  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.
