Questions: Scope Ambiguity and Logical Form

Question 1

The sentence 'Every student read a book' is ambiguous. On the reading where one particular book was read by everyone, the logical form is:

• A. ∀x(Student(x) → ∃y(Book(y) ∧ Read(x, y)))
• B. ∃y(Book(y) ∧ ∀x(Student(x) → Read(x, y)))
• C. ∀x∀y(Student(x) ∧ Book(y) → Read(x, y))
• D. ∃x(Student(x) ∧ ∃y(Book(y) ∧ Read(x, y)))

Answer: (B)

The reading where one specific book is shared by all students requires the existential quantifier over books to take *wide scope* — it is asserted first, binding a particular book y, and then the universal quantifier over students falls within its scope. This gives ∃y(Book(y) ∧ ∀x(Student(x) → Read(x, y))): there is a book that every student read. Option A is the other reading (narrow-scope ∃): each student is guaranteed a book, but the books may differ. The surface English string is genuinely ambiguous between these two logical forms.

Question 2

John does not know that Maria is the tallest spy. Yet he has been watching Maria for years, believing her to be dangerous. On the de re reading of 'John believes the tallest spy is dangerous,' which of the following is true?

• A. John must know that Maria is the tallest spy for the de re reading to apply
• B. The de re reading is false because John does not associate the description 'tallest spy' with Maria
• C. The de re reading is true: John has a belief directly about Maria (the individual), even without knowing she is the tallest spy
• D. De re and de dicto readings agree here since the description is uniquely satisfied

Answer: (C)

The de re reading attributes a belief *about a particular individual* — it says there is an individual x (namely Maria, who happens to be the tallest spy) such that John believes of x that she is dangerous. John need not know the description 'tallest spy' applies to her; the de re reading only requires that his belief is directed at that individual, however he identifies her. The de dicto reading would require John to have a belief *about the description itself* — believing that whoever satisfies 'tallest spy' is dangerous. The two readings differ precisely in whether the description scopes outside (de re) or inside (de dicto) John's belief context.

Question 3

On the de re reading of a belief attribution, the believer is expected to have the individual in mind under the relevant description.

• T. True
• F. False

Answer: False

This is the key misconception about the de re/de dicto distinction. The de re reading requires only that there is a particular individual about whom the belief is held — it does not require the believer to identify that individual under any specific description. John's belief can be de re about Maria even if he knows her only as 'the woman in the blue coat' and has no idea she is the tallest spy. The description in the sentence ('the tallest spy') is used by the *attributor* to pick out the individual; it need not be how the believer represents her.

Question 4

The scope ambiguity in natural language sentences like 'Everyone loves someone' can usually be resolved by careful attention to word order and syntax.

• T. True
• F. False

Answer: False

This is the central lesson of scope ambiguity: natural language surface syntax radically underdetermines logical form. English word order and syntactic structure do not reliably indicate quantifier scope — both readings of 'Everyone loves someone' have identical surface syntax, yet they express logically distinct propositions with different truth conditions. This is why logical form representation is needed as a distinct level of semantic analysis: it makes explicit what natural language leaves systematically ambiguous. Context and pragmatics can sometimes resolve the ambiguity, but this is extra-linguistic information, not syntactic.

Question 5

Why does it matter for belief attributions whether a description takes wide scope or narrow scope relative to the belief operator?

Model answer: Scope determines what the belief is *about*. Under narrow scope (de dicto), the belief is about whoever satisfies the description — John believes there is a tallest spy and that person is dangerous, so his belief would be false if there is no tallest spy. Under wide scope (de re), there must be an actual individual who satisfies the description, and John's belief is directly about that individual, not about the description — even if John doesn't know the description applies. The two readings attribute different cognitive states to John, with different implications about his knowledge, the content of his belief, and what would verify or falsify it.

This matters practically for discussions of intentionality, opacity, and the semantics of propositional attitude verbs. Two people can have 'the same' belief sentence attributed to them but be in entirely different epistemic states, depending on how the descriptions scope. Logical form representation makes these differences precise and testable.