This manuscript develops a framework for compositional semantics and begins illustrating its fruitfulness by applying it to a spectrum of core linguistic data, such as with quantifiers, attitude ascriptions, relative clauses, conditionals, and questions. A key innovation is to introduce variables for assignment functions into the syntax; semantic values are treated systematically in terms of sets of assignments, theoretically interpreted as representing possibilities. The framework provides an alternative to traditional “context- index”-style frameworks descending from Kamp/Kaplan/Lewis/Stalnaker. A principal feature of the (...) account is that it systematizes a range of seemingly disparate linguistic shifting phenomena, such as with quantifiers, intensionality, and context-sensitivity under modals and attitude verbs. The treatment of the syntax/semantics standardizes quantification across domains (individuals, worlds, assignments) via a generalized (type-flexible, cross-categorial) binder-index resulting from type-driven movement. The account affords a unified analysis of the context-sensitivity of expressions such as pronouns, epistemic modals, etc., in the spirit of contextualist theories, while compositionally deriving certain recalcitrant shifting/binding data and providing a framework for theorizing about differences in tendencies for local/global readings. Applications to modal expressions, relative clauses, (non-)quantified noun phrases, ‘if’-clauses, and interrogatives are explored. I show how certain independently motivated syntactic analyses can be implemented in the assignment-variable framework. Nominal quantifiers are treated as introducing quantification over assignments, binding relative pronouns in headed relative constructions as well as other types of pronouns such as donkey pronouns. ‘If’-clauses are treated as free relatives, interpreted as plural definite descriptions of possibilities. Interrogative sentences denote a set of possible answers, conceived as sets of possibilities. Additional shifting data are compositionally derived, e.g. concerning “interrogative flip,” information-sensitivity, indexical shift, and donkey anaphora. Applications to weak crossover and weak vs. strong quantifiers are considered. The account affords a uniform compositional semantics for ‘if’-clauses in diverse conditional constructions (adjoined to NP/VP/IP/CP, with/without a main clause modal or ‘then’), and for individual correlative clauses adjoined to DP/IP/CP. The result is a unified approach to the syntax/semantics of interrogatives, conditionals, and relativization. The semantics avoids introducing added interpretive principles or composition rules such as for quantification, binding, movement (e.g., Predicate Abstraction, Predicate Modification, Event Identification, Trace Conversion). The semantics is fully compositional. (shrink)