We present a new method for characterizing the interpretive possibilities generated by elliptical constructions in natural language. Unlike previous analyses, which postulate ambiguity of interpretation or derivation in the full clause source of the ellipsis, our analysis requires no such hidden ambiguity. Further, the analysis follows relatively directly from an abstract statement of the ellipsis interpretation problem. It predicts correctly a wide range of interactions between ellipsis and other semantic phenomena such as quantifier scope and bound anaphora. Finally, although the analysis itself is stated nonprocedurally, it admits of a direct computational method for generating interpretations. This article is available through the Computation and Language E-Print Archive as cmp-lg/9503008, and also appears in Linguistics and Philosophy 14(4):399–452. cmp-lg/9503008 Ellipsis and Higher-Order Unification 1

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into two principal paradigms: model-theoretic syntax (MTS), which

Category theory is a very general formalism, but there is a certain special way that physicists use categories which turns out to have close analogues in topology, logic and computation. A category has objects and morphisms, which represent things and ways to go between things. In physics, the objects are often physical systems, and the morphisms are processes turning a state of one physical system into a state of another system — perhaps

General relativity may seem very different from quantum theory, but work on quantum gravity has revealed a deep analogy between the two. General relativity makes heavy use of the category nCob, whose objects are (n − 1)-dimensional manifolds representing ‘space ’ and whose morphisms are n-dimensional cobordisms representing ‘spacetime’. Quantum theory makes heavy use of the category Hilb, whose objects are Hilbert spaces used to describe ‘states’, and whose morphisms are bounded linear operators used to describe ‘processes’. Moreover, the categories nCob and Hilb resemble each other far more than either resembles Set, the category whose objects are sets and whose morphisms are functions. In particular, both Hilb and nCob but not Set are ∗-categories with a noncartesian monoidal structure. We show how this accounts for many of the famously puzzling features of quantum theory: the failure of local realism, the impossibility of duplicating quantum information, and so on. We argue that these features only seem puzzling when we try to treat Hilb as analogous to Set rather than nCob, so that quantum theory will make more sense when regarded as part of a theory of spacetime. 1

This work expounds the notion that (structured) categories are syntax free presentations of type theories, and shows some of the ideas involved in deriving categorical semantics for given type theories. It is intended for someone who has some knowledge of category theory and type theory, but who does not fully understand some of the intimate connections between the two topics. We begin by showing how the concept of a category can be derived from some simple and primitive mechanisms of monadic type theory. We then show how the notion of a category with finite products can model the most fundamental syntactical constructions of (algebraic) type theory. The idea of naturality is shown to capture, in a syntax free manner, the notion of substitution, and therefore provides a syntax free coding of a multiplicity of type theoretical constructs. Using these ideas we give a direct derivation of a cartesian closed category as a very general model of simply typed λ-calculus with binary products and a unit type. This article provides a new presentation of some old ideas. It is intended to be a tutorial paper aimed at audiences interested in elementary categorical type theory. Further details can be found in [Cro93]. 1 1