  Natural language semantics has recently sought to combine the complementary
strengths of formal and distributional approaches to meaning. More
specifically, proposals have been put forward to augment formal semantic
machinery with distributional meaning representations, thereby introducing the
notion of semantic similarity into formal semantics, or to define
distributional systems that aim to incorporate formal notions such as
entailment and compositionality. However, given the fundamentally different
'representational currency' underlying formal and distributional approaches -
models of the world versus linguistic co-occurrence - their unification has
proven extremely difficult. Here, we define a Distributional Formal Semantics
that integrates distributionality into a formal semantic system on the level of
formal models. This approach offers probabilistic, distributed meaning
representations that are also inherently compositional, and that naturally
capture fundamental semantic notions such as quantification and entailment.
Furthermore, we show how the probabilistic nature of these representations
allows for probabilistic inference, and how the information-theoretic notion of
"information" (measured in terms of Entropy and Surprisal) naturally follows
from it. Finally, we illustrate how meaning representations can be derived
incrementally from linguistic input using a recurrent neural network model, and
how the resultant incremental semantic construction procedure intuitively
captures key semantic phenomena, including negation, presupposition, and
anaphoricity.
