  Various neural network architectures rely on pooling operators to aggregate
information coming from different sources. It is often implicitly assumed in
such contexts that vectors encode epistemic states, i.e. that vectors capture
the evidence that has been obtained about some properties of interest, and that
pooling these vectors yields a vector that combines this evidence. We study,
for a number of standard pooling operators, under what conditions they are
compatible with this idea, which we call the epistemic pooling principle. While
we find that all the considered pooling operators can satisfy the epistemic
pooling principle, this only holds when embeddings are sufficiently
high-dimensional and, for most pooling operators, when the embeddings satisfy
particular constraints (e.g. having non-negative coordinates). We furthermore
show that these constraints have important implications on how the embeddings
can be used in practice. In particular, we find that when the epistemic pooling
principle is satisfied, in most cases it is impossible to verify the
satisfaction of propositional formulas using linear scoring functions, with two
exceptions: (i) max-pooling with embeddings that are upper-bounded and (ii)
Hadamard pooling with non-negative embeddings. This finding helps to clarify,
among others, why Graph Neural Networks sometimes under-perform in reasoning
tasks. Finally, we also study an extension of the epistemic pooling principle
to weighted epistemic states, which are important in the context of
non-monotonic reasoning, where max-pooling emerges as the most suitable
operator.
