Abstract: Word embeddings generated by neural network methods such as \textit{word2vec} (W2V) are well known to exhibit seemingly linear behaviour, e.g. the embeddings of analogy \emph{``queen is to woman as king is to man''} approximately describe a parallelogram. This property is particularly intriguing since the embeddings are not trained to achieve it. Several explanations have been proposed, but each introduces assumptions that do not hold in practice. We derive a probabilistically grounded definition of \textit{paraphrasing} and show it can be re-interpreted as \textit{word transformation}, a mathematical description of \emph{``$w_x$ is to $w_y$''}. From these concepts we prove existence of the linear relationship between W2V-type embeddings that underlies the analogical phenomenon, and identify explicit error terms in the relationship.