  Answering logical queries over incomplete knowledge bases is challenging
because: 1) it calls for implicit link prediction, and 2) brute force answering
of existential first-order logic queries is exponential in the number of
existential variables. Recent work of query embeddings provides fast querying,
but most approaches model set logic with closed regions, so lack negation.
Query embeddings that do support negation use densities that suffer drawbacks:
1) only improvise logic, 2) use expensive distributions, and 3) poorly model
answer uncertainty. In this paper, we propose Logic Embeddings, a new approach
to embedding complex queries that uses Skolemisation to eliminate existential
variables for efficient querying. It supports negation, but improves on density
approaches: 1) integrates well-studied t-norm logic and directly evaluates
satisfiability, 2) simplifies modeling with truth values, and 3) models
uncertainty with truth bounds. Logic Embeddings are competitively fast and
accurate in query answering over large, incomplete knowledge graphs, outperform
on negation queries, and in particular, provide improved modeling of answer
uncertainty as evidenced by a superior correlation between answer set size and
embedding entropy.
