A Note On Higher Order Grammar

Both syntax-phonology and syntax-semantics interfaces in Higher Order Grammar (HOG) are expressed as axiomatic theories in higher-order logic (HOL), i.e. a language is defined entirely in terms of provability in the single logical system. An important implication of this elegant architecture is that the meaning of a valid expression turns out to be represented not by a single, nor even by a few "discrete" terms (in case of ambiguity), but by a "continuous" set of logically equivalent terms. The note is devoted to precise formulation and proof of this observation.