ∧ ∀x[professor(x) → assigned(x,y)]]. The narrow-scope reading (each professor may have assigned a different paper) is ∀x[professor(x) → ∃y[paper(y) ∧ assigned(x,y)]]. These have genuinely different truth conditions. Option C is wrong — quantifier scope is resolved at Logical Form (LF), which can diverge from surface order. Option D is wrong — the two readings can differ in truth value.

Question 2

In formal semantics, why are quantified noun phrases like 'every student' assigned type ⟨⟨e,t⟩, t⟩ rather than type e (the type for individual entities)?

• A. Because 'every student' refers to all students collectively, forming a plural entity
• B. Because quantified NPs don't denote individual entities — they denote functions that take a property and return a truth value
• C. Because the type ⟨e,t⟩ would cause a type mismatch with predicates
• D. Because students are abstract objects, not concrete individuals, in formal ontology

Answer: (B)

This is the central insight of generalized quantifier theory. 'Every student' does not pick out any particular individual or collection of individuals — it denotes a function: take any property P and return true if every student has P. This is a function from properties (type ⟨e,t⟩) to truth values (t), giving overall type ⟨⟨e,t⟩, t⟩ — a higher-order object. This type distinction from ordinary individual-denoting names (type e) is what makes quantified expressions behave differently in compositional semantics.

Question 3

The sentence 'Everyone loves someone' is systematically ambiguous because the two quantifiers can take scope in either order, yielding two readings with different truth conditions.

• T. True
• F. False

Answer: True

On the wide-scope reading for 'someone': ∃y[∀x[loves(x,y)]] — there is some specific person that everyone loves. On the narrow-scope reading: ∀x[∃y[loves(x,y)]] — for each person, there is (possibly a different) someone they love. These have different truth conditions: the first entails the second, but not vice versa. Formal semantics locates this ambiguity at Logical Form (LF) — the level at which scope relations are explicitly represented, which may differ from surface word order.

Question 4

In formal semantics, 'nearly every student' denotes the set of most students — the collection of individuals who have the property of being a student.

• T. True
• F. False

Answer: False

This is the most tempting misconception about quantified expressions. 'Every student' does not denote a set or collection of individuals — that would give it type ⟨e,t⟩. Instead, it denotes a generalized quantifier: a function from properties to truth values, type ⟨⟨e,t⟩, t⟩. 'Every student passed' is true if and only if the property 'passed' holds of every individual in the student domain. The key distinction is between the quantifier as a second-order object and the extension of the noun 'student' as a set. Confusing these leads to type errors in compositional derivations.

Question 5

What is lambda abstraction, and why is it needed to handle variable binding in a sentence like 'every student thinks he will pass,' where the pronoun 'he' is bound by the quantifier?

Model answer: Lambda abstraction is a formal operation that creates a function by marking a variable as an argument position: λx[φ(x)] denotes a function that takes an individual x and returns the truth value of φ(x). In 'every student thinks he will pass,' the pronoun 'he' functions as a variable ranging over students — it takes its value from the quantifier, not from a fixed referent. Lambda abstraction represents this as: every student [λx [x thinks x will pass]], making explicit that the same x introduced by the quantifier fills both positions inside the clause. Without lambda abstraction, the compositional machinery would have no formal way to represent that a pronoun is bound by a quantifier rather than referring independently. Lambda abstraction is the bridge that allows the same compositional system handling ordinary predication to also handle anaphoric binding without separate mechanisms.

Lambda abstraction allows the compositional system to treat bound pronouns as variables without introducing new theoretical apparatus. The pronoun becomes a placeholder, and lambda marks the abstraction over that placeholder, allowing the quantifier to supply the value through the same functional application operation used everywhere in compositional semantics.