  With the advent of large language models (LLMs), the trend in NLP has been to
train LLMs on vast amounts of data to solve diverse language understanding and
generation tasks. The list of LLM successes is long and varied. Nevertheless,
several recent papers provide empirical evidence that LLMs fail to capture
important aspects of linguistic meaning. Focusing on universal quantification,
we provide a theoretical foundation for these empirical findings by proving
that LLMs cannot learn certain fundamental semantic properties including
semantic entailment and consistency as they are defined in formal semantics.
More generally, we show that LLMs are unable to learn concepts beyond the first
level of the Borel Hierarchy, which imposes severe limits on the ability of
LMs, both large and small, to capture many aspects of linguistic meaning. This
means that LLMs will continue to operate without formal guarantees on tasks
that require entailments and deep linguistic understanding.
