  Conceptual combination performs a fundamental role in creating the broad
range of compound phrases utilized in everyday language. This article provides
a novel probabilistic framework for assessing whether the semantics of
conceptual combinations are compositional, and so can be considered as a
function of the semantics of the constituent concepts, or not. While the
systematicity and productivity of language provide a strong argument in favor
of assuming compositionality, this very assumption is still regularly
questioned in both cognitive science and philosophy. Additionally, the
principle of semantic compositionality is underspecified, which means that
notions of both "strong" and "weak" compositionality appear in the literature.
Rather than adjudicating between different grades of compositionality, the
framework presented here contributes formal methods for determining a clear
dividing line between compositional and non-compositional semantics. In
addition, we suggest that the distinction between these is contextually
sensitive. Utilizing formal frameworks developed for analyzing composite
systems in quantum theory, we present two methods that allow the semantics of
conceptual combinations to be classified as "compositional" or
"non-compositional". Compositionality is first formalised by factorising the
joint probability distribution modeling the combination, where the terms in the
factorisation correspond to individual concepts. This leads to the necessary
and sufficient condition for the joint probability distribution to exist. A
failure to meet this condition implies that the underlying concepts cannot be
modeled in a single probability space when considering their combination, and
the combination is thus deemed "non-compositional". The formal analysis methods
are demonstrated by applying them to an empirical study of twenty-four
non-lexicalised conceptual combinations.
