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In formal semantics, quantifiers like 'all' and 'some' are treated as generalized quantifiers—functions from properties to truth values. Scope relationships determine the logical form of sentences: whether 'some' or 'all' takes wider scope changes truth conditions (e.g., 'Everyone loves someone' is ambiguous depending on which has wider scope). Formal accounts explain scope ambiguity, scope islands, and licensing of negative polarity items through the syntax-semantics interface.
Work through scope ambiguities with multiple quantifiers, computing truth conditions under each scope reading. Use lambda calculus to represent different scope assignments and verify when readings are equivalent or distinct.
From your study of lambda calculus for linguistics, you can represent the meaning of a sentence as a function application: the meaning of a verb phrase is a function, the meaning of a noun phrase argument is an input, and the result is a truth value or a proposition. From Montague semantics, you know how to compose meanings compositionally: complex expressions are interpreted by combining the interpretations of their parts according to their syntactic structure. From your work on quantifiers and scope, you know that expressions like *every*, *some*, and *no* are not referring expressions (they don't pick out specific individuals) but operators that relate sets. This topic puts those tools together to handle the formal treatment of scope ambiguity — one of the most revealing puzzles in the semantics-syntax interface.
The key theoretical move is generalized quantifier theory: quantifiers are not treated as second-order predicates in Frege's sense but as functions from properties to truth values. "Every student left" is analyzed as: the quantifier *every student* denotes a function that, given the property *left*, returns true iff every individual in the student-set has that property. In lambda notation: `λP. ∀x[student(x) → P(x)]`. This type-theoretic treatment lets quantifiers compose uniformly with the rest of the grammar. "Some professor wrote every book" now involves two quantifiers, each of type `⟨⟨e,t⟩,t⟩`, and their relative scope — which takes wider scope — determines the truth conditions.
The ambiguity in "Every student read some book" is the canonical example. Under the surface scope reading (every > some): for every student, there is (possibly a different) some book that they read. Under the inverse scope reading (some > every): there is one particular book such that every student read it. These are genuinely different truth conditions, not just paraphrases. The mechanism that generates inverse scope in formal accounts is Quantifier Raising (QR): at Logical Form (LF), quantifiers can covertly move out of their surface position, leaving a trace, and bind a variable from an adjoined position. The relative scope of two quantifiers is determined by their positions at LF, not their surface order. This is why scope is not a simple left-to-right affair — it depends on movement in the covert syntactic component.
Scope islands are the empirical constraints that prevent QR from applying arbitrarily. A quantifier cannot move out of certain syntactic boundaries: relative clauses, *that*-clauses, questions, and adjuncts tend to be scope islands, meaning a quantifier trapped inside one cannot take scope over something outside. "The man who bought every book left" does not have a reading where *every book* takes scope over the entire sentence — it is frozen inside the relative clause. These constraints are not arbitrary; they correlate with the same boundaries that block overt syntactic movement (island constraints from wh-movement studies), suggesting that QR is a genuine syntactic operation constrained by the same grammar.
Negative polarity items (NPIs) like *any* and *ever* show one of the most striking scope-sensitive licensing patterns: they require a downward-entailing (DE) context — a context where inferences go from sets to subsets. "She didn't eat anything" is fine because negation creates a DE context. "She ate something" does not license *any* because affirmation is upward-entailing. Formally, a DE context is one where if P entails Q, then DP-operator(Q) entails DP-operator(P). Generalized quantifier theory makes this formally precise and explains why *every* creates a DE context in its restrictor but not its nuclear scope — which correctly predicts that "Every student who ever studied linguistics passed" is fine (*any/ever* licensed in the DE restrictor) while "Every student passed any exam" is not.